Strong Discontinuities and Continuum Plasticity Models

Strong Discontinuities and Continuum Plasticity Models�

The Strong Discontinuity Approach

J� Oliver M� Cervera O� Manzoli

E�T�S� Enginyers de Camins� Canals i Ports� Technical University of CataloniaModul C�� Campus Nord UPC

Gran Capita s�n� �� Barcelona� Spain

Abstract

The paper presents the Strong Discontinuity Approach for the analysis and simulation of strongdiscontinuities in solids using continuum plasticity models� Kinematics of weak and strong discon�tinuities are discussed� and a regularized kinematic state of discontinuity is proposed as a mean tomodel the formation of a strong discontinuity as the collapsed state of a weak discontinuity �witha characteristic bandwidth�� induced by a bifurcation of the stress�strain �eld� which propagatesin the solid domain� The analysis of the conditions to induce the bifurcation provides a criticalvalue for the bandwidth at the onset of the weak discontinuity and the direction of propagation�Then a variable bandwidth model is proposed to characterize the transition between the weak andstrong discontinuity regimes� Several aspects related to the continuum and� their associated� dis�crete constitutive equations� the expended power in the formation of the discontinuity and relevantcomputational details related to the �nite element simulations are also discussed� Finally� somerepresentative numerical simulations are shown to illustrate the proposed approach�

� Introduction

Strong discontinuities are understood here as solutions of the quasi�static solid mechanics problemexhibiting jumps in the displacement �eld across a material line �in D problems� or a material surface�in general D problems� which from now on will be named the discontinuity line or surface� Thecorresponding strains� involving material gradients of the displacements� are then unbounded at thediscontinuity line or surface and remain bounded in the rest of the body�

The strong discontinuity problem can be regarded as a limit case of the strain localization one�which has been object of intensive research in the last two decades �Rots et al�� Ortiz et al�� Ortiz and Quigley� �� de Borst et al�� Lee et al�� and where the formation of weak dis�continuities� characterized by continuous displacements but discontinuous strains which concentrate

�

or intensify into a band of �nite width� is considered� As the width of the localization band tends tozero and the value of the strains jump tends to in�nity the concept of strong discontinuity is recovered�

Plasticity models have been often analyzed in the context of strain localization and related topics�the slip lines theory �Chakrabarty� �� for rigid�perfectly plastic models is a paradigm of the use ofplasticity models to capture physical phenomena involving discontinuities� the observed shear bands inmetals can also be explained by resorting to J plasticity models in the context of strain�localizationtheories and weak discontinuities �Needleman and Tvergard� �� Larsson et al�� etc��

Regarding strong discontinuities and their modeling via plasticity models� the topic has been tackledby di�erent authors in the last years� In one of the pioneering works �Simo et al�� the strongdiscontinuity analysis was introduced as a tool to extract those features that make a standard con�tinuum �stress�strain� plasticity model compatible with the discontinuous displacement �eld typicalof strong discontinuities� This work was later continued in �Simo and Oliver� �� Oliver� ��a�Armero and Garikipati� �� Armero and Garikipati� �� Oliver� ��a� Oliver� ��b� Oliver et al�� Oliver et al�� where di�erent aspects of the same topic were examined� as well as in �Larsson et al�� Runesson et al�� in a slightly �regularized� di�erent manner�

This paper aims to clarify the following questions concerning the capture of strong discontinuitiesusing plasticity models�

� Under what conditions typical elasto�plastic �in�nitesimal strains based� continuum constitutiveequations� once inserted in the standard quasi�static solid mechanics problem� induce strongdiscontinuities having physical meaning and keeping the boundary value problem well posed ��

� What is the link of the strong discontinuity approach� based on the use of continuum �stress�strain� models� with the discrete discontinuity approach which considers a non�linear fracturemechanics environment and uses stress vs� displacement�jump constitutive equations to modelthe de�cohesive behaviour of the discontinuous interface �Hillerborg� �� Dvorkin et al�� Lofti and Ching� ��

� What is the role of the fracture energy concept in this context �

� What are the connections of the strong discontinuity approach to the discontinuous failure theories�Runesson and Mroz� �� Runesson et al�� Ottosen and Runesson� �� Steinmann and Willam� ��Stein et al�� aiming at the prediction of the bifurcations induced by continuum constitutiveequations �

Total or partial answers to these questions are given in the next sections� For the sake of simpli�city two dimensional problems �plane strain and plane stress� are considered although the proposedmethodology can be easily extended to the general D cases�

The remainder of the paper is structured as follows� Section deals with the kinematics of thediscontinuous problem and di�erent options are analyzed� In Section the target family of elastoplasticconstitutive equations is described and the corresponding B�V� problem is presented in Section � InSection � the bifurcation analysis of general plasticity models is sketched and some interesting resultsare kept to be recovered in subsequent sections� In Section � the strong discontinuity analysis isperformed and crucial concepts as the strong discontinuity equation� the strong discontinuity conditionsand the discrete consistent constitutive equation are derived� In Section � a variable bandwidth modelis presented as a possible mechanism to link weak to strong discontinuities and to provide a transitionbetween them� In Section � the expended power concept in the formation of a strong discontinuityis examined and the conditions for recovering the fracture energy concept as a material property areestablished� Some details regarding the �nite element simulation in the previously de�ned context are

� In the rest of this paper� the option of modelling strong discontinuities via continuum constitutive equations willbe referred to as the strong discontinuity approach�

then given in Section �� Sections �� and �� are devoted to present some numerical simulations tovalidate the proposed approach� Finally� Section � closes the paper with �nal remarks�

� Weak and strong discontinuities� kinematics

Let us consider a bidimensional body � whose material points are labeled as x� and a material ��xedalong time� line S in �� with normal n �see �gure ��a�� which from now on will be called the discon�tinuity line� Let us also consider an orthogonal system of curvilinear coordinates � and � such thatS corresponds to the coordinate line � � � �S �� fx�� g�� Let us denote by f�e�� e�gthe physical �orthonormal� base associated to that system of coordinates and let r�� and r�� be the corresponding scale factors such that ds� � r� d� and ds� � r� d�� where ds� and ds� are�respectively� di�erential arc lengths along the coordinate lines � and �� We shall also consider thelines S� and S� which coincide with the coordinate lines � � �� and � � �� respectively� enclosinga discontinuity band� �h �� fx�� g� whose representative width h�� from now onnamed the bandwidth� is taken as h�� r��

� � �� Let us �nally de�ne �� and �� as theregions of �n�h pointed to by n and �n� respectively �see �gure ��a� so that � � �� h�

�� Kinematic state of weak discontinuity

Let us consider the displacement �eld u de�ned� in rate form� in � by�

�u�x� t� � ��u�x� t� �H�h�x� t� �� u��x� t� ��

where t stands for the time� and �� stands for the time derivative of �� u�x� t� and ��u��x� t� arecontinuous C� displacement �elds and H�h�x� t�� from now on named the unit ramp function� is alsoa continuous function in � de�ned by�

H�h �

��

� x � ��

� x � ��

��

�� x � �h

� �

Clearly H�h exhibits a unit jump� as di�erence from its values at S� and S� for the same coordinateline � ��H�h �� H�h�� H�h�� From the de�nition of H�h in equation � � thecorresponding gradient can be computed as�

rrrrrrrrrrrrrrH�h � �r�

�H�h

��e�e�e�e�e�e�e�e�e�e�e�e�e�e� �

�r�

�H�h

��e�e�e�e�e�e�e�e�e�e�e�e�e�e� � ��h

�h�

�e�e�e�e�e�e�e�e�e�e�e�e�e�e�

h�� r��

h�� r�� h��

��

where ��h is a collocation function placed on �h ��h � � if x � �h and ��h � � otherwise�� Fromequations �� and �� the kinematically compatible rate of strain �� can be computed as�

��x� t� � rrrrrrrrrrrrrrs �u � rrrrrrrrrrrrrrs ��u�H�h rrrrrrrrrrrrrrs�� u�� z �� continuous�

��h

�

h�� u�� e��

s

� �z � �� discontinuous�

��

where superscript ��s stands for the symmetric part of �� Equation�� states that the rate of strain�eld �� is the sum of a regular �continuous� part� ��x� t�� plus a discontinuous part� ��x� t�� whichexhibits jumps in S� and S� �see �gure ��a�� Equations �� and �� de�ne what will be referred to askinematic state of weak discontinuity which can be qualitatively characterized by discontinuous� butbounded� �rate of� strain �elds�

h ( η )

e∧η

e∧ξ

S+ +( ξ = ξ )

S ξ =( 0 )

ξ

n

Ω+

Ω−

( ξ = ξ −)S

−

Ωη

a)

ξ

ξS

+S

−

ε.h

u.

S

u.

.1h

u ⊗ n( )s

S

ξ

n

Ω+

Ω−

Ωη

b)

ξ

ξ

ε.

u.

S

u.

ξ

h ( η )

S

n

Ωη

c)

ξ

ξ

ε.

u.

S

.1

hu ⊗ n( )s

u.

δS u

.⊗ n( ) s

t

t

Ωh

Figure �� Kinematics� a� Kinematic state of weak discontinuity� b� Kinematic state of strong discon�tinuity� c� Regularized kinematic state of discontinuity�

�� Kinematic state of strong discontinuity

We can now de�ne the kinematic state of strong discontinuity as the limit case of the one describinga weak discontinuity when the band �h collapses to the discontinuity line S �see �gure ��b�� Inother words� when S� and S� simultaneously tend to S �that is� with some abuse in the notation�� h�� and� thus� �h � S in �gure ��a�� In this case the unit ramp function� � becomes a step function HS �HS�x� � � �x � �� and HS�x� � � �x � �� and the rate of thedisplacement �eld �� reads�

�u�x� t� � ��u�x� t� �HS �� u��x� t� ��

the corresponding compatible rate of strain being�

��x� t� � rrrrrrrrrrrrrrs �u � rrrrrrrrrrrrrrs ��u �HS rrrrrrrrrrrrrrs�� u�� z �

�� bounded�

� �S�� u�� n�s� �z � �� unbounded�

��

where �S is a line Dirac�s delta�function placed in S� Now the �rate of� strain �eld �� can be decom�posed into �� exhibiting at most bounded discontinuities� and the unbounded counterpart �S�� u��n�

s�Thus� by contrast with the weak discontinuity case� the strong discontinuity kinematic state can becharacterized by the appearance of unbounded �rate of� strain �elds along the discontinuity line S�

�� Regularized kinematic state of discontinuity

Finally� we consider a kinematic state de�ned by the following rates of displacement and strain �elds�

�u�x� t� � ��u�x� t� �HS �� u��x� t� ��

��x� t� � rrrrrrrrrrrrrrs ��u�HS rrrrrrrrrrrrrrs�� u�� z �

�� regular�

��S�

h�� u�� n�s� �z � ��

��

where �S is a collocation function placed in S � �S�x� � � �x � S� �S�x� � � otherwise��Comparison of equations �� and �� with equations �� to �� suggests the following remarks�

REMARK �� The kinematic state de�ned by equations �� and �� can be consideredrepresentative of a kinematic state of weak discontinuity of bandwidth h�� see �gure��c� in the following sense�

The velocity �eld �u in equation �� exhibits a jump of value �� u�� across the discontinuityline S� whereas in equation �� the jump appears between both sides �S� and S�� ofthe discontinuity band �h� If the bandwidth h�� is small with respect to the typicalsize of �� the former is representative of the later�

The �� counterpart of the rate of strain �eld �� di�ers from the corresponding onein equation �� in that a step function HS is considered in the later instead of theunit ramp function H�h in the former� On the other hand the term �� in equation�� coincides with the value of �� in equation�� evaluated at the points of S �notethat h�� h�� see equation �� and that �e�� n�� see �gure ��a�� Inboth cases they are representative of the corresponding values in equation �� if thebandwidth h�� is relatively small in comparison to the typical size of ��

REMARK � � When the bandwidth h�� tends to zero the kinematic state de�ned byequations �� and �� approaches a kinematic state of strong discontinuity as can bechecked by comparison with equations �� and �� and realizing that when h�� then �S�h�� S ��

�

REMARK �� The rate of the strain �eld �� is not kinematically compatible with thedisplacement �eld �� in the sense that rrrrrrrrrrrrrrs �u � �� since rrrrrrrrrrrrrrHS � �S � n � ��S�h�� n�Compatibility is only approached when the bandwidth tends to zero as commented above��

In the remainder of this paper we will consider equations �� and �� as the description of a kinematicstate of weak discontinuity which approaches a kinematic state of strong discontinuity when the band�width h tends to zero �� Observe that� now� a kinematic state of weak discontinuity is characterized bya discontinuous �rate of� displacement �eld �� jumping across a material line S and an incompatible�and discontinuous across S� �rate of� strain �eld �� whose amplitude along S is characterized by thebandwidth h�� see �gure ��c��

� The elastoplastic constitutive equations

In the rest of this work we will consider the classical elasto�plastic constitutive equations which can bewritten as�

�� C � �� p��p � m��q � �H�q�

�� q� � �� q � �y

m�� q� ��

��

where �� and ��p are the stress� total strain and plastic strain tensors� respectively� C is the elasticconstitutive tensor �C � � � � � � � I� � and I being� respectively� the rank�two and rank�fourunit tensors and � and � the Lame�s constants�� q is the stress�like internal hardening variable� is the plastic multiplier� is the yield function� �y is the yield stress� H is the hardening�softeningparameter� and m� and m are�respectively� the plastic ow tensor and the normal to the yield surface!q �� f�� q� � �g �m � m� for associative plasticity�� The model is supplemented by theloading�unloading �Kuhn�Tucker� and consistency conditions�

�Kuhn� Tucker� � � �� q� � � �� q� � �

�Consistency� �� q� � � if �� q� � ��

in such a way that the elastic and plastic behaviors are characterized by�

� � � � � � �� C � �� Elastic�

� �

��

� � � � � � � �� C � �� Elastic unloading�

� � � �

� � � � �q � �� C � �� Neutral loading� � � �q � �� CCCCCCCCCCCCCCep � �� P lastic loading�

��

where the tangent elasto�plastic constitutive tensor� CCCCCCCCCCCCCCep� and the plastic multiplier can be computedas�

CCCCCCCCCCCCCCep � C�C �m� �m � C

H �m� � C �m��

�m � C � ��

H �m� � C �m��

�We could have started by de�ning a kinematic strain of weak discontinuity by means of equations �� and �� insteadof equations �� and �� However� the introduction made here can help to identify the incompatible kinematic state ��and �� as representative of the compatible� and consequently more familiar� kinematic state dened in Section �� andgure �a

�

Ω − n

Ω S

Ω +

ν

∂ σΩ Γ Γ=u�

Figure � Boundary value problem�

� The boundary value problem

Let us now consider the boundary of the body �� see �gure � with outward normal �� and let "u � ��and "� � �� "u�"� � �� "u�"� � �� be parts of the boundary subjected to the usual essential andnatural conditions� respectively� With the previously stated concepts in hand we can now formulatethe boundary value problem as follows�

FIND � ��u�x� t� � � � �� IRndim

��u��x� t� � �� IRndim��

such that u�x� t� � �u�x� t� �HS ��u��x� t�� and��

��nS

�x� t� � �� IRnstrs

��nS

�x� t� � �� IRnstrs

��S �x� t� � S � �� IRnstrs

��

where �� is the time interval of interest� ndim and nstrs are� respectively� the dimension of the bodyand the number of relevant stresses of the problem �ndim � and nstrs � for D plane�strain casesand ndim � and nstrs � for plane�stress cases�

SUCH THAT �

rrrrrrrrrrrrrr � ��nS

� f � �

rrrrrrrrrrrrrr � ��nS

� f � �

�equilibrium equation� ��

��nS

� ICICICICICICICICICICICICICIC��nS

� ��

��nS

� ICICICICICICICICICICICICICIC��nS

� ��

��S � ICICICICICICICICICICICICICICS �h��

h�� u�� n�s

i�� constitutive equation� ��

where f are the body forces� ICICICICICICICICICICICICICIC stands for the tangent constitutive tensor �ICICICICICICICICICICICICICIC � C or ICICICICICICICICICICICICICIC � CCCCCCCCCCCCCCep dependingon the loading conditions �� and �� rrrrrrrrrrrrrrs ��u�HS rrrrrrrrrrrrrr

s�� u�� is the regular �bounded� part of the rate ofstrain� subjected to the following�

BOUNDARY CONDITIONS �

�

u � u��x� t��nS

� �� t��x� t�

��nS

� n � ��nS

� n � ��S � n

x � "ux � "�x � S

��

�prescribed displacements��prescribed tractions��traction continuity�

��

where u� and t� are the prescribed boundary displacements and tractions� respectively�It is worth noting that equation �� states the continuity of the traction vector across the dis�

continuity line S� in the sense that it takes the same value not only at both sides of S but also atthe discontinuity line itself� As it will be shown in next sections this last condition provides an addi�tional equation with respect to the regular continuum problem which allows the determination of thedisplacement jump ��u��

� Bifurcation analysis� Onset and propagation of the discontinuity�

We will now focus on the problem of the bifurcation of the stress�strain �elds in the neighborhood ofa given material point P in S� constrained by the rate form of the traction continuity condition ��

n � ��nS

� n � ��S ��

where the material character of S � �n � �� has been considered �� The problem can be stated asfollows� �nd under what conditions the stress�strain �elds� continuous in a neighborhood of P ��

�nS�

��S � ��nS � ��S � bifurcate into discontinuous rate of strain �elds� ��nS

� �� and ��S � �� h�� u�� n�s�

such that �see equation ��nS

� ICICICICICICICICICICICICICIC�nS

� ��

��S � ICICICICICICICICICICICICICICS �h��

h�� u�� n�s

i � ��

subjected to condition �� This problem has been widely analyzed in the context of the failureanalysis of solids �see �Runesson et al�� for a complete analysis� so it will only be sketched here�Substitution of equations � �� into equation �� leads� after some algebraic manipulation� to�

�n � ICICICICICICICICICICICICICICS � n�� z �QQQQQQQQQQQQQQ�n�

�� u�� h n � �ICICICICICICICICICICICICICIC�nS

� ICICICICICICICICICICICICICICS � � ��

where QQQQQQQQQQQQQQ�n� is the localization tensor �Steinmann and Willam� �� On the light of equation � �� wecan now consider di�erent possibilities for the onset of bifurcation�

a� The stress state ��nS

� ��S � is elastic� In this case ICICICICICICICICICICICICICIC�nS

� ICICICICICICICICICICICICICICS � C� according to equations ��and equation � �� readsQQQQQQQQQQQQQQe�n�� u�� whereQQQQQQQQQQQQQQe � n�C�n is the elastic acoustic tensor which isshown to be non singular �det�QQQQQQQQQQQQQQe� � �� Runesson et al�� Therefore� �� u�� and bifurca�tion is precluded since then from equation ��

�nS� ��S � �� and ��

�nS� ��S from equations � ��

b� The stress state ��nS

� ��S � is plastic� Let us consider only bifurcations implying unloading or

plastic neutral loading at �nS and loading at S �� Thus� ICICICICICICICICICICICICICIC�nS

� C and ICICICICICICICICICICICICICICS � CCCCCCCCCCCCCCep from equations�� Now both possibilities �elastic unloading or plastic neutral loading in �nS� should beexplored� However� it can be shown �Runesson et al�� that the second possibility is mostcritical �it is �rstly reached in the context of decreasing values of the hardening parameter��

�No distinction is made here between ��nS

and ��nS

The reasoning following below is independent of the choice�Justication for this assumption will be given in Section �see footnote ��

�

Therefore� only plastic neutral loading in �nS and loading at S will be considered here� For thiscase equation � �� can be rewritten as�

n �CCCCCCCCCCCCCCep � n � �� u�� h n � �C� CCCCCCCCCCCCCCep� � �� h n � C�m��m�C� ��H�m��C�m

� �nS

h n �C �m� � �

where the structure of CCCCCCCCCCCCCCep in equation �� and the value of the plastic multiplier �nS

in equa�tion �� have been considered� Since plastic neutral loading is characterized by a null plasticmultiplier �

�nS� �� equation � � �nally reads�

QQQQQQQQQQQQQQep � �� u��

where QQQQQQQQQQQQQQep � n �CCCCCCCCCCCCCCep � n is the elasto�plastic localization tensor�

Equation � � establishes that� for the discontinuity to be initiated �� u�� the elasto�plastic local�ization tensor has to be singular� i�e��

det �QQQQQQQQQQQQQQep�n�H��

In equation � � the dependence� for a given stress state� of the elasto�plastic localization tensor on thenormal n and the hardening�softening parameter H is emphasized� Now� we can consider the set ofvalues of H for which equation � � has at least one solution for n�

G � fH � IR j � n � IRnndim � jjnjj � � � det�QQQQQQQQQQQQQQep�n�H�� g � ��

If G is not empty we can consider the maximum value in this set as the critical one de�ning thebifurcation �Hcrit � max �H � G�� The corresponding solutions for n in equation � � de�ne thepossible directions of propagation of the discontinuity� ncrit� at point P�

ncrit � fn � IRnndim � jjnjj � � � det�QQQQQQQQQQQQQQep�n�Hcrit�� g � ��

For the considered D plane strain and plane stress problems explicit solutions can be given as follows�Let us consider the local orthonormal base fn� t� �e�g where n and t are the normal and tangentvectors to S �see �gure ��b� and �e� � n � t is the out�of�plane unit vector and let mij and m�

ij��i� j � fn� t� g� be the components of m and m� in this local base� Let us also consider the unitvectors �e� and �e� corresponding to the in�plane principal directions of m and m� � and mi and m�

i

�i � f�� g� m� m�� m�� m�

�� the in�plane principal values� and m� � m�� and m�� m�

�� thecorresponding out�of�plane principal values� Let �nally � be the inclination angle of n with respect to�rst principal direction �e� such that n � cos� �e� � sin� �e�� The corresponding values of Hcrit and�crit are presented in Table ��

REMARK �� The preceding bifurcation analysis provides the conditions for the onset andprogression of the discontinuity� Indeed� considering a discontinuity line S propagatingacross the body �� and a given material point P� the �rst ful�llment at P� for a certaintime of the analysis tP � of the condition H�P� tP � � Hcrit�P� tP � implies that� a� Thesolution of the mechanical problem involves a jump in the rate of the displacement �eld at

� It is implicitly assumed that the plastic �ow vector m� and the tensor normal to the yield surface m have the sameprincipal directions This is clearly true for associative plasticity �m� �m� and also for the most frequently used yieldand potential functions in �D non associative plasticity �Lubliner� ��

�For practical purposes� the values of Table � are computed as follows� �� The angle �crit �which is� in turn�determined from the values sin��crit in the table� can be computed in terms of the principal values of m and m� ��Then� the vector n and� therefore� the local base fn� t� �e�g can be determined �� Finally� the explicit values of Hcrit�in terms of the components of m and m� in such local base� can be calculated

�

Plane strain

sin��crit �m� �m�

�� m�

��m� �m�

�� m�

�� m�

�� m�� m�

��m�

��

� �m��m�� m��m�

��

Hcrit � E��

nm�

tt �mtt � � m�� m�� m�� mtt�

o

Plane stress

sin��crit �m��m��m��m� �m�

��m�

��

� �m��m�� m��m�

��

Hcrit �E m�tt mtt

Table �� Results of the D bifurcation analysis for elasto�plastic constitutive models�

P �since H � G and� thus� �� u��P � � from the bifurcation analysis� and� therefore� the stressand strain �elds bifurcate� b� The discontinuity line S has reached P at that time tP � andthe normal ncrit � n��crit�� provides the direction of progression of S from P towards otherpoints in its neighbourhood� Moreover� since the discontinuity line is assumed a material��xed� line� the obtained value for n�P� tP � � ncrit should be considered frozen beyond tP �c� The bifurcation analysis has no sense at P for subsequent times� since the stress andstrain �elds will not remain continuous anymore��

� Strong discontinuity analysis�

Substitution of equations �� and �� into equation �� allows to write the following evolution equationfor the strains�

�� z �bounded

��Sh

�� u�� n�S� �z �unbounded for h��

� C�� z �bounded

� m��

Let us examine under what conditions equation � �� is consistent with the appearance of a strongdiscontinuity characterized by �� u�� and the limit case h� ��

We observe that the regular part of the strain �� is bounded� by de�nition� and that the rate ofthe stress �� has also to remain bounded to keep its physical signi�cance� Thus� for �� u�� not to vanishwhen the bandwidth h tends to zero the unbounded term �S

h�� u�� n�S has to cancel out with some

other unbounded term in the equation� In other words� the factor �Sh

has to appear in the last term ofequation � �� the simplest choice being ��

� �S

�

h� �

� � � � x � �nS � �

h� � x � S

� ��

which states that elastic loading� unloading or plastic neutral loading � � �� see equation �� occursin �nS whereas plastic loading occurs in S �� We now observe that equation � �� implies a particular

Equation �� has to be fullled strictus sensus only when h � �� that is� at the strong discontinuity regimeHowever� it will be held even in the weak discontinuity regime �h �� explored in Section �

This justies the choice made in Section � �see footnote ��

��

structure of the hardening�softening parameter� substitution into equation �� leads to�

H � ��

�q � h ��

��

�q� �z ��H

� � h �H �x � S � ��

Parameter �H in equation � �� will be referred to as the intrinsic or discrete hardening�softening para�meter and it will be considered a material property�

REMARK �� Equation � �� states the localized character of the plastic ow once thediscontinuity appears� i�e�� once the discontinuity is triggered in a given point of S� plasticstrain rate is only allowed to develop at this point whereas its neighborhood at �nS ex�periences elastic loading or unloading � � ��

REMARK �� Equation � �� shows that as long as the strong discontinuity regime isapproached �h � �� the hardening�softening parameter H tends to zero� Thus the strongdiscontinuity regime is only consistent with the part of the hardening�softening branch withnull slope��

We can now rewrite equation � �� restricted to points of S and considering equations � �� and � ��as�

�� z �bounded

��

h�� u�� n�S � C�� z �

bounded

��

h

��H

�q m�� x � S ��

and we realize that as the strong discontinuity regime is approached �h � �� the unbounded termshave to cancel out each other leading to�

�� u�� n�S � ��H

�q m��S � ��

�� Strong discontinuity condition

Equation �� that will be referred to as the strong discontinuity equation� establishes the evolution ofthe jump in the strong discontinuity regime and can be now specialized for the considered D problems�

� Plane strain

Let us now focus on the D plane�strain problem considering� at any point of S� the orthonormalbase fn� t� e�g de�ned in Section �� In this base the �rate of� the displacement jump can bewritten as �� u�� un�� n� �� ut�� t� where ��un�� and ��ut�� are the normal and tangential componentsof the displacement jump at S� and equation �� reads� in terms of components�

�� un�� ut��

�� ut��

��

��H

�q

�� m�

nn m�nt �

m�nt m�

tt �� m�

��

��S

� �

where m�ab� a� b � fn� t� g are the components of the plastic ow tensor m� in the chosen base�

Equation � � can be regarded as a system of four non trivial equations with two unknowns�� un�� ut�� so that two equations involving only the ow tensor componentsm�

ij can be extracted�They clearly are�

m�ttS

� � � m��S

� � ��

��

� Plane stress

Plane stress cases have to be studied in the projected space obtained by elimination of the out�of�plane components of the stresses and the strains� In this case equation �� reads� in terms ofcomponents� �

�� un�� ut��

�� ut��

��

��H

�q

�m�

nn m�nt

m�nt m�

tt

�S

��

Here the system �� includes three equations with the two unknowns �� un�� and �� ut�� so that thefollowing condition emerges�

m�ttS

� � ��

Equations �� and �� which will be named strong discontinuity conditions� are clearly necessaryconditions for the formation of a strong discontinuity� They are not� in general� ful�lled at the initialstages of the plastic ow and preclude� in most of cases� the formation of an strong discontinuity justat the bifurcation stage�

REMARK �� It is illustrating to realize that substitution of conditions �� and ��into the values of Hcrit in Table � gives� both in the plane strain and plane stress cases�Hcrit � �� This result can be justi�ed as follows� a� Equation � �� holds at any stageof the problem since it comes from equations �� and � �� which hold for all the stagesof the analysis� b� The strong discontinuity regime is characterized by the limit caseh � � which implies that� for �� u�� in equation � �� and loading cases �ICICICICICICICICICICICICICICS � CCCCCCCCCCCCCCep

S��

then deth�n � CCCCCCCCCCCCCCep

S� n�

i� det

hQQQQQQQQQQQQQQ�n�H�

i� �� c� According to equation � �� at the strong

discontinuity regime h � � � H � �� whereby dethQQQQQQQQQQQQQQ�n�H�

i��H��

� �� d� Therefore�

H � � belongs to the set G �see equation � �� of solutions for H of equation � �� which isgiven by the values H � Hcrit� In other words� the solution �� u�� of the strong discontinuityproblem lies in the null space of the perfectly plastic �H � �� localization tensor ��

REMARK �� In particular Hcrit � � is a necessary condition to induce a strong discon�tinuity� If that condition occurs at the bifurcation stage the bifurcation could take placeunder the form of a strong discontinuity� In the general case �Hcrit � �� bifurcation willtake place under the form of a weak discontinuity and the strong discontinuity conditions�� or �� must be induced in subsequent stages� In Section � a procedure to model thetransition from the weak to the strong discontinuity regimes is proposed� �

REMARK �� Bifurcation analysis of plastic models shows that� for the associative case�m �m�� it occurs that Hcrit�� Moreover� for most of the stress states it is strictlyHcrit � � and� according to previous remarks� bifurcation can not take place in the formof a strong discontinuity� On the contrary� for non associative plasticity �m � m�� itoften happens that Hcrit�� which could suggest that� since H � � belongs to theset of admissible values G in equation � �� such value of the stresses is compatible with abifurcation in the strong discontinuity fashion� However� the necessary strong discontinuityconditions �� or �� and the subsequent necessary conditionHcrit�� clearly precludesuch possibility� Actually� this only refers to the bifurcation in a strong discontinuity fashionand not to the possibility of bifurcating under a weak discontinuity form and developing astrong discontinuity in subsequent stages� �

� This result was rstly stated in reference �Simo et al� ��

�

�� Discrete constitutive equation

From equations �� and �� and the consistency condition for loading cases � � �m � �� q � �� thestrong discontinuity equation �� can be written�

�� u�� n�S ��H

hm��S � � ��S �

im��S � ��

which is regarded in conjunction with the traction continuity equation ��

t�nS

� ��nS

� n � ��S � n ��

Equations �� and �� constitute� for any point of S� a system of nine non trivial algebraic equa�tions which states� for the general D case� the implicit dependence of nine unknowns �the six stresscomponents ��S and the three jump components ��u�� on the traction vector t

�nS�

��S � FFFFFFFFFFFFFFht�nS

�t�i

��

��u�� IIIIIIIIIIIIIIht�nS

�t�i

��

REMARK �� Equation �� de�nes a discrete �traction� vs� jump� constitutive equa�tion at the interface S� It is worth noting that it emerges naturally �consistently� fromthe continuum �stress�vs�strain� elasto�plastic constitutive equation described in Section when the strong discontinuity kinematics is enforced� Thus� it is not strictly necessaryneither to derive nor to make e�ective use of such discrete constitutive equation for mod�eling and numerical simulation purposes� In fact� the numerical solution scheme shown inSection � does not include the derivation of such equation and deals only with the standardelasto�plastic constitutive equation of Section as the source constitutive equation��

�� Example I� J� �Von Mises associative plasticity in plane strain

This case is characterized by the following expressions for the yield surface and the plastic ow tensor�

�� q� � �� q � �y � �� q

�� kSSSSSSSSSSSSSSk �

m �m� � ��

q��

SSSSSSSSSSSSSSkSSSSSSSSSSSSSSk

��

where SSSSSSSSSSSSSS and �� stand for the deviatoric stresses and the e�ective stress� respectively� Specialization ofequations � � and �� for this case leads to�

�� un�� ut��

�� ut��

��

��H

�SSSSSSSSSSSSSS � �SSSSSSSSSSSSSS

SSSSSSSSSSSSSS � SSSSSSSSSSSSSS

�S

�� Snn Snt �

Snt Stt �� S��

��S

��

From equation �� it is immediately obtained that S��S � SttS � � and then� due to the deviatoriccharacter of SSSSSSSSSSSSSS �TrfSSSSSSSSSSSSSSg � Snn�Stt�S�� also SnnS � �� Therefore the only non zero component

of SSSSSSSSSSSSSS is Snt �SSSSSSSSSSSSSS � Snt �n� t� �Snt �t�n�� and then �SSSSSSSSSSSSSS � �SSSSSSSSSSSSSS��SSSSSSSSSSSSSS � SSSSSSSSSSSSSS� � �Snt�Snt so that �nally we obtainfrom equation �� the additional relationships �� un�� and �� ut�� H� �Snt

S� Hence� equation ��

is equivalent to the following system��nnS � � � SnnS � ��ttS � � � SttS � �

�ntS � SntS � ��S � � � S��S � �

� �

�� un�� ut��

��H

��

�

where � �the mean stress� and � remain as unknowns� They can be determined by resorting tothe two equations provided by the traction continuity condition �� which for this D case read��nnS � �nn�nS � � and �ntS � �nt�nS � � �

�� Example II� �D Rankine associative plasticity

The yield surface and ow tensor are now�

�� q� � �� q � �ym �m� � �p� � �p�

��

where �� stands for the maximum in�plane principal stress �� and �p� is the associated unitvector in the corresponding principal direction which is inclined the angle � with respect to n ��p� �cos� n � sin� t�� In the base fn� t� e�g equation �� now reads�

�� un�� ut��

�� ut��

��

��H� ��S

�� cos�� sin� cos� �sin� cos� sin��

� � �

��

where the resultm � �� has been considered� From the component �� of equation �� we obtainsin�� so that � � � and n � �p�� Thus� n is the �rst principal direction� then �ntS � � andthe discontinuity line S develops perpendicularly to the �rst principal stress� Since sin � � � from

component �� of that equation we obtain �� ut�� and� �nally� �� un�� S

�Hfrom component ��

Therefore� equation �� can be equivalently rewritten as��nnS � ��S � ��ntS � �

��

�� un��

��H

��

�� ut��

where �� the �rst principal stress� remains as an unknown that can be determined through the tractioncontinuity condition �nnS � �nn�nS � ��

REMARK �� Equations �� and �� with � � �nn�nS and � � �nt�nS are specializa�tions of the general form �� for the considered J �plane strain� and Rankine plasticityproblems� Observe that the discrete constitutive equation �� states that only the tan�gent component of the jump ��ut�� can develop ��un�� so that with this type of J plasticity equations the generated strong discontinuity is a slip line �this result was alsofound in �Simo et al�� Oliver� ��a� Armero and Garikipati� �� On the contraryequation �� states that with Rankine�type plasticity models only Mode I �in terms of Frac�ture Mechanics� strong discontinuities can be modeled since the tangent component of thejump ��ut�� Obtaining such explicit forms of the discrete constitutive equations is not sostraight�forward for other families of elastoplastic models� This makes specially relevant amethodology to approach strong discontinuities that does not require the explicit statementof such equations as pointed out in REMARK ��

A variable bandwidth model

The bifurcation and strong discontinuity analyzes performed in Sections � and � above� provide signi��cant information about the mechanism to induce strong discontinuities� This can be summarized asfollows�

�

Strong discontinuityWeak

discontinuityContinuity

hB

(elasto-plastic, h ≠ 0)

SD

B

(plastic, h = 0)Strong discontinuity

(elastic)

YhContinuity(plastic)SD

qy−=Φ σσ )(ˆ

η

SO

Ω

c)

b)

HHh yB =

Hh

Hk

Hk

σy

)(qh

α

αB

)(σcritH

Hy

B

Y

critHH ,,Φ̂

S\Ω

S

HqhqH )(ˆ

)( =∂∂−=

∂Φ∂=

αααB

SD

k

Hy

SD

B

)(ˆ ασ qy−=Φ

Y

q

h

Bq SD

q

Bh

k

a)

⎩⎨⎧

≥≤

=B

By

qqforHqh

qqforHqH

)()(

B

SD

yσ

O

)( Bqy −σβ

Weak discontinuity

Figure � Variable bandwidth model� a� Bandwidth law b� Propagation of the discontinuity c� Harden�ing�Softening law and bandwidth evolution�

��

� Bifurcation of the stress�strain �elds is a necessary condition for the inception of a discontinuityin the displacement �eld� In the context of a variable hardening �or softening� law that bifurcationwill take place� for a given material point� when the condition H�� Hcrit�� is ful�lled for the�rst time� In general Hcrit�� will be non zero �REMARK ��

� Bifurcation will not� in general� produce the strong discontinuity� The necessary condition �toinduce a strong discontinuity�Hcrit�� will not� in general� be ful�lled at the bifurcation stage�REMARKS �� and �� and bifurcation will take place under the form of a weak discontinuity�

� If the hardening�softening parameterH is expressed in terms of the intrinsic hardening�softeningparameter �H �considered a material property� and the bandwidth h� according to equation � ��i�e�� H � h �H� then the bandwidth characterizing the weak discontinuity at the bifurcation willbe given by hcrit � �Hcrit� �H� � ��

Therefore� if the aim of the model is to capture strong discontinuities an additional ingredienthas to be introduced which provides� a� the transition of the bandwidth from the value hcrit � ��at the bifurcation� to the value h � �� in a subsequent time and b� the ful�llment of the strongdiscontinuity conditions �� or �� In �gure what has been termed variable bandwidth model�Oliver et al�� Oliver et al�� Oliver� �� is sketched� It can be described in the followingsteps�

�� Bandwidth law� A certain variation of the bandwidth h� in terms of the stress�like variableq � �� y�� is postulated �see Figure a� �� The bandwidth varies from h � hB� for a certainvalue q

Bof the stress�like variable� which is attained at the bifurcation point B� to drop to h � �

for another value �known and considered a model property� �� qSD

� �qB� �y�� attained at the

strong discontinuity point SD� In fact� for computational purposes the minimum value of h islimited to a very small regularization parameter k instead of zero �see Section � for more details��The value hB is computed when the bifurcation is detected from the value of Hcrit in Table � ashB � Hcrit� �H�

� Hardening�softening parameters� The model is considered ruled by two distinct hardening�softeningH�q� parameters which relate the stress�like internal variable q and the hardening function� � �y � q to the strain�like internal variable � through�

��q��

��

� ��

�� H�q� ��

Before the bifurcation �q � qB� the standard continuum hardening�softening law is char�

acterized by the continuum hardening�softening parameter Hy�� which is considered a

material property �see Figure a��

After the bifurcation �q � �qB� �y�� the discrete hardening�softening parameter �H rules

the cohesive�decohesive behaviour at the discontinuous interface� However� equation � ��and the previously described bandwidth law provide the evolution of the correspondingcontinuum hardening�softening parameter as H�q� � h�q� �H� From this and from equation�� the corresponding continuum hardening�softening law q � � can be readily obtainedfrom integration of�

�q

�� h�q� �H ��

�� In the gure the h� q law has been plotted being linear However other possibilities for the h�q� curve �parabolic�exponential etc� could have been alternatively considered

��More precisely� for softening models what is considered a model property is the relative position of qSD

in theinterval �q

B� �y�� which is characterized by the value � � �� such that q

SD� q

B� ��y � q

B�

��For the sake of simplicity in gure � this parameter is considered constant and negative �strain�softening� althoughmore sophisticated non linear hardening or softening laws could have been considered

��

In Figure b the corresponding h�q�� q � H� � and �� curves are sketched� Observe that�according to equation �� the curve H�� supplies the slope of the hardening�softening curve��

� Characteristic points� Continuous and Discontinuous regimes� In Figure b also a typical evol�ution of the values of Hcrit� obtained from Table �� along the analysis is plotted� For a givenmaterial point yielding begins at point Y of Figure b� in which the hardening�softening para�meter takes the value Hy� While Hcrit � Hy bifurcation is precluded and the behaviour iscontinuous� As soon as Hcrit � Hy the bifurcation point B is detected� the corresponding valuesof n��crit� are computed from Table � which� once introduced in the rest of the model� warrantthat bifurcation at point B takes place under the appropriate loading �at S� and unloading �at�nS� conditions �see Figure b�� Also at this point the value hB � Hcrit� �H� which states theinitial value of the bandwidth law of Figure a� is computed� Since in general hB � �� pointB corresponds to the onset of a weak discontinuity whose bandwidth is enforced to decrease bythe bandwidth law of Figure a beyond this point� As soon as the value q � q

SDis attained

at point SD and� according to the bandwidth law� h � k � � the strong discontinuity regimeis reached and the strong discontinuity conditions �� or �� are naturally induced� Finally�beyond point SD the strong discontinuity regime develops keeping the bandwidth h and thecontinuum hardening�softening parameter H in a null �k�regularized� value�

REMARK �� Since consistency with the results obtained from the bifurcation and strongdiscontinuity analyzes is kept along the process the obtained results warrant that a� Bi�furcation takes place under the appropriate loading�unloading conditions� thus not leadingto a two materials approach �Oliver et al�� and b� The rate of the stresses remainbounded along the whole process keeping their physical signi�cance� �

Translation of this variable bandwidth scheme in terms of the status of the material points of thebody is �nally sketched in Figure c� where a discontinuity line S that advances across the body � isrepresented� At a given time of the analysis most of the material points of the body � are in elasticstate� Material points that are in plastic state de�ne what in Non�linear Fracture Mechanics has beentermed the Fracture Process Zone �Bazant and Oh� �� Those points that lie in the Y �B branchof the curve in Figure b� de�ne a continuous plastic �hardening or softening� zone� Points in theB� SD branch of the curve de�ne the weak discontinuity part of the discontinuity line S to which azone� whose bandwidth is de�ned by the corresponding bandwidth law h�q�� is associated in Figurec� Finally� material points remaining in the branch beyond point SD� in Figure b� de�ne the strongdiscontinuity part of S� In particular� point O� in Figure c� states the end of that segment of S whosematerial points have completely released stresses�

Expended power� Fracture energy�

Let us now deal with the external mechanical power supplied to the body � of �gure along thedeformation process� Neglecting the kinetic energy� and taking into account the existence of a displace�ment jump across the discontinuity line S� the externally supplied mechanical power can be written as

��

the sum of the contributions in �� and �� nS��

Pext �R�nS f � �u d��

R�u��

�� u d"

�R�nS ��nS

� �� d� �RS n � ��

��nS

� �u��nS

d" �RS n � ��

��nS

� �u��nS

d"

�R�nS ��nS

� �� d� �RS n � ��S � � �u

��nS

� �u��nS

�� z � �u

d"

�

Z�nS

��nS

� �� d�� z �Pint�nS

�

ZS��S � �� u�� n�s d"� �z �

PintS

��

where the traction vector continuity condition �n � ��nS

� n � ��nS

� n � ��S � has been considered� We

observe in equation �� that Pint�nS

and PintS

are volumetric and surface counterparts of the supplied

external power� respectively� Thus� we can understand PintS

as the part of the external power internallyspent in the formation of the jump �� u�� at the discontinuity interface S� Therefore� taking into accountequation �� we can write Pint

S� after some algebraic manipulation� as�

PintS

�

ZSh ��S � �C�� S � �� d"� �z �

�O�h�

�ZS

��H

�q ��S �m� d" ��

We now observe that the �rst integral of the right�hand�side of equation �� is bounded and tends tozero with the bandwidth h� Thus� if the bandwidth is small with respect to the representative size of �it can be neglected� Let us now specialize the problem to the cases ful�lling the following conditions�

a� The function �� in equation �� is an homogeneous function �of degree one� of the stresses�� In this case� in virtue of Euler�s theorem for homogeneous functions� it can be written�

��

b� Associative plasticity �m� �m � ��

c� Strain�softening �which implies that q remains in the bounded interval �� y��

We also observe that� for loading processes � � �� equation �� implies that � �� and� thus��y � q � � � �� m � �� see equations �� and �� So that� �nally� equation �� canbe written as�

PintS

�RS ��S � �� u�� n�s d"

� �RS

��H�q ��S �m d" � �

RS

��H�q ��y � q� d" �

�RS

��t

h �

�H��y � q��

i� �z �

��q�

d" �RS

��t��q� d"

��

Let us now compute the energy WSspent at S along any loading process leading to the formation

of a strong discontinuity� The complete loading process can be characterized by the evolution of thestress�like variable q ranging from q � � at the unloaded initial state �t � �� to q � �y at the �nalstate �t � t�� where the stresses are completely released�

WS �

Z t�

�PintS

dt �

Z t�

�

h ZS

�

�t��q� d"

idt �

ZS

h Z t�

�

�

�t��q� dt

i� �z �

Gf

d" ��

�� This is a requirement fullled by many usual yield functions �Von�Mises� Tresca� Mohr�Coulomb� Drucker Prager�Rankine� etc �Khan� ��

��

The kernel of the last integral of equation �� can be now identi�ed as the energy spent� per unit ofsurface� in the formation of the strong discontinuity which� in the context of the non�linear fracturemechanics� is referred to as the fracture energy Gf � In view of equations �� and �� it can bewritten�

Gf �

Z t�

�

�

�t��q� dt �

Z q��y

q��

�

�q��q� dq � ��y��

�

��y�H

��

so that� �nally� equation �� can be solved for the intrinsic hardening�softening parameter� �H� interms of the material properties �y and Gf as�

�H � ��

��yGf

��

REMARK �� Results �� and �� have been obtained for an arbitrary loading process�The material property character of the resulting fracture energy� lies crucially onto this factsince the value of Gf in �� is independent of the loading process� This result� in turn�comes out directly from equation �� namely� ��S � �� u��n�s is an exact time di�erential��S � �� u�� n�s � �

�t��q�� Notice that this is not a completely general result since it has

been obtained under the conditions a�� b� and c� above��

REMARK �� The existence of the fracture energy as a bounded and positive materialproperty is then restricted to associative plasticity models with strain softening accordingto conditions b� and c�� In fact� there is no intrinsic restriction for non�associative strain�hardening constitutive equations to induce strong discontinuities� In that case the intrinsichardening�softening parameter �H would have to be positive according to the condition�H � �H�h� �� However� this scenario does not ensure neither the existence of thefracture energy� as a material property independent of the loading process� nor a boundedvalue for the energy W

Sin equation �� since in that case q � �� On the other

hand� the positiveness of �H would lead to a cohesive �instead of decohesive� character ofthe resulting discrete constitutive equation at the interface� �

� Finite element simulation� Computational aspects

The ingredients of the approach presented above can now be considered for the numerical simulation ofstrong discontinuities� via �nite elements� It was pointed out in REMARK �� that the discrete �stress�jump� constitutive equation �� obtained from the strong discontinuity analysis is not in fact used fornumerical simulation purposes but� on the contrary� it emerges naturally from the continuum stress�strain constitutive equation when the strong discontinuity kinematics is enforced� In consequence�a standard �nite element code for D elasto�plastic analysis only needs some few modi�cations toimplement the present model� Essentially these are�

� Standard C� �nite elements have to be modi�ed in order to make them able to capture jumps inthe displacement �eld� In references �Oliver� ��b� Oliver� ��b� details about a family of suchelements� which has proved very e#cient� can be found� They are based in an enhancement ofthe strain �eld of the standard underlying element by adding a discontinuous incompatible modefor the displacements� Also an extra�integration point is considered where the speci�c kinematicand constitutive properties of the interface S are modeled �see �gure ��

� The standard elasto�plastic constitutive model has to be slightly modi�ed to include the harden�ing�softening law � ��

��

ie

1

je

ke

Se

Sh-

S h+

MS

h

k e

ie

je

1/h

IntegrationPoint # 1 Integration

Point # 2

a) b)

e

Figure � Finite element with embedded discontinuity� a� Discontinuous shape function� b� Discon�tinuous strain �eld and additional sampling point�

� Computation of the bifurcation condition H � Hcrit and the corresponding direction of propaga�tion of the discontinuity has to be included� For D cases results in Table � can be used� Alsothe bifurcation bandwidth of equation �� and the bandwidth evolution of equation �� have tobe computed according to the values Hcrit in Table ��

� In a strain driven algorithm� equation �� has to be numerically integrated to obtain the strain�eld at any given time of the analysis� In fact the rate of the strain �eld at S�

��S � ��

h�q�� u�� n�s ��

can not be analytically integrated due to the appearance of h�q�� u�� which is given in equation�� In the examples shown below the following mid�point rule �second order accuracy� has beenused�

ht� t�

� ��

hh��t��t� ��u��t��t�� z �

ht� t

�h��t� ��u��t�� z �ht

i

��St� t � ��St �$�� ht� t

�

�$ ��u�� n�s ��

where subscripts ��t��t and ��t refer to evaluation at the end of two consecutive time steps and$�� t��t � ��t are the corresponding increments�

� In order to avoid ill�conditioning in equation �� when h � �� the evolution of h given byequations �� is limited to h � �hcrit� k� where k � is a very small regularization para�meter� Typically� k is taken about �� times the size of the �nite element� In references�Oliver� ��a� Oliver� ��b� Oliver� ��b� the objectivity �independence� of the results withrespect to such regularization parameter is shown� provided it is small with respect to the typical�nite element size�

�� A rst illustrative example� uniaxial tension test�

A very simple� but illustrative� example is now examined in order to assess the capacity of the approachto induce strong discontinuities and to reproduce the theoretical predictions of the strong discontinuity

�

δ

l

a

θn

xy

F/a

15.0ˆ

0.5

67.16

0.120

0.2

=

−=

−=

=

=

y

y

y

y

y

q

H

lH

E

a

l

σ

σ

σ

σ

a) b)

c) d)

Figure �� Uniaxial tension test and J plasticity �plane strain��

analysis �essentially� the discrete constitutive equation at the interface�� A J �Von�Mises� model ofassociative plasticity is taken as target constitutive equation and the results are checked via an uniaxialtension test under plane strain conditions� In �gure ��a the loading and geometrical features of theproblem are presented� A linear bandwith law with � � �� has been taken� Since the stress �eld isuniform� the discontinuity must be seeded somewhere� therefore� the lower left corner element of theunstructured �nite element mesh of quadrilateral elements of �gure ��a is chosen for this purpose� In�gure ��b the deformed shape at the �nal stage of the analysis is shown� It can be checked there thatthe deformation corresponds to an almost rigid body motion of the upper part of the specimen slippingalong a straight slip�line� which starts at the aforementioned element and crosses the band of elementshighlighted in �gure ��c� In this �gure the contours of the total displacements group in the patch ofelements that capture the discontinuity �� stating the sharp resolution of the jump� In �gure ��d theslip�line deformation mode is emphasized by displaying the displacement vectors of the nodes of the�nite element mesh�

In �gure � the evolution of di�erent variables of the problem is shown in a non�dimensional fashion�Figures ��a� ��b� ��c ��d and ��e are obtained using a Poisson ratio � � � whereas �gure ��f correspondsto di�erent values of � �� and � � ��

Figure ��a shows the bandwidth evolution� h� at a certain element of the discontinuity path� interms of the total displacement � of �gure ��a� The relevant part of the curve is the one going fromh � hB � at the bifurcation point B� to h � k � �� l at the inception of the strong discontinuity� thestrong discontinuity point SD�

Figure ��b shows the evolution of the vertical component of the stress at the interface� �yyS� and

outside the interface �yy�nS

� Notice that they di�er beyond the bifurcation point B� Also the evolution

of the out of plane deviatoric stress S��S and the normal deviatoric stress SnnS is shown in that �gure�

��For post�processing purposes only displacements of the regular underlying elements are displayed Displacementscorresponding to the elemental discontinuous incompatible modes referred to in Section � are not displayed

�

0 .00 0 .02 0 .04 0 .06δ/l

0 .00

0 .01

0 .02

0 .03

εyy

B

Ω\S

S

SD

Y

0 .00 0 .02 0 .04 0 .06δ/l

-0 .40

0 .00

0 .40

0 .80

1 .20

s tress

σy

Y B SD

S 33S

S nnS

σyyS

σyyΩ\S

0 .00 0 .02 0 .04 0 .06δ/l

-0 .02

-0 .01

0 .00

0 .01

[|u |]l

[|u |]t

[|u |]n

SD

SDBY

0.005 0.010 0.015 0.020 0.025δ/l

-35

-30

-25

-20

-15

-10

-5

0

H σy

H c rit/σyH /σy

Y

Y

BSD

0 .00 0 .02 0 .04 0 .06δ/l

0 .00

0 .10

0 .20

0 .30

0 .40

0 .50

0 .60

0 .70

F l σy

YB SD

Y≡ B≡ SD υ = 0.5υ = 0.0

0.00 0 .02 0.04 0.06δ/ l

0 .00

0 .15

0 .30

h l

Y

B

SD

h B

a) b)

c) d)

f)e)

Figure �� Uniaxial tension test and J plasticity �plane strain�� Evolution of some variables

l

a

F/a

xy

δ

n

0.0ˆ

0.0

67.16

0.120

0.2

=

=

−=

=

=

y

y

y

y

y

q

H

lH

E

a

l

σ

σ

σ

σ

a) b)

c) d)

Figure �� Uniaxial tension test and Rankine plasticity�

0.00 0 .02 0.04 0.06[|u |]n/l

0 .00

0 .40

0.80

1 .20

σnnS

σy

Y≡ B≡ SD

H l/σy

1

0 .00 0 .02 0 .04 0 .06δ/l

-0 .01

0 .00

0 .01

0.02

[|u |]l

[|u |]n

[|u |] t Y≡B≡SD

a) b)

Figure �� Uniaxial tension test and Rankine plasticity� Evolution of some variables

It is worth noting that the strong discontinuity conditions S�� and Snn � � coming out from theanalysis in section �� are not ful�lled at B but� however� they are naturally induced at the onsetof the strong discontinuity regime SD�

Components of the vertical strain �yySand �yy

�nSare plotted in �gure ��c� Observe that whereas

the strain at the interface �yySgrows continuously as corresponds to a plastic loading process� the

contrary occurs in the rest of the body �nS � and the regular strain �yy�nS

� ��yy decreases elastically�

beyond the bifurcation point B� The remaining strain �yy�nS

at the end of the analysis corresponds to

the plastic strain generated at the continuous plastic�softening regime �between points Y and B in the�gure��

In �gure ��d evolutions of the normal� ��un�� and tangential� ��ut�� components of the jump areplotted� Observe that there is a slight initial evolution of the normal jump � �� un�� during theweak discontinuity regime� path B� SD in the �gure� but beyond point SD the evolution stops as itis predicted by the strong discontinuity analysis �see equation �� stating the slip�line character ofthe induced strong discontinuity�

Figure ��e shows the evolution of the computed critical softening parameter Hcrit� in accordance toTable �� and the one of the continuum softening parameter H emerging from the the values of Hy andthe imposed bandwidth law� Both curves intersect at the bifurcation point B where the bifurcationcondition H � Hcrit is accomplished� Beyond this point the evolution of h determines the evolution ofthe continuum softening parameter H according to H � h �H� Both curves eventually tend to zero atpoint SD as it is predicted by the theoretical analysis�

Finally� in �gure ��f the load�displacement curves� F � �� are presented for the two limit values ofthe Poisson�ratio �� and � � �� Observe that the curves are di�erent from each other� sincefor the very particular case � � �� yielding� bifurcation and the onset of the strong discontinuity takeplace simultaneously �points Y� B and SD coincide�� and the curve has a straight descending branch�On the contrary� for � � �� the pathsY�B and B�SD� corresponding to the continuous plastic softeningand the weak discontinuity regimes� respectively� are curved and only beyond point SD the descendingbranch is straight� This agrees with the linear character of the discrete constitutive equation �� thatrules the jump at the interface beyond this point�

Now we consider the same specimen but using a Rankine�type plasticity model as it was describedin Section �� Since the principal stress �� is vertical� the expected strong discontinuity is anhorizontal line for any values of the material properties as it is indicated in �gure ��a� This expectedresult comes out also from the numerical simulation� in �gure ��b the deformed �nite element meshcorresponds to a typical mode I split of the body through an horizontal line passing across the elementthat was initially seeded� The set of elements that capture the discontinuity is shown in �gure ��cby the contours of equal total displacement which dark the path crossed by the discontinuity line� In�gure ��d the mode I discontinuity�type is emphasized by the nodal displacement vectors�

Figure ��a shows the normal stress vs� normal displacement�jump at the discontinuity line� namely�the discrete constitutive equation �� In accordance with the theoretical predictions it is a straightline whose slope is characterized by the inverse of the discrete softening parameter �H� Notice that theyielding point Y� the bifurcation point B� and the strong discontinuity point SD are the same since� forthis type of plasticity model� the strong discontinuity conditions �� or �� are automatically ful�lledat any point of the softening branch as can be checked in Section �� equation �� Therefore�from Table �� Hcrit � � and hB � maxf�Hcrit� �H�� kg � k �� and the three characteristic points Y�B and SD coincide with each other� In �gure ��b the evolution of both components of the jump interms of the imposed displacement � is presented� Observe that the tangential component of the jump��ut�� according with equation ��

�� The value of parameter � in Figure �a does not play here any role� since the bandwidth law is constant �h � k �q �qB�

�� Additional numerical simulations�

The numerical simulations presented in this Section correspond to the classical geomechanical problemof an undrained soil layer subjected to central or eccentric loading exerted by a rigid and roughsurface footing� The same problem was considered in reference �Zienckiewicz et al�� where itwas analyzed using an adaptive remeshing strategy to capture the formation of slip lines under perfectplasticity conditions� Here the problem is solved under plane strain conditions and using a J plasticitymodel in the context of the strong discontinuity approach� The bandwidth law is taken linear and suchthat � � �� Geometry and results for the two cases analyzed are shown in Figures � and �� The �niteelement used in the discretizations is a ��noded quadratic triangle supplemented with the incompatibledisplacement referenced to in Section �� Figure � corresponds to the central loading case� Figure�� shows the deformed shape of the �nite element mesh at the �nal stage� In Figure �� the totaldisplacement contours show the existence of two slips lines that initiate at the bottom corners of thefooting and cross each other at a certain point of the symmetry axis� Figure �� shows the displacementvector �eld� From these it is clear that a triangular wedge of soil beneath the footing moves solidarilywith this� vertically downward� This induces the upward movement of two lateral wedges that slide withrespect the rest of the soil layer� which remains almost undeformed� The attained solution resemblesvery closely the classical result obtained using Slip Line Theory �Chen� ��

Figure �� corresponds to the eccentric loading case� the rest of the geometry and properties beingthe same as previously� Figures �� and �� show the deformed shape of the �nite element mesh�the total displacement contours and the displacement vector �eld� respectively� at the �nal stage� Thedi�erence with the previous case is obvious� Now� only one strong discontinuity line develops� witha wedge of soil moving side and upward attached to the footing� and sliding with respect to the restof the layer� The peak load corresponding to the eccentric case is around �% lower than the oneobtained for the symmetric one�

�� Concluding remarks�

Throughout this paper the here called strong discontinuity approach to displacement discontinuitiesinduced by continuum stress�strain elastoplastic constitutive equations has been presented� The mainfeatures of the approach may be summarized as follows�

� A kinematic state of strong discontinuity� characterized by a discontinuous displacement �eldacross a material discontinuity line� and the corresponding �compatible� unbounded strain �eld�is considered as the limit case of a regularized kinematic state of weak discontinuity charac�terized by discontinuous� but bounded� strains� These strains intensify across the discontinu�ity line proportionally to the inverse of the so called bandwidth of the weak discontinuity� insuch a way that when the bandwidth tends to zero the strong discontinuity kinematic stateis recovered� In turn� such a regularized kinematic state of a weak discontinuity can be con�sidered representative of a compatible kinematic state of weak discontinuity with continuousdisplacements and discontinuous strains that intensify �or localize� at a band of the same band�width� This provides a �rst link to the strain localization�type approaches �Ortiz et al�� Needleman� �� de Borst et al�� Zienckiewicz et al�� which essentially deal with thistype of kinematics� When the bandwidth of the localization band tends to zero the strain�localization state turns to be a strong discontinuity�

� The inception of the discontinuity is characterized through the bifurcation analysis which providesthe conditions for the initiation and propagation of such discontinuity� Since the bifurcationanalysis lies on the singularity of the localization tensor it provides a second link to the failureanalysis methods aiming at characterizing the material instabilities in terms of such localization

�

1 2

3 4

F

4 m

2 m

0.8 m

δRigid androughfooting

mKPaG

MPa

GPaE

f

y

.0.2

45.0

0.1

0.1

==

==

νσ

y

y

q

MPaH

σ5.0ˆ

0.1

=−=

Figure �� Numerical simulation of a foundation collapse� Central loading case

�

1 2

3 4

F

4 m

2 m

0.8 m

δRigid androughfooting

mKPaG

MPa

GPaE

f

y

.0.2

45.0

0.1

0.1

==

==

νσ

y

y

q

MPaH

σ5.0ˆ

0.1

=−=

0.08 m

Figure �� Numerical simulation of a foundation collapse� Excentrical loading case

�

tensor �Runesson et al�� Stein et al�� It is shown that� in general� such a bifurcationcan only appear under the form of a weak discontinuity �non�zero bandwidth� and� if strongdiscontinuities are to be modeled� an additional ingredient is required�

� The variable bandwidth model is then a mechanism devised to induce the strong discontinuityregime from the weak discontinuity one� A bandwidth evolution law� ranging from an initialnon�zero value to zero �k�regularized�� is postulated as a model property in terms of somestress related variable �here the stress�like variable�� Beyond the bifurcation� the continuumhardening�softening parameter is determined as the product of the discrete hardening�softeningparameter times that bandwidth� in such a way that a smooth and consistent transition from theweak discontinuity regime to the �nal strong discontinuity one is obtained�

� The strong discontinuity analysis also provides a very important additional insight on the prob�lem� it is shown that the strong discontinuity kinematics induces from any standard stress�strainconstitutive equation a discrete �traction�vector vs� displacement�jump� constitutive equation atthe interface which is ful�lled once the strong discontinuity regime is reached� This provides anadditional link of the approach to the classical non�linear fracture mechanics and the discrete con�stitutive equation can be then regarded as one of the typical stress�jump constitutive equationsused in fracture mechanics to rule the decohesive behaviour at the interface �Hillerborg� ��The discrete hardening�softening parameter is shown to play an important role in this equationsand it can be readily related� in certain cases� to the fracture energy concept�

� A key point of the approach� as it has been presented here� is that the aforementioned discreteconstitutive equations are neither derived nor used for practical purposes� but they emergenaturally and consistently from the continuum constitutive equation and the entire simulationcan be kept in a standard continuum environment� Therefore� the same continuum constitutiveequation rules the continuous and discontinuous regimes of the problem� Although some of suchdiscrete constitutive equations have been derived in the paper as a matter of example� this doesnot seem in general an easy task for any continuum constitutive equation� Since the proposedapproach does not need such derivation it is not restricted at all by that fact� Some representativenumerical simulations presented in the paper show that the predicted discrete constitutive lawsat the interface are in fact reproduced by the continuum approach�

� The point of stability and uniqueness of the approach has not been addressed here� As far asuniqueness is concerned� in references �Oliver et al�� Oliver� �� and for a simple �D case�the bene�ts of using the strong discontinuity approach in front of the classical strain localizationone were shown and the uniqueness of the solution supplied by the former was proved�

�� Acknowledgment

The third author wishes to acknowledge the �nancial support from the Brazilian Council for Scienti�cand Technological Development�CNPq�

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�

Strong Discontinuities and Continuum Plasticity Models

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