Constitutive Modeling in Earthquake Simulation
Stein Sture and Hon-Yim Ko
Department of Civil, Environmental, and Architectural Engineering
University of Colorado at Boulder
In modern nonlinear coupled motion and pore-water pressure analysis techniques
based on Biot's formulation or general theory of mixture concepts, it is important to
include realistic yet physically transparent constitutive models, which are capable of
accurate simulation of deformation, strength, loading history, nonconservative and
dilatancy behavior, as well as mean effective stress dependence during loading,
unloading and reloading cycles. While a large number of constitutive models for soils
have been developed over the years, many of which are in principle able to simulate
behavior, many of these models are very complex, and it is often difficult to relate the
models' variables, parameters and constants to basic soil properties. Moreover, validation
or verification of model behavior is often lacking. Also, many models are formulated in
two-dimensions, such as axisymmetry, and do not readily transform to plane strain or
three-dimensions. Perhaps more important, many constitutive models, especially those
based on incremental elasto-plasticity are devised in rate form, and little consideration is
given to how these models are integrated within nonlinear finite element or difference
method implementations of standard virtual displacement equations or Biot's or mixture
equations(Jeremic et al., 1997, 1998). These issues, in addition to experimental
observations of constitutive behavior, and philosophical discussions are among the topics,
which are our presentation centers on. The following paragraphs briefly describe recent
and newly resurrected work related to earthquake and constitutive simulations.
While nonlinear finite element analysis implementations of Biot's equations
focused on formulation of field equations, appropriate kinematic formulations, nonlinear
solution algorithms at the system or soil-structure level, and to a lesser extent constitutive
model integration, we believe many analysts may be overlooking important physical
phenomena, such as localization of deformations in narrow or wide shear bands. While
we often observe shear bands in experiments, especially on medium to dense specimens,
shear zones also occur in loose specimens. It is now possible to develop finite element
analysis technology by means of adaptive or moving meshes, that can incorporate
localization phenonena. Shear and related dilatancy behavior clearly result in volume
change, and as a consequence undrained shearing can result in substantial excess pore
water pressure generation in critical shear zones, which may result liquefaction, flow and
failure. It is asserted that this can happen far quicker than predicted via conventional
continuum or element-average analyses, in that the shear-zone region is relatively small,
and excess pore pressure dissipation into the neighboring soil, which may serve as a sink,
happens on a very different time scale than we conventionally consider. While pore
water pressures might increase fast in such an analysis, they may also dissipate quickly.
Clearly, the development of failure zones in sandy soils, including fracture zones in
weakly cemented soils, is strongly linked to interactions between the solid skeleton and
water. The interaction, effective stress level, and dilatancy rate will influence the critical
load level, where localization takes place as well as the orientation of zones of localized
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deformation. The initially undrained deformation condition, which governs subsequent
behavior, has received considerable attention, and we mention the works of Runesson et
al. (1995), Han and Vardoulakis (1995), and Runesson, Larsson and Sture (1998) who
studied deformation, strength and instability phenomena for different ranges of pore fluid
compressibility. It was found that initial shear band formation, pore-pressure build-up,
and soil-structure response are very sensitive to effective stress level, initial density, as
well as basic parameters such as friction, dilatancy and elastic moduli.
In order to gain further insight to cyclic behavior of sandy soils, Gong (2001)
conducted three-dimensional micromechanical computational studies of assemblies of
ellipsoidal particles. Uniform or specified size distributions of particle assemblies were
pluviated into the analysis domain by means of gravity with specified boundaries or
isotropically compressed under either displacement or traction boundary conditions. The
investigation included methods for detecting and defining contacts between the
ellipsoidal particles. The constitutive behavior at the particle contacts was described by
nonlinear elastic Hertz and load-history dependent elastic-plastic slip MindlinDeresiewicz models.
A technique for handling both displacement and traction
boundaries was developed and is described by Gong (2001). Traction boundary
conditions were devised by means of a simulated flexible membrane-system, while the
rigid displacement boundary is implemented either in terms of force or displacement
control. A new integration algorithm was developed specifically for static analysis of
granular materials, which shows increased performance in obtaining accurate and stable
positions for all particles in the assembly. The study included numerical simulations of
true-triaxial cubical, axisymmetric triaxial-prismatic, direct shear, direct simple shear,
pull-out and penetration tests. Numerical experiments were also performed, where the
intermediate principal stress (traction) was varied between the major and minor stresses,
or where different pure displacement or mixed traction-displacement conditions were
applied. The number of ellipsoidal particles present in the various three-dimensional
experiments were in the range of 300 to 1,000. The simulations required nearly 30 hours
on modern, high-clockspeed and large memory PC-machines. The effects of both
smooth and frictional walls (displacement boundaries) were studied. In general, the
results compared favorably with data obtained in related laboratory experiments.
Sensitivity studies involving varying particle properties (Young's modulus, Poisson's
ratio, and Coulomb friction) and time-step sizes are described byGong (2001). It was
found that volumetric change and modulus behavior during load cycles are very sensitive
to particle shape and thus, fabric.
To study behavior of granular soils at low stress levels a set of ground-based and
microgravity-based (Space Shuttle) experiments on clean F75 quartz sand were
conducted under drained conditions at effective stress levels in the range of 0.05 kPa to
20.0 kPa. Batiste (1998, 2001) and Sture et al.,1998) observed increasing dilatancy angles
with decreasing confining stress levels, very high peak friction angles, and very high
elastic moduli. Dilatancy angles in the range of 30 degrees were observed for confining
stresses in the range of 0.05 to 1.30 kPa for initial relative densities in the range of 65%
to 85%, and the dilatancy angles reduced to nearly 20 degrees at 3.5 kPa. Peak friction
angles were also extremely high, as one would expect, and these were in the range of 75
to 48 degrees for the same levels and range of initial densities. The data also indicate that
the critical void ratio is substantially higher at low stress levels, almost in the range of the
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maximum void ratio. Triaxial experiments on loose and very loose specimens (10%,
20% and 30%) resulted in substantial volumetric expansion, rather than contraction, even
at effective confinement levels in the range of 2.5 kPa. Experiments at confinement
levels in the range of 5.0 kPa and higher showed volume contraction during shearing
according to standard concepts. Numerical simulations on assemblies of discrete
ellipsoidal particles prepared at comparable packings showed similar behavior. The fabric
in undeformed and deformed specimens, especially shear localization, were studied using
computer tomography (CT) techniques, which showed high accuracy in detecting
nonuniformities and shear patterns. Multiple symmetrical radial shear bands were
observed in the specimens tested at low stress levels, while nonsymmetrical and far fewer
shear bands were observed in specimens subjected to higher stress levels. Void ratio
distributions within and outside the shear bands were studied, and it was observed that
the variations are of the order of 50%. This again points to shear bands as early
generators of excess pore water pressure in the case of initially loose sands, or pore water
sinks in the case of dilating bands. Moreover, the shear bands may be serving as internal
drainage boundaries.
In constitutive modeling of soils a major goal has been to develop cyclic models,
which in association with the Biot-field equations, are able to accurately predict build-up
of pore water pressure in cycles. In order to achieve this, the models need to include
realistic formulations for dilatancy. Nearly 15 years ago the authors developed a "fuzzy
set" plasticity theory proposed by Klisinski et al. (1987, 1988), which is based on the
mechanical-sublayer concept. Instead of utilizing multiple yield or bounding surfaces a
general surface is introduced that span the stress and membership functions used in fuzzy
set theory. The model formulation and implemenmtation are relatively simple, in that
separate membership functions and related stress spaces are used for the deviatoric and
volumetric components. Kinematic plastic behavior is achieved via simple shifting of the
apex location of the apex of the pyramidal structure representing the stress and
membership function space. Details of this model are contained in Klisinski et al.
(1987, 1988). We are continuing the study of this model and its use in nonlinear finite
element analysis of long-term cyclic loading, mainly because of its relative simplicity in
calculations. Also, we intend to use the experimental observations and micromechanical
simulations described above to tune and verify model behavior. An important feature is
that the fuzzy set plasticity formulation allows for smooth transitions between the elastic
and plastic states, which makes it very advantageous in cyclic simulations. Hysteresis
loops and plastic strain increments for successive cycles can be described in a relatively
simple manner. The concept allows formulation of several different models within the
same mathematical framework, such as models with and without memory, and fading
memory, which may be important to simulate variations in soil skeleton behaviro during
cyclic loading. Overall verification of fully implemented Biot analysis will be conducted
by means of centrifuge model studies.
References
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Alshibli, K.A., Sture, S., Costes, N.C., Frank, M., Lankton, M.R., Batiste, S.N., and
Swanson, R.A., (2000), "Assessment of Localized Deformations in sand Using X-Ray
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Ellipsoidal Particles", Ph.D. Dissertation, University of Colorado at Boulder, 180.
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