Simulation of Earthquake-Induced Permanent Deformations
Majid T. Manzari
The George Washington University
1. Introduction
Simulation of earthquake-induced deformations in geo-structures that contain liquefiable soils is
an important and challenging problem of geotechnical engineering as it involves several key
features of the soil as a multi-phase particulate medium. Some of these features include:
1. The challenging aspects of the stress-strain response of soils such as the dramatic change
of the soil behavior at very low effective stresses and the occurrence of cyclic mobility
2. The uncertainty in the variation of some of the properties of granular soils such as
hydraulic conductivity when an extremely low effective stress is attained.
3. Possible occurrence of strain localization that could lead to a breakdown of the classical
continuum theories for rate-independent constitutive models.
In addition to the challenges associated with the soil modeling, there are a number of issues that
need to be carefully addressed when a numerical method is used to solve the governing
differential equations of a dynamically loaded multiphase continuum. The numerical techniques
used for integration of the governing equations and constitutive laws often play a paramount role
in achieving reasonably fast and highly accurate solutions.
Each of the above issues would be briefly discussed in this paper.
2. Challenges in Constitutive Modeling of Liquefiable Soils
In the past few years, there has been significant progress in the development of critical state
models for cohesionless soils. Unlike the constitutive models used in the early part of 1990, the
new critical state plasticity models do not need a new set of parameters for each encountering
relative density of soil. There are now several critical-state-based constitutive models (e.g.,
Manzari and Dafalias, 1997; Cubrinovski et al., 1995; Li et al., 1999; Li and Dafalias, 2000;
Dafalias and Manzari, 2001) that are capable of simulating the stress-strain behavior of sands
with a single set of parameters for a wide range of relative densities and confining stresses.
Nevertheless there are two important questions that need to be addressed for soils subjected to
cyclic loading:
1) How can one systematically model the large contractive response of sands in a stress
reverse?
2) Can a constitutive model with a unique set of model constants represent the soil behavior at
very small effective stresses?
These issue have been discussed in Dafalias and Manzari (2001) who have used a fabric
dilatancy tensor to take into account the change of soil dilatancy in the reverse loading and have
attempted to address the second question with a modification of the yield surface so that larger
friction angles are attained at very low effective stresses.
q (kPa)
Figure 1 shows the capability of a critical state two-surface plasticity model (Manzari and
Dafalias, 1997; Dafalias and Manzari, 1999 and 2001) in modeling the stress reversal following a
conventional triaxial compression test on Toyoura sand (Verdugo and Ishihara, 1996). The cases
with Zmax = 4 and 0, represent the response of the model with and without the dilatancy fabric
tensor, respectively.
Zmax = 0
Zmax = 4
Experimental Data
4000
3500
3000
2500
2000
1500
1000
500
0
0
1000
2000
p (kPa)
3000
4000
4000
3500
q (kPa)
3000
2500
2000
1500
Zmax = 0
1000
Zmax = 4
500
Experimental Data
0
0
5
10
15
20
Axial Strain (% )
25
30
Figure 1. Modeling of reverse loading by using a dilatancy fabric tensor (Zmax = 4), after
Dafalias and Manzari, 2001. (Experimental data: Verdugo and Ishihara, 1996)
3. Uncertainty in the Properties of a Granular Soil at Very Small Effective Stresses
In addition to a constitutive model for soil skeleton, an effective stress analysis normally
relies on a constitutive law that represents the flow of pore water through the soil. In a
dynamic loading condition, a generalized Darcy’s law has been proposed as the constitutive
law for the flow of the pore water. Therefore, the soil permeability generally appears in the
governing equations used in a dynamic analysis of saturated soils. Equation 1 states the
axiom of mass balance in a soil, which is contributed by the generalized Darcy’s law.
1
∂
∂p
∂
∂
p& + u&i ,i −
(kij
)+
(kij ρ f b j ) −
[kij ρ f (u&&j )] = 0
∂ xi
∂ x j ∂ xi
∂ xi
Γ
(1)
kij, ρf, bj, and p represent the tensor of permeability, mass density of the pore water, body
force in the j direction, and pore water pressure, respectively. Moreover, Γ is the so-called
combined bulk modulus of the soil defined as
1
n 1− n
=
+
(2)
Γ Kf
KS
where n is porosity and Kf and Ks represent the compressibility of pore water and solid
particles, respectively. Equation 1 in combination with the balance of linear momentum
represents the governing equations used in a dynamic analysis of saturated soil. A
temporally and spatially discretized form of these governing equations can be shown as
&&( i +1) − [(θ∆t ) tt Q] t ∆p& ( i +1) =
[α (∆t ) 2 ( tt K L + tt K NL ) + M S ] t ∆u
[(δ∆t ) tt QT +
t +∆t
t +∆t
t (i )
&&(i +1) + [ tt +∆
M (Fi ) ] t ∆u
+ (θ∆t )
+∆t S
t +∆t
t +∆t
t +∆t
R (Si +1)
H ( i ) ] t ∆p& (i +1) =
t +∆t
R (Fi +1)
which are solved to obtain the incremental values of soil acceleration and pore water
pressure. It is therefore obvious that both the soil permeability and Γ are an integral part of
the analysis and an accurate estimation of these parameters are necessary for a successful
simulation of the response of saturated soils to dynamic loading. Measurement of the soil
permeability is normally done at a condition different from the soil condition when it reaches
a liquefaction stage. Despite this fact, an accurate measurement of the soil permeability
during a dynamic event is rather difficult and has not received much attention in the past few
years.
The parameter Γ is usually obtained by using a gross estimate of the compressibility of soil
particles and the pore water. This usually leads to a large value for Γ which may not be
representative of the soil bulk modulus. Moreover, presence of Γ in Equation 1 is more of a
constraint on volume change than a real material parameter. Previous experience in
numerical simulation of nearly incompressible materials shows that an arbitrary treatment of
constraining parameter can lead to erroneous solutions and should be avoided. There is no
reason for not adopting similar remedial techniques in soils that show nearly incompressible
behavior in a fast dynamic loading condition.
Dynamic analyses of several boundary value problems involving soil liquefaction have
shown that these parameters can play significant roles in the success of such analyses.
Therefore, in addition to the constitutive modeling of soil skeleton, efforts are needed to
characterize the dynamic flow of pore water through the soil when soil reaches a low
effective stress.
4. Strain Localization
A possible outcome of earthquake-induced deformations in soils is the localization of shear
strains in a part of soil body, which can potentially lead to catastrophic failures in
geotechnical structures. A possible cause of progressive localization of strains in soils is the
occurrence of strain softening and the ensuing bifurcation phenomenon in the underlying
constitutive behavior of the material. This normally leads to the localization of plastic strains
in certain part of the soil and the elastic unloading outside of the region of localized strains.
Once strain localization occurs, the solution of the boundary value problem cannot be
continued by using a classical continuum theory that is normally used in all effective stress
based analysis of soil liquefaction. In principle, constitutive models incorporating a length
scale (gradient plasticity models), or a modified set of governing equations incorporating a
length scale (micropolar theories), or finite element procedures accommodating discontinuity
of strains or displacements can be used to continue the solution of the boundary value
problem. However, each of these approaches would bring in additional challenges in the
numerical implementation and in the material characterization for determining the additional
parameters that each of the above three alternatives would involve.
Figure 2 shows the displaced finite element mesh at the end of an excavation process in
cohesive frictional material with a friction angle of φ =20o and a dilation angle of ψ = 2.5o.
A classical finite element solution for this problem normally faces significant difficulty in
convergence of the incremental solution. The solution can be continued with a regularization
technique, which adds a length scale to the underlying boundary value problem.
Figure 2: Displaced Finite Element Mesh for an Excavation Analysis with φ =20o, ψ = 2.5o.
5. References:
Cubrinovski, M.; Ishihara, K.; Higuchi, Y. (1995); Verification of a constitutive model for sand
by seismic centrifuge tests , Proceedings of the 1995 1st International Conference on Earthquake
Geotechnical Engineering. Part 2 (of 2), p 669.
Dafalias, Y. F. and Manzari, M. T. (1999). "Modeling of Fabric Effect on the Cyclic Loading
Response of Granular Soils." Proceedings of ASCE 13th Engineering Mechanics Conference,
13-16 June, Baltimore, Maryland.
Dafalias, Y. F. and Manzari, M. T. (2001), A Simple Plasticity Sand Model Accounting For
Fabric Change Effects, submitted to ASCE Journal of Geotechnical and Geoenvironmental
Engineering.
Li, X.S., Dafalias, Y.F. and Wang, Z.L. (1999), State-dependent dilatancy in critical-state
constitutive modeling of sand, Canadian Geot. J., 36, 599-611.
Li, X.S. and Dafalias, Y.F. (2000), Dilatancy for cohesionless soils, Geotechnique, 50(4), 449460.
Manzari, M. T. and Dafalias, Y. F. (1997), A two-surface critical plasticity model for sand,
Geotechnique, 47(2), 255-272.
Verdugo R. and Ishihara, K. (1996), The steady state of sandy soils, Soils and Foundations,
36(2), 81-92.
Acknowledgment:
This paper is based upon the work supported by the National Science Foundation under Grants
9802287 and 9988557; Dr. Clifford J. Astill and Dr. Richard J. Fragaszy are the program
directors for the two projects, respectively. The results shown in Figure 1 are part of a
collaborative research between Professor Y. F. Dafalias of UC Davis and the author.