Computers and Geotechnics 172 (2024) 106430
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Computers and Geotechnics
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Research Paper
Simulation of cone penetration in soil using the material point method
Vibhav Bisht a, Rodrigo Salgado b, *, Monica Prezzi c
a
Software Engineer, Align Technology, 3030 Slater Rd, Morrisville, NC 27560, United States
Charles Pankow Professor in Civil Engineering, Lyles School of Civil Engineering, Purdue Univ., West Lafayette, IN 47907, United States
c
Professor of Civil Engineering, Lyles School of Civil Engineering, Purdue Univ., West Lafayette, IN 47907, United States
b
A R T I C L E I N F O
A B S T R A C T
Keywords:
Generalized interpolation material point
method
MPM
Cone penetration
Critical state model
Realistic simulation of cone penetration in soil presents two main challenges: (1) a numerical scheme that is
accurate under large deformations and that does not resort to overly simplifying assumptions, and (2) a
constitutive model that can accurately predict the stress–strain response of soil under a wide variety of loading
conditions. This paper presents a brief overview of the progress made towards realistic cone penetration simu­
lation. The choice of numerical scheme (the material point method) and constitutive models (bounding surface
models) used for simulating cone penetration in this study are discussed. For sand, 15 cone penetration simu­
lations are performed at different relative densities subjected to different effective stress values. The simulation
results are compared against those from cavity expansion analyses. For clays, cone penetration simulations are
performed at different overconsolidation ratios and initial effective stress. Simulation results are then compared
against cone penetration tests (CPTs) performed in the field. Based on the simulations, cone factor values are
evaluated. The advantages of the current approach and its application to improvement of CPT interpretation are
discussed.
1. Introduction
1.1. Cone penetration simulation
The cone penetration test (CPT) is now firmly established as one the
key site investigation tools available to geotechnical engineers in both
routine and challenging problems. It has features that have made it an
attractive alternative to higher-end laboratory testing or other in situ
tests (Salgado et al., 2022). First, it is economical with respect to so­
phisticated laboratory testing, yet can produce most of the results
desired from high-end testing if its results are interpreted using state-ofthe-art methods. Second, it produces reliable, repeatable results if per­
formed to standard. Third, it is quasi-static, eliminating the challenge of
considering dynamic load effects. However, its interpretation must be
based on rigorous mechanics. Computational simulations of cone
penetration in soil should therefore form the basis for establishing re­
lationships between cone resistance qc and soil state variables or state
properties, but these simulations face challenges.
The key challenges to accurate, realistic simulations of cone pene­
tration in soil are: (1) the occurrence of large displacements, de­
formations, and rotations in the soil, and (2) the evolving loading paths
at points in the soil surrounding the advancing cone penetrometer,
which range all the way from triaxial compression to triaxial extension
paths. Large deformations are problematic because the methods of
analysis that would typically be used for geotechnical analysis, like the
finite element method (FEM), lose accuracy and may not even converge
to a solution after the computational mesh distorts beyond some limit.
Simulation accuracy requires that the constitutive model used to
represent the soil be able to simulate soil response correctly, irrespective
of the loading path to which the soil is subjected.
The CPT has developed as a tool much faster than the ability to model
it theoretically. This is far from uncommon in geotechnical engineering:
the development of methods of testing or the invention of new types of
geotechnical structures or foundation elements typically precede efforts
to model them. Initial methods of interpretation of the CPT were
therefore semi-empirical or based on analyses of an approximate nature.
To remedy the gap in the theoretical basis for CPT interpretation, efforts
started early.
Initial attempts relied on modeling the soil as a rigid-plastic material
with either a Tresca or Mohr-Coulomb yield surface and applying the
notion of a limiting equilibrium to an axially loaded penetrometer and a
surrounding soil mass. The Durgunoglu and Mitchell (1975) model was
* Corresponding author.
E-mail address: rodrigo@ecn.purdue.edu (R. Salgado).
https://doi.org/10.1016/j.compgeo.2024.106430
Available online 29 May 2024
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V. Bisht et al.
Computers and Geotechnics 172 (2024) 106430
the most successful of these early models, but fell short when people
tried to use it. A major shortcoming of the model was its reliance on
perfect plasticity and an interpretation based on estimating “the” fric­
tion angle of the soil. As discussed elsewhere (e.g., Salgado, 2012), the
mobilized friction angle varies across the soil domain in any boundaryvalue problem involving real soil, so any method based on such an oversimplification of soil response is fatally flawed.
The next main thrust in developing a framework for CPT simulations
was cavity expansion theory (e.g., Salgado et al., 1997; Salgado and
Prezzi, 2007; Salgado and Randolph 2001). Cavity expansion theory was
an approximation to the problem based on the requirement that the cone
must expand a cylindrical cavity in the soil in order to advance. These
efforts concentrated on the use of simple elasto-plastic soil models with
Tresca or Mohr-Coulomb yield criteria, although with some modifica­
tions attemping to capture the effects of soil nonlinearities. Salgado and
Randolph (2001b) proposed a framework that can accommodate a
broad range of constitutive models, and cavity expansion analysis based
on that formulation and a relationship between cone resistance and
cylindrical cavity limit pressure produced useful relationships that can
be used in CPT interpretation (Salgado and Prezzi, 2007). However,
cavity expansion process cannot provide a close approximation to the
stress, displacement, and strain fields around an advancing cone
penetrometer.
The strain path method (Teh and Houlsby, 2009) was proposed for
analysis of cone penetration in clay, and it was a reasonable approxi­
mation to the penetration process, but its impact was again limited by a
simple elasto-plastic model with a Tresca yield criterion. Early appli­
cation of the FEM also met with difficulties. In geomechanics applica­
tions, FEM has traditionally followed a Lagrangian approach. This
means that the nodes of a mesh are tracked throughout the analysis, and
it is through node displacements that solutions are obtained. Deforma­
tion follows from relative node displacements, and stress follows from
deformation. After considerable deformation has occurred, mesh ele­
ments may be so distorted that accurate solutions are no longer possible.
In fact, a solution may not even be possible, accurate or not, with the
analysis crashing instead.
The Arbitrary Lagrangian-Eulerian (ALE) FEM approach (Belytschko
and Kennedy, 1978) and the material point method (MPM) (Sulsky
et al., 1994) were proposed to avoid this limitation. In ALE FEM, mesh
distortion is limited by remeshing or node repositioning after some
threshold level of deformation has happened. When nodes are reposi­
tioned or a new mesh formed, we need to map the state variables from
quadrature points on the old mesh to quadrature points on the new
mesh. The remapping can lead to stress states that lie outside the yield
surface for plasticity models with single yield surface or outside the
bounding surface for bounding surface models. Remapping can be
challenging with bounding surface models or complex models with
several variables to map. In contrast to ALE FEM, MPM does not require
remapping of variables because the state variables are carried by the
same material points throughout the computations. The material point
method and its application to cone penetration simulation is discussed
next.
displacements being the basis for the solution. Whereas grid nodes do
move during a computation step, they can be repositioned to their
original locations before the next computation time. It may also be ad­
vantageous to consider parts of the grid to either translate in some di­
rection or to compress one-dimensionally. This means that, whether the
grid remains static, moves in some direction or is unidirectionally
compressed in MPM, it does not get distorted as in FEM, which means
that there are no difficulties associated with mesh distortion.
The second most important implication of the use of material points
instead of Gauss points in an analysis is that the absolute rigor of FEM
with respect to the calculation of integrals over the element domain is
lost. This happens mainly because Gauss integration is no longer
possible, because the material points will, for practical purposes, never
be at the locations required by Gauss integration. This is not fatal, but a
number of remedial measures must be taken to limit any error from this
partial loss of rigor.
This paper presents in detail strategies to realistically simulate cone
penetration in both sands and clays by: (1) using MPM, which is well
suited for simulation of large-deformation problems, and (2) using
advanced bounding surface models with anisotropic hardening that
accurately capture soil behavior under a wide variety of loading con­
ditions. The MPM formulation used in the simulations discussed here is
discussed in detail by Bisht and Salgado (2018), Bisht et al., (2021b),
and Salgado and Bisht (2021). In the following sections, key features of
their implementation is discussed. For alternative MPM formulations
simulating cone penetration, please see the works of Ceccato et al.
(2016), Martinelli and Galavi (2021), Tehrani and Galavi (2018), Tran
and Sołowski (2019), and Yost et al. (2022).
1.2.1. Type of MPM
The original MPM scheme (Sulsky et al., 1994)—in which the ma­
terial points are actual points, lacking dimensionality—produces large
stress oscillations when material points cross element boundaries due to
the discontinuity of the gradient of shape functions across elements.
Numerous MPM schemes can be found in the literature that attempt to
reduce these stress oscillations through the construction of higher-order
shape functions (Bardenhagen and Kober, 2004; Sadeghirad et al.,
2011,2013; Steffen et al., 2008; Zhang et al., 2011). A common
approach is to assign the material “point” a domain—typically a rect­
angle in two-dimensional analyses (Bardenhagen and Kober, 2004;
Sadeghirad et al., 2011,2013). Further distinctions can be made
depending on how the domain is chosen to evolve. In this study, the
uniform Generalized Interpolation Material Point Method (uGIMP)
(Bardenhagen and Kober, 2004) has been adopted. In uGIMP, the par­
ticle domain does not evolve with material deformation. The relatively
simple particle domain evolution scheme offers two key benefits: (1) the
scheme is robust, and (2) it uses less computational power than methods
in which the material point domain is tracked more accurately. Since
particle domains are not tracked in uGIMP, gaps and overlaps between
the material point domains may occur for extremely large deformations.
Despite this limitation, uGIMP has been shown to provide similar results
to those of the convected particle domain interpolation method (Sade­
ghirad et al., 2011)–in which particle domains are partially tracked–for
(1) jacked piles simulations (Lorenzo et al., 2018), and (2) standard
MPM simulations of the cone penetration problem (Bisht et al., 2021a).
1.2. Modeling cone penetration using the MPM
The material point method can be best understood as a variant of the
finite element method in which Gauss points are replaced by so-called
material points. A material point (“MP”) may be a point (as in clas­
sical MPM), but it may also be a small area (for two-dimensional anal­
ysis) or small volume (for three-dimensional analysis). In contrast with
Gauss points, material points may move within and even across ele­
ments. This key difference between FEM and MPM has a number of
computational consequences.
The most important and most useful implication of the use of ma­
terial points instead of Gauss points as used in ALE FEM is that the
approach is a pure Lagrangian approach, with material point
1.2.2. Boundary conditions
Imposition of Dirichlet boundary conditions in MPM is straightfor­
ward: velocities are applied directly on nodes for which boundary
conditions are imposed. For enforcement of Neumann boundary con­
ditions, the required tractions have to be computed. Two choices arise:
(1) computing traction magnitudes at the material point centroids using
traditional GIMP shape functions; (2) computing traction magnitudes at
the material point edge (Bisht and Salgado, 2018; Nairn and Guilkey,
2015) using shape functions proposed by Sadeghirad et al. (2011).
Computing traction at the material point centroids leads to stress
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Computers and Geotechnics 172 (2024) 106430
equations resulting from the governing equations of motion. The addi­
tional equations involving strain tensor components result from the
imposition of the incompressibility constraint on integration points.
Incompressibility or near incompressibility appears in Tresca solids in
the plastic range and at critical state for any soil model. Volumetric
locking often appears as strong spatial oscillations in the values of
fundamental variables—such as stress—that appears as a checkerboard
pattern when these variables are plotted as contour plots in the problem
domain.
In the finite element method, volumetric locking can be avoided by
using higher-order elements, which provide a sufficient number of
variables to overcome any excess equations from the incompressibility
constraint. Using traditional (Lagrangian) higher-order elements is not
an option in MPM, because the corresponding shape functions can take
negative values for some regions in an element, which in turn can lead to
negative mass at the corresponding nodes from a material point located
in such regions. The attending complications are discussed in detail in
Andersen and Andersen (2010) and Bisht et al., (2021a).
An effective way to deal with volumetric locking in MPM is the use of
the so-called non-linear B method (Hughes, 1980; Simo et al., 1985).
The method is based on the fact that volumetric locking can be pre­
vented by reducing the number of constraints imposed by incompres­
sibility in each element. For example, for linear quadrilateral (Q4)
elements, volumetric locking can be prevented if a reduced, single-point
quadrature rule is used (see Sloan and Randolph (1982)). The use of a
single Gauss point in reduced integration for Q4 elements is sufficient to
prevent volumetric locking in FEM, but this strategy is not applicable in
MPM, because the number of material points present within an element
at any given point cannot be controlled.
For MPM, Bisht et al., (2021a) used a large-deformation formulation
of the B method with the deformation gradient F split into a volumetric
and a deviatoric component, with the volumetric component calculated
only at the element center. The computational algorithm that results
requires that the gradient of the material point shape functions (material
point mapping functions) must be calculated also at the element center.
Fig. 1. Application of tractions to a body in uGIMP MPM: the tractions are first
applied on the edges of the material points (represented by thick lines) and then
mapped to the nodes of the element containing the material point.
oscillations near the corresponding boundary. These oscillations can be
avoided by computing traction at the material point edges. Therefore,
traction terms have been computed at the material point edges in this
study (Fig. 1).
1.2.3. Mesh and discretization
A structured irregular background grid (Woo and Salgado, 2017)
divided into moving and compressible zones (Kafaji, 2013) has been
used in this study. The grid is built using linear quadrilateral (Q4) ele­
ments. The grid offers three benefits: (1) element search, which needs to
be performed at each time step in MPM, can be performed quickly; (2)
use of a fine mesh near the cone-soil interface and a coarse mesh else­
where is possible; and (3) the moving and compressible zones can be
translated and compressed, respectively, in such a way as to enable the
fine mesh to remain near the cone-soil interface throughout the pene­
tration process.
An aspect of irregular discretization is that a material point may flow
into a grid element for which its domain is too large. In this case, the
material point domain is split into 4 material points with equal domain
sizes such that energy, momentum, and mass is conserved (Ma et al.,
2009). Splitting ensures that a sufficient number of material points per
element is maintained for adequate quadrature.
One limitation of the discretization strategy adopted in this study is
that Q4 elements are unable to exactly discretize a conical surface,
resulting in elements that contain both cone and soil material points.
The response of such elements is dominated by the material points
belonging to the cone, since the cone is assumed to be rigid, whereas soil
is compressible. The resulting error should decrease with finer dis­
cretization as the boundary between cone and soil can be resolved more
accurately. Bisht et al., (2021a) performed cone penetration in Tresca
material and compared the cone factor values obtained using uGIMP
with an irregular structured grid (used in this study) against cone factor
values obtained by simulations performed by Ceccato (2015), who used
an unstructured grid using the same material model parameters. Near
the end of penetration, both simulations computed a cone factor of 9.7.
A modified deformation gradient F at the material point P—obtained
from combining the deviatoric component of the deformation gradient
with the volumetric component of it at the element center—is then used
in stress integrations.
(P)
1.2.5. Time integration algorithm
An explicit time integration scheme with the Update Stress Last
(USL) algorithm (Wallstedt and Guilkey, 2008) has been used in this
study. For explicit time integration, the critical time step Δtcrit,wave (from
wave propagation) can be calculated using the Courant-Friedrichs-Lewy
(CFL) stability condition (Courant et al., 1967):
( )
h
(1)
Δtcrit,wave = min
cd
where h is the element size, and the speed of the compression wave is
given by:
√̅̅̅̅̅̅̅̅̅̅
cd = Ec /ρ
(2)
for dry soil, and
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
Ec + Kw /η
cd =
(1 − η)ρs + ηρw
(3)
for saturated soil, with Ec being the modulus (typically Young’s modulus
or the constrained modulus), η the porosity, Kw the bulk modulus of
water, and ρ the material density.
For saturated soil, there is another timescale of interest: that of
excess pore-pressure dissipation due to consolidation. Mieremet et al.
(2016) showed that the critical time step Δtcrit,cons. for consolidation is
1.2.4. Volumetric locking
In numerical solutions to problems in mechanics involving incom­
pressible or nearly incompressible materials, a phenomenon known as
volumetric locking may occur. Volumetric locking is the insufficiency of
variables provided by the grid discretization to allow solution of all
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Computers and Geotechnics 172 (2024) 106430
Table 1
Parameter values used in stress-integration algorithm.
Parameter
Value
− 4
STOL
FTOL
10
10− 9
LTOL
10− 6
MAXITS
20
Table 2
Constitutive model parameters for Ottawa Sand.
Remarks
Parameters for
Number
Parameter
Value
Maximum local stress error for any sub-increment
The maximum tolerance within which a point is considered to
be at the yield surface
Determines whether a point at the yield surface follows a
purely plastic path or if it undergoes elastic unloading
The maximum iterations that will be performed to determine
the intersection at the yield surface for a point undergoing
transition from an elastic to a plastic state
Small-strain shear modulus
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
Cg
ng
611
0.437
0.47
0.00065
0.15
0.05
1.21
1.9
2.2
0.081
0.20
0.71
0.35
2.20
0.240
0.81
1.2
0.78
1.31
0.31
0.39
0.780
Elastic shear modulus with Ramberg-Osgood
degradation
Poison’s ratio
Yield surface
Bounding, dilatancy and CS surfaces
State parameter and CSL in e-p’
given by:
Δtcrit,cons. = 2k[ηρsat + (1 − 2η)ρw ]/ηγw
Shape of bounding, dilatancy, CS surfaces in
the π plane
Plastic modulus
(4)
where γ w is the unit weight of water, k is hydraulic conductivity of the
soil, and ρsat is the saturated mass density of the soil. Thus, the critical
time step Δtcrit for saturated soil can be determined using:
)
(
(5)
Δtcrit = min Δtcrit,wave , Δtcrit,cons.
Flow rule
Dilatancy
Fabric dependent scalar
Fabric effect multiplier in H
Intercept of CSL
For explicit time integration, the stable time step is typically chosen
to be smaller than the critical time step by a factor κ known as the
Courant number (Courant et al., 1967):
Δtstable = κΔtcrit
allows the use of a larger scaled time step Δtms :
√̅̅̅
Δtms = ηΔtstable
(8)
μ
c2
D0
α
kh
Γc
1.2.8. Soil modeling using two-surface plasticity
The constitutive models used in the analyses reported later were two
two-surface plasticity models: the Loukidis and Salgado (2009)
(“LS2009″) for sand and the Chakraborty et al. (2013) (”CLS2013″)
model for clays. Both models rely on the critical state-based, bounding
surface plasticity framework, and share many features.
The LS2009 model—which is based on a model originally proposed
by Manzari and Dafalias (1997) and subsequently modified by Dafalias
and Manzari (2004)—is formulated in the critical-state framework, i.e.,
it takes into account the current state of the soil using the state
parameter ψ (Been and Jefferies, 1985), and expresses a number of key
quantities in terms of ψ . Features of soil mechanical response are
expressed through relationships that refer to four surfaces defined in
general stress space: the yield surface, the dilatancy surface, the
bounding surface, and the critical-state surface. In simple terms, the
peak shear strength ratio q/p’ increases with increasing size of the
bounding surface, which in turn increases with increasing negative
distance from the soil state point in e-lnp’ space to the critical-state line:
the more dilative the soil the greater its peak shear strength. In contrast,
contractive soil has a peak shear strength equal to the critical-state shear
strength, so the bounding surface coincides with the critical state sur­
face. The bounding surface for a dilative soil starts larger than the
critical state surface, changes size depending on the evolution of the
state parameter, and then, at sufficiently large shear strains, collapses to
the critical-state surface.
The dilatancy surface enables the model to describe the volumetric
response of the soil: sand contracts if the stress state lies within the
dilatancy surface and dilates if the stress state lies outside the dilatancy
surface. Through this surface, the model can capture the transition from
contractive to dilative response that is referred to as phase trans­
formation. Flow is nonassociated. The yield surface conceptually rep­
resents the same threshold of elastic response as in classical plasticity,
but it is very small because soil expresses inelastic response starting at
1.2.6. Contact
The cone is treated as a rigid body, whereas the soil is deformable.
Although several contact algorithms have been proposed in the MPM
literature (e.g. González Acosta et al., 2021; Hamad et al., 2017; Huang
et al., 2011; Ma et al., 2014), the contact forces between the cone and the
soil were computed using the contact algorithm proposed by Barden­
hagen et al. (2000) based on our prior experience with its use on
penetration problems. For computing the tangential friction force τ, the
Coulomb friction model was used:
τ⩽μN
m
Mcc
kb
kd
λ
ξ
c1
ns
h1
h2
elim
should remain at the yield surface, see Prager, 1961); and (3) selfconsistency (under elastic deformation, the hypo-elastic rate formula­
tion must be exactly integrable to deliver a hyper-elastic relation, see
Bruhns et al., 1999). It should be noted that the commonly used Jau­
mann stress rate does not satisfy the self-consistency criterion, resulting
in an oscillatory response in simple shear (Dienes, 1979; Zbib and
Aifantis, 1988).
In the USL approach, Wallstedt and Guilkey (2008) showed that κ ≈
0.4 yields suitable results for dynamic problems. However, for quasistatic problems such as cone penetration, a higher Courant number
(κ ≈ 0.7) can be used. Additionally, for quasi-static problems, artificially
increasing the material density using a mass scaling parameter η:
(7)
γ1
ν
*after Loukidis and Salgado (2009).
(6)
ρscaled = ηρ
α1
(9)
where μ is the coefficient of friction between soil and the cone pene­
trometer, and N is the normal force.
1.2.7. Stress integration algorithm
We have used an explicit stress-integration scheme with subincrementation and error control (Sloan, 1987; Sloan et al., 2001).
The algorithm breaks down the strain increment into several subincrements and computes the corresponding stress sub-increments
using the Modified Euler scheme. The algorithm contains several pa­
rameters that can be modified to increase the accuracy of the stress subincrements at the cost of computational speed. A brief description of the
parameters and their values chosen in this study are given in Table 1.
The modified kinetic logarithmic rate (Jiao and Fish, 2017,2018) has
been used as the objective stress rate in this study, and the Hencky strain
has been used as the strain measure. This combination of stress–strain
measures has been chosen because they satisfy a few key requirements:
(1) objectivity (the stress rate is frame-invariant, see Lubliner, 2008); (2)
yield stationarity (if the stress rate is zero, a point at the yield surface
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Computers and Geotechnics 172 (2024) 106430
and rate of deformation is reasonable, because cone resistance is more
significantly affected by compression loading paths, whereas the effect
of large rigid-body rotations is more impactful in loading paths
approaching simple shear.
Table 3
Constitutive model parameters for Boston Blue Clay.
Parameters for
Number
Parameter
Value
Poisson’s ratio
Small-strain shear modulus
Elastic moduli with degradation
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
ν
0.25
250
5
0.036
1.138
0.187
0.53
1.305
0.2
0
2.7
1
0.95
0.31
1.1
Normal Consolidation Line
Stress Anisotropy
Shear Strength
Dilatancy surface
Flow rule
Hardening
Cg
ζ
κ
N
λ
K0,NC
Mcc
ns
kb
ρ
D0
c2
ξ
h0
1.2.9. Coupled MPM Formulation
Following the classical theory of poroelasticity by Biot (1941),
several noteworthy advancements in poroelasticity and its application to
geomechanics have been made, (e.g., Selvadurai, 1996,2007; Selvadurai
and Suvorov, 2016; Verruijt, 2015; Wang, 2000). In this study, the
governing equations are obtained based on the theory of mixtures (De
Boer, 2012) and uses the concept of volume fractions. In this approach,
each constituent of the mixture is smeared over the entire domain with
reduced density to create a homogenized continuum. Each individual
constituent in the mixture must satisfy the governing equations.
A 2-phase, 1-point formulation is used to discretize saturated soil, i.
e., soil and water are discretized using a single set of material points. The
primary advantage of a 1-point formulation is that it is less taxing on
computational resources as opposed to using a 2-point formulation (soil
and water are discretized using 1 material point each). A review of the
applicability and advantages of the 1-point and 2-point formulations can
be found in Ceccato et al. (2018).
*after Chakraborty et al. (2013).
very small strains. The yield surface hardens kinematically, and the
bounding surface and dilatancy surface harden isotropically. The model
relies on degradation of the small-strain initial shear modulus to accu­
rately capture clay response at small-strain levels. The model can cap­
ture some complex features of soil response, including stress–strain
nonlinearity from small strains, initial fabric anisotropy, phase trans­
formation and response under various loading paths.
The LS2009 model for sand requires 22 input parameters. In the
simulations discussed later, we use the model parameters for Ottawa
Sand given by Loukidis and Salgado (2009) (Table 2).
The CLS2013 model has a critical state line (“CSL”) that is parallel to
the normal consolidation line (“NCL”), from which soil unloading and
reloading in e − lnpʹ space occurs along straight unloading–reloading
lines. The CSL is assumed to be unique, implying that specimens having
the same void ratio will reach the same pʹ at critical state when loaded
under undrained conditions, irrespective of whether the shearing fol­
lowed an undrained triaxial compression or undrained triaxial extension
loading path. The model defines three surfaces in stress space: the yield
surface, the dilatancy surface, and the bounding surface. They function
essentially in the same way as in LS2009 for sand. The yield surface is
also small for CLS2013, and thus a plastic process is operative near the
cone throughout the entire penetration process. The bounding surface is
made up of two distinct, but connected surfaces: a shear bounding sur­
face and a flat cap. The flat cap is used to capture the yielding of clays in
isotropic compression or compression at small ratios of shear stress to
mean effective stress. The model relies on degradation of the smallstrain initial shear modulus to accurately capture clay response at
small-strain levels. The model also contains parameters to capture the
strain-rate dependent shear strength in clays and the transition of clays
beyond critical state to a residual state.
For the analysis performed here, the CLS2013 model for clay requires
15 parameters (Table 3). In the simulations discussed later, we use the
calibrated values of model parameters for Boston Blue Clay (BBC) that
are specified in Chakraborty et al. (2013) without modification, with
one exception: we ignore strain-rate dependence in the clay response for
computational reasons. Ignoring strain-rate dependence is justified
because the strain-rate effects are small, since cone penetration is a
quasi-static process. Additionally, for BBC, the shear bounding surface
and the critical state surface are assumed to coincide based on data from
Ladd and Varallyay (1965).
The values of parameters for LS2009 model for sands and CLS2013
model for clays described in Table 2 and Table 3 respectively were
originally calibrated using a small-strain formulation. For very large
rates of rotation, it is conceivable that recalibration of the models would
make a nonnegligible difference, but for the calculation of cone resis­
tance, direct use of the model calibrated in terms of stress rate and strain
rate in a formulation in terms of modified kinetic logarithmic stress rate
1.2.10. A computation time step in an MPM simulation
During a time step in an MPM simulation, we follow the steps out­
lined below.
1. Initialize
a. Initialize all variables at material points and grid nodes; set time
t =0
b. Determine an appropriate time increment Δt
2. Compute shape functions and shape function gradients
For each material point P:
a. Compute GIMP functions S(IP) and their gradients S,j
(IP)
at particle P
and at element center C.
b. If P is at a traction or pressure boundary, compute CPDI shape
functions at P
3. For coupled problems, compute updated momentum of water
For each node I:
a. Compute nodal mass of the water mw and weighted nodal mass of
(I)
̃ (I)
water m
w :
m(I)
w =
∑
S(IP) m(P)
w
P
∑
(P)
̃ (I)
m
S(IP) m(P)
w =
w /η
P
b. Compute nodal momentum Pwi due to water and porosity(I)
̃(I) due to water:
weighted nodal momentum P
wi
P(I)
wi =
∑
(P)
S(IP) m(P)
w vwi
̃(I) =
P
wi
∑
(P)
S(IP) η− 1 m(P)
w vwi
P
P
where vwi is velocity of water
c. Compute external forces Fext,wi and internal forces Fint,wi at the
(I)
(I)
nodes due to water:
∑
∑
(I)
(P)
(P)
Fext,wi
= FΓp +
S(IP) η− 1 m(P)
S(IP) fdragi
V (P)
w bi −
P
5
P
V. Bisht et al.
Computers and Geotechnics 172 (2024) 106430
∑ (IP)
(I)
(P)
Fint,wi
=
S,j δij p(P)
w V
Table 4
An overview of the numerical setup used for the cone penetration simulations.
P
Implemented scheme
∫
where FΓp = dΩ N(I) pwi dΩ is the force from the pore pressure pw at
the boundary Γ of system domain Ω, b is the body force per unit
(
)
(P)
(P)
(P)
(P)
mass, fdragi = − η2(P) γ w k− 1(P) vwi − vsi
is the drag force due to
Type of MPM
Time integration
algorithm
Mesh
soil–water interaction derived assuming laminar flow, γw is the
unit weight of water, k is the hydraulic conductivity of the soil, δij
is the Kronecker-delta function, V is volume, N(I) is the finite
element top-hat shape function,
(I)
̃˙ =
d. Compute rate of change of weighted nodal momentum P
wi
(I)
updated nodal momentum Pwi
̃(I) m(I)
̃ (I)
=P
w due to water.
wi w /m
(I)CT
e. If I is a contact node, compute contact forces fwi due to water
(I)
(I)
(I)CT
(2000,2001) and update the nodal momentum Pwi ←Pwi +fwi Δt
due to water
4. Compute updated momentum of soil
For problems with no water, ignore forces due to water (terms with
subscript w)
For each node I:
∑
(I)
(P)
a. Compute nodal mass from the soil:ms = P S(IP) ms
∑ (IP) (P) (P)
(I)
b. Compute nodal momentum due to the soil: Psi = P S ms vsi
(I)
Fint,mi
=
(P)
v(P)
si ←vsi + Δt
(I)
P
∑ ʹ(P) (IP)
∑
(IP)
(P)
σij S,j V (P) +
p(P)
w S,j δij V
(I)
(I)
(I)
(I)
(I)
(I)CT
e. If I is a contact node, compute contact forces fsi
due to soil using
the contact algorithm proposed by Bardenhagen et al.
(2000,2001) and update the nodal momentum Psi ←Psi +
(I)
(I)CT
fsi Δtdue to soil
5. Update material point state
For each material point P
2.1. Domain Discretization
(P)
a. Compute the velocity gradient vi,j (C) at the element center
(P)
b. Compute the deformation gradient Fij (for simplicity, we use
Fig. 2 shows the discretization for simulation of cone penetration in
sand. The mesh dimensions were chosen such that insignificant differ­
ences (<2% difference in qc values) were observed when meshes with
larger dimensions were used. The grid was initially discretized into 2 × 2
material points per element. Owing to the material point splitting
strategy used in this study, the number of material points varies through
the course of the simulation, increasing from approximately 6500
initially to ~13000 near the end of penetration. The essential boundary
conditions are imposed at the bottom, on the right, and at the line of
symmetry by forcing the required velocity terms to be zero. Neumann
boundary conditions are applied to the top of the soil.
lowercase subscripts to represent both the reference and current
configurations)
c. For coupled problems, compute the pore pressure increment using
∂pw
Kw
∂t = η [(1 −
I
Table 4 summarizes the numerical setup used for performing the
cone penetration simulations in sand (and also clay simulations, which
are discussed later). The sand simulations were performed assuming
drained conditions. In the sub-sections below, we justify the use of these
values.
Ṗsi Δt.
(I)
∑
(I)
S(IP) P(I)
si /ms
2. Penetration in sand
Ṗsi = − Ṗwi +Fext,mi − Fint,mi due to soil and update Psi ←Psi +
(I)
I
6. Update grid and reset
a. Adjust (translate/compress/reset) background grid
b. Update time t←t + Δt.
c. If t < tend , go to step 2; otherwise exit
P
∫
where FΓt = dΩ N(I) ti dΩis the external force contribution from
tractions t.
d. Compute
rate
of
change
of
nodal
momentum
(I)
∑
(I)
S(IP) Ṗsi /m(I)
s
(P)
x(P)
si ←xsi + Δt
∑
∑
(P) (IP)
(P) (IP)
m(P)
+
m(P)
w bi S
s bi S
P
Modified kinematic logarithmic rate and Hencky strain
critical state.
the mixture:
P
Clay: μf = 0.30 (from Bisht
tms : scaled time increment; MPs: material points; hmin,w : minimum element
width; dc : cone diamater; K0 : coefficient of lateral earth pressure; vcone : pene­
tration velocity of cone; μf : coefficient of friction; δcs : interface friction angle at
c. Compute external forces Fext,mi and internal nodal forces Fint,mi for
(I)
Sand: Fixed tms = 2 × 10− 5 s
Clay: Fixed tms = 2 × 10− 5 s
Sand: hmin,w = 6.33 mm,
which is ~the size of the
shear band
Clay:hmin,w =
4.46 mm (dc /8)
Sand: vfinal = 10 cm/s
Clay:vfinal = 20 cm/s
(
)
Sand:μf = 0.31 δcs = 17.40
et al., 2021b)
Explicit with automatic sub-stepping and error control
Stress integration
algorithm
Objective stress
rate and strain
measure
using the contact algorithm proposed by Bardenhagen et al.,
Fext,mi = FΓt + FΓp +
Structured irregular grid
with moving and
compressible zones with
initially 2 × 2 MPs per
element
{
0 − vfinal
t ≤ 1s
vcone =
vfinal
t > 1s
Coulomb friction model (
Bardenhagen et al., 2000);
Penetration
velocity
Contact algorithm
(I)
(I)
(I)
̃(I) ←P
̃(I) + P
̃˙ Δt. Compute
Fext,wi − Fint,wi due to water and update P
wi
wi
wi
(I)
uGIMP
Explicit − USL
Remarks
η)∇.vs + η∇.vw ], where Kw is the bulk modulus of
water
d. Using the objective stress rate, update the stresses and internal
variables using Fij
(P)
)
(
(P)
e. Update the material point volume V (P) ← detFij V (P) and soil
2.2. Minimum element size for sand
matrix density. For coupled problems, update the porosity
)
(
)
(
(P)
η(P) ←1 − 1 − ηt = 0(P) / detFij .
For sand, the minimum element size was chosen to be approximately
equal to the width of the shear band. Data from experimental tests
performed on sand (e.g., Alshibli and Hasan, 2008; Alshibli and Sture,
1999; Desrues and Viggiani, 2004) suggest that a shear band thickness of
(P)
f. Update the material point velocity of the soil matrix vsi and po­
(P)
sition xsi :
6
V. Bisht et al.
Computers and Geotechnics 172 (2024) 106430
Fig. 2. Schematic of cone penetration simulation in sand.
roughly 8 − 20D50 (approximately 6–14 mm) is formed after sustained,
localized shearing. Accordingly, the minimum element size (width w ×
height h) is taken to be 6.33 mm × 5.95 mm (8.8D50 × 8.3D50 or dc /6 ×
dh /3; with dh being the height of the cone tip). A smaller minimum
element size of dc /12 × dh /6 resulted in a moderate ∼ 5% reduction in
qc values.
2.4. Initial stress field
Five vertical surcharge values were used in the cone penetration
simulations (25, 50, 75, 100, and 200 kPa). By vertical “surcharge” we
mean the vertical stress applied at the top of the soil. The initial vertical
effective stress σʹv for any point within the model was then determined by
adding the applied surcharge to the self-weight of the soil. For the initial
horizontal effective stress σʹh , Salgado (2022) suggests that the lateralearth pressure K0 can be taken as 0.4–0.5 for sand ranging from high
to low relative densities. In our simulations, K0 = 0.45 was assumed.
The initial stress field was applied at the “zeroth” time step by specifying
the required stresses at the material point centroids.
2.3. Time increment
To compute the time increment for performing explicit analysis, we
note that the maximum modulus yields the smallest time step. We es­
timate the maximum small-strain shear modulus for the LS2009 sand
model using Hardin and Richart (1963):
Gmax = Cg
(2.17 − e)2 ʹng 1− ng
p pA
1+e
2.5. Contact forces
(10)
Han et al. (2018) performed interface direct shear tests for smooth
steel on Ottawa 20–30 sand and measured the interface friction angle
δcs = 17.40 , which corresponds to a friction coefficient of μf = 0.31.
Additional simulations performed by varying the friction coefficient
μf = 0.2 − 0.4 resulted in small changes in qc values (< 5% variation
across all simulations).
where Cg = 611 and ng = 0.437 are model parameters, pA = 100 kPa is
the reference stress, e is the void ratio, and pʹ is the mean effective stress.
Since pʹ cannot be determined a priori, a reasonably high value of pʹ =
10 MPa was used. For DR = 90%, we get a void ratio e = 0.524.
Plugging these values into Eq. (10), we get Gmax = 813 MPa and con­
strained modulus Ec,max = 1974 MPa.
Assuming κ = 0.7 and η = 25, and noting that the minimum element
size hmin used in this study was 5.95 mm, a scaled time step Δtms = 2.0 ×
10− 5 s was computed. Simulations performed using smaller η values
yielded nearly identical results (<1% difference in qc values). Hence, a
fixed time step Δtms = 2.0 × 10− 5 s was used in all the simulations.
2.6. Cone penetration velocity
In the field, cone penetration is typically performed at a penetration
velocity vcone = 2 cm/s. In this study, a higher cone velocity vcone was
used to reduce the simulation time. The vcone was increased linearly from
0 to 10 cm/s for t⩽1 s, and then kept constant at 10 cm/s for t > 1 s. For
7
V. Bisht et al.
Computers and Geotechnics 172 (2024) 106430
qc (MPa)
qc (MPa)
0
0
2
4
6
8
0
4
0
4
8
12 16 20
0
10
8
qc (MPa)
12
16
0
4
8
12
16
Depth (mm)
100
200
300
400
500
600
0
5
10 15 20 25
Depth (mm)
100
200
DR=30%
DR=60%
300
DR=90%
400
500
600
Fig. 3. Cone resistance (qc ) vs. penetration depth curves in sand obtained using MPM simulations for different initial vertical effective stress and relative densities.
comparison purposes, a simulation with vcone = 5 mm/s was performed
that yielded nearly identical results (<2% difference in qc values) to
those computed using vcone = 10 cm/s.
horizontal effective stress values (22.5 kPa⩽σh ⩽450 kPa). By performing
regression on the cavity expansion analyses results, coupled with anal­
ysis linking cone resistance to limit cavity pressure, they proposed the
following relationship to compute qc for sand:
( ʹ )0.841− 0.0047DR
qc
σ
= 1.64exp[0.1041ϕc + (0.0264 − 0.0002ϕc )DR ] h
pA
pA
(11)
2.7. Results
Cone penetration simulations were performed in sand at 3 different
relative densities (30%, 60%, 90%) and 5 different surcharge values (25
kPa, 50 kPa, 75 kPa, 100 kPa, 200 kPa) for a total of 15 simulations. The
cone resistance qc vs penetration depth d curves are shown in Fig. 3. The
curves are relatively smooth and show some expected results: (1) for a
fixed relative density, qc increases, and the penetration distance
required for its complete mobilization decreases with increasing sur­
charge values; and (2) for a fixed surcharge value, qc increases, and the
penetration distance required for its complete mobilization increases
with increasing penetration depth.
Salgado and Prezzi (2007) performed a series of cavity expansion
analyses for a wide range of relative densities (0⩽DR ⩽100) and initial
where ϕc is the critical-state friction angle in degrees, and pA = 100 kPa
is a constant.
Table 5 compares the steady-state qc values obtained from the MPM
simulations against qc values determined estimated using cavity
expansion theory (Eq. (11)) for all test cases. The results indicate that
cavity expansion analysis provides larger qc values compared to MPM,
particularly for loose sands, for which the overprediction can be as large
as 25%. For a low surcharge value of 25 kPa, MPM and cavity expansion
output qc values that differ by 17.6%, 2.4%, and 14.0% for loose,
medium-dense, and dense sands respectively. Tehrani and Galavi (2018)
8
V. Bisht et al.
Computers and Geotechnics 172 (2024) 106430
3.2. Numerical setup
Table 5
A summary of the results from cone penetration test simulations performed in
this study.
Sand
Classification
DR
(%)
Surcharge
(kPa)
qc MPM
(MPa)
qc CE
(MPa)
Error
(%)
Loose
30
25
50
75
100
200
1.4
2.0
2.6
3.1
4.9
1.7
2.5
3.2
3.8
6.1
17.6
20.0
18.8
18.4
19.7
Medium-Dense
60
25
50
75
100
200
4.0
5.1
6.0
6.9
9.7
4.1
5.6
6.8
7.9
11.4
2.4
8.9
11.8
12.7
14.9
Dense
90
25
50
75
100
200
8.6
11.9
14.0
15.7
21.2
10.0
12.6
14.5
16.2
21.3
14.0
5.6
3.4
3.1
0.5
Fig. 5 shows the discretization for simulation of cone penetration in
clay. Initially, each element contained 2 × 2 material points. The bottom
boundary was fixed, and the two side boundaries were roller boundaries
restricting horizontal movement. All three boundaries were assumed to
be impermeable. The number of material points varied from ~15000
initially to ~ 25000 near the end of penetration. In contrast to sand, the
strategy of choosing a minimum element size that approximates the
width of the shear band was unavailable for clays due to the wide range
of shear band thickness, varying from 0.1 mm to 2 cm (Chakraborty
et al., 2013; Kang et al., 2015; Lin and Penumadu, 2006; Moore and
Rowe, 1988; Vardoulakis, 2002), reported in the literature for clays.
Bisht et al., (2021b) performed several cone penetration simulations in
Boston Blue Clay and reported that a minimum element size of a mesh
size of dc /8 (4.46 mm) resulted in reasonably accurate qc values, and this
value has been used in our analyses. Given the low hydraulic conduc­
tivity at the test site, cone penetration in Boston Blue Clay is expected to
occur under completely undrained conditions. The simulations were
performed with a time step size of Δts = 2 × 10− 5 s with a mass scaling
parameter fms = 200. The cone was assumed to be partially rough (μ =
0.3) and was advanced at a rate of 0.2 m/s to reduce the computational
runtime. The constitutive model (Chakraborty et al., 2013) used in this
study accounts for strain-rate effects. To prevent the artificially higher
penetration rate from affecting the simulated cone penetration response,
the model parameters accounting for strain-rate dependence were set to
zero. As discussed earlier, this assumption is justifiable because: (1)
strain-rate effects are expected to be small given that cone penetration is
a quasi-static process, (2) low hydraulic conductivity of the clay at the
test site ensures that penetration will occur under undrained conditions
for penetration velocities of 0.02 m/s and higher. Additionally, pene­
tration at a slower rate of 0.1 m/s gave near identical qc values.
In the simulations, the geostatic stresses and pore pressures were
assigned to the material points in the beginning of the zeroth time step.
The initial stress field was assumed to be constant with depth over the
few diameters of travel path required for a steady state qc value to be
obtained. This approximation is acceptable because the variation in
stresses with depth due to gravity is small compared to the initial stress
values.
DR = Relative Density, MPM = material point method, CE = cavity expansion
theory, Error = 1 – (qc MPM / qc CE).
simulated cone penetration using MPM in Baskarp sand at 3 different
relative densities—30, 50, and 90%— and a surcharge of 25 kPa. They
compared the qc values obtained using MPM against those obtained
from cavity expansion analyses (Eq. (11)) and found that they were in
very good agreement. This is interesting to note because Eq. (11) was
calibrated
for
initial
horizontal
effective
stress
values
(22.5 kPa⩽σ h ⩽450 kPa) with K0 = 0.45, implying initial vertical effec­
tive stress values 50 kPa⩽σʹv ⩽1000 kPa. The results suggest that Eq. (11)
may provide reasonably accurate results even beyond the range of
values against which it was calibrated.
One advantage that MPM simulations provide over cavity expansion
analysis is that we can analyze the values of field variables. Fig. 4 plots
the mean stress values near the end of cone penetration for different
relative densities at an initial vertical effective stress of 200 kPa. A
typical “pressure bulb” pattern is formed below the cone, with the mean
stresses generated for dense sands being significantly greater than those
for loose sand. Some oscillations in the mean stress values are observed,
particularly near the cone-soil interface. As noted earlier, these oscilla­
tions occur because elements near the cone-soil interface may contain
material points belonging to soil and cone. The cone, being rigid, largely
determines the deformation of the element, which affects the stress
computations for the material points belonging to soil.
3.3. Results
Cone penetration resistance vs depth curves at depths of 5.6 m and
6.0 m are shown in Fig. 6. A steady-state cone penetration resistance is
reached after roughly 2 cone diameters of penetration. This is consistent
with the results of Moug et al. (2019), who simulated cone penetration
at the same test site using FLAC (Itasca, 2016) and observed that a
steady-state was reached after 2–6 cone diameters of penetration.
A comparison of the simulated cone resistance values and the cone
resistances measured in the field is provided in Table 7. The simulated
values fall within the range of the measured cone resistances at depths of
6.0 m and 9.6 m. At a depth of 5.6 m, the cone resistance computed using
MPM is 5.4% higher than the cone resistance measured in the field (from
CPT-1). This variation is small, and could be due to errors in estimation
of the in situ stress state, errors in the simulation strategy (e.g., the shape
of the cone or the friction coefficient between the soil and clay), or errors
in the constitutive model. Notably, the difference in the simulated CPT
value using MPM and CPT-1 (42 kPa at 5.6 m) is less than the difference
in cone resistances between CPT-1 and CPT-2 (222 kPa at 5.6 m).
3. Penetration in clay
In this section, we investigate the ability of MPM to simulate cone
penetration by performing simulations for a Boston Blue Clay (BBC)
deposit located at a test site near Newbury, Massachusetts. Clay prop­
erties at the test site have been characterized by Landon (2007). The
numerical setup used for the simulation closely follows the work of Bisht
et al., (2021b). A summary of the numerical setup used is provided in
Table 4. Key features and a description of the site properties follow.
3.1. Initial field conditions
Landon (2007) collected soil samples at several depths to determine
the properties of the soil at the Newbury test site. Additionally, two cone
penetration tests were performed that were spaced roughly 10 m apart.
Key soil properties and cone resistances are provided in Table 6. The
groundwater level was at a depth of 1.7 m. The hydraulic conductivity
varied from k = 10− 10 to 10− 9 m/s.
3.4. Cone factor
For cone penetration under undrained conditions, the undrained
shear strength su is often estimated in practice using:
qc = Nk su + σ v
9
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V. Bisht et al.
Computers and Geotechnics 172 (2024) 106430
Fig. 4. Mean stress values for cone penetration in sand for initial vertical effective stress equal to 200 kPa and various relative densities plotted at a penetration depth
of 550 mm (end of penetration).
extension (UDTXE), and undrained simple shear (UDSS) shearing sim­
ulations were performed using the CLS2013 model. The soil properties
were set to the values estimated by Landon (2007) to estimate Nk. The
results of the simulations are provided in Table 8.
The su values obtained from UDTXC, UDTXE, and UDSS represent the
wide range of su values that could be observed depending on the stress
path to which an element is subjected. Cone factors are obtained by
substituting the su values and qc values obtained from MPM simulations
into Eq. (12). To determine a single value of the cone factor for a given
depth, the cone factors from the three test conditions are averaged, and
Nk,avg = 15.4, 15.7, and 14.4 are obtained at depths 5.6 m, 6.0 m, and
9.6 m respectively. These values lie in the upper range of cone factor
values reported in the literature.
Three important points should be noted when interpreting the Nk
values:
Table 6
Boston Blue Clay properties at Newbury (after Landon, 2007).
Depth (m)
5.6
6.0
9.6
σʹv0 (kPa)
OCR
CPT-1 (kPa)
CPT-2 (kPa)
66.7
69.9
96.6
4.3
3.3
2.2
778
718
730
556
565
580
where Nk is known as the cone factor. Obtaining a singular Nk value
through Eq. (12) presents challenges because undrained shear strength
values may vary for each element depending on the stress path that the
element has experienced. Consequently, a range of Nk values have been
reported in the literature. Salgado (2022) provides a comprehensive
overview of Nk values that have been reported: values range from 8.5 to
24.5, but most fall within the 9–15 range, and outliers are likely due to
partially drained penetration or other experimental error.
Given that soil elements surrounding a cone penetrometer that is
advancing through the soil may experience stress paths ranging from
triaxial compression to triaxial extension, single-element loading simu­
lations—undrained triaxial compression (UDTXC), undrained triaxial
(1) There is a large variation between the Nk values obtained from
UDTXC (12.2–13.3) and UDSS (13.2–14.6) on one hand to those
obtained from UDTXE (17.6–19.2) on the other. If more elements
that more significantly impact the calculated qc value were sub­
jected to stress paths more closely resembling UDTXC or UDSS
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V. Bisht et al.
Computers and Geotechnics 172 (2024) 106430
Fig. 5. Schematic of cone penetration simulation in Boston Blue Clay. (For interpretation of the references to colour in this figure legend, the reader is referred to the
web version of this article.)
than UDTXE,1 a weighted average Nk,avg value would be more
appropriate, and this average would be lower than the un­
weighted average.
(2) Nk,avg is likely overestimated because the qc values obtained using
MPM were obtained using a stair-stepped discretized cone (see
Fig. 5).
(3) As suggested by Salgado et al. (2022), it is not evident which su
should be used to determine Nk . The best possible approach may
lie in not using Nk,avg at all, instead using the Nk that yields su that
is required by the design method.
4 Summary and conclusions
This paper reviews our ability to realistically simulate cone pene­
tration in both sand and clay. Bounding surface models capable of
reproducing stress–strain response for a wide variety of loading condi­
tions and different types of soils were used to simulate soil response. This
capability of bounding surface models comes at the cost of complexity:
the LS2009 model (Loukidis and Salgado, 2009) for sand requires 22
input parameters, whereas the CLS2013 (Chakraborty et al., 2013)
model for clay requires 15 parameters if parameters required to capture
strain-rate-dependence and residual state are ignored (22 if included).
This complexity may impede the use of some numerical schemes that are
used for simulating large deformation problems, such as ALE, which
require the mapping of state variables from old Gauss Point locations to
new Gauss Point locations. This mapping is an approximation, which,
for highly complex models, can lead to issues in achieving an “equilib­
rium” stress-state once the variables are mapped.
To circumvent this issue, the material point method, which does not
require the mapping of state variables, was adopted. However, this too
comes at a cost: the numerical integration in MPM is not guaranteed to
yield the correct integral even for small-deformation elastic problems. In
this study, we discussed several strategies that were adopted to mini­
mize errors in MPM while maintaining computational feasibility, such as
the use of an irregular moving-compressible grid, the B strategy for
mitigating volumetric locking, use of an adaptive sub-incrementation
Although it is difficult to provide an exact value to the amount of
over-estimation as a result of stair-stepped discretization, a reasonable
assumption would be that qc values for a stair-stepped discretized cone
would lie roughly between that of a perfectly discretized cone and a flat
cone. Simulations performed with a flat cone under identical conditions
provided qc values that were ~8% higher than qc values obtained using a
stair-stepped discretization. Correcting the qc values for this difference
would lead to an unweighted average cone factor Nk,avg = 13.7.
1
This indeed appears to be the case, as elements located below the cone tend
to experience loading more akin to TXC, and they, intuitively, are more im­
pactful than elements to the sides, which would experience loading paths more
associated with TXE.
11
V. Bisht et al.
Computers and Geotechnics 172 (2024) 106430
Fig. 6. Cone resistance qc vs. normalized penetration depth d/dc curves obtained from MPM simulations for penetration at depths of: (a) 5.6 m; (b) 6.0 m; and (c) 9.6
m (the two red lines indicate cone resistances obtained from the two in situ CPT tests).
prefer to use an Nk, value corresponding to the undrained shear
strength—UDTXC, UDTXE, or UDSS— required by the design method
being used.
MPM appears to offer a solid base on which to build methods of
interpretation of CPT results. To broaden this base, extension of analyses
to different types of sands and clays and their mixtures is desirable.
Methods of interpretation must then account for other factors, such as
spatial variability of soil state variables, which can also be advanced by
designing suitable MPM simulations.
Table 7
Comparison of CPT resistances measured in the field (after Landon (2007)) and
computed using MPM simulations.
Depth (m)
5.6
6.0
9.6
*
CPT-1 (kPa)
CPT-2 (kPa)
CPT (MPM)
778
718
730
556
565
580
820
704
650*
after (Bisht et al., 2021c).
CRediT authorship contribution statement
Table 8
Nk values determined using single-element loading simulations.
Depth (m)
5.6
6.0
9.6*
*
UDTXC
UDTXE
UDSS
Vibhav Bisht: Writing – review & editing, Writing – original draft,
Software, Methodology, Conceptualization. Rodrigo Salgado: Writing –
review & editing, Writing – original draft, Methodology, Conceptuali­
zation. Monica Prezzi: Writing – review & editing, Writing – original
draft, Methodology, Conceptualization.
Nk,ave
su (kPa)
Nk,TXC
su (kPa)
Nk,TXE
su (kPa)
Nk,SS
55.0
44.6
38.8
13.0
13.3
12.2
38.0
30.9
27.0
18.8
19.2
17.6
49.6
40.6
35.9
14.4
14.6
13.2
15.4
15.7
14.4
Declaration of competing interest
after (Bisht et al., 2021c).
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.
scheme with error control for constitutive model integration, and the
accurate imposition of tractions.
Cone penetration simulations were performed for sands at different
relative densities and vertical effective stresses. Cone resistance values
obtained from MPM were found to be smaller than those predicted using
cavity expansion analysis for all cases, indicating that cavity expansion
analysis may slightly overpredict cone resistances. For clays, the MPM
cone penetration simulations were compared with field results for Bos­
ton Blue Clay at different depths and OCR values, and the penetration
values were found to be, for practical purposes, within the range of the
CPT values recorded in the field. An average cone factor of 13.7 was
computed using the cone resistances computed using MPM and su values
obtained from single-element simulations performed under UDTXC,
UDTXE, and UDSS conditions. Although the cone factor Nk,avg = 13.7 is
within the range of values, 9–15, found in the literature, this value is
likely an overestimate of the cone factor, because of limitations in cone
discretization within MPM. However, instead of Nk,avg, a designer may
Data availability
Data will be made available on request.
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