ISSUES IN MATHEMATICAL MODELING OF STATIC AND DYNAMIC LIQUEFACTION AS A NON-LOCAL INSTABILITY PROBLEM Ronaldo I. Borja Stanford University ABSTRACT The stress-strain behavior of a saturated loose sand undergoing flow liquefaction is characterized by a peak strength, followed by softening and a residual state. This behavior is very much similar to that of a sand body undergoing strain localization. Both liquefaction and strain localization phenomena may be considered as manifestations of material instability; however, whereas there have been significant advances made in strain localization simulations, progress in the modeling of soil liquefaction phenomena as an instability problem has been slow. This paper describes essential elements of a strain localization model that may well be adopted for the analysis of soil liquefaction phenomena. Pressing issues addressed in this paper include bifurcation and strain softening responses. INTRODUCTION The stress-strain behavior of a saturated loose sand undergoing flow liquefaction is characterized by a peak strength, followed by softening and a residual state (Ishihara 1993). The peak strength is particularly prominent at large initial confining stresses but tends to decrease as the initial confining stresses decrease. Prior to peak strength the soil deformation is relatively homogeneous. However, liquefaction, or flow failure, occurs almost immediately at peak strength and is characterized by a non-homogeneous specimen response. This behavior is very much similar to that of an initially homogeneously deforming dense sand specimen prior to the formation of a shear band. The difference lies in the fact that a soil undergoing flow liquefaction collapses by "implosion" rather than by shearing on a planar band. However, both phenomena entail a softening response and thus can be considered as manifestations of an unstable material behavior. Current understanding of the formation of shear bands as material instability suggests that there are three essential elements of a strain localization model: (a) a pre-localized model; (b) a localization condition; and (c) a post-localized model. The pre-localized model captures the initial homogeneous response and defines the range of stable material behavior. The localization condition signifies a critical stage at which multiple solutions could possibly exist, and defines the limit of the stable material response. The post-localized model describes the strain-softening response that follows the onset of strain localization. Each of these elements must be clearly identified and incorporated into the mathematical model or the solution could exhibit spurious mesh sensitivity. Similarly, essential elements of a mathematical model for flow liquefaction instability may be identified as follows: (a) a pre-liquefaction model; (b) a liquefaction condition; and (c) a flowliquefaction model. The pre-liquefaction model captures the homogeneous material response and
defines the range of stable material behavior. The liquefaction condition defines the onset of instability in a loose soil matrix. The flow-liquefaction model defines the strain-softening response following the onset of flow-liquefaction instability. Like the strain localization model defined in the previous paragraph, liquefaction is a non-local phenomenon and requires modeling of the interaction of a macroscopic point in relation to the response of the neighboring points. A detailed description of each component of the liquefaction model is thus presented in this paper. An assumption is made throughout that the nonlinear material behavior may be described by theory of plasticity and that the actual numerical calculations are carried out using the finite element (FE) method. PRE-LIQUEFACTION MODEL The pre-liquefaction model captures the homogeneous soil deformation response and initial development of pore water pressure either by static or cyclic loading. Here, the constitutive model is best formulated in terms of effective stresses and used together with a coupled solid deformation-fluid flow FE model. There are many FE codes with these features that are now currently available. A main issue concerns the selection of a simple but robust constitutive model. Prediction of the excess pore water pressure depends both on the constitutive model used in the simulation and on the rate of application of the external load in relation to the hydraulic conductivity of the soil medium. When cyclic loading is involved, the constitutive model also must capture the stress-induced anisotropy. Balancing simplicity and robustness in a constitutive model is a big challenge to the analyst. A simple constitutive model is desirable because it requires fewer material parameters and is easier to implement. Yet, the limitations of available laboratory testing procedures and existing numerical techniques are easily reached by the requirement that the constitutive model be realistic enough to include important features of soil behavior, such as the stress-induced anisotropy for cyclic loading, nonlinear elasticity, volumetric and deviatoric yielding, and nonlinear hardening. Without a robust stress-point integration algorithm, no constitutive model can survive in any nonlinear FE code. An example of a pre-liquefaction constitutive model that has been successfully implemented into a FE code is the anisotropic bounding surface plasticity model recently formulated and implemented by the author and co-workers (Borja et al. 2001). This model has the following features: (a) ellipsoidal bounding and loading surfaces, F and f, respectively, formulated in a general 3D framework; (b) homology properties, implying that the consistency condition on f is the same as the consistency condition on F; (c) energy-conserving nonlinear elasticity with pressure-dependent elastic bulk and shear moduli; and (d) bilogarithmic compressibility law for the bounding surface. The new homology property (b) arises from a reformulation of previous bounding surface plasticity theories and allows an efficient implementation using the standard return-mapping algorithm of classical plasticity. This latter feature well illustrates that the numerical implementation aspect is an important consideration when developing a constitutive model.
The material parameters required by the above constitutive model are very similar to those used in critical state theory and are obtainable from conventional laboratory tests. Additional parameters reflecting the variation of the plastic modulus can be determined from standard modulus degradation curves. From an algorithmic standpoint, an implicit stress-point integration scheme is desirable because it provides numerical accuracy and stability that are very critical to ensure convergence of the nonlinear problem. The solution of the nonlinear equations generally requires two levels of Newton iterations: a local level for the calculation of the plastic multiplier, and a global level for the calculation of the nodal degrees of freedom. These are standard features of a general return mapping algorithm. CONDITIONS FOR STABILITY Like the problem of strain localization into a shear band, conditions must be specified to define the onset of flow liquefaction instability in granular materials. An extensive discussion of this issue has been presented by Lade (1999). Two stability postulates may readily be noted: (1) Stability postulate of Drucker (1959): According to this postulate, stability requires that the second increment of plastic work is greater than or equal to zero, i.e., δσ ij δε ij p 0, (1) where δσ ij is the incremental Cauchy stress tensor and δε ij p is the incremental plastic strain tensor. For associative plastic flow, positive values of the second increment of plastic work are associated with the ascending branch of the stress-strain response, whereas negative values are associated with the descending branch. (2) Stability condition of Hill (1958): The condition for stability is given by the inequality δσ ij δε ij 0 or δp ia δf ia 0, (2) where δε ij is the incremental total strain tensor, P ia is the increment of the first Piola-Kirchhoff stresses, and F ia is the increment of the deformation gradient. The finite deformation version of the stability condition underscores the importance of geometric non-linearity effects on the bifurcation analysis (Bigoni 1999; Borja 2002). Hill's and Drucker's stability conditions are the same when the elastic component of the strain tensor is zero. Note that both stability conditions depend solely on the state of stress as well as on the preinstability constitutive model. In other words, theoretically, the pre-instability constitutive model should be sufficient to predict the loss of stability, or bifurcation. For non-associated plasticity, neither Drucker's nor Hill's stability postulates guarantees stability in the ascending branch of the stress-strain response. In fact, it is well known that bifurcation into a planar band is possible with a non-associated plasticity model even in the hardening regime of the load-displacement response (Rudnicki and Rice 1975). For a non-associated plasticity model
an exclusion condition similar to Hill's stability postulate was formulated by Raniecki and Bruhns (1981) using the notion of a comparison solid. Their exclusion condition takes the form δε ij c ijkl δε kl 0 or δf ia A iajb δf jb 0 (3) where c ijkl is the tangential moduli tensor of a so-called "linear comparison solid" (the two-point tensor A iajb = P ia / F jb is the first tangential moduli tensor of the comparison solid when the problem is formulated in the finite deformation regime). Note that condition (3) only excludes bifurcation but does not pinpoint the exact bifurcation point. For bifurcation into a planar band, Borja (2002) identified the bifurcation points for the cases of continuous (plastic loading on both sides of the band) and discontinuous (plastic loading outside/elastic loading inside the band) shear band bifurcation modes. Furthermore, Peric et al. (1992) examined the effects of kinematical constraints on the initiation of strain localization. Hill's stability postulate emphasizes the role played by the constitutive tangent operator on the instability characteristics of a material. However, because his postulate considers the possibility of a non-unique velocity field across a particular surface of discontinuity, it does not lend itself to a straightforward geometric interpretation when applied to liquefaction-type instability. Alternately, Lade (1999) utilized Drucker's stability postulate and extended the idea to the nonassociated regime. He showed that for non-associated plasticity Drucker's stability condition is violated in the stress region where the stress rate and the plastic strain increment form an obtuse angle. As noted by Lade, instability is possible even in the work-hardening regime where the first increment of plastic work, σ ij δε ij p, is positive. Furthermore, instability does not occur at low deviator stresses, but is likely to occur at high deviator stresses where the yield surface is inclined toward the origin. From a modeling standpoint, potential instability may be checked at each stage of the solution by tracking the evolution of the constitutive tangent operator at each Gauss integration point, if one follows Hill's stability postulate, or by tracking the evolution of δσ ij and δε ij p at each Gauss integration point and identifying the precise instant at which the second increment of plastic work becomes zero for the first time, if one follows Drucker's stability postulate. This is analogous to a shear band bifurcation analysis where strain localization is detected according to the sign of the determinant of the so-called elastoplastic acoustic tensor (Borja 2002). Unlike solid materials, however, loose granular materials behave in a less predictable way, and the plastic potential function defining the plastic strain increment may not be accurate enough to be used for detecting bifurcation. Alternately, therefore, a surface in stress space is sometimes used as a triggering criterion for flow-liquefaction instability. For saturated sands, Vaid and Chern (1985) identified a boundary separating the stable and unstable (softening) states, commonly known as the flow liquefaction surface (FLS). One possible form of FLS is provided by a Mohr-Coulomb surface truncated at low deviator stresses. The surface must be truncated since flow-liquefaction instability is not expected to occur at low deviator stresses. However, it must be noted that the FLS provides a redundant piece of information since instability is theoretically determined either by the vanishing of the second increment of plastic work, or by the vanishing of the determinant of the acoustic tensor, which are both functions of the pre-liquefaction constitutive model. As noted earlier, however, the pre-
liquefaction constitutive model may not be accurate enough to predict the correct bifurcation stress states in real soils, and so a FLS is still a useful limiting criterion to describe flowliquefaction instability. POST-BIFURCATION AND SOFTENING RESPONSE Beyond the bifurcation point the material develops large strains that eventually lead to failure. The softening response that follows the bifurcation point is usually the most difficult aspect of soil behavior to model. The plastic softening modulus H is not constant, and generally varies with the plastic strain. However, most of the large strains generally develop in the residual stress state, so in this respect the precise variation of the plastic softening modulus may not be as important in the analysis as actually getting to the residual stress state itself. Still, it must be noted that there is a limit to the maximum possible negative value that the plastic softening modulus H may take, given by H > f ij c e ijkl g kl, (4) where f ij and g kl are the stress gradients to the yield and plastic potential functions, respectively, and c e ijkl is the elastic tangential moduli tensor. Condition (4) guarantees a non-negative plastic multiplier whenever the material is yielding plastically. For rate-independent materials a softening response could lead to a loss of strong ellipticity. The strong ellipticity condition as probably first put forth by Hadamard (1903) implies that elastic wave speeds are real and nonzero. Provided that the constitutive equation can be written in a linearized form δσ ij = c ijkl δε kl, where c ijkl is the tangential moduli tensor, the strong ellipticity condition may be extended to the inelastic regime. For waves traveling in an elastoplastic medium the loss of strong ellipticity is given by the condition det A ik = 0, (5) where A ik = n j c ijkl n l is commonly known as the elastoplastic acoustic tensor associated with a plane of unit normal n i. Condition (5) refers to the formation of a stationary discontinuity, or standing wave. For a softening response the elastoplastic acoustic tensor could become negative-definite, in which case, the local partial differential equation could change from hyperbolic (in the case of a dynamic problem) to elliptic, and waves will not propagate. In this case the numerical solution of the boundary-value problem will suffer from spurious mesh sensitivity, rendering the numerical solution meaningless. In order to avoid this difficulty, regularization procedures are often employed to preserve the character of the partial differential equation. An example is viscoplastic regularization which provides the constitutive equation a rate-dependent component. It may be argued, however, that viscoplasticity is not as commonly used as elastoplasticity for describing the constitutive behavior of granular materials. Therefore, an alternative regularization approach has been proposed by Borja et al. (2000) based on an additive decomposition of stresses into viscous and inviscid parts, where the latter part is evaluated from conventional rate-independent
plasticity theory. This formulation was shown to be effective in nonlinear ground response analysis applications. More elaborate regularization techniques circumventing the loss of strong ellipticity include gradient-dependent plasticity enhancements and a Cosserat continuum description. However, the author believes that these more elaborate procedures are much too complicated to be used for modeling soil liquefaction phenomena. OTHER ISSUES The above discussions are by no means complete; however, they elucidate the challenges that the analyst must face when modeling soil liquefaction phenomena as an instability problem. Other issues of relevance include the changed constitutive response at the onset of flow liquefaction instability and the associated re-consolidation effects. With respect to the first aspect, it is well known that collapse of the soil structure changes the constitutive properties of the soil matrix, so it is possible for the constitutive model to also "bifurcate" and "metamorphose" to better capture the soil response at post-bifurcation. An analogy may again be made to strain localization problems: once a shear band forms, the constitutive response of the specimen is dominated by what happens inside the shear band, whose constitutive properties may be significantly different from those of the intact material (Borja and Regueiro 2001). With respect to the process of reconsolidation, it appears that the mathematical challenges in modeling this phenomenon are not as great because it involves a re-hardening (i.e., stable) response. Of course, the challenge lies as to what configuration and constitutive state the re-hardening process will have to start from since it is preceded by soil collapse, which is a far more complex process to model. REFERENCES Bigoni, D. (1999). Bifurcation and instability of non-associative elastoplastic solids." CISM Lecture Notes: Material Instabilities in Elastic and Plastic Solids, H. Petryk (Coord.), Udine, Sept. 13-17. Borja, R.I., Lin, C.H., Sama, K.M., and Masada, G.M. (2000). Modeling non-linear ground response of non-liquefiable soils." Earthquake Engng Struct. Dyn., 29, 63-83. Borja, R.I., Lin, C.H. and Montáns, F.J. (2001). "Cam-Clay plasticity, Part IV. Implicit integration of anisotropic bounding surface model with nonlinear hyperelasticity and ellipsoidal loading function." Comput. Meth. Appl. Mech. Engrg., 190, 3293-3323. Borja, R.I. and Regueiro, R.A. (2001). "Strain localization of frictional materials exhibiting displacement jumps." Comput. Meth. Appl. Mech. Engrg., 190, 2555-2580. Borja, R.I. (2002). "Bifurcation of elastoplastic solids to shear band mode at finite strain." Comput. Meth. Appl. Mech. Engrg., in press. Drucker, D.C. (1959). A definition of a stable inelastic material." J. Appl. Mech., 26, 101-106.
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