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European Congress on Computational Methods in Applied Sciences and EngineeringECCOMAS 2000

Barcelona, 11-14 September 2000 ECCOMAS

COMPUTATIONAL MODELING OF POST-LOCALIZATION BEHAVIOR

OF ELASTO-PLASTIC SOLIDS WITH STRONG DISCONTINUITY

Ronaldo I. Borja*

and Timothy Y. Lai

*Department of Civil and Environmental Engineering

Terman Engineering Center, Room 244, Stanford University

Stanford, CA 94305, USAE-mail: [email protected]

Department of Civil and Environmental Engineering

Building 540, Room 104, Stanford UniversityStanford, CA 94305, USA

E-mail: [email protected]

Key words: Bifurcation, displacement jump, elasto-plasticity, strain localization, strong

discontinuity.

Abstract. This paper discusses some modeling and algorithmic aspects of post-localization

analysis in elasto-plastic materials exhibiting strong discontinuity or displacement jump.Conditions for the onset of displacement jump are reviewed for a rate-independent elasto-

plastic solid, and a class of constitutive models appropriate for cohesive-frictional materials is

proposed describing the evolution of displacement jump in the post-localization regime. The

theory relies on the hypothesis that once localized deformation is detected, the constitutive

relations for continuum-like deformation are suspended in favor of the relation between

tractions and the plastic slip on the zone of discontinuity. The post-localization constitutive

relations are enforced via a finite element formulation based on standard Galerkin

approximation, which yields mesh-independent finite element solutions. An attribute of this

finite element formulation is that no static condensation is required on the finite element level,

and that the post-localization constitutive relation is amenable to exact integration andconsistent linearization. Numerical examples are presented describing the implications of the

proposed post-localization constitutive model, focusing on the role played by the dilatancy

angle on the accelerated softening response.

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Ronaldo I. Borja and Timothy Y. Lai

1 INTRODUCTION

Analysis of localized deformation in geologic materials has attracted much attentionrecently due to dramatic failures that often accompany the response of earth structures

undergoing intense shearing. Zones of localized deformation, in the form of shear bands, are a

common feature of brittle rock masses and dilatant soils that have failed under compressive

stresses. Once these shear bands have formed the material outside the zone of localized

deformation unloads, and any constitutive relations for continuum-like deformation cease to

become active in favor of the constitutive relations between tractions and the plastic

deformation inside the shear band.1 The onset of localized deformation is dictated by a

bifurcation analysis from an initially homogeneous response; thereafter, the bifurcated

behavior takes effect and often dominates the structural response.

Within the context of bifurcation analysis in elasto-plastic solids with smooth yieldsurface, the onset of displacement jump, or strong discontinuity, in a material instantaneously

undergoing homogeneous deformation may be predicted through a discontinuous bifurcation

analysis.2,3 The mathematical formulation relies on the hypothesis that the material outside

the band unloads elastically while plasticity persists on the surface of discontinuity.2

Bifurcation analysis seeks a configuration where stresses are favorable to initiate a

displacement jump, with the orientation of the shear band obtained as a by-product of the

analysis. The criterion for the onset of displacement jump is the vanishing of the determinant

of the so-called elastic-perfectly plastic acoustic tensor.2,4

Following the onset of displacement jump, a bifurcated response ensues in which

deformation is now concentrated to the surface of discontinuity. Here, theory of distribution

may be used to handle the jump in the displacement field, which naturally leads to a

displacement gradient field that varies according to the delta function across the surface of

discontinuity.2 This paper focuses on the softening behavior at post-bifurcation, in particular,

the constitutive modeling aspects relating tractions and plastic slip on the surface of

discontinuity using the finite element formulation based on standard Galerkin formulation

recently advanced in Ref. 5. For this purpose, a class of elasto-plastic constitutive models

appropriate for cohesive-frictional materials, as presented in Refs. 6 and 7, with yield and

plastic potential functions defined by hyperplanes in six-dimensional stress space, is

considered. Numerical examples are presented demonstrating the implications of the post-

localization constitutive model considered in this paper, as well as the absolute mesh-

insensitivity of the resulting finite element solutions.

2 LOCALIZATION CONDITION

We consider an elasto-plastic continuum as shown in Fig. 1. We assume that the

constitutive response prior to localization is described by a yield function F and a plastic

potential function Q, with stress gradientsfand q, respectively. Further, we let ce denote the

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elastic tangential moduli tensor having the property of major and minor symmetry, as well as

positive-definiteness. The condition for the onset of displacement jump along a potential slip

surface Sdescribed by a spatially varying unit normal vectorn is given by

det(A) = 0, A = n. cep.n (1)

where

cep = ce ce:q f:ce/(f: ce:q) (2)

Here,A is the elastic-perfectly plastic acoustic tensor obtained from the continuum elastic-

perfectly plastic tangential moduli tensor, cep, which is devoid of the plastic modulus term H,

see a discussion in Ref. 6. Note that although the plastic modulusHdoes not appear explicitly

in Eq. (1), the determinant expression is still a function of the stress state, and is thereforeinfluenced by the value ofHused to predict that particular stress state.6

x

+

S

n

Figure 1: Elasto-plastic solid cut by a shear band.

Bifurcation analysis seeks a configuration where stresses are favorable to initiate a

displacement jump, based on the satisfaction of the eigenvalue problem implied in Eq. (1). A

by-product of the eigenvalue analysis is the unit vectorn characterizing the orientation of the

shear band at the point where Eq. (1) is first satisfied. In addition, a unit eigenvectorm may

be derived from the eigenvalue problem defining the instantaneous direction of the velocity

jump at the instant of bifurcation. The unit vectors n and m are shown pictorially in Fig. 2(a).At the instant of bifurcation the vectorm makes an angle , called angle of dilatancy, relative

to the shear band. The angle characterizes the constitutive behavior of the shear band at

post-localization and should not be confused with the continuum angle of dilation used to

describe the volume change behavior of an intact continuum.

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n

m

+

S

= 90 ~

n

+= 90

(b) S

p

(a)

Figure 2: Gradient tensors for: (a) damage potential functionR; (b) damage yield function G.

3 MODEL FOR BIFURCATED RESPONSE

For the shear band to continue to evolve the traction vector across the surface of

discontinuity must remain continuous and non-singular. This condition is guaranteed provided

that the damage potential function R on the shear band is such that its stress gradient, r, is

related to the slip tensor, sym(mn), according to the equation [see Fig. 2(a)]

r sym(mn) (3)

Note that R is the counterpart of the continuum plastic potential function Q in the post-

localization regime. Similarly, the consistency condition on the band requires that the damage

yield function G on the shear band is such that its stress gradient,g, satisfies the relation

g sym(pn), p = t+ n tan (4)

where t is the unit tangent vector to the band and is the mobilized friction angle on thesurface of discontinuity [see Fig. 2(b)].

The damage model described above has been cast within the framework of finite element

analysis in Ref. 5 using standard Galerkin approximation, an approach that renders element-

level static condensation completely unnecessary from the point of view of finite element

analysis based on assumed enhanced strains, see Ref. 7. Our goal in this paper is to elaborate

the role played by the post-localization constitutive model on the predicted behavior of a

damaged continuum. To this end, we consider the following damage function characterizing

the yield behavior of a material on the shear band:

G = sym(tn):T [c (nn):Ttan ] = 0 (5)

where T is the Cauchy stress tensor, c is the mobilized cohesion, and is the mobilized

friction angle on the surface of discontinuity. The function described above has a stress

gradientgthat satisfies the relation (4). Eq. (5) implies that yielding on the band takes place

whenever the resolved tangential shear stress reaches a certain maximum value, where the latter

depends on the mobilized cohesion and friction angle on the surface of discontinuity.

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Next we describe the evolution of the slip tensor, sym(mn), characterizing the plastic

component of deformation on the shear band. Recall that the vectorn defines the geometry of

the shear band, and so is fixed. Also, recall that the vector m defines the instantaneous

direction of velocity jump on the surface of discontinuity, and therefore its initial value at the

onset of localization is dictated by the eigenvector at the onset of bifurcation. Fig. 2(a) shows

the relative orientations of these two unit vectors. For simplicity in discussion, let us consider

planar deformation so that the surface of discontinuity is restricted to the plane of Fig. 2(a).

We therefore have the following scenario: The dilatancy angle is given by its initial value 0determined from the bifurcation analysis at the onset of localization. We can let this angle

remain constant, in which case, the material continues to dilate on the shear band, or allow this

angle to decay such that at large plastic slips approaches zero. This latter idea is captured

by an evolution equation for the dilatancy angle of the exponential form

= 0exp(k), (6)

where is the accumulated magnitude of displacement jump,5,7 and k 0 is a parameter

describing the rate at which decays with relative slip on the band. Ifk = 0, then a constant

dilatancy angle is recovered; as k, then the shear band closes in immediately after the

onset of localization. The following section shows the implications of this post-localization

constitutive relation on the predicted load-displacement response of elasto-plastic solid

specimens subjected to compressive stresses under a condition of plane strain.

4 NUMERICAL EXAMPLE

Since the focus of the simulations is the post-localization behavior, we will simply re-analyze the plane strain compression example presented in Ref. 5 (Example 4.2) and perform

parametric studies focusing on the bifurcated response. The problem consists of an elasto-

plastic solid specimen 1m3m, discretized using three different meshes as shown in Fig. 3.

The material response prior to localization is modeled by a non-associated elasto-plastic

Drucker-Prager yield criterion with dilative plastic flow. The notations and material

parameters are the same as those used in Ref. 5, withE= 20,000 kPa, = 0.40, H= 100 kPa,

= 0.495, b = 0.30, and = 17.143 kPa. At post-localization the mobilized friction angle is

assumed to be = 30, while the softening parameter on the shear band is taken as H = 500

kPa. At the onset of localization the predicted shear band orientations are the same for all the

three meshes, as shown in Fig. 3, where the instantaneous dilatancy angle is 0= 18.7.

Three different values of k were used in the simulations: 0, 50, and 1.0106/meter. As

pointed out in the previous paragraph, these correspond to constant, exponentially decaying,

and immediately zero dilatancy following the onset of localization, respectively. Fig. 4 shows

the force-displacement plots as a function ofk, while Fig. 5 shows the variation of dilatancy

angle with accumulated plastic slip . Note that for a constant plastic softening modulus H on

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the band, some strength is gained by allowing the dilatancy angle to decrease with plastic

flow (k 0); equivalently, keeping the dilatancy angle constant accelerates the softening

response of the specimen (k= 0). The load-displacement responses predicted by all the three

meshes at post-localization are identical to machine precision, demonstrating objectivity of the

finite element solutions to mesh refinement and insensitivity to mesh alignment. For the case

of constant dilatancy angle (k = 0), the deformed meshes predicted by the finite element

solutions at the end of the simulations are shown in Fig. 6.

(a) (b) (c)

Figure 3: Undeformed finite element meshes: (a) regular, unstructured coarse mesh; (b) irregular mesh; (c)regular, structured finer mesh.

25

50

75

100

0100 20 30 40 50 60

DISPLACEMENT, mm

LOAD,kN/m

k= 0

k= 50/m

k= 1.0e+6/m

ONSET OF

LOCALIZATION

Figure 4: Influence of parameterkon post-localization responses of cohesive-frictional material.

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0

5

10

15

20

100 20 30 40 50 60

PLASTIC SLIP, mm

DILATANCYANGLEPSI,deg

k= 0

k= 50/m

k= 1.0e+6/m

Figure 5: Variation of dilatancy angle with total accumulated plastic slip .

(a) (b) (c)

Figure 6: Deformed finite element meshes: (a) regular, unstructured coarse mesh; (b) irregular mesh; (c) regular,structured finer mesh.

5 CONCLUSION

This paper discusses some modeling and algorithmic aspects of post-localization analysis

in elasto-plastic solids exhibiting strong discontinuity or displacement jump. Specifically, a

finite element formulation based on the standard Galerkin approximation was used to enforce

the constitutive relations between the tractions and relative displacements on the band. An

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advantage of this formulation is that no static condensation is required on the finite element

level, resulting in more robust implementation capable of accommodating different post-

localization constitutive relations. Preliminary results of numerical simulations suggest that

softening of a specimen of cohesive-frictional material deforming in simple plane strain

compression and undergoing shear banding in the form of displacement jump is accelerated by

the value of the dilatancy angle on the surface of discontinuity. All things being the same, the

larger the dilatancy angle, the steeper the softening branch of the load-displacement curve at

post-localization. As discussed in Ref. 5, this result is obtained from a finite element solution

that exhibits objectivity with respect to mesh refinement and insensitivity to mesh alignment.

ACKNOWLEDGEMENT

Financial support for this research was provided by the G3S Division of the National

Science Foundation under Contract No. CMS-9700426, through the program of Dr. Priscilla P.

Nelson.

REFERENCES

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W.T. Koiter, ed., North-Holland Publ. Co., 207-220 (1976).

[2] J.C. Simo, J. Oliver, and F. Armero, An analysis of strong discontinuities induced by

strain-softening in rate-independent inelastic solids, Computational Mechanics, 12, 277-

296 (1993).

[3] J.R. Rice and J.W. Rudnicki, A note on some features of the theory of localization of

deformation,Int. J. Solids Structures, 16, 597-605 (1980).

[4] F. Armero and K. Garikipati, Recent advances in the analysis and numerical simulation

of strain localization in inelastic solids,Proc. Computational Plasticity IV, D.R.J. Owen,

E. Oate, and E. Hinton, eds., CIMNE, Barcelona, Spain, 547-561 (1995).

[5] R.I. Borja, A finite element model for strain localization analysis of strongly

discontinuous fields based on standard Galerkin approximation, Comput. Methods Appl.

Mech. Engrg., (2000), in press.

[6] R.I. Borja, R.A. Regueiro, and T.Y. Lai, FE modeling of strain localization in soft rock,

J. Geotech. and Geoenvironmental Engrg., ASCE, 126, 335-343 (2000).

[7] R.I. Borja and R.A. Regueiro, Strain localization of frictional materials exhibitingdisplacement jumps, Comput. Methods Appl. Mech. Engrg.., (2000), in press.

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