A framework for geometrically nonlinear continuum damage mechanics

A framework for geometrically nonlinear continuumdamage mechanics

Paul Steinmanna, *, Ignacio Carol b

aInstitut fuÈr Baumechanik und Numerische Mechanik, UniversitaÈt Hannover, Appelstr. 9a, 30167 Hannover, GermanybDepartament D'Enginyeria del Terreny, ETSECCPB, Universitat PoliteÁcnica de Catalunya, E-08034 Barcelona, Spain

Received 5 February 1997; received in revised form 30 June 1997; accepted 28 August 1997

Abstract

The objective of this work is the derivation of a framework for geometrically nonlinear continuumdamage mechanics which allows for the description of damage by second order tensors. To this end, socalled ®ctitious undamaged or rather microscopic con®gurations are considered which are related to themacroscopic con®gurations by linear tangent maps which allow for the interpretation as damagedeformation gradients. The corresponding Finger tensors de®ned in the macroscopic con®guration arethen understood as damaged metrics for measuring the free energy for given strains. Thereby, theunderlying motivation is provided by the hypothesis of strain energy equivalence between microscopicand macroscopic con®gurations. Based on the standard dissipative material approach the constitutiveframework is completed by stresses, damage stresses, a damage condition and the associated evolutionlaws for the damage metric and the internal hardening variable. The relationship of the present damagemetric based theory to the classical understanding of damage as an area reduction is highlighted.Finally, a simple prototype model problem is presented which allows for an e�cient and accuratealgorithmic treatment. Numerical examples demonstrate the applicability of the proposed framework.# 1998 Elsevier Science Ltd. All rights reserved.

1. Introduction

It is a desirable feature of a continuum damage theory to provide su�cient freedom to

capture the anisotropic nature of damage, thus the state of damage has at least to be described

by a second order tensor. Therefore this contribution aims at the development of a

geometrically nonlinear formulation of tensorial second order continuum damage. Thereby,

International Journal of Engineering Science 36 (1998) 1793±1814

0020-7225/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved.PII: S0020-7225(97)00116-X

PERGAMON

* Corresponding author. Present address: Lehrstuhl fuÈ r Technische Mechanik, UniversitaÈ t Kaiserslautern, D-67653Kaiserslautern, Germany. Tel.: +49-631-2052421; fax: +49-631-2052128; e-mail: [email protected].

even if the nominal strains are considered as small in a particular application, the full blownup apparatus of the geometrically nonlinear theory is justi®ed since the amount of damagemeasured in an appropriate norm is ®nite.In order to de®ne tensorial second order damage Betten [1] and Murakami [29, 30]

introduced the notion of ®ctitious undamaged con®gurations. Following these approaches, thewell known concept of deformed and reference, or rather undeformed, macroscopiccon®gurations of a material body within the geometrically nonlinear continuum theory issupplemented by the concept of ®ctitious undamaged microscopic con®gurations which areattained by unloading with appropriate damage deformation gradients.Classically, damage is described by the assumption of a reduced e�ective area within a cross

section of the damaged material. Nevertheless, in strong contrast to the classical approachesmentioned above, the present damage theory is based on the notion of a second order damagemetric and its e�ects on the stored strain energy. Thereby, as the fundamental assumption, thestorage of strain energy due to either nominal or e�ective strains is measured by either thedamage or the energy metric based on the hypothesis of strain energy equivalence betweenmicroscopic and macroscopic con®gurations.Moreover, in contrast to previous approaches, the formulation of damage evolution is

consequently based on the framework of standard dissipative materials, a concept proposed byHalphen and Nguyen [15], by introducing a dissipative surface and exploiting the postulate ofmaximum dissipation. Here the formalism of standard dissipative materials ®rst introduces thedistinction between equilibrium and damage stresses in the macroscopic and microscopiccon®gurations. Moreover, the structure of the dissipation inequality suggests the introductionof a dissipation surface which de®nes an admissible elastic domain in the space of macroscopicdamage stresses. Likewise, the postulate of maximum dissipation renders an associatedevolution rule for the damage metric. Subsequently, an attempt to relate all those concepts tothe classical interpretation of damage is given.Finally, as an illustration of the developed methodology, the proposed theory is elucidated

for the simplest subcase of brittle isotropic materials. To this end a St. Venant-typehyperelastic model is equipped with the simplest choice for the damage surface which turns outto measure the stored energy of the material. Then two di�erent laws for the thermodynamicalforce characterizing the hardening of the material are proposed. Based on these choices theresulting exponential-type algorithmic setting of the stress update takes a remarkable simpleformat which completely circumvents a local iteration procedure. The di�erent responsepredictions are then examined for the example of a homogeneous deformation state of uniaxialextension.

1.1. A brief account on previous approaches

The appropriate choice of the physical nature of mechanical variables describing the damagestate of a material and their tensorial representation is since long under discussion. In the earlyworks by, for example, Kachanov [18], Rabotnov [33] and Hayhurst and Leckie [16], scalarvariables were used to characterize the reduction of net area fraction in uniaxial stress±strainrelationships. The simplest and still most widely used extension of this concept to multiaxial

P. Steinmann, I. Carol / International Journal of Engineering Science 36 (1998) 1793±18141794

conditions is a single scalar damage coe�cient which reduces in the same proportion the netarea fraction in all directions of space, see Refs. [22, 24, 26, 34].As a ®rst attempt to describe the anisotropic nature of damage, vectorial damage variables

have been considered by various authors, see Refs. [11, 13, 17, 20, 21, 37]. However, vectorsexhibit conceptual di�culties to be used as a general representation of damage. Besides purelytheoretical arguments, e.g. Ref. [23], it is immediate to verify that isotropic damage cannot beobtained as a particular case of the vectorial representation, which seems only adequate for afamily of microcracks all with the same orientation.The characterization of anisotropic damage by second order tensors was advocated by

Vakulenko and Kachanov [38], Dragon and Mroz [14], Kachanov [19], Murakami andOhno [28], Cordebois and Sidoro� [10], Betten [2], Oda [31], Cowin [12], Murakami [29, 30]and Chow and Wang [9] among others.Nevertheless, other more complicated approaches can also be found in the literature using

fourth or even eight order damage tensors (see, for example, Refs. [8, 32, 36]). A systematicdiscussion and comparison of damage tensors of the various orders can be found in Ref. [25].A general framework to formulate elastic degradation and damage in small strain regardless ofthe speci®c nature of the damage variables was proposed by Carol et al. [5], and other aspectsof these models such as localization properties and recovery of sti�ness were addressed inRefs. [6, 35]. An alternative framework is provided by the microplane theory, under whichsecond and fourth order damage tensors have been derived as integrals of the damage onplanes of various physical orientations [4, 7].

2. Strain and energy metrics

To set the stage for the subsequent developments, we ®rstly discuss the essential kinematicalrelationships of geometrically nonlinear continuum damage mechanics. As usual, the nonlineardeformation map x= jj(X):B04R3 maps particle positions X in the macroscopic referencecon®guration B0 to their actual positions x in the deformed macroscopic con®guration B. ThenF= HXjj with J = det F>0, denotes the deformation gradient de®ning a linear map ofelements in the tangent space TB0 into elements of the tangent space TB

x � jjj�X�:B0 ÿ4B and F � rXjjj�X�:TB0 ÿ4TB: �1�Fig. 1 contains, in a symbolic graphical representation, the four di�erent con®gurationsinvolved in the present framework, namely the undeformed and deformed (damaged)macroscopic con®gurations, B0 and B, and the undeformed and deformed (®ctitious,undamaged) microscopic con®gurations, B0 and B, which will be introduced later. Thekinematic tensor ®elds referring to the tangent spaces of these di�erent con®gurations aredepicted within the trapezoids.Expressed with respect to the natural and dual base vectors of a convected coordinate

parametrization x= x(yI) and yI=yI(X) we have the following representation

F � gI GI where gI � @yIx and GI � rXyI: �2�

P. Steinmann, I. Carol / International Journal of Engineering Science 36 (1998) 1793±1814 1795

Next, the change of the square of the length of a line element due to the deformation is givenvia the Lagrange strain tensor E in TB0 and the Almansi strain tensor e in TB

ds2 ÿ dS2 � 2dX � E � dX � 2dx � e � dx, �3�whereby the covariant strain tensors E and e take the interpretation as the di�erence ofcovariant metric tensors in TB0 and TB

E � 12 �Cÿ G� � 1

2 �gIJ ÿ GIJ�GI GJ and e � 12 �gÿ c� � 1

2 �gIJ ÿ GIJ�gI gJ: �4�Here, G and g denote the euclidean metric tensors in TB0 and TB with metric coe�cientsGIJ=GI �GJ and gIJ=gI �gJ, respectively, corresponding to the second order identity tensor in acartesian coordinate system. The Cauchy±Green and the Finger tensor C = Ft �F andc = [F �Ft]ÿ1 may be referred to as elastic metric tensors in TB0 and TB, respectively. Based onthese considerations we tend to denote the strain tensors E and e as the material and spatialstrain metric tensors.Next, in order to compute scalar measures of the covariant strain metric tensors as

arguments of the free energy C we introduce in addition contravariant metric tensors,geometrically related via pull-back and push-forward, which will be denoted energy metrictensors in the sequel

B 2 TB0 and b � F � B � Ft 2 TB: �5�Thereby, the material and spatial energy metric tensors B and b of an undamaged materialtrivially correspond to the inverse Gÿ1 of the euclidean metric tensor in TB0 and the inverse

Fig. 1. Strain, energy and damage metrics referring to the tangent spaces of the di�erent macroscopic and

microscopic con®gurations.


cÿ1 of the elastic metric tensor in TB

B � bIJGI GJ and b � bIJgI gJ with bIJundamaged� G IJ: �6�

The introduction of B embodies the central idea of this contribution: In order to compute thestrain energy stored in a hyperelastic body for a given amount of strain, say the strain metricE= eIJG

I GJ, typically the invariants of E are computed as the traces of the mixed-varianttensor E �Gÿ1=eIJG

JKGI GK

�I1 � E:Gÿ1 � eIJGJI,

�I2 � �E � Gÿ1 � E�:Gÿ1 � eIJGJKeKLG

LI,

�I3 � �E � Gÿ1 � E � Gÿ1 � E�:Gÿ1 � eIJGJKeKLG

LMeMNGNI: �7�

Here Gÿ1=GIJGI GJ denotes the inverse of the euclidian metric tensor and plays the role ofthe energy metric. Thereby, Gÿ1 is considered as a measure of the perfect integrity of thematerial, i.e. no damage has occurred. In the sequel the idea of an undamaged material will beabandoned. As a measure of the degrading integrity of the material due to the evolution ofdamage, the energy metric will be allowed to decrease into the damage metric. Thus, to indicatethat the energy metric is allowed to change, i.e. to degrade, and to contrast this variable metricfrom the constant inverse euclidian metric Gÿ1, it is denoted by B1.Next, as a restriction on the formulation of the free energy function C, it is requested that C

remains invariant under a superposed deformation map, a di�eomorphism, ~jjj(x): R34R3, notnecessarily a rigid body motion, onto the current con®guration given by jj(X):B04R3 and if, inaddition, the energy metric b is transformed in the sense of a push-forward by thecorresponding relative deformation gradient FÄ =Hx~jjj. This is the requirement of covariance andembodies the less restrictive notion of objectivity if ~jjj is a rigid body motion. Now, choosing ~jjjas the inverse deformation map jjÿ1 results in FÄ =Fÿ1 for the relative deformation gradient.Then the spatial strain metric e is transformed into the strain metric associated with thecomposed deformation map ~jjjwjj as eÄ=FÄ ÿt �e �FÄ ÿ1=E. Accordingly, the spatial energy metric bis pushed forward by FÄ resulting in bÄ=FÄ �b �FÄ t=B. Thus, as a consequence of the covariance ofthe free energy C with respect to superposed di�eomorphisms, which will here be identi®edwith the inverse of the deformation gradient F $ GL+, the free energy may generally beexpressed in terms of the material and spatial strain and energy metric tensors as

C � C�E, B� � C�e, b�: �8�Following standard representation theorems for e.g. isotropic tensor functions with twosymmetric second order tensor arguments (see, for example, Ref. [3]), the free energy might beexpressed by a set of ten basic invariants containing traces of certain combinations of E and B.Clearly, in the simplest context of an isotropic undamaged material this set boils down to the

1 The overbar for bIJ in Fig. 1 indicates that the coe�cients bIJ of the material damage metric B and its spatialcounterpart b are degraded with respect to the undamaged case, characterized by the initial coe�cients bIJ, due tothe evolution of damage.


three basic invariants �I1; �I2 and �I3, thus an isotropic free energy function is given as

C � C� �I1, �I2, �I3�: �9�Invariants of a second order tensor, say E, may be de®ned as the traces of the ®rst threepowers of that tensor. Expressed in covariant coe�cients eIJ with respect to the contravariantbase vectors GK these traces involve the contravariant metric coe�cients GIJ, i.e. the constantenergy metric for an undamaged material. Since the central idea of the present damageframework is the degradation of the metric coe�cients as a measure of evolving damage, theconstant energy metric coe�cients bIJ=GIJ will be substituted in the sequel by variable damagemetric coe�cients bIJ. Therefore, the invariants �I1; �I2; �I3 are given in their expanded formatexplicitly involving the energy metric B.Thus, tolerating the redundancy of de®ning separate energy metric tensors for the undamaged

case for the sake of distinction, taking into account the de®nition of the strain and the energymetric tensors as covariant or contravariant tensor ®elds and abbreviating the coe�cients of Eand e into eIJ=

12[gIJÿGIJ], the three basic invariants are ®nally expanded as

�I1 � E:B � e:b � eIJbJI,

�I2 � �E � B � E�:B � �e � b � e�:b � eIJbJKeKLb

LI,

�I3 � �E � B � E � B � E�:B � �e � b � e � b � e�:b � eIJbJKeKLb

LMeMNbNI: �10�

Please note that the standard set of principal invariants I1, I2, I3 de®ned as the coe�cients ofthe characteristic equation of a second order tensor is fully interchangeable with the set ofinvariants �I1; �I2; �I3 de®ned by the traces of the ®rst three powers of that tensor. The reasonfor choosing the latter set of invariants is that they involve the energy metric in a moretransparent way as, for example, the determinant I3 of a second order tensor.

3. Damage metrics and microscopic con®gurations

Damage of materials is characterized by the progressive change in the microscopic internalstructure of materials which in turn leads to the deterioration of the mechanical properties.These structural changes include the nucleation, growth and coalescence of spatially distributedmicrocracks or microvoids under various loading together with the process of their coalescenceto macrocracks or macrovoids. The shape, orientation and evolution of these microdefectsdepend signi®cantly on the direction of stress and strain, therefore the process of damage isgenerally characterized by anisotropy.In response to the same external load the existence of these microdefects results on the one

hand in an increase of the stress level in the remaining e�ective material and on the other handin a decrease of the stored strain energy in the damaged material when compared to theresponse of the virgin undamaged material.To describe this phenomenon within the constitutive setting of a hyperelasto-damage

framework, we allow the energy metrics B in TB0 and b in TB to degrade in order to measurea reduced stored strain energy of the damaged material


d

dtkGI � B � GJkR0 and

d

dtkgI � b � gJkR0: �11�

The degraded energy metrics will be denoted as damage metrics and follow an evolution lawwhich we will specify lateron, thus we substitute the initial coe�cients bIJ by the decreasingcoe�cients bIJ:

B ÿ4 �bIJGI GJ * b ÿ4 �bIJgI gJ with �bIJdamaged

6� G IJ: �12�

The damage metrics are assumed to be symmetric and positive de®nite, thus there always existlocal con®gurations B0 and B, see Fig. 1 with associated linear tangent maps F $ [I, o] andf $ [I, o] with det F>0 and det f>0 such that the following push-forward relationship holds

B � �F � �B � �F t with �B � �bIJ �GI �GJ and b � �f � �b � �ft with �b � �bIJ �gI �gJ: �13�Thereby, in the locally de®ned con®guration B0 the damage metric B=Gÿ1 coincides with theeuclidean metric with coe�cients bIJ=GI �GJ. The ®ctitious con®gurations B0 and B which areassumed undamaged but otherwise mechanically equivalent will be addressed as microscopiccon®gurations in the sequel.Fictitious undamaged con®gurations have been proposed for example by Betten [1] and

Murakami [30] based on the classical but alternative assumption of a reduced e�ective areawithin a cross section of the damaged material. Observe carefully, that in contrast to theclassical approach the present damage metric based theory will be motivated by considering thee�ects of damage on the stored strain energy.The linear tangent maps F and f relating the microscopic and macroscopic tangent spaces are

denoted as material and spatial damage deformation gradients and do not necessarily follow asthe gradients of an associated global vector ®eld, thus the microscopic con®gurations aregenerally incompatible. Nevertheless, anholonomic, i.e. incompatible co-ordinate systems withbase vectors GI and gI are de®ned in the microscopic con®gurations via the anholonomic Pfa�transformations F and f

�F � GI �GI:T �B0 ÿ4TB0 and �f � gI �gI:T �B ÿ4TB: �14�So far the relationship between the tangent spaces TB0 and TB to the damaged macroscopiccon®gurations B0 and B has been established by the macroscopic deformation gradient F.Correspondingly, the relationship between the tangent spaces TB0 and TB0 (respectively, TBand TB) to the undamaged microscopic and the damaged macroscopic con®gurations B0 andB0 (respectively, B and B) has been identi®ed to be governed by the damage deformationgradient F (respectively, f). Now it only remains to set up a relationship between the tangentspaces TB0 and TB to the undamaged microscopic con®gurations B0 and B. Here we assumethat the undamaged microscopic tangent spaces TB0 and TB are related in the same way asthe damaged macroscopic tangent spaces TB0 and TB, thus we assume that the microscopicand the macroscopic con®gurations are subjected to the identical deformation gradient which isa common assumption in homogenization theories

F � gI GI�: �gI �GI: �15�


As a consequence, the damage deformation gradients F and f are no independent quantitiesbut are related geometrically by the macroscopic deformation gradient

�f � F � �F � Fÿ1 � � gI GI� � �GJ �GJ� � � �GK �gK� � � gI �gI�: �16�Therefore, they possess the same determinant J=det F=det f>0 and, moreover, since Bcorresponds to the euclidean metric in TB0 the damage metric b=cÿ1 in the local con®gurationTB coincides with the inverse of the Finger tensor, the energy metric tensor, which isassociated to the macroscopic deformation gradient F

�b � F � �B � Ft: �17�Equivalently, we may introduce the e�ective strain metrics E in TB0 and e in TB as the pull-back of the nominal strain metric tensors E in TB0 and e in TB to the microscopiccon®gurations

�E � �Ft � E � �F � eIJ �GI �GJ and �e � �ft � e � �f � eIJ �gI �gJ: �18�Then, the requirement of covariance of the free energy C to superposed di�eomorphisms whichare now identi®ed with the inverse of the damage deformation gradients F $ GL+ andf $ GL+, renders the fundamental statement of strain energy equivalence between themicroscopic and macroscopic con®gurations

C � C�E, B� � C� �E, �B� and C � C�e, b� � C� �e, �b�: �19�This statement of strain energy equivalence deserves some comments. With respect to thedamaged macroscopic con®gurations the stored strain energy is expressed in terms of nominalstrain metric tensors corresponding to standard strain measures and damage metric tensorscorresponding to reduced energy metric tensors. Vice versa, with respect to the ®ctitiousundamaged microscopic con®gurations the stored strain energy is expressed in terms of thee�ective strain metric tensors corresponding to reduced strain measures and damage metrictensors corresponding to standard energy metric tensors. Observe that the proposedhyperelasto-damage formulation is required to obey the following property at zero nominalstrain which is a characteristic di�erence as compared for example to hyperelasto-plasticity

C ÿ40 for E ÿ4o and C ÿ40 for e ÿ4o: �20�For an account on the formulation of hyperelasto-plasticity based on metric tensors please referto Miehe [27]. The relationships of the di�erent kinematical quantities introduced so far are forconvenience summarized in Fig. 1.

4. Standard dissipative material approach

In an attempt to cover a wide range of dissipative phenomena within the samethermodynamically sound framework, Halphen and Nguyen [15] proposed the theory ofstandard dissipative materials based on a constitutive assumption on the free energy in terms ofkinematical and internal variables, the exploitation of the Clausius±Duhem inequality to de®ne


the thermodynamical forces, the introduction of a dissipation surface and the postulate ofmaximum dissipation to derive the associated evolution laws for the internal variables.In the sequel, we will follow the formalism advocated by Halphen and Nguyen [15] to

develop the constitutive setting of hyperelasto-damage within a geometrically nonlinearcontinuum description. Thereby, in order to capture the in¯uence of the microdefect density onthe hardening behaviour, it appears physically meaningful to consider besides the strain anddamage metrics, a scalarvalued internal hardening variable b as well in the free energy C perunit volume of the reference con®guration B0

C � Cstr�E, B� �Chrd�b�: �21�Then, exploiting the Clausius±Duhem inequality expressed in terms of the second Piola±Kirchho� stress SS and the material rate of the strain metric E in TB0

D � SSS: _Eÿ _C ��Sÿ @C

str

@E

�: _Eÿ @C

str

@B: _Bÿ @C

hrd

@b_br0 �22�

renders the hyperelastic part of the constitutive law for the contravariant second Piola±Kirchho� stress SS, which might be considered as the material equilibrium stress and, as novelquantities, dissipative stresses thermodynamically conjugated to B and b which are introducedas the covariant material damage stress DD and the hardening stress D in TB0

SSS � tIJGI GJ � @Cstr

@Eand DDD � �dIJG

I GJ �defÿ @Cstr

@Band D� �def @C

hrd

@b: �23�

Accordingly, push-forward to the spatial macroscopic con®guration B renders the constitutivelaw for the contravariant Kirchho� stress tt, i.e. the spatial equilibrium stress and the covariantspatial damage stress dd

ttt � tIJgI gJ �@Cstr

@eand ddd � �dIJgI gJ � ÿ @C

str

@b: �24�

The derivation of the constitutive relationships in Eq. (24) is given by the following line ofarguments for the contravariant spatial Kirchho� stress

ttt � F � @Cstr

@E� Ft � F � @C

str

@eIJGI GJ

� �� Ft � @C

str

@eIJgI gJ �

@Cstr

@e�25�

and for the covariant spatial damage stress

ÿddd � Fÿt � @Cstr

@B� Fÿ1 � Fÿt �

�@Cstr

@bIJGI GJ

�� Fÿ1 � @C

str

@bIJgI gJ � @C

str

@b: �26�

On the other hand, pull-back to the ®ctitious undamaged microscopic con®gurations B0 and Brenders the e�ective and the microscopic damage stresses

�SSS � tIJ �GI �GJ, �ttt � tIJ �gI �gJ and �DDD � �dIJ �GI �GJ, �ddd � �dIJ �gI �gJ: �27�The relationships of the di�erent stress quantities introduced so far are for convenience


summarized in Fig. 2. Thereby, the squares in Fig. 2 denote again in a symbolic graphical

representation the four di�erent con®gurations, namely the undeformed and deformed

(damaged) macroscopic con®gurations B0 and B and the undeformed and deformed (®ctitious,

undamaged) microscopic con®gurations B0 and B, involved in the present framework. The

stress like tensor ®elds referring to the tangent spaces of these di�erent con®gurations are

depicted within the trapezoids.

With these de®nitions the remaining dissipation inequality writes in the material setting

D�DDD, D� � DDD: _Bÿ D _br0: �28�Next, a dissipation or rather damage surface F is introduced in terms of the damage stress, the

damage metric and the hardening stress which de®nes an admissible elastic cone A for which

no damage evolution is possible

A � f�DDD, D�jF � F�DDD, D;B� � j�DDD;B� ÿ DR0g: �29�Then the postulate of maximum dissipation requires for the dissipative stresses in the

admissible elastic cone

D�DDD, D�rD�DDD*, D*� 8�DDD*, D*� 2A �30�and renders the associated evolution laws for the damage metric B and the internal variable bwith g de®ning a Lagange or rather damage multiplier

Fig. 2. Equilibrium and damage stress referring to the tangent spaces of the di�erent macroscopic and microscopic

con®gurations.


_B � g@F@DDD� gN and _b � ÿg @F

@D� g with N � @j

@DDDand M � @j

@B�31�

together with the Kuhn±Tucker complementary conditions which are supplemented by adamage consistency condition

gr0, FR0, gF � 0, g _F � 0: �32�Here, the gradients of the damage surface F with respect to the damage stress DD and thedamage metric B have been denoted by N and M, respectively. Next, exploiting the damageconsistency condition in the case of damage loading determines the damage multiplier g as

g _F � 0* g � N:HBE: _E

�MÿN:HBB�:NÿHbb: �33�

Here we introduced the following abbreviations for the Hesse type second derivatives of thefree energy

HEE � @2Cstr

@E@E, HBE � @

2Cstr

@B@E, HBB � @

2Cstr

@B@B, Hbb � @

2Chrd

@b@b: �34�

Finally, the material rate of the second Piola±Kirchho� stress is related to the material rate ofthe strain metric by the fourth order hyperelastic-damaged tangent operator Led

_SSS �Led: _E with Led �Lel ÿ HEB:NN:HBE

�N:HBB ÿM�:N�Hbb: �35�

Here the Hesse type second derivative of the free energy with respect to the strain metric E hasbeen identi®ed with the purely hyperelastic fourth order tangent operator Lel=HEE.

5. Relationship to classical damage theory

In this section the formal connection of the present geometrically nonlinear continuumdamage theory to its classical geometrically linear counterpart is highlighted. Thereby, wefollow lines advocated by Murakami [30] to extend the classical interpretation of damage as areduction of nominal area elements by a scalar factor [1ÿ d] to the case of second orderdamage. Nevertheless, the interpretation as an area reduction is only formal and shall not bepursued in our subsequent damage metric based developments.As a point of departure, we introduce the oriented surface elements dA �J � �Jÿ1 dA in TB0

and da �J � �Jÿ1 da in TB weighted by the determinant �Jÿ1 of the inverse damage deformationgradients. The motivation for this construction will become clear when the resulting tractionvector is computed in terms of weighted stress measures. Then, the Nanson formula may beapplied either to relate the weighted oriented material surface element dA �J in TB0 to theoriented material surface element dA in TB0 in terms of the material damage deformationgradient F or, equivalently, to relate the weighted oriented spatial surface element da �J in TB tothe oriented spatial surface element da in TB in terms of the spatial damage deformation


gradient f, see Fig. 3

d �A � �F t � dA �J and d �a � �f t � da �J: �36�From this consideration the classical interpretation of damage as an area reduction in the®ctitious undamaged con®guration, here either B0 or B, is formally retrieved

d �A�: �IÿD�t � dA �J and d �a�: �Iÿ d �t � da �J: �37�Thereby, the area reduction is given by mixedvariant material and spatial damage tensorsD $ [o, I] and d $ [o, I]

D � Iÿ �F and d � Iÿ �f with d � F �D � Fÿ1: �38�Note, that the mixedvariant material and spatial damage tensors D and d reduce to the samespherical tensor D= d = d I with the scalar damage measure d $ [0, 1] and F=f=[1ÿ d]Iunder the assumption of isotropic damage. Consequently, in this case the damage metricsfollow simply as B= [1ÿ d]2Gÿ1 and b= [1ÿ d]2b.Next, we introduce the weighted nominal ®rst Piola±Kirchho� type stress PP in TB0 and the

weighted nominal Kirchho� type stress pp in TB as

PPP � �JFÿ1 � ttt and ppp � �JJ ÿ1ttt: �39�Clearly, the motivation for these new stress measures is provided by the observation that theprojection with the respective weighted surface elements renders the in®nitesimal force elements.

Fig. 3. Weighted surface elements and stress measures referring to the tangent spaces of the di�erent macroscopicand microscopic con®gurations.


Equivalently, from the requirement that the in®nitesimal force elements as computed by thetraction vectors times the area elements in the damaged and ®ctitious undamagedcon®gurations coincide, see Fig. 3, we might introduce the weighted e�ective Piola±Kirchho�type stress tensors �PPP in TB0 and �ppp in TB by

PPPt � dA �J � �PPPt � d �A and pppt � da �J � �pppt � d �a: �40�Therefore, the analogy to the classical understanding of geometrically linear continuumdamage theory may be established in terms of the weighted e�ective and nominal stresses whichare related by a tensorial [1ÿ d] relationship

�PPP � �IÿD�ÿ1 �PPP and �ppp � �Iÿ d �ÿ1 � ppp: �41�Please note once again that this correspondence is only formal and that we will not use themixedvariant second order damage tensors D and d in the sequel.

6. Prototype model problem

Although the proposed theory is general enough to allow for more sophisticated orthotropicdamage states, in this section we will outline the simplest possible constitutive model based onthe present continuum damage framework for the sake of demonstration. Thereby it turns outthat the formulation boils down to isotropic damage if the initial damage metric B(t = t0) is aspherical second order tensor. The performance of the resulting model is ®nally illustrated forthe academic example of a homogeneous deformation state of uniaxial extension for the sakeof demonstration.

6.1. St. Venant material with damage

In the sequel we assume that the free energy Cstr is an isotropic function of the twosymmetric tensor arguments in TB0: nominal strain metric E and damage metric B. Pleaseobserve that this assumption does not rule out anisotropic response for the equilibrium stress inTB0. The set of ten basic invariants of these two arguments is then reduced due to thecovariance or rather strain energy equivalence statement which requires that Cstr is as well anisotropic function of the two symmetric tensor arguments in TB0: e�ective strain metric E andenergy metric B. Since the energy metric B coincides by de®nition with the euclidean metric inTB0, the free energy is a function of the three basic invariants �I1; �I2 and �I3 of E with�In=[E �B]n:I and thus the response for the e�ective stress in TB0 is isotropic.The St. Venant material is the simplest hyperelastic model in B0 and is given in terms of the

two invariants �I1 and �I2 and the two LameÂ constants l and m as

Cstr � 12l� �E: �B�2 � m� �E � �B�:� �B � �E� � 1

2l�E:B�2 � m�E � B�:�B � E�: �42�The St. Venant model is valid for small elastic strains which is a realistic assumption for manybrittle materials. Nevertheless, the full blown up apparatus of a geometrically nonlinear theoryis justi®ed since the amount of damage measured in an appropriate norm of the damage metric


is ®nite. Based on this particular choice for Cstr, the equilibrium second Piola±Kirchho� stressSS and the damage stress DD in B0 follow as

S � l�E:B�B� 2mB � E � B and ÿ DDD � l�E:B�E� 2mE � B � E: �43�Apparently, SS is not necessarily an isotropic function of E as soon as the damage metric Bdeparts from Gÿ1. It is for example easily proved that SS is at most an orthotropic function of Eunder the assumption of a symmetric damage deformation gradient F. Within the geometricallylinearized continuum theory a similar modi®cation of the elastic secant sti�ness based on thenotion of a second order integrity tensor has earlier been proposed by Valanis [39].

In the subsequent developments the following relationship between the damage stress and thedamage metric will play a fundamental role

ÿDDD:B � 2Cstr�DDD, B�* 2@Cstr

@DDD� ÿB: �44�

Guided by the above observation, the simplest possible damage surface F is introduced as thedi�erence of the ®rst invariant of the damage stress and the damage metric and the hardeningstress, thus, the strain energy has to exceed a certain value before damage evolution is possible

F � ÿDDD:Bÿ D � 2Cstr�DDD, B� ÿ DR0: �45�Consequently, the associated evolution laws for the damage metric B and the internal variableb follow as

_B � ÿgB and _b � g with N � ÿB and M � ÿDDD: �46�For the hardening stress we will investigate the following two di�erent laws

D�b� � D0 or D�b� � �D0 � �D1 ÿ D0��1ÿ exp�ÿhb��exp�ÿ2b� �47�Here D0 and D1 denote the initial damage threshold and the damage saturation strength, h isan exponential hardening modulus.

Finally, to derive the tangent operator, the purely hyperelastic part of the tangent operator,i.e. the Hesse type second derivative of the free energy is given by the fourth order tensor

Lel �HEE � lB B� m�B � B� B B� �48�and, moreover, the following relationships are easily proved

HEB:B � 2SSS and B:HBB:B � 2Cstr: �49�Based on these preliminaries, the hyperelastic-damaged tangent operator follows as

Led � lB B� m�B � B� B B� ÿ SSS SSSCstr � @bD=4

: �50�

Remark: Besides the standard dyadic product , we employed the following symbolicnotation for nonstandard dyadic products of second order tensors a and b as expressed inindex notation when referred to a Cartesian co-ordinate system


�a �b�ijkl � �a�ik�b�jl and �ab�ijkl � �a�il�b�jk:

6.2. Integration algorithm

For the stress update algorithm we note that this particular simple damage evolution law forthe damage metric may be integrated in closed form within a time step Dt=n�1tÿn t to give anexponential approximation, the internal hardening variable is updated by the Euler backwardmethod

n�1B � exp�ÿDg�nB and n�1b �n b� Dg: �51�Within a strain driven setting, the elastic trial state may be de®ned as follows

eF � 2eCstr ÿe D with eCstr � Cstr�n�1E, nB� and eD � D�nb�: �52�In the case of loading characterized by the algorithmic loading condition eF>0 the algorithmicconsistency condition takes the following format

n�1F�Dg� � 2eCstrexp�ÿ2Dg� ÿn�1 D�Dg��: 0: �53�Thereby, the structure of the chosen hardening stress laws in its algorithmic form

n�1D � D0 orn�1D

exp�ÿ2Dg� � �D0 � �D1 ÿ D0��1ÿ exp�ÿhn�1b��exp�ÿ2nb� �54�

allows for a closed form solution for the incremental damage multiplier and thus completelycircumvents a local iteration procedure

Dg � ÿ 1

2log

�D0

2eCstr

�or Dg � ÿ 1

hlog 1ÿ 2eCstrexp�2nb� ÿ D0

D1 ÿ D0

� �ÿn b: �55�

Finally it shall be observed that the algorithmic hyperelasto-damaged tangent operator Laed

coincides with the continuum tangent operator Led as given in the previous subsection sincen�1 _B � D_gn�1B and n�1 _b � D_g.

6.3. Example

In this academic example we investigate the qualitatively di�erent response predictions of thetwo di�erent hardening stress laws within the model problem discussed above. To this end, asquare domain with side length L = 1 � 10 inch is subjected to a state of uniaxial extensioncharacterized by the deformation gradient F= le1 e1 with principal stretch lr1 and eI thebase vectors of a ®xed cartesian coordinate system. The total deformation is applied inprescribed displacement increments of L/100. The relationship between the elastic constants ischosen as to model a gray cast iron with l = 150 � 102 ksi and m= 100 � 102 ksi.First we examine the model with constant hardening stress D(b) = D0=1. The resulting load

is plotted versus the principal stretch l in Fig. 4. After an initial stage of purely elastic response


we observe a sudden exponential decay in load carrying capacity of the material which

typically characterizes a brittle material. Nevertheless the limit of vanishing load carrying

capacity is only reached asymptotically. This observation coincides neatly with the evolution of

the damage metric B= BdIJeI eJ displayed in Fig. 5. Here, after a sudden transition in the

damaged regime the metric coe�cient B is reduced to less than 10% at a principal stretch of

l= 1.5. The reduction of the stored energy to the constant value D0/2 = 0.5 in the elasto-

damaged phase is clearly visible in Fig. 6. The accompanying evolution of the hardening

variable b is illustrated for completeness in Fig. 7.

Next we oppose these results to the model with non-constant hardening stress and the

parameter choice D0=1, D1=100 and h = 2, thus rendering D(b) = [1 + 99(1ÿ exp(ÿ2b)]]exp(ÿ2b). The resulting load is plotted versus the principal stretch l in Fig. 8. Here, after the

initial stage of purely elastic response we observe a smooth transition into the nonlinear

regime. After reaching peak load the resistance of the material decays smoothly until the load

carrying capacity is completely exhausted. This observation coincides neatly with the evolution

of the damage metric B= BdIJeI eJ displayed in Fig. 9. Here the metric coe�cient B is

smoothly reduced to 0% of its initial value at a principal stretch of approximately l= 1.44.

The characteristic evolution of the stored energy which equals half of the hardening stress D(b)and is completely exhausted at l = 1.44 is depicted in Fig. 10. Obviously, for vanishing damage

metric the hardening variable b tends to in®nity as is displayed in Fig. 11.

Fig. 4. Constant hardening stress law: resulting load vs stretch.


Fig. 5. Constant hardening stress law: damage metric vs stretch.

Fig. 6. Constant hardening stress law: stored energy vs stretch.


Fig. 7. Constant hardening stress law: hardening variable vs stretch.

Fig. 8. Non-constant hardening stress law: resulting load vs stretch.


Fig. 9. Non-constant hardening stress law: damage metric vs stretch.

Fig. 10. Non-constant hardening stress law: stored energy vs stretch.


Clearly, the actual choice of the hardening stress law D(b) heavily in¯uences the resultingresponse prediction of the model and will constitute an issue of major research if the presenttheory is applied to model a speci®c material.

7. Summary and conclusion

The main contribution of this work has been the development of a geometrically nonlinearframework for second order continuum damage capable of describing anisotropic damagestates. Thereby, the underlying mechanism to incorporate the e�ects of damage has beenprovided by the hypothesis of strain energy equivalence between microscopic and macroscopiccon®gurations.As a point of departure, we ®rst introduced microscopic con®gurations characterizing the so

called ®ctitious undamaged material based on standard terminology in nonlinear continuummechanics. Then the relationship of the microscopic to the macroscopic con®gurations isestablished by linear tangent maps which are interpreted as damage deformation gradients. Asa consequence, as the keypoint of the development, the corresponding Finger tensors de®ned inthe macroscopic con®guration constitute damage metrics for the measurement of the freeenergy for given nominal strains. In the current context the hypothesis of strain energyequivalence is considered as a manifestation of spatial covariance of the free energy withrespect to superposed di�eomorphisms, which are here identi®ed with the damage deformationgradients.

Fig. 11. Non-constant hardening stress law: hardening variable vs stretch.


The straightforward exploitation of the concept of standard dissipative materials has thencompleted the constitutive framework by stresses and damage stresses de®ned in the di�erentcon®gurations together with a hardening stress conjugated to an internal hardening variable.Likewise, a damage condition and the associated evolution laws for the damage metric and theinternal hardening variable are a natural outcome of this procedure. As an interesting aspectthe present damage metric based theory has formally been related to the classicalunderstanding of damage as an area reduction.As a demonstration of the developed methodology a simple prototype model problem was

proposed and an e�cient and accurate exponential-type algorithmic treatment has beenelaborated.Finally, based on the prototype model problem the in¯uence of two di�erent hardening

stress laws has been studied in a numerical example of homogeneous uniaxial tension. Therebyit turned out that the choice of the hardening law has a dominating in¯uence on the resultingresponse prediction.In conclusion it is believed that this contribution clari®ed issues of how to formulate second

order continuum damage within a geometrically nonlinear setting in a kinematically andthermodynamically consistent way. Nevertheless, the application of the advocated theory tomodel speci®c materials, especially if anisotropic damage states and/or couplings with plasticityare relevant, will necessarily constitute an area of major importance in future research.

Acknowledgements

Partial ®nancial support from DGICYT-MEC (Madrid, Spain) under research projectsPB93-0955 and PB95-0771 is gratefully acknowledged. The ®rst author also wishes to thankthe Technical University of Catalonia (UPC) for funding his visits at ETSECCPB in 1996 and1997, during which this work was developed.

References

[1] J Betten, Net-stress analysis in creep mechanics, Ing Arch 52 (1982) 405±419.[2] J Betten, Damage tensors in continuum mechanics, J Mech Theor Appl 2 (1983) 13±32.

[3] Boehler JP. In: Boehler JP, editor. Applications of tensor functions in solid mechanics. Wien: Springer-Verlag,1987.

[4] I Carol, ZP Bazant, PC Prat, Geometric damage tensor based on microplane model, J Eng Mech ASCE 117

(1991) 2429±2448.[5] I Carol, E Rizzi, K Willam, A uni®ed theory of elastic degradation and damage based on a loading surface, Int

J Solids Struct 31 (1994) 2835±2865.[6] I Carol, K Willam, Spurious energy dissipation/generation in sti�ness recovery models for elastic degradation

and damage, Int J Solids Struct 33 (1996) 2939±2957.[7] Carol I, Bazant ZP. Damage and plasticity in microplane theory. Int J Solids Struct 1997, in press.[8] Chaboche JL. Description thermodynamique et phenomenologique de la viscoplaticite cyclique avec endomm-

agement. PhD Thesis. Univ. Paris VI, 1978.[9] CL Chow, J Wang, An anisotropic damage theory of continuum damage mechanics for ductile fracture, Eng

Fract Mech 30 (1987) 547±563.


[10] Cordebois JP, Sidoro� F. Endommagement anisotrope en eÂ lasticiteÂ et plasticiteÂ . J Mec Theor Appl 1982;No.SpeÂ cial:45±60.

[11] LS Costin, Damage mechanics in the post-failure regime, Mech Mater 4 (1985) 149±160.[12] SC Cowin, The relationship between the elasticity tensor and the fabric tensor, Mech Mater 4 (1985) 137±147.[13] L Davidson, A Stevens, Thermomechanical constitution of spalling elastic bodies, J Appl Phys 44 (1973) 667±

674.[14] A Dragon, Z Mroz, A continuum model for plastic±brittle behavior of rock and concrete, Int J Eng Sci 17

(1979) 121±137.

[15] B Halphen, QS Nguyen, Sur les MateÂ riaux Standards GeÂ neÂ raliseÂ s, J Mec 14 (1975) 39±62.[16] DR Hayhurst, FA Leckie, The e�ect of creep constitutive and damage relationships upon the rupture time of a

solid circular torsion bar, J Mech Phys Solids 21 (1973) 431±446.

[17] DR Hayhurst, B Storakers, Creep rupture of the andrade shear disk, Proc R Soc London Ser A: 349 (1976)369±382.

[18] LM Kachanov, Time rupture process under creep conditions, Izv Acad Nauk SSSR OTN 8 (1958) 26±31 (inRussian).

[19] ML Kachanov, Continuum model of medium with cracks, ASCE J Eng Mech Div 106 (1980) 1039±1051.[20] D Krajcinovic, GU Fonseka, The continuum damage theory of brittle materials. Part 1: General theory, ASME

J Appl Mech 48 (1981) 809±815.

[21] D Krajcinovic, Continuum damage mechanics, Appl Mech Rev 37 (1984) 1±6.[22] FA Leckie, The constitutive equations of continuum creep damage mechanics, Phil Trans R Soc London 288

(1978) 27±47.

[23] Leckie FA, Onat ET. Tensorial nature of damage measuring internal variables. Proc. IUTAM Symp. at Senlis.France, 1980.

[24] Lemaitre J, Chaboche JL. Aspects pheÂ nomeÂ nologiques de la rupture par endommagement. J Mec Appl 1978;2.

[25] VA Lubarda, D Krajcinovic, Damage tensors and crack density distribution, Int J Solids Struct 30 (1993)2859±2877.

[26] Mazars J, Lemaitre J. Application of continuous damage mechanics to strain and fracture behavior of concrete.In: Shah SP, editor. Application of fracture mechanics to cementitious composites. Northwestern University:

NATO Advanced Research Workshop, 4±7 Sept 1984:375±378.[27] C Miehe, A theory of large-strain isotropic thermoplasticity based on metric transformation tensors, Arch Appl

Mech 66 (1995) 45±64.

[28] Murakami S, Ohno N. A continuum theory of creep and creep damage. In: Ponter ARS, Hayhurst DR, editors.Creep in structures. Berlin: Springer, 1981.

[29] Murakami S. Anisotropic damage in metals. In: Boehler JP, editor. Failure criteria of structured media.

Rotterdam: AA Balkema, 1986.[30] S Murakami, Mechanical modeling of material damage, ASME J Appl Mech 55 (1988) 280±286.[31] M Oda, A method for evaluating the e�ect of crack geometry on the mechanical behavior of cracked rock

masses, Mech Mater 2 (1983) 163±171.

[32] M Ortiz, A constitutive theory for the inelastic behavior of concrete, Mech Mater 4 (1985) 67±93.[33] Rabotnov YN. Creep problem in structural members. Amsterdam: North-Holland, 1969.[34] L Resende, A damage mechanics constitutive theory for the inelastic behavior of concrete, Comp Methods

Appl Mech Eng 60 (1987) 57±93.[35] E Rizzi, I Carol, K Willam, Localization analysis of elastic degradation with application to scalar damage, J

Eng Mech ASCE 121 (1995) 541±554.

[36] JC Simo, JW Ju, Strain- and stress-based continuum damage models. I. Formulation, Int J Solids Struct 23(1987) 821±840.

[37] W Suaris, SP Shah, Rate-sensitive damage theory for brittle solids, J Eng Mech ASCE 110 (1984) 985±997.

[38] AA Vakulenko, ML Kachanov, Continuum theory of cracked media, Mekh Tverdogo Tela SSSR 6 (1971) 159±166 (in Russian).

[39] KC Valanis, A theory of damage in brittle materials, Eng Fract Mech 36 (1990) 403±416.


A framework for geometrically nonlinear continuum damage mechanics

Documents

Transcript of A framework for geometrically nonlinear continuum damage mechanics