Heliyon 9 (2023) e14465
Contents lists available at ScienceDirect
Heliyon
journal homepage: www.cell.com/heliyon
Review article
Extensive overview of soil constitutive relations and applications
for geotechnical engineering problems
Kennedy C. Onyelowe a, b, *, Ahmed M. Ebid c, Evangelin Ramani Sujatha d,
Farid Fazel-Mojtahedi e, Ali Golaghaei-Darzi f, Denise-Penelope N. Kontoni a, g,
Nabaz Nooralddin-Othman h
a
Department of Civil Engineering, School of Engineering, University of the Peloponnese GR-26334, Patras, Greece
Department of Civil Engineering, Michael Okpara University of Agriculture, Umudike, Nigeria
c
Department of Structural Engineering, Future University in Egypt, New Cairo, Egypt
d
School of Civil Engineering, SASTRA Deemed University, Thanjavur, India
e
Department of Infrastructure Engineering, University of Melbourne, Australia
f
Department of Civil Engineering, Sharif University of Technology, Tehran, Iran
g
School of Science and Technology, Hellenic Open University, GR-26335 Patras, Greece
h
Department of Geotechnical Engineering, Faculty of Engineering, Koya University, Sulaymaniyah, Kurdistan Region, Iraq
b
A R T I C L E I N F O
A B S T R A C T
Keywords:
Constitutive relations
Constitutive applications
Stress-strain behavior
Soil constitutive modeling
Finite element method
And finite difference method
A state-of-the-art review has been conducted in this work on soil constitutive modeling, which has
emphasized on: soil type, ground-water conditions, loading conditions, structural behavior,
constitutive relation discipline, and dimensions. By extension also, the soil constitutive applica­
tions were reviewed on the bases of: single discipline dealing with soil mechanical properties
constitutive modeling which included slope stability problems, bearing capacity, settlement of
foundations, earth pressure problems, soil dynamics, soil structure interaction, thermal and hy­
drological conditions; bi-discipline (coupled problems) which solve problems related to ther­
momechanical (freeze/thaw conditions), smoothed particle hydrodynamics (SPH) and
hydromechanical (consolidation, collapse and liquefaction) conditions in soils and rocks and
multi-discipline constitutive models which solve complex problems related to thermohydromechanical (THM) conditions in soils and rocks. This work has shown that smoothed
particle hydrodynamics (SPH) and hydromechanical (HM) models, which belong to bi-discipline
or coupled conditions are better suited for geotechnical applications, generally, while thermohydromechanical (THM) models, which belong to multi-discipline are better suited to solving
freeze/thaw and thermal piles problems and these are proven with high performance and
flexibility.
1. Introduction
Soil is a material, which is diverse in behavior and lies on the surface of the earth, which may comprise organic and weathered
* Corresponding author. Department of Civil Engineering, School of Engineering, University of the Peloponnese GR-26334, Patras, Greece.
E-mail addresses: konyelowe@mouau.edu.ng, konyelowe@gmail.com (K.C. Onyelowe), ahmed.abdelkhaleq@fue.edu.eg (A.M. Ebid), sujatha@
civil.sastra.edu (E. Ramani Sujatha), ali.golaghaei@sharif.edu (A. Golaghaei-Darzi), kontoni.denise@ac.eap.gr (D.-P.N. Kontoni), nabaz.
nooralddin@gmail.com (N. Nooralddin-Othman).
https://doi.org/10.1016/j.heliyon.2023.e14465
Received 7 April 2022; Received in revised form 5 March 2023; Accepted 7 March 2023
Available online 15 March 2023
2405-8440/© 2023 The Authors.
Published by Elsevier Ltd.
This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Heliyon 9 (2023) e14465
K.C. Onyelowe et al.
materials, as well as water and/or air. However, under various conditions of stress, it behaves in a heterogeneous pattern i.e., during
loading, unloading, and reloading state conditions. Moreover, it exhibits non-linear behavior pattern below the plastic limit with
stress-dependent stiffness [1]. A good understanding of all this complexity states and the knowledge of the soil behavior is important in
geotechnical analysis and design of geostructures.
In a geotechnics engineering project, one of the most sensitive assignments is to select the appropriate failure or deformation of the
soil model that can represent the reality of the soil state. With the development of numerical models and analysis techniques such as
finite elements methods (FEM) and finite difference methods (FDM) for geotechnics problems, it is possible to analyze and forecast the
behavior of soil and soil-structure interaction problems. Coincidentally, these analyses are based on the representation of the relations
between stresses and strains for the endless modeling materials. In geotechnical engineering, the relationships between stresses and
strains labeled as constitutive soil modeling are just mathematical equations (expressions) that outline the closest behavior of soil and
rock deformations and investigate its failure mechanism. Modeling is fundamentally concerned with finding solutions to real-world
problems and implies approximations, which are essentially a simplification of reality. To understand the complexity of soil
behavior and implementation of constitutive modeling, Brinkgreve [2] illustrated five (5) basic aspects of the behavior of soils.
Additionally, the three-dimensional deformation of the soil subjected stress is unavoidably complex. Therefore, there are lots of va­
rieties to explicate failure state condition under such stresses. For this reason, numerous researchers in the decades proposed simple to
advanced soil modeling techniques known as the stress-strain and failure behavior of soils. Meanwhile, these models contain certain
advantages and limitations, which depend largely on their application. On the other hand, Chen and Baladi [3] suggested three basic
criteria for the evaluation of models. The first criteria are the theoretical evaluation of the models with respect to the basic principles of
continuum mechanics to ascertain their consistency with the theoretical requirements of continuity, stability and uniqueness. Second
is the experimental evaluation of the models with respect to their suitability to fit experimental data from a variety of available tests
and the ease of the determination of the material parameters from standard test data. The final criteria are the numerical and
computational evaluation of the models with respect to the facility which can be implemented in computer calculations [1]. However,
the objective of this paper is to extensively review classical and advanced soil constitutive models, which have been implemented by
several geotechnical software, to present a better understanding to young modelers in their projects, and also identify the applicability
and limitation swifter. The choice to adopt these two major categories of relations and application with adjoining topics is to guide the
readers straight to what area they need information on. There have been lots of efforts made in this area to suggest constitutive models
for solving complex geotechnical engineering problems. For this reason, researchers face great difficulty in accessing the lots of
available literature and in making the choice of the best approach for different geotechnical situations. Hence this research review
reduces this task and helps researchers have a grasp of the foundational knowledge in choosing modeling techniques suitable for their
particular problems.
2. Classifying the constitutive relations
2.1. According to soil type
2.1.1. Soft clay (Typically consolidated)
There are designing and constructing problems in geotechnical/civil engineering structures like road embankments over soft soils
owing to high compressibility, high water content, and low shear strength of the soils. Such soft soil deposits basically show
considerable time-dependent performance, and they are more troubling to handle. Time-dependent performance represents the ratedependent behavior or creep of soil, and not the consolidation procedure. Numerous models have been proposed for illustrating the
soft soils’ behavior within the range of simple to very complex elastic-viscoplastic models. Though, still, no constitutive model is
extensively accepted for completely describing the performance of soft soil, particularly those representing considerable time-based
behavior. The models were classified by Liingaard et al. [4] to cope with the time-dependent performance of soil into three consid­
erable groups, which are empirical models, rheological models, and general stress-strain-stress rate-strain rate models. The first one
includes 3D models, normally in incremental forms. This class of models involves an extensive range of time-based or inviscid elas­
toplastic models (like Modified Cam Clay or Cam Clay), viscoelastic-viscoplastic models, and elastic-viscoplastic models [5–7]. There
are several groups of elastic-viscoplastic models, including overstress type elastic-viscoplastic models and non-stationary flow surface
type elastic-viscoplastic models. Several efforts have been expended over more than 30 years. However, visco-plastic type analysis is
common among geotechnical engineers.
2.1.2. Hard clay (over consolidated)
Over-consolidation is one of the most critical factors considerably affecting the over-consolidated (OC) clays’ mechanical per­
formance. Previously, considerable development was made to develop constitutive models to predict the clays’ over-consolidation
performance [8–19]. The mechanical behavior of OC clays can be adequately described by numerous elastoplastic models through
some parameters. However, model capacity is increased frequently to reduce simplicity. Recently, further consideration has been paid
to the hypoplastic models [20–27]. Benefiting from the nonlinear tensorial function, the over-consolidation behavior can be described
by the hypoplastic models without resources to the concepts in elastoplastic theory like plastic potential, yield surface, and differ­
entiation between plastic and elastic behaviors, as well as loading criterion [27–29]. Moreover, by increasing the over-consolidation
degree, volume dilatancy happens more efficiently, representing a lower stress ratio corresponding to the transformation phase from
the volume shrinkage to the dilatancy. Dilatancy happens with no shrinkage under shear loading for some heavy OC clays. The
Modified Cam Clay (MCC) model could be prolonged to describe the dilatancy features of OC soil. Thus, it could explain the specific
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performance of OC clay. The Unified Hardening (UH) model is one of the most illustrative results in this regard [16,17,30,31]. A
unified hardening parameter was added for the model for over-consolidated clay under the MCC model framework indicating the
power of the model to explain some of OC clay features, including shear dilatancy, shear contraction, strain hardening, and softening
for the over-consolidation degree of zero. Then, it naturally returns to the MCC model. Thus, the MCC-based constitutive model can
simulate the OC clay soil behavior effectively.
2.1.3. Granular soil
A classic deformation pattern is followed by clayey soil, normally denoted as isotach material, based on time [32–34]. Though, this
classic pattern is not followed by granular soils denoted as non-isotach materials [35,36]. Typically, constitutive modeling of the
granular soils’ time-based performance is oriented by the following methods: (1) empirical models [37] through differential equations
or closed-form solutions that are effective only under specific circumstances; (2) rheology models, like μ (i) models [38,39],
Bingham-based and Herschel–Bulkley models [40], where the viscous deformation occurs only when the yield limit is exceeded by the
shear stress; (3) general elasto-viscoplastic models [41] describing the viscoplastic flow’s constant rate; and (4) models taking into
account the acceleration effects [42]. Though, the last model suffers from a shortcoming since the frictional component cannot
describe the nonlinear behavior since it is too simple. Moreover, the viscous component has higher complexity for practical use.
Modeling the granular materials’ viscous and frictional behaviors within the hypoplastic framework is an alternative method. This
model makes an explanation of viscous and frictional behavior feasible. It is assumed that the viscous and frictional stresses coexist in
granular material [43]. Many of the aforementioned constitutive models could not investigate the negative void ratio and, subse­
quently, the high-stress condition. To solve this shortcoming, the critical state framework has been used in some recent work [44–47].
2.1.4. Cemented soil
Normally, natural soils are cemented weakly for their responses; cementation bonds have a key role. AC soils (Artificially
cemented) exist through ground enhancement procedures. The responses are highly affected by cementation bonds. This kind of soil,
based on its mechanical performance, falls in an intermediate area between soils and rocks. Considerable efforts have been made [48],
indicating that cemented soils also reveal the behavior the same as other materials with elastoplastic behavior. However, the strains
are primarily recoverable and more minor relatively within the yield locus. A transformation occurs in shear there from brittle to
ductile behavior by increasing the confining stress. A general recognition will exist to model the cemented soils’ behavior while
considering the performance of a corresponding reconstituted state [49,50]. Gens and Nova [49] provided the framework for modeling
of elastoplastic behavior of bonded soils. This model has unique features, including the yield surface with a form and shape similar to
the uncemented soil. With the introduction of a stress history tensor, a model was developed by Adachi and Okano [51] and Adachi and
Oka [52] to describe the strain-softening reaction of the soft rocks. It should be noted that combining soft clay with cement is a
well-known ground enhancement method since it has a lower cost and enhances the consolidation rate. Recently, some of the essential
constitutive models were reviewed for cement-treated clay [53]. A straightforward model was also developed in their work in terms of
a prolonged version of the Mohr-Coulomb model for simulating the cement-mixed clay behavior. They used triaxial laboratory tests for
calculating the model parameters. However, the suggested models heavily rely on characteristics of the cement-treated soil, including
elastic modulus, residual strength, and peak strength. Moreover, the plastic deviatoric strains are also responsible equivalent to the
residual and peak states. In fact, high-quality triaxial test data contribute to calculating the model parameters. Briefly, most existing
models are oriented by bounding surface plasticity, a critical state framework, and the kinematic yield surface concept, combining
breakage of the soil-cement structure over-loading [54–62].
2.1.5. Frozen soil
This type of soil is a multiphase substance and an important strength feature in frozen soil mechanics. Unfrozen water, solid mineral
particles, ice particles, and gas components are included in frozen soil, along with mechanical features like the modulus and strength of
elasticity varying extensively. Moreover, the complication of the mechanical properties is determined by the existence of microcracks
distributed randomly. It is greatly significant theoretically and practically to assess the constitutive relations and dynamic mechanical
characteristics of frozen soil. Frozen soils are sensitive to temperature considering numerous triaxial compression tests [63]. They
assessed the dilatant-hardening model for predicting the frozen sand’s strength. With developments in thermodynamics, damage
micromechanics, and mechanics, they gradually explain the failure and deformation characteristics. A damage model (stochastic) was
developed based on some experiments on warm ice-rich frozen clay, utilizing the probability and continuous damage. Using the
Mohr-Coulomb failure criterion, the damages in the soil element were judged [64]. To discuss the frozen soil’s volume yield surface,
Zhu et al. [65] used generalized plastic mechanics. Then, an elastic framework was provided for frozen soil in this work. A yield
function was presented by Lai et al. [66] utilizing the laboratory data based on Drucker’s postulate. Furthermore, an elastoplastic
framework was developed for frozen soil by deducing the cross-anisotropic damage variables. A relatively simple elastoplastic model
was proposed to evaluate the nonlinear mechanical performance of frozen silt in terms of the related flow rules [67]. This kind of soil is
regularly exposed to impact loading along with conventional quasi-static loading. Recently, research has been performed on dynamic
mechanical features focusing on the effects of temperature and strain rate. Besides, a constitutive model was developed tentatively,
comprising higher anisotropy and strain-rate sensitivity. Unfortunately, a wide range was exhibited by the experimental results due to
structural differences and various frozen permafrost components [68]. The SHPB (Split Hopkinson Pressure Bar) was utilized by Hu
and Wang [69] and Ma [70] to assess the dynamic performance of frozen soils at various strain rates and temperatures. Then, an elastic
framework was presented [71], supposing the damage value based on the Weibull distribution. It was claimed that temperature
damage is unique emotional damage establishing a model through temperature damage to describe the effects of the increased
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transient temperature on the dynamic performance. Furthermore, the ZWT model [72] was modified considering the accumulated
damage process. These models are extensively utilized to describe the constitutive relationships of concrete and metal, respectively.
For predicting the frozen soil well’s dynamic stress-strain curves, an energy-based constitutive model was also developed [73].
However, the framework could not explain the failure process due to the pathway independence of the energy method. To more
accurately describe the behavior of frozen soil, the damage progress procedure of frozen soil exposed to the impact loading must be
studied. Damage evolution in materials will happen during the loading process. The process becomes considerably complex with time
and other factors. Therefore, further investigation should be focused on this problem in the future.
2.1.6. Soil reinforced with fibers
Geo fiber is a synthetic material that is utilized for improving the soils’ engineering properties by the provision of extra resistance to
tensile stress and shear. Fiber-reinforced soil can be regarded as a composite material. There are studies on the mechanical perfor­
mance of soils with geo-fiber reinforcement comparatively. Different investigations have been performed on short geo fiber-reinforced
soils [74–78]. The synthetic fibers have primary effects on the soil mass, which reduce the increased tensile strain. Similarly, artificial
confinement is presented to the sand in the exact directions similar to that of saturated sand in undrained circumstances. Here, pore
water pressures are made owing to the restrained volumetric strains. Moreover, the stress state is modified within the sand mass.
Saturated soil is regarded as a continuum attained by superposing two continua, a fluid (the water in the pores) and a solid (the soil)
mechanically. It is assumed that they both are onto the same volume of space while undergoing the same strains. The stresses are
summed to achieve the equilibrium conditions. Obviously, regarding the saturated soil, the interaction between effective stresses and
pore water pressure influences only the isotropic part of the stress tensor since there is no stress for water. The anisotropic structured
clay model (ASCM) was established with the MCC as its foundation. The performance of intact clays connected to their natural
structure is considered by the ASCM [79]. The mechanical behavior of stiff clay, softly structured clay, and artificially reinforced clay
can be predicted by the model. Therefore, ASCM is a good framework for investigating the behavior of the soil reinforced with fibers.
2.1.7. Soft rock
Creep is the tendency to deform under mechanical stresses permanently. It can happen by long-term high-stress exposure, which is
still less than the yield strength. Geological disasters may be induced by accumulating the rocks’ creep damage. Actual projects
commonly contain soft rocks with considerable plastic deformation features. Particularly, numerous investigations were performed on
the constitutive creep model for soft rock in terms of theoretical analysis or creep tests. A constitutive creep model was proposed by
Zhou et al. [80] for salt rock regarding fractional derivatives. The burgers creep model was adopted by Cao et al. [81], revealing the
rock’s creep features in Jinchuan, China. It is indicated that the time-based performance of soft rock is significantly important in
designing structures over soft rock. Therefore, long-or short-term creep tests were used in several experiments, though evaluation of
the time-based behavior of soft rock is complicated and challenging [82–85].
Furthermore, the inelastic creep deformation was found to induce increased long-term compressive strength [86,87]. It is worth
noting that the deep saturated and dry conditions have very different creep rules. The saturated and dry conditions have very different
rheological properties under a high level of stress, particularly when occurring serious unloading, damage, and weakening of the rock
quality [88]. Water also has non-neglectable effects. There is a mathematical model for explaining the creep behavior in the Tuzkoy
region [89]. They used the results derived from the creep tests performed on the rock salt mine specimens. Moreover, a framework
describes the nonlinear creep of soft rock based on the results of the step-loading creep test and triaxial shear test [90]. Several studies
have been performed on the coupling effects of seepage, temperature, stress, and chemical on creep behavior [91,92]. A novel
computing technique was proposed by Yang et al. [93] to estimate the scale of the collapsing area of the tunnel face under various
degrees of saturation. They used the nonlinear Hoek–Brown framework and the association between the strength and saturation in
terms of the multi-block failure mechanism. This model could be a good choice for investigating soft rock behavior and further
development in the future.
2.1.8. Intact and fractured rock
A rock mass is a complicated fractured medium that comprises multiple cracks and joints. Based on its strain-stress behavior,
designing and assessment are performed in rock engineering by modern researchers [94–97]. Constitutive models have been estab­
lished for rocks considering rheology, nonlinearity, anisotropy, and other features over a solid theoretical basis for engineering
practice [98–101]. Though, during coal mining, tunnel excavation, and support, rocks are normally loaded and unloaded frequently.
Hysteretic behavior is represented by rock masses under such circumstances, and the deformability and strength are closely associated
with the loading history and stress state. Thus, under conventional triaxial and uniaxial loading circumstances, the necessities of
engineering practice in more complicated circumstances are not met by current constitutive models [102]. The deformability and
strength of rocks have been investigated from two aspects: one included building frameworks for rock flaws and interfaces under
various loading circumstances. For instance, the deformation of rock joints was simulated by Phueakphum and Fuenkajorn [103]
under cyclic loading through physical models. A sliding crack model was built by David et al. [104] for hysteresis and nonlinearity in
the uniaxial curve of the rock for stress-strain behavior. The other method was phenomenological, where the initiation, extension, and
merging procedures are neglected by the workers related to microcracks or micro-voids. The deformation features of different rocks
were studied by Liu and He [105] under cyclic loading. The deformability and strength of flaws in the rock are described by the first
approach. Though, it is still not able to direct the engineering practice, in which flaws are randomly distributed. The other method
cannot represent the progress mechanism of rock flaws; however, it guides the engineering practice regarding the strength and
deformability of rocks as a whole.
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2.2. According to ground-water
2.2.1. Dry soil (Saturated soil)
An assemblage of particles is included in soils of various shapes and sizes, forming a skeleton containing voids full of water and gas
or air. Thus, the term "soil’’ represents a combination of assorted mineral grains with different fluids. Generally, the soil must be
considered as a single (dry soil), two (saturated soil), or multiphase (partly saturated soil) material with states explained by the
displacements (velocities) and stresses within each phase [106,107]. During deformations, irreversible motions like slips at grain
boundaries, formations of voids by particles resulting from a packed configuration, as well as the integrations of such irreversible
motions are experienced by the solid particles forming the soil skeleton. When the microscopic origin and the particulate nature of the
involved phenomena are not considered, using phenomenological equations, the performance of the different phases creating the soil
medium is explained. Therefore, the conceptual model in multiphase theories includes each phase (or constituent) entering through its
averaged properties attained when the particles were smeared out in space. Furthermore, the constituent’s particulate nature is
explained based on phenomenological laws since the particles collectively behave as a continuum. Thus, soil comprises a solid skeleton
with interactions with the pore fluids [34].
2.2.2. Unsaturated soil
The conventional completely saturated soil models are the basis of most models for partially saturated soils. They comprise the
impacts of suction and partial soil saturation on the mechanical performance by explicitly or implicitly prolonging the 3D yield along
with plastic potential surface expressions of the completely saturated models to 4 dimensions. Here, suction or a type of equivalent
suction is introduced as an extra stress state variable. Normally, this is obtained by the adoption of specific expressions for increasing
the isotropic yield stress and apparent cohesion by increasing the suction [108–110]. Moreover, other assumptions are presented about
the effects of suction on the soil within the yield surface, such as the elastic one [108]. The previous models mainly are oriented by a
critical state framework particularly, on the altered Cam clay model or its derivatives. Geiser et al. [111] used Desai’s [112] disturbed
state concept. This method allows hardening compared to the simpler critical state models, owing to both deviatoric and volumetric
strains, more realistic volumetric strains before peak stress circumstances, as well as stress path-dependent strength. Alonso et al. [110]
presented the Barcelona basic model (BBM), considering the loading–collapse (LC) yield function concept along with the suction
increase yield function considered as the first elastoplastic models for partially saturated soil. Several elastoplastic frameworks were
established for describing the unsaturated soil’s coupled hydro-mechanical performance. Considering the adopted state variables,
these models can be generally classified into three classes (1) net suction and stress [113]; (2) effective suction and stress [114,115];
(3) degree of saturation and effective stress [116,117]. The validity of the present frameworks is calibrated carefully for unsaturated
soil by using the experimental test results under monotonic loading. However, by these models, it was not verified that their accuracy is
still assured when exposed to cyclic loading. The unsaturated soil’s dynamic behaviors have recently attracted researchers’ and en­
gineers’ attention owing to the severe damage to unsaturated slopes and embankments followed by an earthquake in some studies
[118–120]. The near-saturated sand’s mechanical performance via some undrained cyclic and monotonic triaxial experiments is
provided in the study of Ishihara et al. [121]. Tsukamoto et al. [122] conducted some cyclic triaxial tests controlled by stress on
unsaturated sand exposed to undrained conditions and investigated the silty sand’s cyclic resistance with various grain configurations.
Some strain-controlled cyclic triaxial experiments were performed by Unno et al. [123] on unsaturated soils subjected to unvented and
undrained conditions to evaluate the cyclic mechanical performance of such soils. It should be noted that the present experimental
findings revealed the possibility of liquefying in the unsaturated soil under a relatively higher saturation degree. It is worth noting that
the soil water retention curve (SWRC) is one of the critical features of unsaturated media that simply shows the relationship between
soil suction and water content [124–127]. By investigating the previous literature, it is clear that any condition that causes a change in
the soil structure or pore fluid chemistry could cause a change in different soils’ (especially the clay soils) water retention capacity
[128,129]. Many previous constitutive models for unsaturated soil only consider the distilled water condition in their formulation.
However, due to the inevitable effects of pore fluid salinity on the hydromechanical behavior of unsaturated soil [130–134], it is
essential to investigate this phenomenon in future research further.
2.2.3. Drained/undrained soil
There are different design properties in porous materials, such as soils under undrained and drained circumstances. The pore water
can drain out quickly from the soil matrix under drained conditions. However, in the undrained condition, the pore water cannot drain
out, or there is a much quicker loading rate than the rate for draining out the pore water. In soil, the presence of either an undrained or
a drained condition is based on soil type (sand, silt, gravel, and clay), geological formation (embedded sand layers in clay, fissures),
and loading rate [135]. According to the experiments, there is almost always the drained condition for coarse-grained material like
sands and gravels exposed to static or monotonic loading. The reason is the material’s larger permeability to drain out the pore water
very fast. It is worth noting that the quick loading rate can make undrained loading circumstances for the saturated loose sands leading
to liquefaction under seismic loading. Moreover, under earthquakes and quick static loads, an undrained condition is always created by
the lower material permeability for silts and clays. In the analysis of quasi-static conditions, undrained or drained characteristics are
assumed on the basis of loading speed and the soil hydraulic conductivity. When these physical properties are not extreme, it is
essential to consider the performance of complete consolidation. However, it is difficult to evaluate the point to decide on the more
complex result. The situation becomes complicated by inertia terms in dynamic analysis. Here, the completely drained assumptions are
never reasonable. Though, undrained performance provides a reasonable supposition under such fast loads as earthquakes. For the
significant permeability, this assumption is invalid, and the impacts of drainage may be considered on several occasions. Therefore, it
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is essential to use the drained parameters (E , ν , φ ) in analyzing coarse-grained substances (sands and gravels). When liquefaction
under unloading conditions, using residual strength parameters of liquefied soil per Boulanger and Idriss [136] guidelines or adopting
advanced constitutive models can be typical approaches where the drained parameters are the input parameters. The failure behavior
is well captured generally in drained circumstances; however, the operative stress pathway followed in undrained materials may
significantly deviate from observations. Hence, in an undrained analysis, it is preferred to utilize undrained shear parameters with a
friction angle of zero. Prior to the local shear, the stiffness behavior (or deformation) is modeled poorly. For perfect plasticity, softening
or strain-hardening effects of the soil should not be included in the model. For instance, The Modified Cam-clay predicts more realistic
undrained shear strength in comparison to the Mohr-Coulomb model [137].
′
′
′
2.3. According to a type of loading
2.3.1. Monotonic and cyclic loading
From the most straightforward curve-fitting analytical expressions to the more complicated elastoplastic constitutive relationship
for soils, the common disadvantage is related to either the monotonic or cyclic loading of a phenomenon. Therefore, existing
constitutive models pertinent to monotonic loading could not be used in cyclic loading models. Or in other words, the constitutive
relationship used for soil characterization must be of fundamental nature, i.e., must be equally applicable to monotonic or cyclic,
drained or undrained, or any other type of loading. Monotonic and cyclic loading was primarily applied using the triaxial test on the
sand specimen. This is because of the liquefaction phenomenon, which is happened under cyclic or seismic loading and makes the
uncemented soil strength decrease. This is why many previous types of research try to investigate the soil behavior under both cyclic
and monotonic loading [138–140]. Therefore, many elastoplastic frameworks have been presented for saturated soils to reproduce the
soils’ mechanical behavior exposed to cyclic loading [141–146]. A super-load concept was proposed for describing the effects of the
soil structure observed commonly in soils deposited naturally (the main reason for the significant difference in soils from place to
place) [143]. A complicated constitutive model was proposed by Zhang et al. [147] for soils. They presented a novel method for
describing the anisotropy induced by stress along with the super-load and the sub-loading concepts. As a rotating hardening elasto­
plastic model, the model is able to consider the effects of the anisotropy induced by the density, stress, and soil structure. Few pub­
lications exist on the unsaturated soil constitutive models exposed to cyclic loading. An elastoplastic model was proposed by Yang et al.
[148] based on the plastic bounding surface model for loess soils under cyclic loading in unsaturated conditions. A coupled
hydro-mechanical elastoplastic model in the bounding surface plasticity framework for unsaturated soil has been presented by Khalili
et al. [149,150]. Gao and Zhao [151] present a proper bounding surface model to investigate the fabric effect on sandy soil behavior
based on Gao et al. [152] and Papadimitriou et al. [153].
2.3.2. Impact loading
Constitutive models according to impact loading are mainly devoted to frozen soils. Frozen soil combines anisotropic and het­
erogeneous four-phase complex parts, i.e., solid mineral particles, liquid water, viscoplastic ice, and gaseous inclusions [154]. Almost
23% of the Earth is coated with frozen soil, and this kind of soil is seasonally the most common of these. Due to the sensitivity of ice
particles to temperature variation, the mechanical properties of frozen soil are more complex than those of ordinary soil and need the
proposal of advanced constitutive models [155–159]. However, there is not much research to investigate the dynamic mechanical
features of frozen soil under high-speed impact loading as a result of the defects that appear in experiments and the homogeneity of the
specimens. Nevertheless, much research exists that experimentally investigates the mechanical features of frozen soil from freezing
temperature, confining pressure, cyclic loading, loading rate, and loading path aspects. Moreover, equivalent constitutive models have
been established for characterizing the frozen soil deformation [160–162].
The minimum creep strain rate, peak shear strength, and failure strain of frozen soil were considerably affected by the strain rate
and ice content significantly in loading tests of stress rate and constant strain [163]. An elastoplastic framework for frozen soil was
invented via a nonlinear Lade–Duncan yield criterion and non-correlated flow criterion [164]. A two-stress state variable model was
established by Amiri et al. [165], comprising cryogenic suction and solid phase stress for describing deformation features of frozen
soils, particularly the phenomena of strength weakening and ice segregation owing to the pressure melting. According to Xu et al.
[166], under cyclic triaxial circumstances, frozen soil represented cyclic hardening along with a criterion of cyclic failure. It was also
indicated that [167] the layer propulsion can represent the dynamic destruction of frozen soil exposed to the impact loading revealing
an association between the strain and instantaneous temperature rise for obtaining the effective elastic modulus’ expression via SHPB
tests. A framework was established for describing the frozen soil’s dynamic mechanical performance in terms of a plastic constitutive
framework taking into account the damage evolution and strain-rate effect and the law of micro-voids and micro-cracks of frozen soil
[168]. The frozen soil’s wave impedance increments within a shorter period [169]. They presented viscoelasticity and a damaged
constitutive model defining the damage variable through longitudinal wave velocity. Therefore, presently SHPB apparatus has mainly
been adopted in the experiments on dynamic mechanical features of frozen soil exposed to impact loading.
2.3.3. Seismic loading
The other type of loading includes dynamic or seismic loading. However, the previous subsections are also a dynamic loading type,
but this section especially focuses on seismic loading due to its specific characteristics. The cycling loads-unloads-reloads sequences are
included in earthquake-caused vibratory ground motions, mainly owing to the shear waves’ upward propagation from underlying rock
formations. Thus, the soils’ cyclic stress-strain performance seems to be very important to reliably predict the seismic response.
Numerous studies have been performed so far [170–175]. Several studies have been performed on the dissipative capacity of soils, and
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the models have been created either by adapting the damping hysteretic nature or utilizing the corresponding viscous damping hy­
pothesis [176,177]. The "Backbone curve” concept (or skeleton curve) determining the shear stress amplitude as a function of the shear
strain amplitude (Christensen, 1982)is used in most of these models, and the reloading and unloading branches of the hysteretic loops
are made via the Masing criterion [175]. The stationary cycle rules are prolonged for modeling irregular cyclic deformation possessing
a variable amplitude [173]. Masing’s works are the base of these rules, which took into account a plasticity model of
elastic-perfectly-plastic elements with various yield stresses. Furthermore, the inelastic behavior is represented by the soils’ cyclic
shearing with strain amplitudes of 10− 5, as well as a hysteretic association between the shear stress and strain. The strain–stress as­
sociation for a considered soil is based on the density of the soil, the confining pressure, and the strain amplitude. The saturated soil’s
response to the cyclic deformation is also based on the drainage circumstances. In the drained conditions, the cyclic shearing of dry or
saturated soil leads to the soil’s gradual compaction. In undrained conditions, by the shearing of saturated soil, the effective pressure is
reduced, changing the soil’s shear stiffness, thus affecting the strain–stress association. It is worth noting that, under strong ground
motion, soil behavior tends to be nonlinear. Therefore, in this respect, the constitutive model that could consider the nonlinearity of
soils during seismic loading should have been used [178–181].
2.4. Structural behavior
2.4.1. Linear and non-linear
Soil as a complex material has non-linear behavior and regularly represents time-based and anisotropic behavior under stress.
Usually, the soil has different behavior in principal loading, reloading, and unloading. It behaves nonlinearly well under failure
conditions with stress-based stiffness. Soil experiences plastic deformation, which is not consistent in dilatancy. Minor strain stiffness is
experienced by soil at very lower stress and strain reversal. In the simple elastic-perfectly plastic Mohr-Coulomb model, such a general
performance could not be considered. However, the model has advantages making it a satisfactory case as a soil model. The neoHookean constitutive relation is generalized in linear elasticity through Hooke’s law to a linear stress-strain relation in finite elas­
ticity for isotropic materials. An affine relation between the left Cauchy-Green Tensor B and the Cauchy stress is generally used for
rubberlike materials that are not compressible. Assuming a linear association between the Green-St and the second Piola–Kirchhoff
stress tensor S is another possibility. Venant strain tensor E [182–184]. These two relations are decreased to Hooke’s law for infini­
tesimal deformations. It should be noted that the response of most substances deformed in simple shear or straightforward extension is
not qualitatively mimicked by a linear relationship between S and E. The nominal slope (or the first Piola–Kirchhoff) stress-strain curve
is normally a non-incrementing function of strain for finite deformations of various elastic materials when they are distorted either in
simple shear or tension [185,186]. Though, for isotropic materials, a linear relation between E and S foresees an increase in this slope
by deformation, therefore, the instability load. End chronic [187,188], elastic [189,190], micromechanical, and several elastic-plastic
have been proposed models with different sophistication degrees or complexity. The most auspicious models are elastic (-visco)-plastic
models. According to the experience, fundamental aspects of natural soil behavior are not modeled by the simple nonlinear elastic
stress-strain models such as the hyperbolic model. However, they are practically popular. Furthermore, several "empirical’’ models
[191] are still widely used. These models are based on analytical relations in terms of experience or experimental observations between
quantities of direct interest (like the rate of pore-pressure build-up in a cyclic test) with no rigorous constitutive formulation frame­
work such as elastoplasticity.
2.4.2. Time independence (Elasto-plastic)
In the conventional elastoplasticity theory, the yield surface, potential plastic function, and hardening law should be assumed. An
elastoplasticity constitutive model was established by Boulon and Nova [192], with the principal features including the ability to
recreate the impacts of damaging the sand particles. A model was presented by Gennaro and Frank [193] in terms of a Mohr-Coulomb
failure criterion such as phase transformation (dilatancy and compaction) and ultimate state; An elastoplasticity constitutive model
was proposed by Ghionna and Mortara [194] considering the CNS results (constant normal stiffness) of direct shear apparatus. The
similarity between the interface and soil’s behavior was discussed by Liu et al. [195], along with the perception of critical state soil
mechanics in the formulation of the elastoplasticity constitutive model of interface. An elastoplasticity constitutive model was pro­
posed by Sun and Wang [196] for the calcareous sand structure interface taking into account the effects of grain crushing on the
interface. Furthermore, Zong-ze et al. [197] presented a rigid-plastic interface model. A nonlinear perfect plastic-elastic model of the
interface was presented by Luan and Wu [198]. Until now many elasto-plastic constitutive models have been proposed in literature
from frozen to unfrozen or saturated to unsaturated soils [199–202].
2.4.3. Time-dependent (Visco-plastic)
Within the framework of viscoelasticity, several materials have been modeled in previous literature. Among the common models
are the Maxwell, the Voigt-Kelvin, and the Spring-Voigt three-parameter models for linear evaluation. The linear Spring-Voigt model
was reported to explain the soil’s dynamic nature [203]. By introducing the distribution concept of relaxation time into the linear
model, various time-based behavior of soil can be modeled. The time dependency of clay was proved by Murayama and Shibata [204]
in high-frequency regions taking into account the relaxation time distribution. Non-linear viscoelastic and viscoplastic models were
presented by Murayama and Shibata [204] in terms of the original model. A simple asymptotic body (SAB) was presented by DI et al.
[205] for simplifying the viscoelastic framework for soil which can be categorized into three-parameter models. The linear viscoelastic
method is effective in small strains. Though, the properties contain both viscoplasticity and viscoelasticity in the large strain range.
Oka et al. [206] explained the performance of a viscoelastic-viscoplastic framework for clay, considering the dynamic performance of
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clay for various strain levels. The viscous performance of clay was investigated in the field of Micro-rheology. The creep performance of
clay in terms of the rate process theory was described successfully by Singh and Mitchell [207] and Singh and Mitchell [208]. The rate
process theory was used for the shear force higher than the thermal energy to create an exponential type of non-linear flow law be­
tween the strain rate and the shear force acting on each flow unit. Viscoelastic modeling is essential for describing both the plastic
nature of the soil and the viscous nature. A viscoplastic theory was proposed by Perzyna [209], generalizing the viscoplasticity theory’s
linear theory of Hohenemser and Prager [210]. A linear prolonged viscoplastic model considering the plasticity and fluid model of
Bingham was presented by Hohenemser and Prager [210]. According to Yong and Japp [211], it is possible to utilize the viscoplasticity
theory to the clay’s dynamic behavior. An elasto-viscoplastic theory was proposed by Adachi and Okano [51] first for clay, taking into
account Perzyna’s theory as well as the original Cam-clay model. A real-time elastic-viscoplastic model was presented by Sekiguchi
[212]. This model was presented initially as a creep model comprising failure. Time-dependent models were also derived [213,214],
known as non-stationary models. It is worth noting that these models involve time, which explicitly violates the objectivity principle.
An elastic-viscoplastic model was proposed by Yin and Graham [215] in terms of the flow surface and the modified Cam-clay model.
Recently, new visco-plastic constitutive models have been used for creep analysis in soft rocks or soils that some of them mentioned in
the following [216–218].
2.5. According to discipline
2.5.1. Bi-discipline (Coupled)
2.5.1.1. Thermo-mechanical. Recently, investigations have been performed on the effects of temperature on soils from different as­
pects. Studying the thermal impacts on soils’ engineering features dates back to the late 30s. The first research was reported by Gray
[219] on the thermal effects on soil consolidation. In this field, the state-of-art motivating factors are the growth of industrial tech­
nologies like a thermal technique for improving soil stability [220], changes in temperature of soil during storing and sampling [221],
temperatures around higher voltage buried cables [222], underground storage of thermal energy [223],temperature-induced by
augmented motions of some landslides [224,225], and neutralizing the reduction in permeability owing to remolding of soil followed
by installing drains by a mandrel [226]. It was indicated that the engineering properties of soils, such as compressibility, shear
strength, and permeability, are highly affected by increasing temperature to less than the boiling temperature of the water. Graham
et al. [227] presented various constitutive laws to simulate the soil’s behavior in increased temperatures occasionally oriented by
thermodynamic principles. However, they mainly have the CCM (Cam-clay) template and critical state concepts. Coupled behavior
was considered in some models, such as viscosity, thermal, hydraulic, mechanical, and chemical aspects [228,229]. Former studies
have mainly revealed the normal consolidation line movement to lower void ratio values at increased temperatures. Typically, the
changes in the location of the normal consolidation line (NCL)owing to the increased temperature are regarded as a parallel movement
to lower void ratio values in the compression plane [230]. Though, such assumptions are not always by the detailed assessment of
experimental data. Considering the modified Cam-clay model, an innovative method was proposed [231] for the structured soils’
deformation analysis. To obtain the generalization of the present model and pretend the performance of unsaturated soils observed
experimentally, Bishop’s suction and stress are introduced as independent stress parameters to modify the hardening agent and the
yield conditions to consider the suction role. A similar approach is used to predict an isothermal framework for estimating the un­
saturated clays’ thermo-elastoplastic performance in triaxial stress space. By this model, the thermo-elastoplastic model is extended for
completely saturated clays presented by Hamidi et al. [225]. A non-related temperature-based flow rule was used in the current model
for simulating the clays’ mechanical performance on changes in suction and temperature. Recently, constitutive models of this topic
have been used to investigate the behavior of clay soils and rocks and, specifically, the structure variation during the thermo­
mechanical type of loading condition. Also, pile-soil interaction is the other recent development field for these models [232–237].
2.5.1.2. Hydro-mechanical (HM). The macro behavior is controlled by the coupled HM mechanism owing to the interaction of the two
grain-scale phenomena, (1) rearrangement and grain sliding, (2) ruptures of a liquid bridge [238,239]. Noticeable hydro-mechanical
coupling is caused by these grain-scale phenomena at the continuum level with extensive investigations in various suction-controlled
[240,241] as well as constant water content [242,243]. Mainly, by increasing the suction, the shear strength [244] is increased along
with the yield limit [245] and dilatancy [246]. All these properties include the volume change related to the irreversible changes in
saturation [247] that are induced by the coupled mechanical and drying or wetting procedures. Such volume alteration is regarded as
an essential feature of unsaturated soils [248]. A constitutive model should be reflected by these critical features of partially saturated
soil for capturing the transition between unsaturated and saturated circumstances [249]. For instance, several papers have reported
the impacts of suction on the stress-strain relations [250]. Particularly, suction was utilized in a loading-collapse section for capturing
the plastic compression owing to the collapse behavior induced by wetting. The normal consolidation line (NCL) is changed with
suction, in which the compression index decreases [251,252] or increases by incrementing the suction [253]. However, these models
did not consider the relation between the degree of saturation and suction. Thus, it becomes difficult to regenerate the model responses
dependent on various saturation trends. For instance, there is very different cohesion induced by the liquid bridge distributions be­
tween particles in the three saturation trends [254]. In this regard, an NCL was used [255] with the soil compression index changing
with the effective saturation degree. However, an NCL was proposed by Alonso et al. [240] based on both saturation degree and
suction. Further hardening constitutive laws were suggested [256,257] regarding the coexistence of saturation and strain rates. These
papers revealed that saturation and suction degrees have an indispensable role in model collapses of partially saturated soils induced
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by wetting or drying. Recent work in this area mainly considers the modeling of unsaturated soil behavior under different
hydro-mechanical loading. The new constitutive models have the advantages of more straightforward model calibration and subse­
quent model parameters from different laboratory tests [258–261].
2.5.2. Multi-discipline
2.5.2.1. Thermo-hydro-mechanical (THM). Recently, the THM performance of partially saturated soils has been very attractive in
engineering, particularly in nuclear waste disposal. For a long time, it has been well-known that the variation of temperature highly
affects the hydro-mechanical features of porous media. Several attempts have been devoted to deeply comprehending the performance
in various aspects. Creating the THM mathematical model is one of the critical features of a quantitative study of coupled, transient
Thermo-Hydro-Mechanical performance. It investigates the mechanisms leading the THM performance and its evolution, formulation
of the PDE (partial differential equations), and discretizing of the leading equation system for numerical solutions. The finite element
model was developed [262–264] along with its solution process for heat transfer beside the nonlinear HM procedure in unsaturated
and saturated porous media. A completely coupled THM mathematical model was also presented [265–268], considering the
experimental findings and former studies. Several efforts have been allocated to numerically model the coupled
Thermo-Hydro-Mechanical procedures in environmental engineering problems, mainly the numerical simulation of the THM behavior
in engineered clay barriers. A general mathematical model was provided [269–271] to analyze Thermo-Hydro-Mechanical problems in
partially saturated soils with possible pollutant transport. Moreover, using the model was extended to numerically analyze THM
performance and damage phenomena of concrete at higher temperatures. Besides, it has also been attempted for mathematical
modeling. Another most vital feature is the experimental work on the constitutive connection for describing the
thermo-hydromechanical performance quantitatively. Hueckel and Borsetto [272] extended the modified Cam-Clay model to model
the saturated soils considering the temperature impacts. A thermo-mechanical model was presented by Cui et al. [273] for saturated
clays in terms of the Cam-Clay model. Significant contributions were made by Gens and Alonso [274] to experimentally assess the
constitutive modeling of unsaturated soils. Though, fewer studies have been performed on developing the THM constitutive model for
partially saturated soils. In the constitutive model of Gens et al. [275] for unsaturated soils, a generalization of the Cam-Clay yield
surface was considered in the suction, stress, and temperature space. Recently different THM models have been used to investigate the
saturated or unsaturated soil behavior that can be used for a specific problem [276–279].
2.6. According to dimensions
2.6.1. 1D
1D relaxation experiments for saturated soils, especially clay, can be conducted in swelling and creep areas. Here, swelling is not
similar to the unsaturated soils. It happens in saturated conditions and is led by the time-based expansion of the saturated clay plates’
skeleton with a negative charge owing to moving the water with dipole attraction to the clay surface [280]. In clay soils, the swelling
was found in odometer circumstances [281]. Three distinct phases of swelling were defined by Sivapullaiah et al. [282]: primary
swelling, inner void swelling, and secondary swelling. The early experimental works were performed on a sand-bentonite mixture and
reconstituted illite [283]. A 1D elastic visco-plastic model was contemporarily proposed by Yin and Graham [283], considering the
equivalent time idea in the one-dimensional straining conditions while considering the clayey soils’ viscous performances. This EVP
model was esteemed and utilized for modeling and analyzing soft soils [284–286]. A novel constitutive 1D elastic visco-plastic model
was adopted by Tong and Yin [286], taking into account the swelling (1D EVPS) model for describing and simulating the nonlinear
stress-time performance of soils.
2.6.2. 2D and 3D
The fractional plastic flow rule and a three-dimensional yield function matching together are vital to establishing a threedimensional fractional framework for soils. Presently, yield function can be made in the 3D stress space in two methods: (1) Direct
technique where the yield surface is expressed explicitly based on the stress components [286–289], (2) Indirect technique, in which
the yield surface is made under triaxial compression circumstances in a transitional space to reflect yield features of soils under true
three-dimensional stress circumstances [290,291]. Regarding general stress components σij using the present fractional flow rule, it is
not possible to obtain clear expressions of the fractional derivative to the common 3D yield functions made by the specifically
mentioned approaches [292]. Therefore, to develop a three-dimensional fractional plasticity model, a three-dimensional fractional
plastic flow rule is eagerly needed with no restriction of the coordinate basis of the yield surface. Different identification methods are
now available, armed with completely-field measurements, including the Reciprocity Gap Method [293], Equilibrium Gap Method
[294], Finite Element Model Updating [295], and the Constitutive Equation Gap Method [296]. There is a common restriction in such
approaches as they are limited to 2D applications when analyzing two-dimensional full-field data. Though, three-dimensional impacts
should not be ignored in cases such as for thick samples constructed of an elastoplastic material. Wu et al. [297] used stereo-correlation
systems on different sides of a specimen for extracting the plane averaged through the thickness strain components. The framework
presented in Réthoré [298] can be prolonged to 3D kinematics. It should be noted that Rossi et al. [299] recently prolonged the Virtual
Field Method to include 3D data.
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Fig. 1. Cam clay and modified cam clay yield surface in (a) 2D and (b) 3D space. Cam clay over-consolidation, normal, and critical state line with
equivalent model parameters [308,309].
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2.7. According to theory frameworks
2.7.1. Critical state models
In soil mechanics, critical state theory was a considerable breakthrough for developing simple and extensively utilized constitutive
models and Modified Cam Clay (MCC) or Cam Clay (CC) [300,301]. Critical state theory is presently accepted as relevant to clays,
however, a debate exists continuously about sands owing to the grain-crushing impacts [302,303]. The Critical State concept refers to
the ultimate state in which shearing can occur indefinitely with no changes in volume or effective stress. Once the critical state is
′
initiated, shear distortions (εs ) happen with no alterations in mean effective stress (p ), specific volume (ν) or deviatoric stress (q). This
means that the critical state limits the changes in effective stress, deviator stress, and specific volume that can occur during a test. For a
′
given soil, all critical states create a unique line known as the Critical State Line (CSL) determined by the equations q = Mp and ν = Γ−
′
′
′
′
λ ln(p ) in the space (p ,q, ν), where Γ, M, and λ are soil constants. It is worth noting that M and λ is the slope of CSL in q- p and ν - ln(p )
plane. By the first equation, the magnitude of the deviatoric stress q is determined, which is required to maintain the continuous soil
′
flowing as the product of the mean effective stress p and a frictional constant M (capital μ). The second equation indicates that by
increasing the logarithm of the mean effective stress, the specific volume ν filled by the unit volume of flowing particles is decreased
[304–307]. Fig. 1 shows the schematic illustration of these parameters. MCC or CC model variation was represented in two (Fig. 1a, c)
or three (Fig. 1b) dimensional spaces, which explicitly shows the difference in yield surface (Fig. 1a). The NCL shows points at which
soil specimens are normally consolidated, while the CSL shows where soil specimens reach critical states.
Various improvements were presented to this theory including taking into account fabric effects or using the nonlinear formulas for
the critical state line [310,311]. Within the critical state theory framework, a proper constitutive model is also searched for OC clays
[312]. Roscoe and Burland [301] presented the modified Cam-clay (MCC) model as the soil models’ prototype in this context.
Several models were presented in terms of the MCC model for overcoming some of its disadvantages like excessive dilatancy, the
sharp transition from elasticity to plasticity, and the overestimation of the shear resistance of OC clays [313,314]. Considering the
restrictions of the classical theory of plasticity, several studies have tried to look for better hardening rules for modeling cyclic behavior
and smooth elastic-plastic transition from the 1960s [315]. There are two successful and common theories in this regard including (a)
The theory of multi-surface plasticity presented by Mróz [316] and Iwan [317]; and (b) The theory of bounding surface (or two
surfaces) plasticity caused by Dafalias and Popov [318], and Krieg, [319]. These concepts were developed initially for metals.
However, they found applications quickly in modeling geomaterials [320–324]. For the structured soils, some of the conventional
elastoplastic models are compensated by integrating multi-surface kinematic bubble and bounding surfaces models such as stress
history dependency of the material, stiffness nonlinearity in the elastic domain, and reduced size of yield surface caused by bond
degradation. Constitutive models were developed by incorporating the above features [55,56,59,114]. In most of the soil models
oriented by critical state soil mechanics, it is assumed that the soil structure’s isotropic behavior avoids redundant complexity of
mathematical details owing to the anisotropic behavior. Nevertheless, some models are oriented by critical state concepts considering
the soil’s anisotropic nature [325]. The critical state-based soil models are not able to explain the memory of immediate stress history,
the small strain stiffness, and stiffness changes in the estimated response of soils under a non-monotonic loading. However, such soil
model (multi-surface and bounding models) enhances the critical state elasto-plastic models to describe the complex soil performance
under cyclic unloading and loading by incrementing the coupling level between deviatoric and volumetric components of soil
Fig. 2. Relationship between, Artificial intelligence (AI), Machine learning (ML), and Deep Learning (DL).
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performance.
As a kinematic-hardening model, the bubble model consists of a small yield surface enclosing an elastic domain that can move
within but never outside of the outer boundary. By using the bubble model, Cam clay can be extended to describe small strain stiffness,
stiffness degradation with strain in clays that have been overconsolidated, moderate memory of recent stress history, and hysteresis
when loaded cyclically [326,327]. It is challenging for most bubble models to be incorporated into finite element programs. Moreover,
they do not take into account the cementation-caused cohesion intercept. In this regard, some of the critical properties of the clay
structured artificially are not believed mainly the crushing in soil cementation structures. Thus, these models are merely appropriate
for naturally structured soils. However, it can be utilized for describing the constitutive performance of lightly cemented soils.
Nevertheless, the cohesion intercept is considerable for the cement-treated clay [328,329].
2.7.2. Artificial intelligence (machine learning and, deep learning)
Artificial intelligence (AI) is a computer system’s ability to mimic human cognitive functions like problem-solving and learning. A
computer system utilizes math and logic through AI for simulating reasoning, which is used by people for learning from new infor­
mation and making decisions. Moreover, machine learning means computers learn from data utilizing algorithms to conduct a task
without an explicit program. A complex structure of algorithms is used by deep learning. It is modeled on the human brain and allows
the processing of unstructured data like images, documents, and text. Indeed, deep learning is a specialized subset of machine learning
as a subset of artificial intelligence [330–333]. The difference between these words is represented in Fig. 2.
The conventional physics-based models’ limitations have been overcome by advances in the modeling of practical problems uti­
lizing deep learning or machine learning (ML) methods [334–338]. Thus, an alternative method is offered for modeling deep neural
networks (DNN) and complex multiphase systems, which is also a promising method for constitutive modeling [339–345]. For
instance, the practicability of constitutive modeling utilizing neural networks has been demonstrated in pioneering work in various
applications, for example, Ghaboussi et al. [346] used it for modeling concrete, Ellis et al. [347] for modeling sands, Liang and
Chandrashekhara [348] for modeling hyper-elastic materials, Shen et al. [349], and Furukawa and Yagawa [350] employed it for
modeling viscoelastic materials. Presently, recurrent neural networks (RNNs) have been used to model multiscale multi-permeability
poroelasticity. These models are effective for multiscale-plasticity, history-based phenomena, and multiscale one-dimensional bars
[351,352]. Xu et al. [337] recently proposed research in this area by proposing a new neural-network architecture, known as the
Cholesky-factored symmetric positive definite neural network (SPD-NN), for modelling constitutive relations in computational me­
chanics. In this method, the SPD-NN trains a neural network rather than predicting the stress of the material directly, to estimate the
tangent stiffness matrix’s Cholesky factor and calculate the stress in incremental form.
Deep learning (DL) method such as long short-term memory (LSTM) and convolutional neural network (CNN) has been recently
utilized for simulating different features of materials. It has attracted significant attention because of its solid non-linear mapping
capacity [353–355]. Microscopic information can be extracted by CNN and its variants directly from images. They can also present
features from the studied material’s images such as evaluation of the porous media’s permeability and prediction of brittle material
fracture evolution [356,357]. A new deep learning-based modeling strategy was proposed in this regard, where the convolutional
neural network (CNN) was utilized as an image identification algorithm for extraction of the particle information (particle size dis­
tribution PSD and morphology) based on a granular sample’s image [333]. Then, using the bidirectional long short-term memory
(BiLSTM) neural network, the model was trained able to reproduce mechanical behaviors and induce fabric evolutions of the specimen
with equivalent particle information.
It should be noted that the plastic and the elastic strain are not generally distinguished or partitioned by machine learning models.
Thus, Xu et al. [337] introduced a smooth transition function to solve this problem and create a finite transition zone between the
plastic and elastic ranges for an incremental constitutive law made from supervised learning. Machine learning methods were
introduced in previous studies for deducing the yield function and then deducing the distortion hardening or optimal linear mechanism
to minimize the yield function discrepancy [353]. On the other hand, Wang et al. [358] introduced a reinforcement learning algorithm
for deducing the plasticity models’ optimal configuration among all the available options to make completely interpretable plasticity
models. However, these former methods cannot deduce novel hardening or softening mechanisms unbeknownst previously to mod­
elers. To solve this problem, a deep learning framework was proposed as a method for deducing the solutions for Hamilton–Jacobi
equation governing the hardening-softening mechanism by reorganizing the yield function as a developing level set [359].
Training of ML-based models begins with the selection of hyperparameters, and the performance of these models is primarily
determined by the hyperparameters. In order to obtain a well-performing ANN-based model, studies frequently focus on optimizing the
architecture, that is, the number of hidden layers and hidden neurons. Most commonly, users adjust the number of hidden layers and
neurons by trial and error, utilizing their knowledge of the domain and relying heavily on their subjective experience to make ad­
justments. There is currently a lack of methods that are used to prevent overfitting problems in ML-based constitutive models and,
therefore, improve their robustness. In ML-based models, two common methods of preventing overfitting problems are weight decay
and dropout. A penalty function is added to the objective function during weight decay, which constraints the model’s capacity. In
dropout, neurons are randomly deactivated to prevent co-adaptation from becoming excessive [360]. ANNs, Genetic Programming
(GP), and Evolutionary Polynomial Regression algorithms can be integrated with weight decay [361,362], whereas dropout algorithms
are tailored to ANNs [363]. Currently, ML-based constitutive models are constructed with modules (e.g., activation functions), as well
as learning strategies (e.g., optimizer, overfitting prevention, and robustness improvement), that lag behind the development in the ML
field [342]. It is therefore necessary to introduce efficient training methods into the ML domain in order to develop a more robust
constitutive model based on ML.
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3. Applications of constitutive relations
Soil is a complex natural material that exhibits varied behavior. Constitutive relations used to model the geotechnical behavior of
different types of soil widely range from basic linear elastic models to more complex relations that are required to model hardening and
consolidation. Constitutive relations are mathematical representations of the stress-strain behavior of the soil under different loading
conditions. The constitutive models vary from simple to complex relationships to represent the various geotechnical conditions. The
common models used for the simulation of soil behavior are Hooke’s model, Mohr-Coloumb’s model, hyper and hypo elastic model,
cam clay model, etc., and can be extended for various geotechnical applications.
3.1. Single discipline
3.1.1. Mechanical (Soil strength)
3.1.1.1. Slope stability. Slope stability analysis is vital to design safe slopes for any engineering structure like an embankment, natural
slope, landfill, etc. Experimental investigations do not provide an adequate understanding on the internal deformation of failed slopes
or the evolution strain of slope failure [216]. Numerical methods can be very helpful in not only analyzing slope failures but also in
helping to identify the failure response under various loading, drainage and environmental conditions. The finite element method
(FEM) is more popularly used for analyzing the stability of slopes and predicting their behavior, particularly for complex slopes under
non-steady state conditions. The success of the analysis depends greatly on the stress-strain response of the material and its mathe­
matical representation of behavior (i.e) constitutive relationship. Linear and non-linear constitutive relations like elastic,
elasto-plastic, hardening, and softening are used to model the slope failure. Shear strength is the most important factor that governs
slope stability and therefore emphasis on strength and related parameters are of significance for modeling slope stability. Constitutive
relations based on the statistical theory of damage have been successfully used to model the strain-softening behavior of both soil and
rock [364]. The availability of commercial software packages like PLAXIS, Geo-slope, FLAC, etc. presents the possibility of modeling
slopes with a combination of constitutive relations. Chang and Yin [365] modeled anisotropic granular soil in the slip surface of a failed
slope and studied the deformation behavior of each plane by varying the mechanical properties of the granular soil in each of the
directions of the mobilized shear planes. The model successfully predicted the behavior of the granular soil in the slip surface under
diverse stress paths. Kalantari et al. [366] compared the performance of modified cam clay and the Mohr-Coulomb model for slopes
made of fine-grained soil stochastically and reported that the modified cam clay model showed a lower reliability coefficient than the
Mohr-Coulomb model. Zou et al. [367] developed a shear constitutive model that described the deformation behavior and failure
characters of the slip zone to address the gap in soil softening models that did not take into account the physical properties and
stress-strain of the slip zone. Their model predicted the shear –displacement response under different normal stresses with a limited
number of experimental data on shear strength and estimated the dynamic factor of safety.
Numerical models are effective in capturing the hydro-mechanical behavior of soil but there is a discernible difference in the
behavior of the natural and modeled slope due to improper representation of the physical process which can be attributed to a number
of factors like selection of poorly suited the constitutive relation, complexity in slope geometry, unknown initial and boundary con­
ditions, or combination of any of the mentioned factors [368]. Mohsan et al. [368] suggest that measured slope behavior can be used
for the selection of constitutive relations and subsequent modeling. Data assimilation or inverse analysis that adjusts the data based on
field conditions can be used to improve the model significantly as established by authors Brinkgreve et al. [2], Liu et al. [259],
Numerous studies compare the modeled displacement obtained from various constitutive relations with observed field data
[369–371]. Mohsan et al. [368] studied the effect of assimilating the observed field displacement measurement in the calculation of
the factor of safety using the Mohr-Coulomb and hardening soil model under unsteady hydraulic conditions and investigated the
non-linearity in the selected models by comparing the conditions close to failure and far from failure. Mohsan et al. [368] concluded
their study by stating that the hardening soil model tends to estimate the factor of safety with a narrow posterior distribution and also
demands less effort on computation than the Mohr-Coulomb soil model indicating that constitutive models with an influence of
strength in the elastic zone are better suited for effective data assimilation.
3.1.1.2. Bearing capacity. Bearing capacity determination is the fundamental input in the design of shallow foundations. Bearing
capacity is often estimated based on saturated soil conditions neglecting the effect of capillarity and matrix suction that results in a
lower bearing capacity. Constitutive relations for the case of unsaturated soil are in general based on completely saturated soils ac­
counting for the effect of the suction and partial saturation. Georgiadis et al. [372] used a three-dimensional model that used
elasto-plastic constitutive relation to understand the collapse behavior of unsaturated soils over a wide range of stresses and suction
adopting a non-linear relationship between suction and shear strength. This approach provided more flexibility. Vanapalli and
Mohamed [373] proposed an equation to determine the bearing capacity of unsaturated soil taking into account the non-linear
behavior of soil based on laboratory tests on the square footing and validated the equation with experimental studies. Zhang et al.
[252] investigated the effectiveness of the Barcelona basic model for unsaturated soils and reported that this model does not effectively
reflect the behavior of unsaturated soil under both elastic and undrained conditions. The limitation was attributed to inconsistency in
the assumption that wetting and drying curves showed a unique function between matric suction and degree of saturation with the
volume change behavior. McMahon and Bolton [374] adopted the energy method for a linear and elastic von Mises geomaterial along
with the flow to investigate the deformation mechanism under a shallow foundation both before and after failure. Wuttke et al. [375]
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studied the failure surface and validation of deformation caused by plastic load using a hardening model (single surface) using
experimental data from tests conducted on small-scale footings placed on partially saturated sand to formulate the elasto-plastic
macro-element in case of shallow footings. The applied load was vertical and concentric. The parameters affecting soil suction
were also investigated. Ghorbani et al. [376] used the extended modified cam clay model to study the load-displacement behavior of
rigid footings that were loaded statically. Their work effectively predicted the changes in both the mechanical and hydraulic properties
and was able to accommodate problems involving both static and dynamic loading. Tang et al. [377] used the elastic – perfectly plastic
Mohr-Coulomb constitutive relation to investigate the bearing capacity in unsaturated soils under different drainage conditions like
constant water content, constant suction and constant influence of suction on effective stress and strength. The hydraulic hysteresis
and the relation of soil water characteristic curve on change in volume were also considered in the study. Lavasan et al. [378] used
finite difference (FDM) package FLAC that applied the elasto-plastic Mohr-Coulomb constitutive relation to study the ultimate bearing
capacity of closely spaced strip footings. Their study compared the failure mechanisms using enhanced limit equilibrium, kinetic
element method and FDM and concluded that inverted arching that enhanced bearing capacity of closely spaced footings did not form
in soil with low friction angles. Evans and Baker [379] studied the bearing capacity of shallow foundation on unsaturated soil using the
constitutive model by adopting a modified cohesion term in the conventional Vesic’s equation and found that bearing capacity was
under estimated in conventional analysis.
3.1.1.3. Settlement of foundations. Settlement calculations are a routine and mandatory part of foundation design. The linear elastic
behavior of soil is often used to determine the settlement of a particularly shallow foundation. But soil exhibits a non-linear behavior
even for strains of very small magnitude [380]. This makes the choice of stiffness challenging in an elastic analysis as it increases with
strain level. Abed and Vermeer [381] investigated the settlement behavior of footings using Barcelona basic model and modified cam
clay model under saturated conditions and observed that theoretical results matched the results obtained from numerical analysis.
Osman and Bolton [382] modeled the non-linear behavior of soil by adopting a scaled stress-strain curve (from triaxial test data to
footing load-settlement curve) using mobilized strength design (MSD). Klar and Osman [383] further extended the MSD using
constitutive relations elastic perfectly plastic, hyperbolic and truncated power law for minimizing energy. This allowed displacement
to change with loading sequence. In the early stages of loading elastic relation was adopted and in the latter stages used Pradtl’s plastic
solution. Vardanega and Bolton [384] proposed a power-law relation to representing the non-linear response of stress-strain in clay
and reported that their model predicted the settlement behavior with an increase in load. Dougherty and Wood [385] used an extended
Mohr-Coulomb model that combines an isotropic elastic component, yield surface adopted from the Mohr-Coulomb model, and flow
rule from the cam clay model and using an asymptotic strain hardening rule to estimate the settlement of footing placed on the sand.
There was good agreement between the load test results and the predicted settlement values. A considerable amount of creep set­
tlement was also observed indicating the sand in the site is over-consolidated. Le et al. [386] used the Barcelona basic model to
represent the mechanical behavior and van Genuchten’s relation to model flow in unsaturated soil to investigate the differential
settlement caused by rainfall infiltration. They also combined the finite element method (FEM) with random fields to account for
heterogeneity in soil. The results show that structures may experience cracking or tilting when supported by rigid foundations resting
on partially saturated soil characterized by varying distribution of pre-consolidation stress due to infiltration of rain water. McMohan
et al. [374] using the energy method investigated the settlement of the foundation adopting von Mises yield criteria and concluded in
their study that the cavity mechanism is more suited to predict small settlements while Prandtl’s mechanism yields a better upper
bound solution. Oh and Vanapalli [387] developed a technique extending the finite element analysis to investigate the bearing ca­
pacity and settlement behavior of partially saturated fine-grained soils and found that settlement behavior is not affected the earth
pressure when the modified total stress analysis was used as the friction angle adopted was zero and Poison’s ratio of 0.495 yielded best
results for the soil under study.
3.1.1.4. Earth pressure. Design of retaining structures unloading combined in part with deviatoric loading. Also arching behind the
retaining walls produce varying load distributions. Federico et al. [388] predicted the coefficient of earth pressure at rest for soils that
are normally consolidated by adopting the elasto-plastic constitutive model. In order to estimate lower values of the earth pressure
coefficient for fixed friction angles, the surface shape of the plastic potential of the modified cam clay model non-associated flow rule
was adopted. Al Atik and Sitar [389] conducted an experimental and analytical study to investigate both the distribution and
magnitude of earth pressure against a cantilever retaining structure that held a sand backfill of dense nature. An elasto-plastic
hardening constitutive relation was used by Bakr and Ahmad [390] to model the earth pressure on a rigid retaining wall subjected
to seismic loads in PLAXIS. The soil model considers the stiffness of soil, stress dependency, reduction in the shear modulus
non-linearly, and hysteretic damping. Design charts portraying the relation between seismic earth pressure and its movement were
derived accounting for the effect of the height of the wall, acceleration level, and frequency of seismic motion.
3.1.1.5. Soil dynamics. Constitutive relations for soil dynamics are fundamental to understanding the dynamic properties of the soil
and also the soil structure interaction on the application of dynamic loads. Considerable efforts have been taken by geotechnical
engineers to describe the hysteretic non-linear behavior in soils, particularly granular soils. Yang and Ling [391] modified the plastic
model to dependency at pressure level and hardening on being subjected to cyclic load which showed good agreement with experi­
mental data but suffered the limitation of lacking the adaptability to consider the impact of change in void ratio. Masing [392]
developed a one-dimensional stress-strain response under cyclic loading of constant stress type by adopting a hyperbola to express the
skeleton curve and constructed the hysteretic curve by double times method but this procedure had a major drawback where the stress
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calculated was greater than the ultimate stress when cyclic loading was of irregular type. Several authors conducted investigations on
Masing type hysteresis loop to analyze the response in earthquake sites [393–395]. Chong [396] adopted the Davidenkov model to
describe the relation between amplitude and friction angle in relation to the non-linear hysteretic behavior. The study shows that the
model is successful in capturing the stress-dependent dynamic response of the soil and also helps to understand the non-linear hys­
teretic stress-strain response.
3.1.1.6. Soil-structure interaction. Soil-structure interaction (SSI) is of vital importance in geotechnical design. The chief concern in SSI
problems is the mechanical behavior of both soil and the material in contact with the soil. Geotechnical design of retaining structures,
cut-off walls, deep foundations and reinforced earth benefit from the knowledge of SSI parameters. Soil-structure interface is an
important parameter in a constitutive relation and is necessary for obtaining realistic solutions matching field performance in prob­
lems involving soil-structure interaction (SSI). The frictional characteristics of the soil interface are of primary importance for theo­
retical analysis and also design. Numerous constitutive models are used to describe the interface behavior. The elastic perfect-plastic
relation has been adopted to model the interface and the tangential stiffness of the interface is set to zero when the shear stress is
greater than the shear strength [397]. Clough and Duncan [398] developed a non-linear model that has also been used widely for
modeling the behavior of interface but this model is not capable of considering the retrained dilation that occurs in the interface of the
dense sand and structure. Plastic models have also been proposed by some authors [399,400] that take into account the coupling of the
shear and normal behavior. The interface behavior of rock joints was modeled using the disturbed state concept (DSC) and this model
considered the coupling behavior of interfaces [401]. The DSC model can address the dilatancy and strain-softening behavior. Hu and
Pu [402] developed a damage model based on the DSC and simulated a rough interface. Liu et al. [195] attempted to model the SSI of
sandy soil through the concept of critical state soil mechanics addressing the variations in density and normal stress. The proposed
method has the capability to model strain hardening, softening, dilatancy, and stress path dependency of the SSI when sheared. Zhou
and Lu [403] proposed a bi-potential SSI model to assess the strain-softening and dilatancy behavior of the soil under the generalized
potential theory. This model has the advantage of deriving the elasto-plastic matrix without deriving plastic potential and yield
surfaces.
Over the years, researchers started concentrating on the dynamic effects of SSI. Torabi and Rayhani [404] proposed a 3D finite
element model to study the seismic response of SSI and validated the model through numerical results of soft clay under the situation.
Results reveal the tall and slender structures under seismic load suffer dynamic effects of SSI predominantly. Knappett et al. [405]
studied the dynamic response of adjacent buildings through the SSI mechanism. The proposed model could effectively predict the
structure behavior and the model could be effective in designing buildings in earthquake-prone urban areas. Saberi et al. [406] studied
the SSI effects on geostructures under both sandy and gravelly stratum under static and dynamic loading with the ability to predict
their complex mechanisms under various loading and stress path conditions. Mercado et al. [407] highlighted the drawbacks of as­
sumptions and simplifications in soil modeling to reduce the complexity. They compared the effects of seismic loads on a tall building
with linear elastic and non-linear inelastic models. Results showed that the non-linear inelastic model was more accurate in the
prediction. This accumulated knowledge on the SSI and its effects could be helpful in the analysis of various geotechnical applications
like the prediction of a pile or piled raft behavior, tunneling, and shoring activities.
Luccioni [408] proposed a constitutive model for the simulation of deep excavation in clay focusing on the deformations around
them. Hejazi et al. [409] compared the effects and the limitations of three materials models, namely the linear-elastic perfectly plastic
Mohr-Coulomb model, the Hardening soil model, and the Hardening soil model with small-strain stiffness. Validation of results from
real-time results reveals that complexity in the model showed the advantage of better prediction of responses. The outcome revealed
that Bourgeois et al. [410] proposed an SSI-based multiphase model to analyze the settlement of piled raft foundations. Upon vali­
dation with numerical analysis, the model could effectively predict the variations in length and diameter of the pile group in the
squared piled raft foundation. Essa and Desai [411] analyze the dynamic response of individual piles subjected to dynamic loads with
the disturbed state concept. The non-linear model helped in the accurate prediction of dynamic and liquefaction effects of soil and
structure interaction.
3.1.2. Thermal conditions
The thermal model discusses the effect of temperature variations in the soil and its effects. Applications of these models include the
prediction of soil freezing behavior and the influence of thermal piles in the surrounding soil. Mikkola and Hartikainen [412] proposed
a model which works on the principles of continuum and macroscopic thermodynamics to simulate the freezing behavior of saturated
soil. The proposed model works on the principle that the freezing caused suction and it causes moving the water to the freezing zone
and this, in turn, causes an increase in volume. However, this method is not verified through experimental results. Amico et al. [413]
proposed a model for variably-saturated soil. The proposed soil model extends its application to unsaturated soil and could monitor the
liquid-to-solid phase transition. Bai et al. [414] established the numerical equation for the soil freezing characteristic curve, which
explains the behavior of unfrozen water content in the frozen soil. The method extends its relationship between frozen temperature and
the pore radius. The equation and the proposed method are utilized with coupled hydro-thermal-vapor model and it established a
reasonable relationship with the numerical simulations. Bi et al. [414] simulated the thermal behavior of frozen soil by modeling the
frost heave action and thereby the increase in thermal conductivity. Further, the model established series and parallel connections
between pores in the soil, grains of soil, frozen ice, and unfrozen water. This method agreed well with the eight samples of silty clay
chosen for verification.
A thermal pile is a foundation component that helps in bearing the structure as well as acting as a thermal (or heat exchanger) to the
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building. It transfers ground heat to the buildings during winter and it transfers the heat from the building to the ground in summer.
Iodice et al. [415] used a coupled thermo-hydro-mechanical model to predict the behavior of thermal pile in normally consolidated
clay with three soil models, namely, the Mohr-Coulomb model, modified cam clay model and Hypoplastic model under thermal cycles.
Results reveal that higher-end complex models like the hypoplastic model are necessary to simulate the thermal behavior of soil. Song
et al. [416] did a long-term evaluation of thermal piles using a thermo-mechanical model under various climatic conditions. Under
each condition, the thermal variations, axial load on the pile, and displacement of the pile head with thermal variation were compared.
The work showed that the thermal imbalance with nonsymmetrical thermal demands is more concerned than the balanced thermal
demands in the surrounding soil. The induced axial forces are less, whereas the head displacement is more at nonsymmetrical thermal
demands.
3.1.3. Hydrological
Hydrological soil models deal with the water flow characteristics, effects of seepage and dewatering, and other effects of changes in
the saturation conditions of the soil. Que et al. [417] developed a constitutive model to study the effect of seepage on a columnar
jointed rock mass by adopting quadrangular, pentagonal, and hexagonal prism models and by altering the column deflection angles.
Numerical validation suggests that the pentagonal prism model predicted the behavior in an effective way, and the relation between
hydraulic conductivity and confining pressure followed the general seepage rule. Zhang et al. [418] worked on the seepage analysis of
the dike due to an increase in water level and due to rainfall. A coupled two-phase smoothened particle hydrodynamic model was used
to simulate dike failure conditions. The same has been validated through the model test on a slope and dike failure under rainfall
conditions. Water was considered as a quasi-incompressible fluid and soil was in its unsaturated conditions. This model could act as a
preliminary study on the seepage problems associated with dikes. Waterproofed curtains will usually be used as barriers to prevent the
entry of water in deep well excavations. However, tampering and damaging or leakages from the barriers are usually possible under
operating conditions. Wu et al. [419] proposed a 3D solid-fluid coupled model to study the effects of such barrier leakage and found
that the leakage induces excess deformation in the longitudinal direction, a drawdown of groundwater outside the excavation zone,
change in the direction of flow. They found that the factors influencing the leakage are the area and location of leakage, depth of
penetration of the pumping well, and the anisotropic nature of the aquifer.
3.2. Bi-discipline (Coupled)
3.2.1. Thermo-mechanical
The application of thermo-mechanical in geotechnical engineering extends to areas such as nuclear waste disposal, geothermal
structures, petroleum drilling, buried cables, and seasonal variations of pavement subgrade and foundations for furnaces [420]. Liu
and Xing [421] derived a newer non-linear soil model from a double-hardening soil model to outline the behavior of saturated clays.
The model predicted the effects of temperature variations and over-consolidation ratio on the saturated clays and the same had been
verified through experimental results. Hamidi and Khazaei [255] proposed a thermal extension of the Modified Cam clay model to
simulate the behavior of saturated clays up to 100 ◦ C in triaxial conditions. Further, the work gave a clear idea of the reduction of void
ratio with temperature increase highly relies on the stress conditions of the soil element. However, this model holds good only on fixed
higher temperatures and it will not explain the isothermal expansive nature of the soil. Zhou and Ng [422] proposed a
thermo-mechanical elastoplastic model for the soil incorporating the structural effects. The aspects of the soil with the structure are
discussed with the help of three parameters, namely pre-consolidation pressure, thermal softening, and degradation of soil structure.
The model is validated through cyclic heating and cooling of loess at various compaction conditions. Zymnis et al. [423] analyzed the
thermal consolidation properties and accumulation of strain subjected to season thermal variation in the Tsinghua Thermo Soil (TTS)
model and calibrated it with the laboratory results of Geneva clay. However, the authors concluded that further validation with other
types of clay is necessary for fully understanding the thermo-mechanical behavior of the TTS model.
3.2.2. Hydro-mechanical
Zhou and Sheng [424] proposed a 13 parameter-based constitutive model to analyze the behavior of partially saturated soil under
various initial density conditions. The modeling uses a sub-loading surface with hardening parameters in Bishop’s effective
stress-effective degree of saturation space to study the impact of initial densities. This model, in turn, will be reduced to modified cam
clay under fully saturated and normally consolidated conditions. A similar approach was attempted by Lei et al. [425] to predict the
unsaturated soil’s response at various over-consolidation ratios. Gholizadeh and Latifi [426] proposed a constitutive model addressing
both saturated and unsaturated soil having friction and cohesion characteristics. The models work with deviatoric and isotropic
mechanisms at multi-surface and conventional plasticity frameworks, respectively. Under the concept of the bounding surface, the
hydraulic part works in the critical state framework. Qiao et al. [427] proposed a micro and macro-structured hydro-mechanical model
for compacted bentonite. The model incorporates mechanical and water-retention models under micro and macro scales, respectively.
It is observed upon validation that elastoplastic behavior is a characteristic of both micro and macrostructure and the water retention
phenomenon is unique for both structures. Tang et al. [428] proposed a hypoplastic material-based coupled hydro-mechanical model
for understanding rainfall-induced landslides. The model is tested with various rainfall types and intensities and the model is verified
with the Baishuihe landslide in South China. This model holds the advantage of having simple and fewer parameters and would be
helpful when there are fewer rainfall data or parameters available. However, the model does not account for the unsaturated con­
ditions and behavior of soil.
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3.2.2.1. Consolidation, swelling and collapse. Constitutive modeling of partially saturated soil with non-linear compression lines tends
to predict the collapse potential inaccurately. Georgiadis et al. [429] proposed a 22-parameter constitutive model which can be ob­
tained through a series of laboratory tests were proposed. The proposed model described the wetting-induced collapse with great
accuracy. Liu et al. [430] investigated stress-induced deformation and time-dependent swelling behavior of expansive soils using
one-dimensional constitutive modeling. The advantage of this proposed model is that the parameters can be obtained through routine
tests under general practice. Despite the simplicity of the model, this leads the way for modeling soil that possesses both high
compressive and swelling behavior.
3.2.2.2. Liquefaction. Liquefaction is a phenomenon where saturated or partially saturated sand loses its shear strength and starts
behaving like a fluid. Liquefaction is a dynamic phenomenon that mandates numerical modeling. Material modeling of liquefaction in
soil has been extensively studied by researchers since the early and late 1980s. However, over time many new and innovative de­
velopments were made in the initial models of liquefaction analysis. Liyanapathirana and Poulos [431] used an effective stress-based
model to ascertain the loss of shear strength and the development of pore pressure upon liquefaction. The new method developed has
the advantage of fewer parameters than the existing method. Moreover, the new methods showed higher accuracy in the prediction of
pore pressure ratio than the traditional equivalent cycle method and the same had been verified with the parameters of the Kobe
earthquake. Mroz et al. [432] proposed a superior sand model to predict liquefaction under saturated conditions in loose sand under
monotonic, cyclic loading. The model predicted plane strain compression behavior under drained and undrained conditions. Najma
and Latifi [433] developed a model for a flow liquefaction study and the results revealed that the stress ratio that triggers liquefaction
depends on void ratio, state parameters, and other material-dependent parameters. This method divides stress paths into two parts,
namely the one which retains the original soil structure and the other which collapses the soil for all cyclic, monotonic, and constant
deviatoric stresses. Choobbasti et al. [434] analyzed the effect of fines content in the liquefaction potential of sand using various
experimental analyses and through Finn’s constitutive model, it is observed that at lower content of fines (up to 20%), the sand
behavior was predominant however with the increase in the percentage of fines, the critical slope decreases as the liquefaction
resistance increased. In the case of liquefaction behavior being cyclic mobility, elasto-plastic constitutive models can be successfully
used but if they are of flow liquefaction, the behavior is complex. Lü et al. [435] used a constitutive relation that allows phase transition
criteria to sense the start of liquefaction and uses both elastoplastic and fluid relations in a framework to model liquefaction. They
observed in their study that the developed model described the behavior of soil in solid and fluid phases smoothly transiting from solid
to liquid phase.
3.3. Multi-discipline
3.3.1. Thermo-hydro-mechanical (THM) model
THM model is a multi-discipline soil model which involves complex relations among the parameters. Classic examples of THM
models include handling nuclear waste, buried high-voltage cables, and heat dissipation embankments. These models generally
involve the effect of suction due to temperature variations in the soil. Collin et al. [436] studied the THM effects on clay barriers with
unsaturated soil conditions to handle nuclear wastes. Chemkhi et al. [437] studied the effect of drying on the deformable soil media
and the stress variations it underwent. Research showed that the deformation velocity of the solids and pressure change in the liquid
phase remained the critical parameters of the coupled model. Liu and Yu [438] applied the THM approach on porous soil under frost
actions and observed that heat transfer caused variation in stress and hydraulic field more predominantly when a change of phase was
observed in the pores. Kang et al. [439] investigated the idea of using coupled THM models to study the freezing-thawing effect of rock
mass in cold conditions for gas storage at lower temperatures. The model predicted the fall of the freezing point with a hike in the
pressure and the relationship is fairly linear. A new parameter called frozen ratio is proposed based on the principles of phase-change
theory and thermodynamics. Zhuang et al. [440] studied the THM model for compressed air energy storage in a jointed rock mass.
Fig. 3. THM model of soil (Gens et al. (2017) [??????].
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They applied the principles of static equilibrium, mass and energy conservation to obtain the governing equation. They found that
compressed air energy storage is not feasible in the jointed rock mass and as it results in energy loss. Gens et al. [441] analyzed the
consequences of thermal loading on the Opalinus clay of Switzerland, which served as the source soil at the disposal site of nuclear
waste. They studied thermal conductivity, thermal-induced pore pressure, and mechanical effects on the soil in connection with the
THM model. Fig. 3 shows the THM model involved in the thermal loading of soil. Yang and Bai [442] coupled the idea of thermo­
dynamics with granular soil hydrodynamics, and predicted the plastic deformation and effective stress on unsaturated soil, thereby
providing a framework on complex geotechnical problems under unsaturated soil conditions. With several works started emerging in
coupled THM soil behavior, this model still remains as one of the complex models to simulate soil conditions.
4. Discussions
4.1. Searching for gap studies
Soils exhibit spontaneous and erratic reactions during loading and unloading and also when exposed to moisture effects during soil
reconstitution and some natural occurrences like percolation, seepage, impregnation, capillary rise, etc. Due to this reason, it is a
difficult geomaterial to handle both at the design and construction stages. The dry, partially saturated, and fully saturated states of soil
are studied with utmost design considerations more especially when stress-strain relationships are concerned due to the effect of
moisture conditions at the three different hydraulic states. In this work, soil constitutive modeling in both relations and applications in
solving geotechnical engineering problems have been extensively reviewed. This has been done with emphasis: on soil type; soft clayconsolidated and hard clay-overconsolidated conditions, granular soil, cemented soil, soft rock, intact and fractured rock, on ground
water; dry soil, saturated soil, and drained/undrained soil conditions, on loading conditions; monotonic and cyclic loading, impact
loading, and seismic loading, on structural behavior; linear and nonlinear, time-independence (elastoplastic conditions), and timedependent (viscoplastic conditions); on constitutive relation discipline; bi-discipline (coupled conditions) like thermomechanical
(TM) and hydromechanical (HM) conditions, multidiscipline like thermohydromechanical (THM) conditions and on dimensions; one,
two and three-dimensional modeling relations. By extension also, the soil constitutive applications were reviewed on the bases of:
single discipline dealing with soil mechanical properties constitutive modeling which included slope stability problems, bearing ca­
pacity, settlement of foundations, earth pressure problems, soil dynamics, soil structure interaction, thermal and hydrological con­
ditions; bi-discipline (coupled problems) which solve problems related to thermo-mechanical (freeze/thaw conditions) and
hydromechanical (consolidation, collapse, and liquefaction) conditions in soils and rocks and multi-discipline constitutive models
which solve complex problems related to thermo-hydromechanical (THM) conditions in soils and rocks. It can be observed from other
related reviews that have been done in recent years that none has been as extensive as this present work and as current in exposing
recent developments in soil constitutive modeling for instance the application of HM models, THM models, and SPH in solving
geotechnical engineering problems.
4.2. Recommend new trends in developing and applying constitutive relations
Constitutive equations are the ones that simply describe the relationship between stresses and strains in geomechanics material
under different environmental or boundary conditions. The existing constitutive relation in previous literature could be categorized
according to soil type, groundwater, type of loading, structural behavior, discipline, dimensions, and so on. From reviewed literature,
it can be determined which relations dealing with soil type, ground water, loading type, behavior, discipline and dimensions, we have
model strengths compared to others. In each item, there are multiple equations that particularly deal with a specific phenomenon. We
have tried to mention some of the well-known equation territories in this paper. Choosing each of the models directly depends on the
geotechnical and geophysical problems, materials, loading types, and many other parameters. However, a comparison has been
conducted in this paper for some of the abovementioned items. Due to the specific characterization of existing constitutive equations,
the designer or constructor could choose an appropriate model group and then, using each subsection’s content and potentially
comparison, find his/her relevant models. We have tried to extensively describe the previous models and some of their advantages and
disadvantages that could help anyone find the best one based on its choice related to the problem territory. But the best attractive part
of this paper is the nearly all recent models (more than 300 constitutive models) in 6 classes that have been extensively reviewed.
Focusing on this information could attract many researchers and let them know about the recent progress in their fields of interest.
5. Conclusions
From the foregoing overview of literature solving geotechnical engineering problems by adopting soil constitutive modeling re­
lations and applications, it can be concluded that;
• In slope stability problems, the hardening soil model tends to estimate the factor of safety with a narrow posterior distribution and
also demands less effort on computation than the Mohr-Coulomb soil model indicating that constitutive models with an influence of
strength in the elastic zone are better suited for effective data assimilation.
• In bearing capacity problems, elasto-plastic Mohr-Coulomb constitutive relation studies the ultimate bearing capacity of closely
spaced strip footings and compares the failure mechanisms using enhanced limit equilibrium, kinetic element method and FDM and
uses inverted arching that enhanced bearing capacity of closely spaced footings which do not form in soils with low friction angles.
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• In settlement problems, Barcelona basic model has been proposed to best solve the mechanical behavior and van Genuchten model
to solve flow in unsaturated soil and investigate the differential settlement caused by rainfall infiltration. It is also noted that these
were combined in the finite element method (FEM) with random fields to account for heterogeneity in soil.
• Validation of results from real-time results reveals that complexity in the model showed the advantage of better prediction of
responses
• A non-linear model helps in the accurate prediction of dynamic and liquefaction effects of soil and structure interaction
• Higher-end complex models like the hypoplastic model are necessary to simulate the thermal behavior of soil
• The work showed that the thermal imbalance with nonsymmetrical thermal demands is more concerned than the balanced thermal
demands at the surrounding soil
• A 3D solid-fluid coupled model has been proposed to best study the effects of barrier leakage though with induced excess defor­
mation in the longitudinal direction, a drawdown of groundwater outside the excavation zone, and change in the direction of flow
• Thermo-mechanical elastoplastic model, thermal consolidation properties and accumulation of strain subjected to season thermal
variation in Tsinghua Thermo Soil (TTS) model and calibrated with the laboratory results of Geneva clay has been proposed to best
model soil thermal coupled conditions. It is equally concluded that further validation with other types of clay is necessary for fully
understanding the thermo-mechanical behavior of the TTS model.
• Hypoplastic material-based coupled hydro-mechanical model for understanding rainfall-induced landslides is also proposed to best
model geophysical flows like landslides. The model is tested with various rainfall types and intensities and the model is verified
with the Baishuihe landslide in South China. This model holds the advantage of having simple and fewer parameters and would be
helpful when there are fewer rainfall data or parameters available. However, the model does not account for the unsaturated
conditions and behavior of soil.
• Finally, hydromechanical (HM) models are better suited for geotechnical applications, while thermo-hydromechanical (THM)
models are better suited to solving freeze/thaw and thermal piles problems and these are proven with high performance and
flexibility.
Author contribution statement
All authors listed have significantly contributed to the development and the writing of this article.
Funding statement
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Data availability statement
No data was used for the research described in the article.
Declaration of interest’s statement
The authors declare no conflict of interest.
References
[1] S.T. Kok, B.B.K. Huat, J. Noorzaei, M.S. Jaafar, S.S. Gue, A case study of passive piles failure in open excavation, DFI J. The J. Deep Found. Inst. 3 (2) (2009)
49–56, https://doi.org/10.1179/dfi.2009.011.
[2] R.B. Brinkgreve, Selection of soil models and parameters for geotechnical engineering application, in: Soil constitutive models (Ed.), Evaluation, Selection, and
Calibration, 2005, pp. 69–98.
[3] W.F. Chen, G.Y. Baladi, Soil Plasticity: Theory and Implementation, Elsevier, 1985.
[4] M. Liingaard, A. Augustesen, P.V. Lade, Characterization of models for time-dependent behavior of soils, Int. J. GeoMech. 4 (3) (2004) 157–177, https://doi.
org/10.1061/(ASCE)1532-3641(2004, 4:3(157).
[5] T. Adachi, F. Oka, An elasto-plastic constitutive model for soft rock with strain softening, Int. J. Numer. Anal. Methods GeoMech. 19 (4) (1995) 233–247,
https://doi.org/10.1002/NAG.1610190402.
[6] B.L. Kutter, N. Sathialingam, Elastic-viscoplastic modelling of the rate-dependent behaviour of clays, Geotechnique 42 (3) (1992) 427–441, https://doi.org/
10.1680/GEOT.1992.42.3.427.
[7] J. Yin, J. Graham, Viscous–elastic–plastic modelling of one-dimensional time-dependent behaviour of clays, Can. Geotech. J. 26 (2) (1989) 199–209, https://
doi.org/10.1139/T89-029.
[8] M.J. Pender, A model for the behaviour of overconsolidated soil, Geotechnique 28 (1) (1978) 1–25, https://doi.org/10.1680/GEOT.1978.28.1.1.
[9] Yannis F. Dafalias, L.R. Herrmann, Bounding surface plasticity. II: application to isotropic cohesive soils, J. Eng. Mech. 112 (12) (1986) 1263–1291, https://
doi.org/10.1061/(ASCE)0733-9399(1986, 112:12(1263.
[10] K. Hashiguchi, Subloading surface model in unconventional plasticity, Int. J. Solid Struct. 25 (8) (1989) 917–945, https://doi.org/10.1016/0020-7683(89)
90038-3.
[11] Akira Asaoka, M. Nakano, T. Noda, Superloading yield surface concept for highly structured soil behavior, Soils Found. 40 (2) (2000) 99–110, https://doi.org/
10.3208/SANDF.40.2_99.
[12] G.R. McDowell, K.W. Hau, A generalised Modified Cam clay model for clay and sand incorporating kinematic hardening and bounding surface plasticity,
Granular Matter 6 (1) (2004) 11–16, https://doi.org/10.1007/S10035-003-0152-8, 2004 6:1.
[13] Yin, Zhen Yu, C.S. Chang, Non-uniqueness of critical state line in compression and extension conditions, Int. J. Numer. Anal. Methods GeoMech. 33 (10) (2009)
1315–1338, https://doi.org/10.1002/NAG.770.
19
Heliyon 9 (2023) e14465
K.C. Onyelowe et al.
[14] X.S. Li, Y.F. Dafalias, Anisotropic critical state theory: role of fabric, J. Eng. Mech. 138 (3) (2012) 263–275, https://doi.org/10.1061/(ASCE)EM, 19437889.0000324.
[15] Y.-P. Yao, L.-M. Kong, A.-N. Zhou, J.-H. Yin, Time-dependent unified hardening model: three-dimensional elastoviscoplastic constitutive model for clays,
J. Eng. Mech. 141 (6) (2014), 4014162, https://doi.org/10.1061/(ASCE)EM, 1943-7889.0000885.
[16] Y.-P. Yao, N.-D. Wang, Transformed stress method for generalizing soil constitutive models, J. Eng. Mech. 140 (3) (2014) 614–629, https://doi.org/10.1061/
(ASCE)EM, 1943-7889.0000685.
[17] Y.P. Yao, W. Hou, A.N. Zhou, UH Model: Three-Dimensional Unified Hardening Model For Overconsolidated Clays, 2015, pp. 451–469, https://doi.org/
10.1680/GEOT.2007.00029, 59(5).
[18] Z. Gao, J. Zhao, Z.-Y. Yin, Dilatancy relation for overconsolidated clay, Int. J. GeoMech. 17 (5) (2017), 6016035, https://doi.org/10.1061/(ASCE)GM, 19435622.0000793.
[19] S. Jocković, M. Vukićević, Bounding surface model for overconsolidated clays with new state parameter formulation of hardening rule, Comput. Geotech. 83
(2017) 16–29, https://doi.org/10.1016/J.COMPGEO.2016.10.013.
[20] A. Das, P.K. Bajpai, A hypo-plastic approach for evaluating railway ballast degradation, Acta Geotechnica 13 (5) (2018) 1085–1102, https://doi.org/10.1007/
S11440-017-0584-7.
[21] C. Peng, X. Guo, W. Wu, Y. Wang, Unified modelling of granular media with smoothed particle hydrodynamics, Acta Geotechnica 11 (6) (2016) 1231–1247,
https://doi.org/10.1007/S11440-016-0496-Y.
[22] C. Peng, S. Wang, W. Wu, H. Yu, sui, Wang, C., & Chen, J. yu, LOQUAT: an open-source GPU-accelerated SPH solver for geotechnical modeling, Acta
Geotechnica 14 (5) (2019) 1269–1287, https://doi.org/10.1007/S11440-019-00839-1.
[23] C. Peng, W. Wu, H. Yu, sui, & Wang, C, A SPH approach for large deformation analysis with hypoplastic constitutive model, Acta Geotechnica 10 (6) (2015)
703–717, https://doi.org/10.1007/S11440-015-0399-3.
[24] W. Wu, S. Wang, G. Xu, J. Qi, D. Zhang, J.R. Kim, Numerical simulation of a CAES pile with hypoplasticity, in: Springer Series in Geomechanics and
Geoengineering, 2019, pp. 242–249, https://doi.org/10.1007/978-3-030-14987-1_29.
[25] D.D. Porcino, V. Diano, T. Triantafyllidis, T. Wichtmann, Predicting undrained static response of sand with non-plastic fines in terms of equivalent granular
state parameter, Acta Geotechnica 15 (4) (2020) 867–882, https://doi.org/10.1007/S11440-019-00770-5.
[26] M. Tafili, T. Triantafyllidis, AVISA: anisotropic visco-ISA model and its performance at cyclic loading, Acta Geotechnica 15 (9) (2020) 2395–2413, https://doi.
org/10.1007/S11440-020-00925-9.
[27] S. Wang, W. Wu, A simple hypoplastic model for overconsolidated clays, Acta Geotechnica (2020), https://doi.org/10.1007/S11440-020-01000-Z.
[28] W. Wu, E. Bauer, D. Kolymbas, Hypoplastic constitutive model with critical state for granular materials, Mech. Mater. 23 (1) (1996) 45–69, https://doi.org/
10.1016/0167-6636(96)00006-3.
[29] D. Kolymbas, I. Herle, Shear and objective stress rates in hypoplasticity, Int. J. Numer. Anal. Methods GeoMech. 27 (9) (2003) 733–744, https://doi.org/
10.1002/NAG.297.
[30] A. Singha, Al Faruque, , M. A., R. Balachandar, Vortices and large-scale structures in a rough open-channel flow subjected to bed suction and injection, J. Eng.
Mech. 138 (5) (2012) 491–501, https://doi.org/10.1061/(ASCE)EM, 1943-7889.0000354.
[31] Z. Wan, C. Song, S. Xue, L. Xie, Elastoplastic constitutive model describing dilatancy behavior of overconsolidated clay, Int. J. GeoMech. 21 (3) (2021),
4021008, https://doi.org/10.1061/(ASCE)GM, 1943-5622.0001947.
[32] S. Kimoto, F. Oka, An elasto-viscoplastic model for clay considering destructuralization and consolidation analysis of unstable behavior, Soils Found. 45 (2)
(2005) 29–42, https://doi.org/10.3208/SANDF.45.2_29.
[33] Y.F. Jin, Z.Y. Yin, Y. Riou, P.Y. Hicher, Identifying creep and destructuration related soil parameters by optimization methods, KSCE J. Civil Eng. 21 (4) (2016)
1123–1134, https://doi.org/10.1007/S12205-016-0378-8, 2017 21:4.
[34] Z. Yin, P. Hicher, Y. Jin, Practice of constitutive modelling for saturated soils. https://books.google.com/books?
hl=en&lr=&id=vqT8DwAAQBAJ&oi=fnd&pg=PR5&dq=Practice+of+Constitutive+Modelling+for+Saturated+
Soils&ots=TT3rqN0gYl&sig=WhH3lYU2zHwwEd7xShCf34QZ7g4, 2020.
[35] A. Augustesen, M. Liingaard, P.V. Lade, Evaluation of time-dependent behavior of soils, Int. J. GeoMech. 4 (3) (2004) 137–156, https://doi.org/10.1061/
(ASCE)1532-3641(2004, 4:3(137).
[36] H. Karimpour, P.V. Lade, Creep behavior in Virginia Beach sand, Can. Geotech. J. 50 (11) (2013) 1159–1178, https://doi.org/10.1139/CGJ-2012-0467.
[37] S. Murayama, Formulation of stress-strain-time behavior of soils under deviatoric stress condition, Soils Found. 23 (2) (1983) 43–57, https://doi.org/10.3208/
SANDF1972.23.2_43.
[38] P. Jop, Y. Forterre, O. Pouliquen, A constitutive law for dense granular flows, Nature 441 (7094) (2006) 727–730, https://doi.org/10.1038/nature04801,
2006 441:7094.
[39] M. Tankeo, P. Richard, É. Canot, Analytical solution of the μ(I) - rheology for fully developed granular flows in simple configurations, Granul. Matter 15 (6)
(2013) 881–891, https://doi.org/10.1007/S10035-013-0447-3/FIGURES/11.
[40 ] H. Chen, C.F. Lee, Runout analysis of slurry flows with Bingham model, J. Geotech. Geoenviron. Eng. 128 (12) (2002) 1032–1042, https://doi.org/10.1061/
(ASCE)1090-0241(2002, 128:12(1032).
[41] C. Di Prisco, S. Imposimato, I. Vardoulakis, Mechanical modelling of drained creep triaxial tests on loose sand, Geotechnique 50 (1) (2000) 73–82, https://doi.
org/10.1680/GEOT.2000.50.1.73.
[42] F. Tatsuoka, M. Ishihara, H. Di Benedetto, R. Kuwano, Time-dependent shear deformation characteristics of geomaterials and their simulation, Soils Found. 42
(2) (2002) 103–129, https://doi.org/10.3208/SANDF.42.2_103.
[43] K. Wang, W.C. Sun, A multiscale multi-permeability poroplasticity model linked by recursive homogenizations and deep learning, Comput. Methods Appl.
Mech. Eng. 334 (2018) 337–380, https://doi.org/10.1016/J.CMA.2018.01.036.
[44] K. Liu, J.H. Yin, W.B. Chen, W.Q. Feng, C. Zhou, The stress–strain behaviour and critical state parameters of an unsaturated granular fill material under
different suctions, Acta Geotechnica 15 (12) (2020) 3383–3398, https://doi.org/10.1007/S11440-020-00973-1/FIGURES/13.
[45] T. Luo, D. Chen, Y.P. Yao, A.N. Zhou, An advanced UH model for unsaturated soils, Acta Geotechnica 15 (1) (2020) 145–164, https://doi.org/10.1007/
S11440-019-00882-Y/FIGURES/24.
[46] Yao, Yang Ping, Liu, L., T. Luo, A constitutive model for granular soils, Sci. China Techn. Sci. 61 (10) (2018) 1546–1555, https://doi.org/10.1007/S11431017-9205-8, 2018 61:10.
[47] Yao, Yang Ping, Wang, N. B., D. Chen, UH model for granular soils considering low confining pressure, Acta Geotechnica 16 (6) (2021) 1815–1827, https://
doi.org/10.1007/S11440-020-01084-7/TABLES/4.
[48] S. Leroueil, P.R. Vaughan, The general and congruent effects of structure in natural soils and weak rocks, Geotechnique 40 (3) (1990) 467–488, https://doi.
org/10.1680/GEOT.1990.40.3.467.
[49] A. Gens, R. Nova, Conceptual Bases For A Constitutive Model For Bonded Soils And Weak Rocks. Geotechnical Engineering Of Hard Soils-Soft Rocks, 1993,
pp. 485–494.
[50]] A. Vatsala, R. Nova, B.R.S. Murthy, Elastoplastic model for cemented soils, J. Geotech. Geoenviron. Eng. 127 (8) (2001) 679–687, https://doi.org/10.1061/
(ASCE)1090-0241(2001)127:8(679.
[51]] T. Adachi, M. Okano, A constitutive equation for normally consolidated clay, Soils Found. 14 (4) (1974) 55–73, https://doi.org/10.3208/SANDF1972.14.4_
55.
[52] Toshihisa Adachi, F. Oka, Constitutive equations for normally consolidated clay based on elasto-viscoplasticity, Soils Found. 22 (4) (1982) 57–70, https://doi.
org/10.3208/SANDF1972.22.4_57.
[53] N.N.S. Yapage, D.S. Liyanapathirana, A review of constitutive models for cement-treated clay, Int. J. Geotech. Eng. 13 (6) (2019) 525–537, https://doi.org/
10.1080/19386362.2017.1370878.
20
Heliyon 9 (2023) e14465
K.C. Onyelowe et al.
[54] K. Kasama, H. Ochiai, N. Yasufuku, On the stress-strain behaviour of lightly cemented clay based on an extended critical state concept, Soils Found. 40 (5)
(2000) 37–47, https://doi.org/10.3208/SANDF.40.5_37.
[55] M. Kavvadas, A. Amorosi, A constitutive model for structured soils, Geotechnique 50 (3) (2000) 263–273, https://doi.org/10.1680/GEOT.2000.50.3.263.
[56] M. Rouainia, D.M. Wood, A kinematic hardening constitutive model for natural clays with loss of structure, Geotechnique 50 (2) (2000) 153–164, https://doi.
org/10.1680/GEOT.2000.50.2.153.
[57] Simon J. Wheeler, A. Näätänen, M. Karstunen, M. Lojander, An anisotropic elastoplastic model for soft clays, Can. Geotech. J. 40 (2) (2003) 403–418, https://
doi.org/10.1139/T02-119.
[58] K. Lee, D. Chan, K. Lam, Constitutive model for cement treated clay in a critical state frame work, Soils Found. 44 (3) (2004) 69–77, https://doi.org/10.3208/
SANDF.44.3_69.
[59] B. Baudet, S. Stallebrass, A constitutive model for structured clays, Geotechnique 54 (4) (2004) 269–278, https://doi.org/10.1680/GEOT.2004.54.4.269.
[60] S. Horpibulsuk, M.D. Liu, D.S. Liyanapathirana, J. Suebsuk, Behaviour of cemented clay simulated via the theoretical framework of the Structured Cam Clay
model, Comput. Geotech. 37 (1–2) (2010) 1–9, https://doi.org/10.1016/J.COMPGEO.2009.06.007.
[61] M. Taiebat, Y.F. Dafalias, R. Peek, A destructuration theory and its application to SANICLAY model, Int. J. Numer. Anal. Methods GeoMech. 34 (10) (2010)
1009–1040, https://doi.org/10.1002/NAG.841.
[62] L.D. Nguyen, B. Fatahi, H. Khabbaz, A constitutive model for cemented clays capturing cementation degradation, Int. J. Plast. 56 (2014) 1–18, https://doi.org/
10.1016/J.IJPLAS.2014.01.007.
[63] C. Swan, C. Ladd, J. Germaine, G. Anderson, Physical mechanisms controlling the strength-deformation behavior of frozen sand: Part 2. Massachusetts inst of
tech cambridge dept of civil engineering. https://apps.dtic.mil/sti/citations/ADA296303, 1995.
[64] S. Li, Y. Lai, S. Zhang, D. Liu, An improved statistical damage constitutive model for warm frozen clay based on Mohr–Coulomb criterion, Cold Reg. Sci.
Technol. 57 (2–3) (2009) 154–159, https://doi.org/10.1016/J.COLDREGIONS.2009.02.010.
[65] Z.W. Zhu, J. Ning, W.M. Physics, A constitutive model of frozen soil with damage and numerical simulation for the coupled problem. Science China Physics,,
Mech. Astron. 53 (2010) 699–711, https://doi.org/10.1007/s11433-010-0169-z.
[66] Y. Lai, L. Jin, X. Chang, Yield criterion and elasto-plastic damage constitutive model for frozen sandy soil, Int. J. Plast. 25 (6) (2009) 1177–1205, https://doi.
org/10.1016/J.IJPLAS.2008.06.010.
[67] Rotta Loria, , A. F., B. Frigo, B. Chiaia, A non-linear constitutive model for describing the mechanical behaviour of frozen ground and permafrost, Cold Reg. Sci.
Technol. 133 (2017) 63–69, https://doi.org/10.1016/J.COLDREGIONS.2016.10.010.
[68] M. Furnish, Measuring Static and Dynamic Properties of Frozen Silty Soils. (No. SAND98–1497). Sandia National Labs., Albuquerque, NM (US); Sandia
National Labs, CA (US), Livermore, 1998. https://www.osti.gov/biblio/697.
[69] S.S. Hu, D.R. Wang, Dynamic constitutive relation of concrete under impact, Explos. Shock Waves 22 (3) (2002) 242–246, https://scholar.google.com/scholar?
hl=en&as_sdt=0%2C5&q=Hu+S+and+Wang+D+%282002%29+Dynamic+constitutive+relation+of+concrete+under+impact.+Explosion+and+Shock+
Waves+3%3A+242–246.&btnG=.
[70] Q.Y. Ma, Experimental analysis of dynamic mechanical properties for artificially frozen clay by the split Hopkinson pressure bar, J. Appl. Mech. Tech. Phys. 51
(3) (2010) 448–452. https://idp.springer.com/authorize/casa?redirect_uri=https://link.springer.com/content/pdf/10.1007/s10808-010-0060-8.pdf&casa_
token=8dSsZ61rB8gAAAAA: X8LSqFz2zfjdHbeMVAAFWsG3mRTeOcNEN9u5orQHk9JMFXpNTrLbyn6n5qqCL-3AGxthWjMCO-JNCX6J.
[71] Zhiwu Zhu, G. Kang, Y. Ma, Q. Xie, D. Zhang, J. Ning, Temperature damage and constitutive model of frozen soil under dynamic loading, Mech. Mater. 102
(2016) 108–116, https://doi.org/10.1016/J.MECHMAT.2016.08.009.
[72] C.H. Chu, L.L. Wang, D. Xu, A nonlinear thermo-viscoelastic constitutive equation for thermoset plastics at high strain rates, in: In Proceedings of the
International Conference on Nonlinear Mechanics, 1985, pp. 92–97, in: https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Chu+C-H%2C+Wang
+L+and+Xu+D+%281985%2C+October%29.+A+nonlinear+thermo-viscoelastic+constitutive+equation+for+thermoset+plastics+at+high+strain+rates.
+In%3A+Chien+WZ+%28ed.%29+Proceedings+of+the+Intern.
[73] Q. Xie, Z. Zhu, G. Kang, A dynamic micromechanical constitutive model for frozen soil under impact loading, Acta Mech. Solida Sin. 29 (1) (2016) 13–21,
https://doi.org/10.1016/S0894-9166(16)60003-4.
[74] P.V. McLaughlin, S.C. Batterman, Limit behavior of fibrous materials, Int. J. Solid Struct. 6 (10) (1970) 1357–1376, https://doi.org/10.1016/0020-7683(70)
90068-5.
[75] R.L. Michalowski, A. Zhao, Continuum versus structural approach to stability of reinforced soil, J. Geotech. Eng. 121 (2) (1995) 152–162, https://doi.org/
10.1061/(ASCE)0733-9410(1995, 121:2(152).
[76] R.L. Michalowski, A. Zhao, Failure of fiber-reinforced granular soils, J. Geotech. Eng. 122 (3) (1996) 226–234, https://doi.org/10.1061/(ASCE)0733-9410
(1996, 122:3(226).
[77] S. Pietruszczak, G.N. Pande, Limit condition for unidirectionally reinforced soils, in: Numerical Models in Geomechanics, 1995, pp. 237–242.
[78] J. Li, D.W. Ding, Nonlinear elastic behavior of fiber-reinforced soil under cyclic loading, Soil Dynam. Earthq. Eng. 22 (9–12) (2002) 977–983, https://doi.org/
10.1016/S0267-7261(02)00122-7.
[79] A. Sawicki, Plastic limit behavior of reinforced earth, J. Geotech. Eng. 109 (7) (1983) 1000–1005, https://doi.org/10.1061/(ASCE)0733-9410(1983, 109:7
(1000).
[80] H.W. Zhou, C.P. Wang, B.B. Han, Z.Q. Duan, A creep constitutive model for salt rock based on fractional derivatives, Int. J. Rock Mech. Min. Sci. 48 (1) (2011)
116–121, https://doi.org/10.1016/J.IJRMMS.2010.11.004.
[81] P. Cao, W. Youdao, W. Yixian, Y. Haiping, Y. Bingxiang, Study on nonlinear damage creep constitutive model for high-stress soft rock, Environ. Earth Sci. 75
(10) (2016), https://doi.org/10.1007/s12665-016-5699-x.
[82] K.T. Chang, L. Ge, H.H. Lin, Slope creep behavior: observations and simulations, Environ. Earth Sci. 73 (1) (2015) 275–287, https://doi.org/10.1007/S12665014-3423-2.
[83] B. Mishra, P. Verma, Uniaxial and triaxial single and multistage creep tests on coal-measure shale rocks, Int. J. Coal Geol. 137 (2015) 55–65, https://doi.org/
10.1016/J.COAL.2014.11.005.
[84] M. Li, H. Zhang, W. Xing, Z. Hou, P. Were, Study of the relationship between surface subsidence and internal pressure in salt caverns, Environ. Earth Sci. 73
(11) (2015) 6899–6910, https://doi.org/10.1007/S12665-015-4405-8.
[85] F. Wu, J.F. Liu, J. Wang, An improved Maxwell creep model for rock based on variable-order fractional derivatives, Environ. Earth Sci. 73 (11) (2015)
6965–6971, https://doi.org/10.1007/S12665-015-4137-9.
[86] W. Chen, H. Konietzky, Simulation of heterogeneity, creep, damage and lifetime for loaded brittle rocks, Tectonophysics 633 (1) (2014) 164–175, https://doi.
org/10.1016/J.TECTO.2014.06.033.
[87] H.Y. Liu, S.R. Lv, L.M. Zhang, X.P. Yuan, A dynamic damage constitutive model for a rock mass with persistent joints, Int. J. Rock Mech. Min. Sci. 75 (2015)
132–139, https://doi.org/10.1016/J.IJRMMS.2015.01.013.
[88] L. Liu, G.M. Wang, J.H. Chen, S. Yang, Creep experiment and rheological model of deep saturated rock, Trans. Nonferrous Metals Soc. China 23 (2) (2013)
478–483, https://doi.org/10.1016/S1003-6326(13)62488-7.
[89] H. Özşen, I. Özkan, & Şensöǧüt, C, Measurement and mathematical modelling of the creep behaviour of Tuzköy rock salt, Int. J. Rock Mech. Min. Sci. 66
(2014) 128–135, https://doi.org/10.1016/J.IJRMMS.2014.01.005.
[90] X. Li, W.G. Cao, Y.H. Su, A statistical damage constitutive model for softening behavior of rocks, Eng. Geol. 143–144 (2012) 1–17, https://doi.org/10.1016/J.
ENGGEO.2012.05.005.
[91] J. Zhu, X. Yang, N. He, Experimental research on coal rock creep deformation- seepage coupling law, Procedia Eng. 26 (2011) 1526–1531, https://doi.org/
10.1016/J.PROENG.2011.11.2334.
[92] Y. Zhang, W.Y. Xu, J.F. Shao, H. Bin Zhao, W. Wang, Experimental investigation of creep behavior of clastic rock in Xiangjiaba Hydropower Project, Water Sci.
Eng. 8 (1) (2015) 55–62, https://doi.org/10.1016/J.WSE.2015.01.005.
21
Heliyon 9 (2023) e14465
K.C. Onyelowe et al.
[93] Z. Yang, X. Yang, J. Zhang, Y.L.-J. South, of C., & 2015, undefined, Upper Bound Analysis of Collapsing Area of Tunnel Face in Broken Soft Rocks under
Different Saturations, 6, Journal of Central South University, Science and Technology, 2015, https://en.cnki.com.cn/Article_en/CJFDTotal-ZNGD201506039.
htm.
[94] C.Y. Zhou, F.X. Zhu, An elasto-plastic damage constitutive model with double yield surfaces for saturated soft rock, Int. J. Rock Mech. Min. Sci. 47 (3) (2010)
385–395, https://doi.org/10.1016/J.IJRMMS.2010.01.002.
[95] A. Taheri, F. Tatsuoka, A new method to simulate stress-strain relations from multiple-step loading triaxial compression test results, Geotech. Test J. 36 (6)
(2013), https://doi.org/10.1520/GTJ20130005.
[96] J.A. Sloan, G.M. Filz, J.G. Collin, Field-scale column-supported embankment test facility, Geotech. Test J. 36 (6) (2013), https://doi.org/10.1520/
GTJ20120229.
[97] O. Pourhosseini, M. Shabanimashcool, Development of an elasto-plastic constitutive model for intact rocks, Int. J. Rock Mech. Min. Sci. 66 (2014) 1–12,
https://doi.org/10.1016/J.IJRMMS.2013.11.010.
[98] E.E. Alonso, M.T. Zandarín, S. Olivella, Joints in unsaturated rocks: thermo-hydro-mechanical formulation and constitutive behaviour, J. Rock Mech. Geotech.
Eng. 5 (3) (2013) 200–213, https://doi.org/10.1016/J.JRMGE.2013.05.004.
[99] J. Duriez, F. Darve, F.V. Donzé, A discrete modeling-based constitutive relation for infilled rock joints, Int. J. Rock Mech. Min. Sci. 48 (3) (2011) 458–468,
https://doi.org/10.1016/J.IJRMMS.2010.09.008.
[100] J. Deng, D. Gu, On a statistical damage constitutive model for rock materials, Comput. Geosci. 37 (2) (2011) 122–128, https://doi.org/10.1016/J.
CAGEO.2010.05.018.
[101] Z.B. Liu, S.Y. Xie, J.F. Shao, N. Conil, Effects of deviatoric stress and structural anisotropy on compressive creep behavior of a clayey rock, Appl. Clay Sci. 114
(2015) 491–496, https://doi.org/10.1016/J.CLAY.2015.06.039.
[102] O. Craciun, S.C.R. Lo, Matric suction measurement in stress path cyclic triaxial testing of unbound granular base materials, Geotech. Test J. 33 (1) (2010),
https://doi.org/10.1520/GTJ102390.
[103] K. Fuenkajorn, D. Phueakphum, Physical model simulation of shallow openings in jointed rock mass under static and cyclic loadings, Eng. Geol. 113 (1–4)
(2010) 81–89, https://doi.org/10.1016/J.ENGGEO.2010.03.003.
[104] E.C. David, N. Brantut, A. Schubnel, R.W. Zimmerman, Sliding crack model for nonlinearity and hysteresis in the uniaxial stress–strain curve of rock, Int. J.
Rock Mech. Min. Sci. 52 (2012) 9–17, https://doi.org/10.1016/J.IJRMMS.2012.02.001.
[105] E. Liu, S. He, Effects of cyclic dynamic loading on the mechanical properties of intact rock samples under confining pressure conditions, Eng. Geol. 125 (2012)
81–91, https://doi.org/10.1016/J.ENGGEO.2011.11.007.
[106] J. Prevost, R. Popescu, Constitutive relations for soil materials, Electron. J. Geotech. Eng. 1 (1996) 1–45. https://www.researchgate.net/profile/RaduPopescu/publication/260294219_Constitutive_Relations_for_Soil_Materials/links/58f2f67aa6fdcc11e569ee50/Constitutive-Relations-for-Soil-Materials.pdf.
[107] Zhongqi Wang, H. Hao, Y. Lu, A three-phase soil model for simulating stress wave propagation due to blast loading, Int. J. Numer. Anal. Methods GeoMech. 28
(1) (2004) 33–56, https://doi.org/10.1002/NAG.325.
[108] E. Alnnso, A. Gens, Yuk Gehling, W. Y, Elastoplastic Model For Unsaturated Expansive Soils, 1994, pp. 11–18.
[109] S.J. Wheeler, V. Sivakumar, An elasto-plastic critical state framework for unsaturated soil, Geotechnique 45 (1) (1995) 35–53, https://doi.org/10.1680/
GEOT.1995.45.1.35.
[110] E.E. Alonso, A. Gens, A. Josa, A constitutive model for partially saturated soils, Geotechnique 40 (3) (1990) 405–430, https://doi.org/10.1680/
GEOT.1990.40.3.405.
[111] F. Geiser, L. Laloui, L.V. Wuhan, Constitutive modelling of unsaturated sandy silt, in: IACMAG, Ninth Intern (Eds.), Conference on Computer Methods and
Advances in Geomechanics. Wuhan, Nov, 1997, pp. 2–7. https://infoscience.epfl.ch/record/95965/files/article-IACMAG97.pdf.
[112] C.S. Desai, Hierarchical single surface and the disturbed state constitutive models with emphasis on geotechnical applications, Int. J. Rock Mech. Min. Sci.
Geomech. Abstracts 3 (32) (1995) 118A. https://www.infona.pl//resource/bwmeta1.element.elsevier-64ed1124-20ad-3d06-bab2-6f328daf7e08.
[113] D.A. Sun, H.B. Cui, H. Matsuoka, D.C. Sheng, A three-dimensional elastoplastic model for unsaturated compacted soils with hydraulic hysteresis, Soils Found.
47 (2) (2007) 253–264, https://doi.org/10.3208/SANDF.47.253.
[114] S.J. Wheeler, R.S. Sharma, M.S.R. Buisson, Coupling of hydraulic hysteresis and stress-strain behaviour in unsaturated soils, Geotechnique 53 (1) (2003)
41–54, https://doi.org/10.1680/GEOT.2003.53.1.41.
[115] M. Lloret-Cabot, S. Wheeler, M. Sánchez, A unified mechanical and retention model for saturated and unsaturated soil behaviour, Acta Geotechnica 12 (1)
(2017) 1–21, https://doi.org/10.1007/s11440-016-0497-x.
[116] Feng Zhang, T. Ikariya, A new model for unsaturated soil using skeleton stress and degree of saturation as state variables, Soils Found. 51 (1) (2011) 67–81,
https://doi.org/10.3208/SANDF.51.67.
[117] A.N. Zhou, D. Sheng, S.W. Sloan, A. Gens, Interpretation of unsaturated soil behaviour in the stress - saturation space, I: volume change and water retention
behaviour, Comput. Geotech. 43 (2012) 178–187, https://doi.org/10.1016/J.COMPGEO.2012.04.010.
[118] A.R.V. Chillarige, N.B. Morgenstern, P.K. Robertson, H.A. Christian, Seabed instability due to flow liquefaction in the Fraser River delta, Can. Geotech. J. 34 (4)
(1997) 520–533, https://doi.org/10.1139/T97-019/ASSET/T97-019.FP.PNG_V03.
[119] J.L.H. Grozic, P.K. Robertson, N.R. Morgenstern, Cyclic liquefaction of loose gassy sand, Can. Geotech. J. 37 (4) (2000) 843–856, https://doi.org/10.1139/
T00-008.
[120] R. Uzuoka, N. Sento, M. Kazama, T. Unno, Landslides during the earthquakes on may 26 and july 26, 2003 in miyagi, Japan, Soils Found. 45 (4) (2005)
149–163, https://doi.org/10.3208/SANDF.45.4_149.
[121] K. Ishihara, Y. Tsukamoto, K. Kamada, Undrained Behaviour Of Near-Saturated Sand In Cyclic And Monotonic Loading. Cyclic Behaviour Of Soils And
Liquefaction Phenomena, 2004, pp. 27–39, https://doi.org/10.1201/9781439833452.CH4.
[122] Y. Tsukamoto, S. Kawabe, J. Matsumoto, S. Hagiwara, Cyclic resistance of two unsaturated silty sands against soil liquefaction, Soils Found. 54 (6) (2014)
1094–1103, https://doi.org/10.1016/J.SANDF.2014.11.005.
[123] T. Unno, M. Kazama, R. Uzuoka, N. Sentos, Liquefaction of unsaturated sand considering the pore air pressure and volume compressibility of the soil particle
skeleton, Soils Found. 48 (1) (2008) 87–99, https://doi.org/10.3208/SANDF.48.87.
[124] D. Fredlund, H. Rahardjo, Soil Mechanics For Unsaturated Soils, 1993.
[125] N. Lu, W.J. Likos, Unsaturated Soil Mechanics, Wiley, 2004. http://10.1.1.57, 8888/dspace/handle/hau/5179.
[126] H. Sadeghi, A. Golaghaei Darzi, Modelling of soil-water retention curve considering the effects of existing salt solution in the pore fluid, in: MATEC Web of
Conferences, 337, 2021, p. 2001, https://doi.org/10.1051/matecconf/202133702001.
[127] H. Sadeghi, A. Golaghaei Darzi, A review of different approaches to analytical modeling of soil-water retention curves, Sharif J. Civil Eng. (2021), https://doi.
org/10.24200/J30.2021.56932.2870, 0(0), 0–0.
[128] K.C. Onyelowe, F.F. Mojtahedi, S. Azizi, H.A. Mahdi, E.R. Sujatha, A.M. Ebid, F.I. Aneke, Innovative overview of SWRC application in modeling geotechnical
engineering problems, Design 6 (5) (2022) 69.
[129] H. Sadeghi, H. Nasiri, Hysteresis of soil water retention and shrinkage characteristics for various molar concentrations, Géotech. Lett. 11 (1) (2021) 1–27,
https://doi.org/10.1680/jgele.20.00047.
[130] R. Chen, Experimental Study and Constitutive Modelling of Stress-dependent Coupled Hydraulic Hysteresis and Mechanical Behaviour of an Unsaturated Soil,
Hong Kong University of Science and Technology, Hong Kong, 2007. https://search.proquest.com/openview/59b1da6219c6f9e47c18dea952cf037d/1?pqorigsite=gscholar&cbl=18750.
[131] G. Musso, E. Romero, G. Della Vecchia, Double-structure effects on the chemo-hydro-mechanical behaviour of a compacted active clay, Geotechnique 63 (3)
(2013) 206–220, https://doi.org/10.1680/geot.SIP13.P.011.
[132] W.M. Ye, F. Zhang, B. Chen, Y.G. Chen, Q. Wang, Y.J. Cui, Effects of salt solutions on the hydro-mechanical behavior of compacted GMZ01 Bentonite, Environ.
Earth Sci. 72 (7) (2014) 2621–2630, https://doi.org/10.1007/s12665-014-3169-x.
22
Heliyon 9 (2023) e14465
K.C. Onyelowe et al.
[133] Y. He, Y.G. Chen, W.M. Ye, B. Chen, Y.J. Cui, Influence of salt concentration on volume shrinkage and water retention characteristics of compacted GMZ
bentonite, Environ. Earth Sci. 75 (6) (2016), https://doi.org/10.1007/s12665-015-5228-3.
[134] Feng Zhang, W.M. Ye, Y.G. Chen, B. Chen, Y.J. Cui, Influences of salt solution concentration and vertical stress during saturation on the volume change
behavior of compacted GMZ01 bentonite, Eng. Geol. 207 (2016) 48–55, https://doi.org/10.1016/j.enggeo.2016.04.010.
[135] J.E. Andrade, K.C. Ellison, Evaluation of a predictive constitutive model for sands, J. Geotech. Geoenviron. Eng. 134 (12) (2008) 1825–1828.
[136] R.W. Boulanger, I.M. Idriss, Closure to “liquefaction susceptibility criteria for silts and clays” by ross W. Boulanger and I. M. Idriss, J. Geotech. Geoenviron.
Eng. 134 (7) (2008) 1027–1028, https://doi.org/10.1061/(ASCE)1090-0241(2008, 134:7(1027.
[137] K.S. Ti, B.B. Huat, J. Noorzaei, M.S. Jaafar, G.S. Sew, A review of basic soil constitutive models for geotechnical application, Electron. J. Geotech. Eng. 14
(2009) 1–18.
[138] Y.F. Dafalias, On Cyclic And Anisotropic Plasticity: I) A General Model Including Material Behavior Under Stress Reversals. Ii) Anisotropic, 1975. https://
search.proquest.com/openview/be9c850dd941eeb2a7f0feb33d933fe9/1.pdf?pq-origsite=gscholar&cbl=18750&diss=y.
[139] Y.F. Dafalias, A Model For Soil Behavior Under Monotonic And Cyclic Loading Conditions, 1979. https://repository.lib.ncsu.edu/bitstream/handle/1840.20/
26760/K1-8.pdf?sequence=1.
[140] Y. Xiong, Ye lin, G. lin, Y. Xie, B. Ye, S. Zhang, F. Zhang, A unified constitutive model for unsaturated soil under monotonic and cyclic loading, Acta
Geotechnica 14 (2) (2019) 313–328, https://doi.org/10.1007/S11440-018-0754-2.
[141] K. Hashiguchi, Elastoplastic constitutive laws of granular materials, in: Constitutive Equations of Soils (Proc. 9th ICFSME, Spec. Session 9), 1977, pp. 73–82.
https://ci.nii.ac.jp/naid/10007802133.
[142] Z. ǒz, V.A. Norris, O.C. Zienkiewicz, An anisotropic hardening model for soils and its application to cyclic loading, Int. J. Numer. Anal. Methods GeoMech. 2 (3)
(1978) 203–221, https://doi.org/10.1002/NAG.1610020303.
[143] A. Asaoka, M. Nakano, T. Noda, Super loading yield surface concept for the saturated structured soils, in: Application of Numerical Methods to Geotechnical
Problems, 1998, pp. 233–242, https://doi.org/10.1007/978-3-7091-2512-0_22.
[144] W. Wu, E. Bauer, A simple hypoplastic constitutive model for sand, Int. J. Numer. Anal. Methods GeoMech. 18 (12) (1994) 833–862, https://doi.org/10.1002/
NAG.1610181203.
[145] Archilleas G. Papadimitriou, G.D. Bouckovalas, Y.F. Dafalias, Plasticity model for sand under small and large cyclic strains, J. Geotech. Geoenviron. Eng. 127
(11) (2001) 973–983, https://doi.org/10.1061/(ASCE)1090-0241(2001, 127:11(973.
[146] Yannis F. Dafalias, M.T. Manzari, Simple plasticity sand model accounting for fabric change effects, J. Eng. Mech. 130 (6) (2004) 622–634, https://doi.org/
10.1061/(ASCE)0733-9399(2004, 130:6(622.
[147] Feng Zhang, B. Ye, T. Noda, M. Nakano, K. Nakai, Explanation of cyclic mobility of soils: approach by stress-induced anisotropy, Soils Found. 47 (4) (2007)
635–648, https://doi.org/10.3208/SANDF.47.635.
[148] C. Yang, Y.J. Cui, J.M. Pereira, M.S. Huang, A constitutive model for unsaturated cemented soils under cyclic loading, Comput. Geotech. 35 (6) (2008)
853–859, https://doi.org/10.1016/J.COMPGEO.2008.08.005.
[149] N. Khalili, M.A. Habte, S. Zargarbashi, A fully coupled flow deformation model for cyclic analysis of unsaturated soils including hydraulic and mechanical
hystereses, Comput. Geotech. 35 (6) (2008) 872–889, https://doi.org/10.1016/J.COMPGEO.2008.08.003.
[150] N. Khalili, M.D. Liu, On generalization of constitutive models from two dimensions to three dimensions, Int. J. Numer. Anal. Methods GeoMech. 32 (17) (2008)
2045–2065, https://doi.org/10.1002/NAG.740.
[151] Z. Gao, J. Zhao, Constitutive modeling of artificially cemented sand by considering fabric anisotropy, Comput. Geotech. 41 (2012) 57–69, https://doi.org/
10.1016/J.COMPGEO.2011.10.007.
[152] Z. Gao, J. Zhao, X.S. Li, Y.F. Dafalias, A critical state sand plasticity model accounting for fabric evolution, Int. J. Numer. Anal. Methods GeoMech. 38 (4)
(2014) 370–390, https://doi.org/10.1002/NAG.2211.
[153] A.G. Papadimitriou, Y.F. Dafalias, X.S. Li, Sand model within anisotropic critical state theory with evolving fabric, Geomech. Micro Macro 1 (2014) 646–651,
https://doi.org/10.1201/B17395-31/sand-model-within-anisotropic-critical-state-theory-evolving-fabric-papadimitriou-dafalias-li.
[154] N.A. Tsytovich, Strength properties of frozen ground: critical and design strengths, Mech. Frozen Ground (1975) 138–180.
[155] V.L. Streeter, E.B. Wylie, F.E. Richart, Soil motion computations by characteristic method, ASCE J. Geotech. Eng. Div. 100 (GT3) (1974) 247–263, https://doi.
org/10.1061/AJGEB6.0000021.
[156] W.B. Joyner, A.T.F. Chen, Calculation of nonlinear ground response in earthquakes, Bull. Seismol. Soc. Am. 65 (5) (1975) 1315–1336, https://doi.org/
10.1785/BSSA0650051315.
[157] P.W. Taylor, T.J. Larkin, Seismic site response of nonlinear soil media, J. Geotech. Eng. Div. 104 (3) (1978) 369–383, https://doi.org/10.1061/
AJGEB6.0000602.
[158] P.P. Martin, H.B. Seed, One-dimensional dynamic ground response analyses, J. Geotech. Eng. Div. 108 (7) (1982) 935–952, https://doi.org/10.1061/
AJGEB6.0001316.
[159] V.A. Osinov, Cyclic shearing and liquefaction of soil under irregular loading: an incremental model for the dynamic earthquake-induced deformation, Soil
Dynam. Earthq. Eng. 23 (7) (2003) 535–548, https://doi.org/10.1016/S0267-7261(03)00072-1.
[160] A.M. Fish, Thermodynamic model of creep at constant stress and constant strain rate, Cold Reg. Sci. Technol. 9 (2) (1984) 143–161, https://doi.org/10.1016/
0165-232X(84)90006-5.
[161] J. Lee, Y.S. Kim, D. Chae, W. Cho, Loading rate effects on strength and stiffness of frozen sands, KSCE J. Civil Eng. 20 (1) (2015) 208–215, https://doi.org/
10.1007/S12205-015-1417-6, 2015 20:1.
[162] P. Zhang, Z.Y. Yin, Y. Zheng, F.P. Gao, A LSTM surrogate modelling approach for caisson foundations, Ocean. Eng. 204 (2020) 107263, https://doi.org/
10.1016/J.OCEANENG.2020.107263.
[163] L.U. Arenson, M.M. Johansen, S.M. Springman, Effects of volumetric ice content and strain rate on shear strength under triaxial conditions for frozen soil
samples, Permafr. Periglac. Process. 15 (3) (2004) 261–271, https://doi.org/10.1002/PPP.498.
[164] L. Yuanming, Y. Yugui, C. Xiaoxiao, L. Shuangyang, Strength criterion and elastoplastic constitutive model of frozen silt in generalized plastic mechanics, Int. J.
Plast. 26 (10) (2010) 1461–1484, https://doi.org/10.1016/J.IJPLAS.2010.01.007.
[165] G. Amiri, G. Grimstad, S. Nordal, S.A.G. Amiri, G. Grimstad, M. Kadivar, S. Nordal, Constitutive model for rate-independent behavior of saturated frozen soils,
Cdnsciencepub.Com 53 (10) (2016) 1646–1657, https://doi.org/10.1139/cgj-2015-0467.
[166] G. Xu, M. Gutierrez, K. Arora, X. Wang, Visco-plastic response of deep tunnels based on a fractional damage creep constitutive model, Acta Geotechnica 17 (2)
(2022) 613–633, https://doi.org/10.1007/S11440-021-01226-5/FIGURES/17.
[167] Z. Zhou, W. Ma, S. Zhang, H. Du, Y. Mu, G. Li, Multiaxial creep of frozen loess, Mech. Mater. 95 (2016) 172–191, https://doi.org/10.1016/J.
MECHMAT.2015.11.020.
[168] T. Fu, Z. Zhu, C. Cao, Constitutive model of frozen-soil dynamic characteristics under impact loading, Acta Mechan. 230 (5) (2019) 1869–1889, https://doi.
org/10.1007/S00707-019-2369-6, 2019 230:5.
[169] Fulai Zhang, Z. Zhu, T. Fu, J. Jia, Damage mechanism and dynamic constitutive model of frozen soil under uniaxial impact loading, Mech. Mater. 140 (2020)
103217, https://doi.org/10.1016/J.MECHMAT.2019.103217.
[170] B.O. Hardin, W.L. Black, Closure to “vibration modulus of normally consolidated clay, J. Soil Mech. Found Div. 95 (6) (1969) 1531–1537, https://doi.org/
10.1061/JSFEAQ.0001364.
[171] H.B. Seed, IDRISS I.M, Soil moduli and damping factors for dynamic response analysis. EERC. https://ci.nii.ac.jp/naid/10022026877, 1970.
[172] B.O. Hardin, V.P. Drnevich, Shear modulus and damping in soils: design equations and curves, J. Soil Mech. Found Div. 98 (7) (1972) 667–692, https://doi.
org/10.1061/JSFEAQ.0001760.
[173] I.M. Idriss, R. Dobry, R.D. Singh, Nonlinear behavior of soft clays during cyclic loading, ASCE J. Geotech. Eng. Div. 104 (12) (1978) 1427–1447, https://doi.
org/10.1061/AJGEB6.0000727.
23
Heliyon 9 (2023) e14465
K.C. Onyelowe et al.
[174] D. Bratosin, Nonlinear behaviour of soils during harmonic vibration. Revue roumaine des sciences techniques, Série de Mécanique Appliquée 31 (2) (1986)
227–234.
[175] K. Ishihara, Soil Behaviour in Earthquake Geotechnics, 1996. http://ndl.ethernet.edu.et/bitstream/123456789/45794/1/Soil. Behaviour in Earthquake
Geotechnics.pdf.
[176] V.P. Drnevich, B.O. Hardin, D.J. Shippy, Modulus and damping of soils by the resonant-column method, ASTM (Am. Soc. Test. Mater.) Spec. Tech. Publ. (1978)
91–125, https://doi.org/10.1520/STP35673S.
[177] M.L. Lin, T.H. Huang, J.C. You, The effects of frequency on damping properties of sand, Soil Dynam. Earthq. Eng. 15 (4) (1996) 269–278, https://doi.org/
10.1016/0267-7261(95)00045-3.
[178] J. Xu, J. Bielak, O. Ghattas, J. Wang, Three-dimensional nonlinear seismic ground motion modeling in basins, Phys. Earth Planet. In. 137 (1–4) (2003) 81–95,
https://doi.org/10.1016/S0031-9201(03)00009-8.
[179] D. Assimaki, W. Li, J. Steidl, J. Schmedes, Quantifying nonlinearity susceptibility via site-response modeling uncertainty at three sites in the los angeles basin,
Bull. Seismol. Soc. Am. 98 (5) (2008) 2364–2390, https://doi.org/10.1785/0120080031.
[180] R. Taborda, J. Bielak, D. Restrepo, Earthquake ground-motion simulation including nonlinear soil effects under idealized conditions with application to two
case studies, Seismol Res. Lett. 83 (6) (2012) 1047–1060, https://doi.org/10.1785/0220120079.
[181] Zhongxian Liu, Y. Qiao, X. Cheng, El Naggar, M. H, Nonlinear seismic response and amplification effect of 3D sedimentary basin based on bounding surface
constitutive model, Soil Dynam. Earthq. Eng. 158 (2022) 107292, https://doi.org/10.1016/J.SOILDYN.2022.107292.
[182] M.B. Rubin, A thermoelastic-viscoplastic model with a rate-dependent yield strength, J. Appl. Mech. 49 (2) (1982) 305–311, https://doi.org/10.1115/
1.3162085.
[183] R.C. Batra, X.Q. Liang, Finite dynamic deformations of smart structures, Comp. Mech. 20 (5) (1997) 427–438, https://doi.org/10.1007/S004660050263, 1997
20:5.
[184] J. Zhai, M. Zhou, Finite element analysis of micromechanical failure modes in a heterogeneous ceramic material system, Int. J. Frac. 101 (1) (2000) 161–180,
https://doi.org/10.1023/A:1007545105723, 2000 101:1.
[185] J.F. Bell, Mech. Solid. 3 (1973), https://doi.org/10.1007/978-3-642-69565-0. /6a/1 (C. Truesdell (Ed.)). Springer Berlin Heidelberg.
[186] R.C. Batra, Linear constitutive relations in isotropic finite elasticity, J. Elasticity 51 (3) (1998) 243–245, https://doi.org/10.1023/A:1007503716826.
[187] Z.P. Bazant, R.J. Krizek, Endochronic constitutive law for liquefaction of sand, ASCE J. Eng. Mech. Div. 102 (2) (1976) 225–238, https://doi.org/10.1061/
JMCEA3.0002104.
[188] K.C. Valanis, H.E. Read, A New Endochronic Plasticity Model For Soils.. Soil Mechanics-Transient And Cyclic Loads, 1982, pp. 375–417, 30, https://scholar.
google.com/scholar?hl=en&as_sdt=0%2C5&q=Valanis%2C+K.C.+and+H.J.E.+Read.+%281982%29%2C+A+New+Endochronic+Plasticity+Model+for+
Soils.+Soil+Mechanics-Transient+and+Cyclic+Loads%2C+G.N.+Pande+and+O.C.+Zienkiewicz%2C+. %2C+Wiley%2C+pp.+375-4.
[189] J.M. Duncan, C.-Y. Chang, Nonlinear analysis of stress and strain in soils, J. Soil Mech. Found Div. 96 (5) (1970) 1629–1653, https://doi.org/10.1061/
JSFEAQ.0001458.
[190] M.D. Coon, R.J. Evans, Recoverable deformation of cohesionless soils, J. Soil Mech. Found Div. 97 (2) (1971) 375–391, https://doi.org/10.1061/
JSFEAQ.0001548.
[191] W.D.L. Finn, K.W. Lee, G.R. Martin, An effective stress model for liquefaction, ASCE J. Geotech. Eng. Div. 103 (6) (1977) 517–533, https://doi.org/10.1061/
AJGEB6.0000434.
[192] M. Boulon, R. Nova, Modelling of soil-structure interface behaviour a comparison between elastoplastic and rate type laws, Comput. Geotech. 9 (1–2) (1990)
21–46, https://doi.org/10.1016/0266-352X(90)90027-S.
[193] V. Gennaro, De, & Frank, R, Elasto-plastic analysis of the interface behaviour between granular media and structure, Comput. Geotech. 29 (7) (2002) 547–572,
https://doi.org/10.1016/S0266-352X(02)00010-1.
[194] V.N. Ghionna, G. Mortara, An elastoplastic model for sand–structure interface behaviour, Geotechnique 52 (1) (2002) 41–50, https://doi.org/10.1680/
GEOT.2002.52.1.41.
[195] H. Liu, E. Song, H.I. Ling, Constitutive modeling of soil-structure interface through the concept of critical state soil mechanics, Mech. Res. Commun. 33 (4)
(2006) 515–531, https://doi.org/10.1016/J.MECHRESCOM.2006.01.002.
[196] J.Z. Sun, Y. Wang, Constitutive model of calcareous sand structure interface, Chin. Q. Mech. 27 (3) (2006) 476–480. https://en.cnki.com.cn/Article_en/
CJFDTotal-SHLX200603015.htm.
[197] Y. Zong-ze, Z. Hong, X.U. Guo-hua, Numerical simulation of the deformation in the interface between soil and structural material, Chin. J. Geotech. Eng.
(1994). https://en.cnki.com.cn/Article_en/CJFDTotal-YTGC403.001.htm.
[198] M. Luan, Y.J. Wu, A nonlinear elasto-perfectly plastic model of interface element for soil-structure interaction and its applications, Rock Soil Mech. Wuhan 25
(2004) 507–513. https://en.cnki.com.cn/Article_en/CJFDTotal-YTLX200404001.htm.
[199] Annan Zhou, D. Sheng, Yield stress, volume change, and shear strength behaviour of unsaturated soils: validation of the SFG model, Can. Geotech. J. 46 (9)
(2009) 1034–1045, https://doi.org/10.1139/T09-049.
[200] P. Koci, C.D. Foster, A fiber-reinforced constitutive model for earthen materials, Acta Geotechnica (2022) 1–20, https://doi.org/10.1007/S11440-022-01564Y.
[201] X. Kong, L. Tang, X. Ling, Y. Zhao, S. Cong, K. Sun, A 3D elasto-plastic constitutive model for clay based on the unified hardening model with insights into the
coupling effect of small-strain stiffness and over-consolidation ratio. International journal of civil engineering, 1–16. https://doi.org/10.1007/S40999-02100697-7/FIGURES/16, 2022.
[202] F. Yu, P. Guo, S.H. Na, A framework for constructing elasto-plastic constitutive models for frozen and unfrozen soils, Int. J. Numer. Anal. Methods GeoMech. 46
(2) (2022) 436–466, https://doi.org/10.1002/NAG.3306.
[203] R.L. Kondner, M.M.K. Ho, Viscoelastic response of a cohesive soil in the frequency domain, Trans. Soc. Rheol. 9 (2) (2000) 329, https://doi.org/10.1122/
1.549003.
[204] S. Murayama, T. Shibata, Flow and stress relaxation of clays, Rheolog. Soil Mechan./Rhéologie et Mécanique Des Sols (1966) 99–129, https://doi.org/
10.1007/978-3-662-39449-6_10.
[205] B. Di, F. Tatsuoka, M. Ishihara, Time-dependent shear deformation characteristics of sand and their constitutive modelling, Soils Found. 42 (2) (2002) 1–22.
https://www.jstage.jst.go.jp/article/sandf1995/42/2/42_2_1/_article/-char/ja/.
[206] F. Oka, T. Kodaka, Y.S. Kim, A cyclic viscoelastic-viscoplastic constitutive model for clay and liquefaction analysis of multi-layered ground, Int. J. Numer. Anal.
Methods GeoMech. 28 (2) (2004) 131–179, https://doi.org/10.1002/NAG.329.
[207] Awtar Singh, J.K. Mitchell, General stress-strain-time function for soils, J. Soil Mech. Found Div. 94 (1) (1968) 21–46, https://doi.org/10.1061/
JSFEAQ.0001084.
[208] A. Singh, J.K. Mitchell, Creep Potential And Creep Rupture Of Soils, 1969.
[209] P. Perzyna, The constitutive equation for work-hardening and rate sensitive plastic materials. In Proc. Vibrational Problems, 281–290. https://ci.nii.ac.jp/
naid/10007807797/, 1963.
[210] K. Hohenemser, W. Prager, Über die Ansätze der Mechanik isotroper Kontinua, ZAMM - J. Appl. Mathem. Mechan./Z. Angew. Math. Mech. 12 (4) (1932)
216–226, https://doi.org/10.1002/ZAMM.19320120403.
[211] R. Yong, R.D. Japp, Stress-strain behavior of clays in dynamic compression. In vibration effects of earthquakes on soils and foundations. ASTM international.
https://www.astm.org/DIGITAL_LIBRARY/STP/PAGES/STP33643S.htm, 1969.
[212] H. Sekiguchi, Rheological Characteristics of Clays, 1, Proc. 9th ICSMFE, Tokyo, 1977, pp. 289–292, https://doi.org/10.1143/JPSJ.43.1327, 43(4).
[213]] A. Dragon, Z. Mróz, A model for plastic creep of rock-like materials accounting for the kinetics of fracture, Int. J. Rock Mech. Min. Sci. Geomech. Abstracts 16
(4) (1979) 253–259, https://doi.org/10.1016/0148-9062(79)91200-2.
24
Heliyon 9 (2023) e14465
K.C. Onyelowe et al.
[214]] T. Matsui, N. Abe, Elasto/viscoplastic constitutive equation of normally consolidated clays based on flow surface theory, in: In International Conference on
Numerical Methods in Geomechanics, 1985, pp. 407–413.
[215]] J.H. Yin, J. Graham, Elastic viscoplastic modelling of the time-dependent stress-strain behaviour of soils, Can. Geotech. J. 36 (4) (1999) 736–745, https://doi.
org/10.1139/T99-042.
[216] J.Z. Zhang, X.P. Zhou, P. Yin, Visco-plastic deformation analysis of rock tunnels based on fractional derivatives, Tunn. Undergr. Space Technol. 85 (2019)
209–219, https://doi.org/10.1016/J.TUST.2018.12.019.
[217] N. Zhang, T. Nagel, Error-controlled implicit time integration of elasto-visco-plastic constitutive models for rock salt, Int. J. Numer. Anal. Methods GeoMech.
44 (8) (2020) 1109–1127, https://doi.org/10.1002/NAG.3049.
[218] M. Alibeikloo, H. Khabbaz, B. Fatahi, T.M. Le, Reliability assessment for time-dependent behaviour of soft soils considering cross correlation between viscoplastic model parameters, Reliab. Eng. Syst. Saf. 213 (2021) 107680, https://doi.org/10.1016/J.RESS.2021.107680.
[219] H. Gray, Progress report on research on the consolidation of fine-grained soils. In proceedings, first international conference on soil mechanics and foundation
engineering, 138–141. https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Progress+report+on+research+on+the+consolidation+of+finegrained+soils.&btnG=, 1936.
[220] A.A. Beles, I.I. Stanculescu, Thermal treatment as a means of improving the stability of earth masses, Geotechnique 8 (4) (2015) 158–165, https://doi.org/
10.1680/GEOT.1958.8.4.158.
[221] D. Hight, Laboratory investigations of sea-bed clays. https://spiral.imperial.ac.uk/bitstream/10044/1/7370/1/David_William_Hight-1983-PhD-Thesis.pdf,
1983.
[222] James K. Mitchell, J.C. McMillan, S.L. Green, R.C. Sisson, Field testing of cable backfill systems. Underground cable thermal backfill, 19–33. https://doi.org/
10.1016/B978-0-08-025387-9.50007-5, 1982.
[223] O. Andersson, G. Hellström, B. Nordell, Heating and cooling with UTES in Sweden: current situation and potential market development, in: N International
Conference on Thermal Energy Storage: 01/09/2003-04/09/2003, 207–215, 2003. https://www.diva-portal.org/smash/record.jsf?pid=diva2, 1006446.
[224] I. Vardoulakis, Dynamic thermo-poro-mechanical analysis of catastrophic landslides, Geotechnique 52 (3) (2002) 157–171, https://doi.org/10.1680/
GEOT.2002.52.3.157.
[225] A. Hamidi, C. Khazaei, A thermo-mechanical constitutive model for saturated clays, Int. J. Geotech. Eng. 4 (4) (2010) 445–459, https://doi.org/10.3328/
IJGE.2010.04.04.445-459.
[226] H.M. Abuel-Naga, D.T. Bergado, S. Chaiprakaikeow, Innovative thermal technique for enhancing the performance of prefabricated vertical drain during the
preloading process, Geotext. Geomembranes 24 (6) (2006) 359–370, https://doi.org/10.1016/J.GEOTEXMEM.2006.04.003.
[227] J. Graham, N. Tanaka, T. Crilly, M. Alfaro, Modified Cam-Clay modelling of temperature effects in clays, Can. Geotech. J. 38 (3) (2001) 608–621, https://doi.
org/10.1139/T00-125.
[228] B. Gatmiri, P. Delage, A formulation of fully coupled thermal–hydraulic–mechanical behaviour of saturated porous media—numerical approach, Int. J. Numer.
Anal. Methods GeoMech. 21 (3) (1997) 199–225.
[229] G. Chen, D. Gallipoli, A. Ledesma, G.J. Chen, ⋅ A. Ledesma, D. Gallipoli, Chemo-hydro-mechanical coupled consolidation for a poroelastic clay buffer in a
radioactive waste repository, Trans. Porous Media 69 (2) (2007) 189–213, https://doi.org/10.1007/S11242-006-9083-2, 2007 69:2.
[230] C. Cekerevac, L. Laloui, Experimental study of thermal effects on the mechanical behaviour of a clay, Int. J. Numer. Anal. Methods GeoMech. 28 (3) (2004)
209–228, https://doi.org/10.1002/NAG.332.
[231] M.D. Liu, J.P. Carter, Str. Cam Clay Model 39 (6) (2011) 1313–1332, https://doi.org/10.1139/T02-069.
[232] Xiaoli Xu, F. Gao, Z. Zhang, Thermo-mechanical coupling damage constitutive model of rock based on the Hoek–Brown strength criterion, Int. J. Damage
Mech. 27 (8) (2017) 1213–1230, https://doi.org/10.1177/1056789517726838.
[233] C. Zhou, C.W.W. Ng, A new thermo-mechanical model for structured soil, Geotechnique 68 (12) (2018) 1109–1115, https://doi.org/10.1680/JGEOT.17.
T.031.
[234] D.M. Zymnis, A.J. Whittle, X. Cheng, Simulation of long-term thermo-mechanical response of clay using an advanced constitutive model, Acta Geotechnica 14
(2) (2019) 295–311, https://doi.org/10.1007/S11440-018-0726-6/Figures/18.
[235] K. Wang, W.C. Sun, Q. Du, A cooperative game for automated learning of elasto-plasticity knowledge graphs and models with AI-guided experimentation,
Comput. Mech. 64 (2) (2019) 467–499, https://doi.org/10.1007/S00466-019-01723-1/FIGURES/16.
[236] W.H. Al-Soudani, , Fattah -, M, Y. Ma, H. Lei, S. Yang, B., -, al, Chunxiang, G., Weijia, Z., Laiku, J., Wang, X., Wang, J., & Zhang, Z, A thermo-mechanical model
for pile-soil interface behavior. IOP Conference Series: Earth Environ. Sci. 861 (7) (2021), 72145 https://doi.org/10.1088/1755-1315/861/7/072145.
[237] A. Golchin, P.J. Vardon, M.A. Hicks, A thermo-mechanical constitutive model for fine-grained soils based on thermodynamics, Int. J. Eng. Sci. 174 (2022)
103579, https://doi.org/10.1016/J.IJENGSCI.2021.103579.
[238] R. Mani, D. Kadau, H.J. Herrmann, Liquid migration in sheared unsaturated granular media, Granul. Matter 15 (4) (2013) 447–454, https://doi.org/10.1007/
s10035-012-0387-3.
[239] F. Bianchi, M. Thielmann, R. Mani, D. Or, H.J. Herrmann, Tensile stress relaxation in unsaturated granular materials, Granul. Matter 18 (4) (2016), https://doi.
org/10.1007/S10035-016-0673-6.
[240] E.E. Alonso, N.M. Pinyol, A. Gens, Compacted soil behaviour: initial state, structure and constitutive modelling, in: Partial Saturation in Compacted Soils:
Geotechnique Symposium in Print, 2011, 2013, pp. 3–18, https://doi.org/10.1680/GEOT.11.P.134.
[241] E.J. Macari, L.R. Hoyos, P. Arduino, Constitutive modeling of unsaturated soil behavior under axisymmetric stress states using a stress/suction-controlled
cubical test cell, Int. J. Plast. 19 (10) (2003) 1481–1515, https://doi.org/10.1016/S0749-6419(02)00018-9.
[242] M. Maleki, M. Bayat, Experimental evaluation of mechanical behavior of unsaturated silty sand under constant water content condition, Eng. Geol. 141–142
(2012) 45–56, https://doi.org/10.1016/J.ENGGEO.2012.04.014.
[243] L. Li, Evaluate Unsaturated Soil Behavior Using Constant Water Content Triaxial Tests, University of Alaska Fairbanks, 2015. https://search.proquest.com/
openview/5e09b66d370c23c6c7bc4f63b7c1dde3/1?pq-origsite=gscholar&cbl=18750.
[244] J. Vaunat, V. Merchán, E. Romero, J. Pineda, Residual Strength Of Clays At High Suctions. Theoretical And Numerical Unsaturated Soil Mechanics, 2007,
pp. 151–163, https://doi.org/10.1007/3-540-69876-0_17.
[245] Y.J. Cui, P. Delage, Yielding and plastic behaviour of an unsaturated compacted silt, Geotechnique 46 (2) (1996) 291–311, https://doi.org/10.1680/
GEOT.1996.46.2.291.
[246] C.W.W. Ng, R.Z.B. Zhou, Effects of soil suction on dilatancy of an unsaturated soil. 559–562. https://doi.org/10.3233/978-1-61499-656-9-559, 2005.
[247] R. Sharma, Mechanical Behaviour of Unsaturated Highly Expansive Clays. Doctoral Dissertation, University of Oxford, 1998. https://ethos.bl.uk/OrderDetails.
do?uin=uk.bl.ethos.285563.
[248] D. Sheng, D.G. Fredlund, A. Gens, A new modelling approach for unsaturated soils using independent stress variables, Can. Geotech. J. 45 (4) (2008) 511–534,
https://doi.org/10.1139/T07-112.
[249] D. Sheng, Review of fundamental principles in modelling unsaturated soil behaviour, Comput. Geotech. 38 (6) (2011) 757–776, https://doi.org/10.1016/J.
COMPGEO.2011.05.002.
[250] K. Stropeit, S. Wheeler, Y. Cui, An anisotropic elasto-plastic model for unsaturated soils. https://eprints.gla.ac.uk/89683/, 2008.
[251] Y.J. Cui, P. Delage, N. Sultan, An elasto-plastic model for compacted soils, A. A, in: E.E. Alonso, P. Delage (Eds.), Proceedings of the First International
Conference on Unsaturated Soils, Balkema, Rotterdam, Netherlands, 1995, pp. 579–584.
[252] Xiong Zhang, R.L. Lytton, Modified state-surface approach to the study of unsaturated soil behavior. Part I: basic concept, Can. Geotech. J. 46 (5) (2009)
536–552, https://doi.org/10.1139/T08-136.
[253] H. Matsuoka, D. Sun, A. Kogane, N. Fukuzawa, W. Ichihara, Stress-strain behaviour of unsaturated soil in true triaxial tests, Can. Geotech. J. 39 (3) (2002)
608–619, https://doi.org/10.1139/T02-031/ASSET/IMAGES/GR.GIF.
25
Heliyon 9 (2023) e14465
K.C. Onyelowe et al.
[254] J.P. Wang, E. Gallo, B. François, F. Gabrieli, P. Lambert, Capillary force and rupture of funicular liquid bridges between three spherical bodies, Powder
Technol. 305 (2017) 89–98, https://doi.org/10.1016/J.POWTEC.2016.09.060.
[255] Annan Zhou, D. Sheng, An advanced hydro-mechanical constitutive model for unsaturated soils with different initial densities, Comput. Geotech. 63 (2015)
46–66, https://doi.org/10.1016/J.COMPGEO.2014.07.017.
[256] R. Tamagnini, An extended Cam-clay model for unsaturated soils with hydraulic hysteresis, Geotechnique 54 (3) (2004) 223–228, https://doi.org/10.1680/
GEOT.2004.54.3.223.
[257] G. Buscarnera, R. Nova, An elastoplastic strainhardening model for soil allowing for hydraulic bonding-debonding effects, Int. J. Numer. Anal. Methods
GeoMech. 33 (8) (2009) 1055–1086, https://doi.org/10.1002/NAG.756.
[258] E. Gholizadeh, M. Latifi, A coupled hydro-mechanical constitutive model for unsaturated frictional and cohesive soil, Comput. Geotech. 98 (2018) 69–81,
https://doi.org/10.1016/J.COMPGEO.2017.11.010.
[259] W. Liu, Q. Yang, X. Sun, Hydro-mechanical constitutive model for overconsolidated unsaturated soils, Eur. J. Environ. Civil Eng. 24 (11) (2018) 1802–1820,
https://doi.org/10.1080/19648189.2018.1488622.
[260] G. Cai, B. Han, M. Li, K. Di, Y. Liu, J. Li, T. Wu, Numerical implementation of a hydro-mechanical coupling constitutive model for unsaturated soil considering
the effect of micro-pore structure, Appl. Sci. 11 (12) (2021) 5368, https://doi.org/10.3390/APP11125368.
[261] D.G. Phan, G.D. Nguyen, H.H. Bui, T. Bennett, Constitutive modelling of partially saturated soils: hydro-mechanical coupling in a generic thermodynamicsbased formulation, Int. J. Plast. 136 (2021) 102821, https://doi.org/10.1016/J.IJPLAS.2020.102821.
[262] R.W. Lewis, J.T. Cross, D.T. Gethin, R.S. Ransing, M.T. Manzari, N.A. Rahman, Recent developments in heat transfer and porous media, WIT Trans. Eng. Sci. 20
(1970). https://www.witpress.com/elibrary/wit-transactions-on-engineering-sciences/20/6618.
[263] R.W. Lewis, C.E. Majorana, B.A. Schrefler, A coupled finite element model for the consolidation of nonisothermal elastoplastic porous media, Transport Porous
Media 1 (2) (1986) 155–178, https://doi.org/10.1007/BF00714690.
[264] W.J. Ferguson, R.W. Lewis, A fully nonlinear analysis of temperature, moisture content, and pressure in a capillary porous body. Numerical Heat Transfer, Part
B: Fundamentals 23 (1) (1993) 91–110, https://doi.org/10.1080/10407799308914891.
[265] H.R. Thomas, The influence of non-linear thermal parameters on moisture content distributions in unsaturated soil, Int. J. Numer. Methods Eng. 26 (1) (1988)
263–279, https://doi.org/10.1002/NME.1620260118.
[266] H.R. Thomas, Y. He, M.R. Sansom, C.L.W. Li, On the development of a model of the thermo-mechanical-hydraulic behaviour of unsaturated soils, Eng. Geol. 41
(1–4) (1996) 197–218, https://doi.org/10.1016/0013-7952(95)00033-X.
[267] Thomas, Hywel Rhys, M.R. Sansom, Fully coupled analysis of heat, moisture, and air transfer in unsaturated soil, J. Eng. Mech. 121 (3) (1995) 392–405,
https://doi.org/10.1061/(ASCE)0733-9399(1995, 121:3(392).
[268] D.A. Sun, D.C. Sheng, H.B. Cui, S.W. Sloan, A density-dependent elastoplastic hydro-mechanical model for unsaturated compacted soils, Int. J. Numer. Anal.
Methods GeoMech. 31 (11) (2007) 1257–1279, https://doi.org/10.1002/NAG.579.
[269] B.A. Schrefler, FE in environmental engineering: coupled thermo-hydro-mechanical processes in porous media including pollutant transport, Arch. Comput.
Methods Eng. 2 (3) (1995) 1–54. https://idp.springer.com/authorize/casa?redirect_uri=https://link.springer.com/content/pdf/10.1007/BF02736173.
pdf&casa_token=JD7zWKURD44AAAAA:gh1ziOe6Of1EPOtOK__NUFPc0OtvuAM6JXT1l1iTqZrAKat7t-gGGEZeMjyOOZkwiTRr5IPSlX9_Jdbp.
[270] Dariusz Gawin, B.A. Schrefler, Thermo-hydro-mechanical analysis of partially saturated porous materials, Eng. Comput. 13 (7) (1996) 113–143, https://doi.
org/10.1108/02644409610151584/FULL/HTML.
[271] D. Gawin, F. Pesavento, B.A. Schrefler, Modelling of hygro-thermal behaviour and damage of concrete at temperature above the critical point of water, Int. J.
Numer. Anal. Methods GeoMech. 26 (6) (2002) 537–562, https://doi.org/10.1002/NAG.211.
[272] T. Hueckel, M. Borsetto, Thermoplasticity of saturated soils and shales: constitutive equations, J. Geotech. Eng. 116 (12) (1990) 1765–1777, https://doi.org/
10.1061/(ASCE)0733-9410(1990, 116:12(1765).
[273] Y.J. Cui, N. Sultan, P. Delage, A thermomechanical model for saturated clays, Can. Geotech. J. 37 (3) (2000) 607–620, https://doi.org/10.1139/T99-111.
[274] A. Gens, E.E. Alonso, A framework for the behaviour of unsaturated expansive clays, Can. Geotech. J. 29 (6) (1992) 1013–1032, https://doi.org/10.1139/T92120.
[275] A. Gens, P. Jouanna, B. Schrefler, Modern Issues in Non-saturated Soils, Springer, Berlin, 1995. https://link.springer.com/content/pdf/10.1007/978-3-70912692-9.pdf.
[276] Y. Rui, K. Soga, Thermo-hydro-mechanical coupling analysis of a thermal pile, Geotech. Eng. 172 (2) (2019) 155–173, https://doi.org/10.1680/
JGEEN.16.00133.
[277] Y. Tang, W. Wu, K. Yin, S. Wang, G. Lei, A hydro-mechanical coupled analysis of rainfall induced landslide using a hypoplastic constitutive model, Comput.
Geotech. 112 (2019) 284–292, https://doi.org/10.1016/J.COMPGEO.2019.04.024.
[278] M.M. Arzanfudi, R. Al-Khoury, L.J. Sluys, G.M.A. Schreppers, A thermo-hydro-mechanical model for energy piles under cyclic thermal loading, Comput.
Geotech. 125 (2020) 103560, https://doi.org/10.1016/J.COMPGEO.2020.103560.
[279] Y. Liu, G. Cai, A. Zhou, B. Han, J. Li, C. Zhao, A fully coupled constitutive model for thermo-hydro-mechanical behaviour of unsaturated soils, Comput.
Geotech. 133 (2021) 104032, https://doi.org/10.1016/J.COMPGEO.2021.104032.
[280] J.K. Mitchell, K. Soga, Fundamentals of Soil Behavior, John Wiley & Sons, Inc, New York, 1993. https://scholar.google.com/scholar?hl=en&as_sdt=0%
2C5&q=Mitchell%2C+J.+K.+%281993%29.+Fundamentals+of+soil+behaviour%2C+Wiley%2C+New+York.&btnG=.
[281] F. Tong, Y. Jian-Hua, Nonlinear creep and swelling behavior of bentonite mixed with different sand contents under oedometric condition, Mar. Georesour.
Geotechnol. 29 (4) (2011) 346–363, https://doi.org/10.1080/1064119X.2011.560824.
[282] P.V. Sivapullaiah, A. Sridharan, V.K. Stalin, Swelling behaviour of soil-bentonite mixtures, Can. Geotech. J. 33 (5) (1996) 808–814, https://doi.org/10.1139/
T96-106-326.
[283] J. Yin, J. Graham, Equivalent times and one-dimensional elastic viscoplastic modelling of time-dependent stress–strain behaviour of clays, Can. Geotech. J. 31
(1) (1994) 42–52, https://doi.org/10.1139/t94-005.
[284] J. Sun, Rheological Behavior of Geomaterials and its Engineering Applications, China Architecture and Building Press, Beijing, 1999, pp. 60–61. https://
scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Sun%2C+J.+%281999%29.+Rheology+of+geomaterials+and+applications%2C+China+
Construction+Publication+House%2C+Beijing.&btnG=.
[285] C. O’Loughlin, Modelling the time-dependent compression of fibrous peat using the EVP model. In Proceedings of the 14th Young Geotechnical Engineering
Conference. https://research-repository.uwa.edu.au/en/publications/modelling-the-time-dependent-compression-of-fibrous-peat-using-th, 2001.
[286] F. Tong, J.-H. Yin, Experimental and constitutive modeling of relaxation behaviors of three clayey soils, J. Geotech. Geoenviron. Eng. 139 (11) (2013)
1973–1981, https://doi.org/10.1061/(ASCE)GT, 1943-5606.0000926.
[287] R.I. Borja, K.M. Sama, P.F. Sanz, On the numerical integration of three-invariant elastoplastic constitutive models, Comput. Methods Appl. Mech. Eng. 192
(9–10) (2003) 1227–1258, https://doi.org/10.1016/S0045-7825(02)00620-5.
[288] J. Duriez, E. Vincens, Constitutive modelling of cohesionless soils and interfaces with various internal states: an elasto-plastic approach, Comput. Geotech. 63
(2015) 33–45, https://doi.org/10.1016/J.COMPGEO.2014.08.001.
[289] A. Panteghini, R. Lagioia, An extended modified Cam-Clay yield surface for arbitrary meridional and deviatoric shapes retaining full convexity and double
homothety, Geotechnique 68 (7) (2018) 590–601, https://doi.org/10.1680/JGEOT.17.P.016.
[290] E.Q. Chowdhury, T. Nakai, Consequences of the tij-concept and a new modeling approach, Comput. Geotech. 23 (3) (1998) 131–164, https://doi.org/10.1016/
S0266-352X(98)00017-2.
[291] Y.P. Yao, W. Hou, A.N. Zhou, UH model: three-dimensional unified hardening model for overconsolidated clays, Geotechnique 59 (5) (2009) 451–469, https://
doi.org/10.1680/GEOT.2007.00029.
[292] W. Sumelka, Fractional viscoplasticity, Mech. Res. Commun. 56 (2014) 31–36, https://doi.org/10.1016/J.MECHRESCOM.2013.11.005.
26
Heliyon 9 (2023) e14465
K.C. Onyelowe et al.
[293] H.D. Bui, A. Constantinescu, H. Maigre, Numerical identification of linear cracks in 2D elastodynamics using the instantaneous reciprocity gap, Inverse Probl.
20 (4) (2004) 993, https://doi.org/10.1088/0266-5611/20/4/001.
[294] D. Claire, F. Hild, S. Roux, A finite element formulation to identify damage fields: the equilibrium gap method, Int. J. Numer. Methods Eng. 61 (2) (2004)
189–208, https://doi.org/10.1002/NME.1057.
[295] D. Lecompte, A. Smits, H. Sol, J. Vantomme, D. Van Hemelrijck, Mixed numerical–experimental technique for orthotropic parameter identification using
biaxial tensile tests on cruciform specimens, Int. J. Solid Struct. 44 (5) (2007) 1643–1656, https://doi.org/10.1016/J.IJSOLSTR.2006.06.050.
[296] E. Pagnacco, A. Moreau, D. Lemosse, Inverse strategies for the identification of elastic and viscoelastic material parameters using full-field measurements,
Mater. Sci. Eng., A 452–453 (2007) 737–745, https://doi.org/10.1016/J.MSEA.2006.10.122.
[297] T. Wu, M. Coret, A. Combescure, Strain localisation and damage measurement by full 3D digital image correlation: application to 15-5PH stainless steel, Strain
47 (1) (2011) 49–61, https://doi.org/10.1111/J.1475-1305.2008.00600.X.
[298] J. Réthoré, A fully integrated noise robust strategy for the identification of constitutive laws from digital images, Int. J. Numer. Methods Eng. 84 (6) (2010)
631–660, https://doi.org/10.1002/NME.2908.
[299] M. Rossi, F. Pierron, M. Rossi, ⋅ F. Pierron, F. Pierron, Identification of plastic constitutive parameters at large deformations from three dimensional
displacement fields, Comp. Mech. 49 (1) (2011) 53–71, https://doi.org/10.1007/S00466-011-0627-0, 2011 49:1.
[300] A. Schofield, P. Wroth, Critical state soil mechanics. https://ndl.ethernet.edu.et/bitstream/123456789/46322/1/23.pdf, 1968.
[301] K. Roscoe, J. Burland, On the generalized stress-strain behaviour of wet clay. https://trid.trb.org/view/124868, 1968.
[302] M. Jefferies, K. Been, Implications for critical state theory from isotropic compression of sand, Geotechnique 50 (4) (2000) 419–429, https://doi.org/10.1680/
GEOT.2000.50.4.419.
[303] M.G. Jefferies, Nor-Sand: a simle critical state model for sand, Geotechnique 43 (1) (2015) 91–103, https://doi.org/10.1680/GEOT.1993.43.1.91.
[304] A.M. Britto, M. Arul, M.J. Gunn, J. Michael, Critical State Soil Mechanics Via Finite Elements, 1987, p. 488.
[305] J. Atkinson, The Mechanics of Soils and Foundations, Second Edition. In the Mechanics of Soils and Foundations, Second Edition, CRC Press, 2007, https://doi.
org/10.1201/9781315273549/MECHANICS-SOILS-FOUNDATIONS-JOHN-ATKINSON.
[306] J.H. Atkinson, P.L. Bransby, The Mechanics Of Soils : An Introduction To Critical State Soil Mechanics, 1978. https://espace.library.uq.edu.au/view/UQ:
731372.
[307] J.A.R. Ortigao, Soil Mechanics In The light Of Critical State Theories : An Introduction. Soil Mechanics In The Light of Critical State Theories, 2020, https://doi.
org/10.1201/9781003077701.
[308] J.J. Zhang, Rock strengths and rock failure criteria, Appl. Petrol. Geomechan. (2019) 85–131, https://doi.org/10.1016/B978-0-12-814814-3.00003-4.
[309] H.S. Yu, P.Z. Zhuang, P.Q. Mo, A unified critical state model for geomaterials with an application to tunnelling, J. Rock Mech. Geotech. Eng. 11 (3) (2019)
464–480, https://doi.org/10.1016/J.JRMGE.2018.09.004.
[310] Yannis F. Dafalias, Must critical state theory be revisited to include fabric effects? Acta Geotechnica 11 (3) (2016) 479–491, https://doi.org/10.1007/S11440016-0441-0.
[311] S. Rachdi, E. Jahangir, M. Tijani, J.F. Serratrice, Critical state constitutive models and shear loading of overconsolidated clays with deviatoric hardening.
Studia Geotechnica et Mechanica. https://doi.org/10.2478/SGEM-2019-0024, 2019.
[312] Y.N. Chen, Z.X. Yang, A family of improved yield surfaces and their application in modeling of isotropically over-consolidated clays, Comput. Geotech. 90
(2017) 133–143, https://doi.org/10.1016/J.COMPGEO.2017.06.007.
[313] A. Gens, D.M. Potts, Critical state models in computational geomechanic*, Eng. Comput. 5 (3) (1988) 178–197, https://doi.org/10.1108/EB023736/FULL/
HTML.
[314] R. Lagioia, A.M. Puzrin, D.M. Potts, A new versatile expression for yield and plastic potential surfaces, Comput. Geotech. 19 (3) (1996) 171–191, https://doi.
org/10.1016/0266-352X(96)00005-5.
[315] Multi-Surface and Bounding Surface Plasticity, Plasticity and Geotechnics. Advances in Mechanics and Mathematics, 13, Springer, Boston, MA, 2006,
pp. 153–196, https://doi.org/10.1007/978-0-387-33599-5_7.
[316] Z. Mróz, On the description of anisotropic workhardening, J. Mech. Phys. Solid. 15 (3) (1967) 163–175, https://doi.org/10.1016/0022-5096(67)90030-0.
[317] W. Iwan, On a class of models for the yielding behavior of continuous and composite systems. 612–617. https://asmedigitalcollection.asme.org/
appliedmechanics/article-abstract/34/3/612/387102, 1967.
[318] Y.F. Dafalias, E.P. Popov, A model of nonlinearly hardening materials for complex loading, Acta Mech. 21 (3) (1975) 173–192, https://doi.org/10.1007/
BF01181053.
[319] R. Krieg, A Practical Two Surface Plasticity Theory, 1975. https://asmedigitalcollection.asme.org/appliedmechanics/article-abstract/42/3/641/388046.
[320] J. Prévost, -H, Mathematical modelling of monotonic and cyclic undrained clay behaviour, Int. J. Numer. Anal. Methods GeoMech. 1 (2) (1977) 195–216,
https://doi.org/10.1002/NAG.1610010206.
[321] J.H. Prevost, Plasticity theory for soil stress-strain behavior, J. Eng. Mech. Div. 104 (5) (1978) 1177–1194, https://doi.org/10.1061/JMCEA3.0002411.
[322] Z. Mróz, V.A. Norris, O.C. Zienkiewicz, Application of an anisotropic hardening model in the analysis of elasto—plastic deformation of soils, Geotechnique 29
(1) (1979) 1–34, https://doi.org/10.1680/GEOT.1979.29.1.1.
[323] Z. Mröz, V.A. Norrist, O.C. Zienkiewicz, An anisotropic, critical state model for soils subject to cyclic loading, Geotechnique 31 (4) (1981) 451–469, https://
doi.org/10.1680/GEOT.1981.31.4.451.
[324] Y. Dafalias, Bounding Surface Formulation Of Soil Plasticity. Soil Mechanics-Transient And Cyclic Loads, 1982, pp. 253–282. https://ci.nii.ac.jp/naid/
10029489830/.
[325]] A.J. Whittle, M.J. Kavvadas, Formulation of MIT-E3 constitutive model for overconsolidated clays, J. Geotech. Eng. 120 (1) (1994) 173–198, https://doi.org/
10.1061/(ASCE)0733-9410(1994)120:1(173.
[326]] A. Al-Tabbaa, D. Muir Wood, Experimentally based" bubble" model for clay. Numerical models in geomechanics, in: Numog Iii. Proceedings Of The 3rd
International Symposium Held At Niagara Falls, Canada. Publication of, Elsevier Applied Science Publishers Limited, 1989. https://trid.trb.org/view/316121.
[327] Y. Tian, M.J. Cassidy, Modeling of pipe–soil interaction and its application in numerical simulation, Int. J. GeoMech. 8 (4) (2008) 213–229, https://doi.org/
10.1061/(ASCE)1532-3641(2008, 8:4(213).
[328] M.D. Liu, J.P. Carter, A structured Cam Clay model, Can. Geotech. J. 39 (6) (2002) 1313–1332, https://doi.org/10.1139/T02-069.
[329] J. Suebsuk, S. Horpibulsuk, M.D. Liu, Modified Structured Cam Clay: a generalised critical state model for destructured, naturally structured and artificially
structured clays, Comput. Geotech. 37 (7–8) (2010) 956–968, https://doi.org/10.1016/J.COMPGEO.2010.08.002.
[330] D.M. Dimiduk, E.A. Holm, S.R. Niezgoda, Perspectives on the impact of machine learning, deep learning, and artificial intelligence on materials, processes, and
structures engineering, Integr. mMatermManuf. innovation 7 (3) (2018) 157–172, https://doi.org/10.1007/S40192-018-0117-8, 2018 7:3.
[331] K.O. Achieng, Modelling of soil moisture retention curve using machine learning techniques: artificial and deep neural networks vs support vector regression
models, Comput. Geosci. 133 (2019) 104320, https://doi.org/10.1016/J.CAGEO.2019.104320.
[332] D.T. Bui, P. Tsangaratos, V.T. Nguyen, N. Liem, Van, & Trinh, P. T, Comparing the prediction performance of a Deep Learning Neural Network model with
conventional machine learning models in landslide susceptibility assessment, Catena 188 (2020) 104426, https://doi.org/10.1016/J.CATENA.2019.104426.
[333] P. Zhang, Y.F. Jin, Z.Y. Yin, Machine learning–based uncertainty modelling of mechanical properties of soft clays relating to time-dependent behavior and its
application, Int. J. Numer. Anal. Methods GeoMech. 45 (11) (2021) 1588–1602, https://doi.org/10.1002/NAG.3215.
[334] Zeliang Liu, C.T. Wu, Exploring the 3D architectures of deep material network in data-driven multiscale mechanics, J. Mech. Phys. Solid. 127 (2019) 20–46,
https://doi.org/10.1016/J.JMPS.2019.03.004.
[335] M. Peigney, A Fourier-based machine learning technique with application in engineering, Int. J. Numer. Methods Eng. 122 (3) (2021) 866–897, https://doi.
org/10.1002/NME.6565.
[336] J. Kneifl, D. Grunert, J. Fehr, A nonintrusive nonlinear model reduction method for structural dynamical problems based on machine learning, Int. J. Numer.
Methods Eng. 122 (17) (2021) 4774–4786, https://doi.org/10.1002/NME.6712.
27
Heliyon 9 (2023) e14465
K.C. Onyelowe et al.
[337] K. Xu, D.Z. Huang, E. Darve, Learning constitutive relations using symmetric positive definite neural networks, J. Comput. Phys. 428 (2021) 110072, https://
doi.org/10.1016/J.JCP.2020.110072.
[338] P. Zhang, Z.-Y. Yin, Y.-F. Jin, Bayesian neural network-based uncertainty modelling: application to soil compressibility and undrained shear strength
prediction, Can. Geotech. J. 59 (4) (2022) 546–557, https://doi.org/10.1139/CGJ-2020-0751.
[339] P. Zhang, Z.Y. Yin, Y.F. Jin, G.L. Ye, An AI-based model for describing cyclic characteristics of granular materials, Int. J. Numer. Anal. Methods GeoMech. 44
(9) (2020) 1315–1335, https://doi.org/10.1002/NAG.3063.
[340] P. Zhang, Z.-Y. Yin, Y.-F. Jin, J. Yang, B. Sheil, Physics-Informed multifidelity residual neural networks for hydromechanical modeling of granular soils and
foundation considering internal erosion, J. Eng. Mech. 148 (4) (2022), 4022015, https://doi.org/10.1061/(ASCE)EM, 1943-7889.0002094.
[341] P. Zhang, Z.Y. Yin, A novel deep learning-based modelling strategy from image of particles to mechanical properties for granular materials with CNN and
BiLSTM, Comput. Methods Appl. Mech. Eng. 382 (2021) 113858, https://doi.org/10.1016/J.CMA.2021.113858.
[342] P. Zhang, Z.Y. Yin, Y.F. Jin, State-of-the-Art review of machine learning applications in constitutive modeling of soils, 2021 28:5, Arch. Comput. Methods Eng.
28 (5) (2021) 3661–3686, https://doi.org/10.1007/S11831-020-09524-Z.
[343] P. Zhang, Z.Y. Yin, Y.F. Jin, T.H.T. Chan, A novel hybrid surrogate intelligent model for creep index prediction based on particle swarm optimization and
random forest, Eng. Geol. 265 (2020) 105328, https://doi.org/10.1016/J.ENGGEO.2019.105328.
[344] P. Zhang, Z.Y. Yin, Y.F. Jin, T.H.T. Chan, F.P. Gao, Intelligent modelling of clay compressibility using hybrid meta-heuristic and machine learning algorithms,
Geosci. Front. 12 (1) (2021) 441–452, https://doi.org/10.1016/J.GSF.2020.02.014.
[345] R.P. Chen, P. Zhang, X. Kang, Z.Q. Zhong, Y. Liu, H.N. Wu, Prediction of maximum surface settlement caused by earth pressure balance (EPB) shield tunneling
with ANN methods, Soils Found. 59 (2) (2019) 284–295, https://doi.org/10.1016/J.SANDF.2018.11.005.
[346] J. Ghaboussi, J.H. Garrett Jr., X. Wu, Knowledge bBased modeling of material behavior with neural networks, J. Eng. Mech. 117 (1) (1991) 132–153, https://
doi.org/10.1061/(ASCE)0733-9399(1991, 117:1(132).
[347] G.W. Ellis, C. Yao, R. Zhao, D. Penumadu, Stress-strain modeling of sands using artificial neural networks, J. Geotech. Eng. 121 (5) (1995) 429–435, https://
doi.org/10.1061/(ASCE)0733-9410(1995, 121:5(429).
[348] G. Liang, K. Chandrashekhara, Neural network based constitutive model for elastomeric foams, Eng. Struct. 30 (7) (2008) 2002–2011, https://doi.org/
10.1016/J.ENGSTRUCT.2007.12.021.
[349] Y. Shen, K. Chandrashekhara, W.F. Breig, L.R. Oliver, Finite element analysis of V-ribbed belts using neural network based hyperelastic material model, Int. J.
Non Lin. Mech. 40 (6) (2005) 875–890, https://doi.org/10.1016/J.IJNONLINMEC.2004.10.005.
[350] T. Furukawa, G. Yagawa, Implicit constitutive modelling for viscoplasticity using neural networksr. International journal for numerical methods in engineering
int, J. Numer. Meth. Engng 43 (1998) 195–219, https://doi.org/10.1002/(SICI)1097-0207(19980930, 43:2.
[351] M. Mozaffar, R. Bostanabad, W. Chen, K. Ehmann, J. Cao, M.A. Bessa, Deep learning predicts path-dependent plasticity, Nat. Acad. Sci. 116 (52) (2019)
26414–26420, https://doi.org/10.1073/pnas.1911815116/-/DCSupplemental.y.
[352] F. Ghavamian, A. Simone, Accelerating multiscale finite element simulations of history-dependent materials using a recurrent neural network, Comput.
Methods Appl. Mech. Eng. 357 (2019) 112594, https://doi.org/10.1016/J.CMA.2019.112594.
[353] A. Zhang, D. Mohr, Using neural networks to represent von Mises plasticity with isotropic hardening, Int. J. Plast. 132 (2020) 102732, https://doi.org/
10.1016/J.IJPLAS.2020.102732.
[354] W. Zhang, H. Li, Y. Li, H. Liu, Y. Chen, X. Ding, Application of deep learning algorithms in geotechnical engineering: a short critical review, Artif. Intell. Rev. 54
(8) (2021) 5633–5673, https://doi.org/10.1007/S10462-021-09967-1/TABLES/2.
[355] P. Zhang, Y. Yang, Z.-Y. Yin, BiLSTM-based soil–structure interface modeling, Int. J. GeoMech. 21 (7) (2021), 4021096, https://doi.org/10.1061/(ASCE)GM,
1943-5622.0002058.
[356] J. Tian, C. Qi, Y. Sun, Z.M. Yaseen, Surrogate permeability modelling of low-permeable rocks using convolutional neural networks, Comput. Methods Appl.
Mech. Eng. 366 (2020) 113103, https://doi.org/10.1016/J.CMA.2020.113103.
[357] M. Schwarzer, B. Rogan, Y. Ruan, Z. Song, D.Y. Lee, A.G. Percus, V.T. Chau, B.A. Moore, E. Rougier, H.S. Viswanathan, G. Srinivasan, Learning to fail:
predicting fracture evolution in brittle material models using recurrent graph convolutional neural networks, Comput. Mater. Sci. 162 (2019) 322–332,
https://doi.org/10.1016/J.COMMATSCI.2019.02.046.
[358] Zhen Wang, Z. Zhu, S. Zhu, Thermo-mechanical-water migration coupled plastic constitutive model of rock subjected to freeze-thaw, Cold Reg. Sci. Technol.
161 (2019) 71–80, https://doi.org/10.1016/J.COLDREGIONS.2019.03.001.
[359] N.N. Vlassis, W.C. Sun, Sobolev training of thermodynamic-informed neural networks for interpretable elasto-plasticity models with level set hardening,
Comput. Methods Appl. Mech. Eng. 377 (2021) 113695, https://doi.org/10.1016/J.CMA.2021.113695.
[360] N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, R. Salakhutdinov, Dropout: a simple way to prevent neural networks from overfitting, J. Mach. Learn.
Res. 15 (1) (2014) 1929–1958. https://www.jmlr.org/papers/volume15/srivastava14a/srivastava14a.pdf?utm_content=buffer79b43&utm_
medium=social&utm_source=twitter.com&utm_campaign=buffer.
[361] Y.F. Jin, Z.Y. Yin, W.H. Zhou, H.W. Huang, Multi-objective optimization-based updating of predictions during excavation, Eng. Appl. Artif. Intell. 78 (2019)
102–123, https://doi.org/10.1016/J.ENGAPPAI.2018.11.002.
[362] A. Krizhevsky, I. Sutskever, G.E. Hinton, Imagenet classification with deep convolutional neural networks, Adv. Neural Inf. Process. Syst. 25 (2012), in: https://
proceedings.neurips.cc/paper/2012/hash/c399862d3b9d6b76c8436e924a68c45b-Abstract.html.
[363] R. Moradi, R. Berangi, B. Minaei, A survey of regularization strategies for deep models, Artif. Intell. Rev. 53 (6) (2020) 3947–3986, https://doi.org/10.1007/
S10462-019-09784-7/TABLES/6.
[364] Y. Lai, S. Li, J. Qi, Z. Gao, X. Chang, Strength distributions of warm frozen clay and its stochastic damage constitutive model, Cold Reg. Sci. Technol. 53 (2)
(2008) 200–215, https://doi.org/10.1016/J.COLDREGIONS.2007.11.001.
[365] C.S. Chang, Z.Y. Yin, Micromechanical modeling for inherent anisotropy in granular materials, J. Eng. Mech. 136 (7) (2010) 830–839.
[366] A.R. Kalantari, A. Johari, M. Zandpour, M. Kalantari, Effect of spatial variability of soil properties and geostatistical conditional simulation on reliability
characteristics and critical slip surfaces of soil slopes, Trans. Geotech. (2023), 100933.
[367] Z. Zou, J. Yan, H. Tang, S. Wang, C. Xiong, X. Hu, A shear constitutive model for describing the full process of the deformation and failure of slip zone soil, Eng.
Geol. 276 (2020) 105766.
[368] M. Mohsan, P.J. Vardon, F.C. Vossepoel, On the use of different constitutive models in data assimilation for slope stability, Comput. Geotech. 138 (2021)
104332.
[369] B.C.B. Hsiung, S.D. Dao, Evaluation of constitutive soil models for predicting movements caused by a deep excavation in sands, Electron. J. Geotech. Eng. 19
(2014) 17325–17344.
[370] A. Sekhavatian, A. Janalizadeh Choobbasti, Comparison of Point Estimate and Monte Carlo probabilistic methods in stability analysis of a deep excavation, Int.
J. Geo-Eng. 9 (1) (2018) 20.
[371] A. Sekhavatian, A. Janalizadeh Choobbasti, Reliability analysis of deep excavations by RS and MCS methods: case study, Innov. Infrastru. Solut. 3 (2018) 1–15.
[372] K. Georgiadis, D.M. Potts, L. Zdravkovic, Numerical Study of the Stability of a Partially Saturated Soil Slope. In X International Symposium on Numerical
Models in Geomechanics (NUMOG X), Rhodes, Greece, Balkema, 2007, pp. 545–550.
[373] S.K. Vanapalli, F.M. Mohamed, Bearing Capacity of Model Footings in Unsaturated Soils. In Experimental Unsaturated Soil Mechanics, Springer Berlin
Heidelberg, 2007, pp. 483–493.
[374] B.T. McMahon, S.K. Haigh, M.D. Bolton, Bearing capacity and settlement of circular shallow foundations using a nonlinear constitutive relationship, Can.
Geotech. J. 51 (9) (2014) 995–1003.
[375] F. Wuttke, B. Kafle, Y. Lins, T. Schanz, Macroelement for statically loaded shallow strip foundation resting on unsaturated soil, Int. J. GeoMech. 13 (5) (2013)
557–564.
[376] J. Ghorbani, M. Nazem, J.P. Carter, Numerical modelling of multiphase flow in unsaturated deforming porous media, Comput. Geotech. 71 (2016) 195–206.
28
Heliyon 9 (2023) e14465
K.C. Onyelowe et al.
[377] Y. Tang, H.A. Taiebat, A.R. Russell, Bearing capacity of shallow foundations in unsaturated soil considering hydraulic hysteresis and three drainage conditions,
Int. J. GeoMech. 17 (6) (2017), 4016142.
[378] A.A. Lavasan, M. Ghazavi, A. von Blumenthal, T. Schanz, Bearing capacity of interfering strip footings, J. Geotech. Geoenviron. Eng. 144 (3) (2018), 04018003.
[379] T.M. Evans, J.D. Baker, An Effective Stress Framework for Bearing Capacity of Shallow Foundations in Unsaturated Soils. In Proceedings of the 3rd
International Conference on Sustainability in Civil Engineering: ICSCE 2020, 26-27 November, Vietnam, Hanoi, 2021, pp. 3–14 (Springer Singapore).
[380] G.T. Houlsby, C.P. Wroth, The variation of shear modulus of a clay with pressure and over consolidation ratio, Soils Found. 31 (3) (1991) 138–143.
[381] A.A. Abed, P.A. Vermeer, Numerical simulation of unsaturated soil behaviour, Int. J. Comput. Appl. Technol. 34 (1) (2009) 2–12.
[382] A.S. Osman, M.D. Bolton, Simple plasticity-based prediction of the undrained settlement of shallow circular foundations on clay, Geotechnique 55 (6) (2005)
435–447.
[383] A. Klar, A.S. Osman, Load–displacement solutions for piles and shallow foundations based on deformation fields and energy conservation, Geotechnique 58 (7)
(2008) 581–589.
[384] P.J. Vardanega, M.D. Bolton, Strength mobilization in clays and silts, Can. Geotech. J. 48 (10) (2011) 1485–1503.
[385] J.P. Doherty, D. Muir Wood, An extended Mohr–Coulomb (EMC) model for predicting the settlement of shallow foundations on sand, Geotechnique 63 (8)
(2013) 661–673.
[386] T.M.H. Le, D. Gallipoli, M. Sanchez, S. Wheeler, Rainfall-induced differential settlements of foundations on heterogeneous unsaturated soils, Geotechnique 63
(15) (2013) 1346–1355.
[387] W.T. Oh, S.K. Vanapalli, Modeling the stress versus settlement behavior of shallow foundations in unsaturated cohesive soils extending the modified total stress
approach, Soils Found. 58 (2) (2018) 382–397.
[388] A. Federico, G. Elia, A. Murianni, The at-rest earth pressure coefficient prediction using simple elasto-plastic constitutive models, Comput. Geotech. 36 (1–2)
(2009) 187–198.
[389] L. Al Atik, N. Sitar, Seismic earth pressures on cantilever retaining structures, J. Geotech. Geoenviron. Eng. 136 (10) (2010) 1324–1333.
[390] J. Bakr, S.M. Ahmad, A finite element performance-based approach to correlate movement of a rigid retaining wall with seismic earth pressure, Soil Dynam.
Earthq. Eng. 114 (2018) 460–479.
[391] S. Yang, H.I. Ling, Calibration of a generalized plasticity model and its application to liquefaction analysis, in: Soil Constitutive Models (Ed.), Evaluation,
Selection, and Calibration, 2005, pp. 483–494.
[392] G. Masing, Eigenspannungen und verfestigung beim messing, in: In Proceedings, second international congress of applied mechanics, 1926, pp. 332–335.
[393] R.I. Borja, C.H. Lin, K.M. Sama, G.M. Masada, Modelling non-linear ground response of non-liquefiable soils, Earthq. Eng. Struct. Dynam. 29 (1) (2000) 63–83.
[394] D.V. Okur, A. Ansal, Stiffness degradation of natural fine grained soils during cyclic loading, Soil Dynam. Earthq. Eng. 27 (9) (2007) 843–854.
[395] C. Phillips, Y.M. Hashash, Damping formulation for nonlinear 1D site response analyses, Soil Dynam. Earthq. Eng. 29 (7) (2009) 1143–1158.
[396] S.H. Chong, Soil dynamic constitutive model for characterizing the nonlinear-hysteretic response, Appl. Sci. 7 (11) (2017) 1110.
[397] G.R. Webb, B.R. Bass, C.H. Goodman, M.A. Goodman, K.M. Land, Reply by authors to RH plaut, and to PC parks and AJ pritchard, AIAA J. 6 (7) (1968),
1438–1438.
[398] G.W. Clough, J.M. Duncan, Finite element analyses of retaining wall behavior, J. Soil Mech. Found Div. 97 (12) (1971) 1657–1673.
[399] J. Ghaboussi, E.L. Wilson, J. Isenberg, Finite element for rock joints and interfaces, J. Soil Mech. Found Div. 99 (10) (1973) 833–848.
[400] A. Gens, I. Carol, E.E. Alonso, A constitutive model for rock joints formulation and numerical implementation, Comput. Geotech. 9 (1–2) (1990) 3–20.
[401] K. Fakharian, E. Evgin, Elasto-plastic modelling of stress-path-dependent behaviour of interfaces, Int. J. Numer. Anal. Methods GeoMech. 24 (2) (2000)
183–199.
[402] L. Hu, J. Pu, Testing and modeling of soil-structure interface, J. Geotech. Geoenviron. Eng. 130 (8) (2004) 851–860.
[403] A. Zhou, T. Lu, Elasto-plastic constitutive model of soil-structure interface in consideration of strain softening and dilation, Acta Mech. Solida Sin. 22 (2) (2009)
171–179.
[404] H. Torabi, M.T. Rayhani, Three dimensional finite element modeling of seismic soil–structure interaction in soft soil, Comput. Geotech. 60 (2014) 9–19.
[405] J.A. Knappett, P. Madden, K. Caucis, Seismic structure–soil–structure interaction between pairs of adjacent building structures, Geotechnique 65 (5) (2015)
429–441.
[406] M. Saberi, C.D. Annan, J.M. Konrad, Implementation of a soil-structure interface constitutive model for application in geo-structures, Soil Dynam. Earthq. Eng.
116 (2019) 714–731.
[407] J.A. Mercado, K.R. Mackie, L.G. Arboleda-Monsalve, Modeling nonlinear-inelastic seismic response of tall buildings with soil–structure interaction, J. Struct.
Eng. 147 (7) (2021), 4021091.
[408] L.X. Luccioni, Numerical Development and Implementation of a Constitutive Model for Clays with Application to Deformations Around a Deep Excavation,
University of California, Berkeley, 1999.
[409] Y. Hejazi, D. Dias, R. Kastner, Impact of constitutive models on the numerical analysis of underground constructions, Acta Geotechnica 3 (2008) 251–258.
[410] E. Bourgeois, P. de Buhan, G. Hassen, Settlement analysis of piled-raft foundations by means of a multiphase model accounting for soil-pile interactions,
Comput. Geotech. 46 (2012) 26–38.
[411] M.J.K. Essa, C.S. Desai, Dynamic soil–pile interaction using the DSC constitutive model, Indian Geotech. J. 47 (2017) 137–149.
[412] M. Mikkola, J. Hartikainen, Mathematical model of soil freezing and its numerical implementation, Int. J. Numer. Methods Eng. 52 (5–6) (2001) 543–557.
[413] M. Dall’Amico, S. Endrizzi, S. Gruber, R.J.T.C. Rigon, A robust and energy-conserving model of freezing variably-saturated soil, Cryosphere 5 (2) (2011)
469–484.
[414] Y. Bai, L. Wang, S. Zhang, N. Xie, H. Chen, Heat transfer characteristics of a natural circulation separate heat pipe under various operating conditions, Int. J.
Heat Mass Tran. 126 (2018) 191–200.
[415] P. Iodice, G. Langella, A. Amoresano, Energy performance and numerical optimization of a screw expander–based solar thermal electricity system in a wide
range of fluctuating operating conditions, Int. J. Energy Res. 44 (3) (2020) 1858–1874.
[416] Song, H., Pei, H., & Cui, H. (2021, October). A comparison study on the long-term thermomechanical performance of energy piles under three climatic
conditions. In IOP Conference Series: Earth and Environmental Science (Vol. 861, No. 7, p. 72128). IOP Publishing.
[417] X. Que, Z. Zhu, Z. Niu, W. Lu, Estimating the strength and deformation of columnar jointed rock mass based on physical model test, Bull. Eng. Geol. Environ. 80
(2021) 1557–1570.
[418] W. Zhang, K. Maeda, H. Saito, Z. Li, Y. Huang, Numerical analysis on seepage failures of dike due to water level-up and rainfall using a water–soil-coupled
smoothed particle hydrodynamics model, Acta Geotechnica 11 (2016) 1401–1418.
[419] Y.X. Wu, S.L. Shen, H.M. Lyu, A. Zhou, Analyses of leakage effect of waterproof curtain during excavation dewatering, J. Hydrol. 583 (2020) 124582.
[420] L. Laloui, Thermo-mechanical behaviour of soils, Revue française de génie civil 5 (6) (2001) 809–843.
[421] E.L. Liu, H.L. Xing, A double hardening thermo-mechanical constitutive model for overconsolidated clays, Acta Geotechnica 4 (1) (2009) 1–6.
[422] C.W.W. Ng, C. Zhou, Plastic Deformations of Unsaturated Fine-Grained Soils Under Cyclic Thermo-Mechanical Loads. In Advances in Laboratory Testing and
Modelling of Soils and Shales (ATMSS), Springer International Publishing, 2017, pp. 14–28.
[423] D.M. Zymnis, A.J. Whittle, X. Cheng, Simulation of long-term thermo-mechanical response of clay using an advanced constitutive model, Acta Geotechnica 14
(2019) 295–311.
[424] A. Zhou, D. Sheng, An advanced hydro-mechanical constitutive model for unsaturated soils with different initial densities, Comput. Geotech. 63 (2015) 46–66.
[425] X.Q. Lei, Z.J. Yang, S.M. He, E.L. Liu, H. Wong, X.P. Li, Hydro-mechanical analysis of rainfall-induced fines migration process within unsaturated soils, J. Mt.
Sci. 14 (12) (2017) 2603–2619.
[426] E. Gholizadeh, M. Latifi, A coupled hydro-mechanical constitutive model for unsaturated frictional and cohesive soil, Comput. Geotech. 98 (2018) 69–81.
[427] Y. Qiao, Y. Xiao, L. Laloui, W. Ding, M. He, A double-structure hydromechanical constitutive model for compacted bentonite, Comput. Geotech. 115 (2019)
103173.
29
Heliyon 9 (2023) e14465
K.C. Onyelowe et al.
[428] Y. Tang, W. Wu, K. Yin, S. Wang, G. Lei, A hydro-mechanical coupled analysis of rainfall induced landslide using a hypoplastic constitutive model, Comput.
Geotech. 112 (2019) 284–292.
[429] K. Georgiadis, D.M. Potts, L. Zdravkovic, Three-dimensional constitutive model for partially and fully saturated soils, Int. J. GeoMech. 5 (3) (2005) 244–255.
[430] P. Liu, R.P. Chen, K. Wu, X. Kang, Effects of drying-wetting cycles on the mechanical behavior of reconstituted granite-residual soils, J. Mater. Civ. Eng. 32 (8)
(2020), 04020199.
[431] D.S. Liyanapathirana, H.G. Poulos, A numerical model for dynamic soil liquefaction analysis, Soil Dynam. Earthq. Eng. 22 (2002) 1007–1015, 9-12.
[432] Z. Mroz, N. Boukpeti, A. Drescher, Constitutive model for static liquefaction, Int. J. GeoMech. 3 (2) (2003) 133–144.
[433] A. Najma, M. Latifi, Predicting flow liquefaction, a constitutive model approach, Acta Geotechnica 12 (2017) 793–808.
[434] A. Janalizadeh Choobbasti, H. Selatahneh, M. Karimi Petanlar, Effect of fines on liquefaction resistance of sand, Innov. Infrastru. Solut. 5 (2020) 1–16.
[435] X. Lü, D. Xue, B. Zhang, Q. Zhong, Experimental studies and constitutive modeling of static liquefaction instability in sand–clay mixtures, Int. J. GeoMech. 22
(9) (2022), 4022149.
[436] F. Collin, X.L. Li, J.P. Radu, R. Charlier, Thermo-hydro-mechanical coupling in clay barriers, Eng. Geol. 64 (2002) 179–193, 2-3.
[437] S. Chemkhi, W. Jomaa, F. Zagrouba, Application of a coupled thermo-hydro-mechanical model to simulate the drying of nonsaturated porous media, Dry.
Technol. 27 (7–8) (2009) 842–850.
[438] Z. Liu, X. Yu, Coupled thermo-hydro-mechanical model for porous materials under frost action: theory and implementation, Acta Geotechnica 6 (2011) 51–65.
[439] Y. Kang, Q. Liu, S. Huang, A fully coupled thermo-hydro-mechanical model for rock mass under freezing/thawing condition, Cold Reg. Sci. Technol. 95 (2013)
19–26.
[440] X. Zhuang, R. Huang, C. Liang, T. Rabczuk, A coupled thermo-hydro-mechanical model of jointed hard rock for compressed air energy storage, Math. Probl
Eng. 2014 (2014).
[441] A. Gens, K. Wieczorek, I. Gaus, B. Garitte, J.C. Mayor, K. Schuster, T. Trick, Performance Of The Opalinus Clay Under Thermal Loading: Experimental Results
From Mont Terri Rock Laboratory (Switzerland). Mont Terri Rock Laboratory, 20 Years: Two Decades of Research And Experimentation on Claystones for
Geological Disposal of Radioactive Waste, 2018, pp. 271–288.
[442] G. Yang, B. Bai, Thermo-hydro-mechanical model for unsaturated clay soils based on granular solid hydrodynamics theory, Int. J. GeoMech. 19 (10) (2019),
4019115.
30