Three-invariant non-coaxial elastoplastic constitutive modeling and its implications on
the localization properties of rocks
Principal Investigator: Ronaldo
I. Borja
Project Sponsor: Division of Chemical Sciences, Geosciences and Biosciences, Office
of Science, Department of Energy
Project Description
Strain localization is a ubiquitous feature of geomaterials
undergoing nonhomogeneous deformation. In soils and
rocks, the zone of localized deformation is generally referred to as either a
fault, shear band, rupture zone, or simply a failure plane, and is well
explained by either fracture mechanics or bifurcation theory. Localized
deformation is typically followed by a reduction in the overall strength of the
material as the loading proceeds. It is thus of
considerable interest and importance to be able to predict when a shear
band forms, how this narrow zone of discontinuity is oriented within the
material, and how the propagation of the
shear band is influenced by the post-localization constitutive responses.
Much experimental work has been conducted to understand the inception of
localized deformation in rocks. It is important to recognize that the overall
material response observed in the laboratory is a result of many different
micromechanical processes such as microcracking in
brittle rocks, mineral particle rolling and sliding in granular soils, and
mineral particle rotation and translation in the cement matrix of soft rocks.
Ideally, any localization/bifurcation model for geomaterials
must capture all of these important micromechanical processes. However, current
limitations in the laboratory testing capabilities and mathematical modeling
techniques inhibit the use of a micromechanical description of their behavior,
and a macromechanical approach, such as that
employing theory of plasticity, is still heavily favored by the geomechanics modeling community.
Strain localization is often viewed as an instability process that can be
predicted in terms of the pre-localization constitutive relations. The material
is assumed to deform homogeneously until its constitutive relations allow a
bifurcation from a smoothly varying deformation field into a highly
concentrated shear band mode. The bifurcation point is detected by a stability
analysis. For modeling purposes the bifurcation point signals the onset of a
localization mode. Therefore, an accurate prediction of the bifurcation point
is very crucial in the simulation of the mechanical behavior of geomaterials. Equally critical is an accurate
representation of the mechanical response following localization.
It is well known that predictions of faulting as a bifurcation from homogeneous
deformation are strongly dependent on the constitutive description of
homogeneous deformation. Unfortunately, standard plasticity models are
inadequate for predicting the bifurcation point. The absence of the third
stress invariant and the inability of a plasticity model to capture the
vertex-like structure exhibited by many brittle materials may have contributed
to the model's poor prediction of the bifurcation point.
In brittle rock masses the existence of a vertex-like structure makes the
constitutive response a function not only of the state of stress but also of
the direction of the stress increment. This may be illustrated by considering a
stress path tangent to a yield surface (loading to the side), which is
predicted by any conventional plasticity model with a smooth yield surface to
result in an elastic response, whereas the presence of a vertex structure at
the same stress point would otherwise predict a plastic response. Vertex
effects play no role for proportional loading, and are insignificant or
non-existent for non-proportional loading with fixed principal stress axes.
However, they are very important for highly non-proportional loading with
rotating principal stress axes and are known to enhance strain localization.
Unfortunately, the vertex effect creates enormous difficulties with the
classical tensorial models of plasticity, and to date
no tensorial models exhibiting the vertex effect are
currently available for large-scale finite element analysis.
In this project, we investigate the effects of the third stress invariant and
vertex-like responses on the prediction of the onset of strain localization in geomaterials in general, and in rocks in particular. Our
analysis will build upon a recently developed stress-point integration
algorithm for a class of three-invariant elastoplastic
constitutive models for geomaterials. Our rationale
for proposing to carry out this investigation contingent upon the availability
of a robust stress-point integration algorithm is that in a majority of cases a
robust algorithm is necessary to describe the discrete evolution of the
stresses and plastic internal variables, and that without this algorithm it may
not be possible to accurately capture the plastic loading paths leading to
strain localization.
Some of the three-invariant plasticity models we will consider include the
smooth versions of the Mohr-Coulomb plasticity model, i.e., the Matsuoka-Nakai and Lade-Duncan
models, recently implemented within the context of return mapping
algorithm by the Stanford Geomechanics group. Widely
used for representing the yield and failure behavior of cohesive-frictional
materials, these models are most conveniently
expressed in terms of principal stresses and predict higher yield/failure
strength in compression than in tension. In addition, we will also introduce
enhancements to these models to capture the curved shape of the yield surface
on the meridian plane. By utilizing constitutive models truly appropriate for geomaterials, we hope to represent their mechanical
responses more realistically and predict the onset of strain localization more
accurately.
In addition to choosing a more accurate characterization of the overall
behavior through the use of more realistic plasticity models, we also plan to
investigate the effect of vertex-like yielding on the localization properties
of brittle rock masses. Our investigation will build upon a non-coaxial
plasticity model formulation that circumvents the assumed coaxiality
between the principal stresses and the principal plastic rate of deformation.
We assume that the non-coaxial part of the deformation rate is related to the
rate of relative stress, which in turn translates the yield surface, thus alleviating
the highly nonlinear consistency condition that results from the classical
vertex plasticity formulations.
To complete the strain localization modeling, we will put the above bifurcation
theory in the framework of the strong discontinuity concept for modeling
post-localization responses. This final step allows us to track the propagation
and evolution of the shear band using an enhanced finite element formulation
that alleviates if not completely circumvents the mesh dependence issues in
finite element modeling. Modeling the post-localization responses is a critical
component of strain localization analysis since shear bands do not appear
instantaneously in the material, but rather, they develop progressively. The
occurrence of plastic slip at the tip of the shear band destroys the symmetry
of the finite element problem, and thus alters the stress state in the next
localizing elements. With a non-coaxial plasticity formulation, we hope to
develop a more realistic finite element description of the localization and
post-localization responses of brittle rock masses particularly at the tip of
the shear band where the influence of vertex yielding is expected to be more
pronounced.