Formulation and computation of geometrically non-linear gradient damage

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING

Int. J. Numer. Meth. Engng. 46, 757–779 (1999)

FORMULATION AND COMPUTATION OF GEOMETRICALLYNON-LINEAR GRADIENT DAMAGE

PAUL STEINMANN∗; †

Department of Mechanical Engineering; University of Kaiserslautern; P.O. Box 3049;D-67653 Kaiserslautern; Germany

SUMMARY

The aim of this contribution is the extension of a small strain and small deformation formulation of gradientenhanced damage to the geometrically non-linear case. To this end, Non-local Stored Energy densities, (NSE)are introduced as primary variables. Fluxes conjugated to the gradients of the NSE are then computed frombalance laws which in the small strain limit correspond to the averaging equation well known in the literature[1–3]. The principal task is then to establish constitutive laws for these newly introduced NSE- uxes. Thereby,four di�erent options are investigated which are motivated from Lagrange and Euler averaging procedurestogether with changes of the metric tensors. Issues of the corresponding FE-formulation and its linearizationwithin a Newton–Raphson procedure are addressed in detail. Finally, the four di�erent formulations arecompared for the example of a bar in tension whereby large strains are truly envisioned. Copyright ? 1999John Wiley & Sons, Ltd.

KEY WORDS: gradient damage; localization, large strains; FE-technology

1. INTRODUCTION

Softening at the continuum level due to damage accumulation mimics deterioration processeswithin the material at the micro scale. E.g. for carbon �lled polymers, which can sustain largestrains up to several hundred per cent, we might think of debonding mechanisms between themolecular chains and=or the carbon �llers. As a consequence of softening, damage and accordinglylarge deformations tend to accumulate within the narrow bands, the so-called localized zones. Inexperiments these localization zones display a �nite width which is related to the micro structureof the material. Upon further loading localized zones then most often form a precursor to the �nalrupture of the material. On the other hand, in a standard continuum description and in particularin the corresponding numerical solution schemes no �nite width is obtained, instead pathologicallymesh-dependent solutions are observed upon re�nement of the discretization.Among the most e�ective remedies against this unphysical behaviour non-standard continuum

theories have been proposed which incorporate higher gradients of those quantities which areresponsible for softening.

∗ Correspondence to: Paul Steinmann, Department of Mechanical Engineering, University of Kaiserslautern, P.O. Box 3049,D-67653 Kaiserslautern, Germany. E-mail: [email protected]† Chair for Applied Mechanics

CCC 0029-5981/99/290757–23$17.50 Received 21 August 1998Copyright ? 1999 John Wiley & Sons, Ltd. Revised 30 January 1999

758 P. STEINMANN

Higher gradients in plasticity have been motivated by dislocations in single crystals in the worksof Aifantis [4; 5] and recently from an alternative point of view by Steinmann [6]. One-dimensionalinvestigations on a higher gradient continuum were performed by Schreyer and Chen [7]. Lasryand Belytschko [8] coined the notion of localization limiters. A variational framework for gradientplasticity was proposed by M�uhlhaus and Aifantis [9], a gradient theory of phenomenologicalplasticity was developed by de Borst and M�uhlhaus [10]. The application of a higher gradientformulation to granular materials was examined by Vardoulakis and Aifantis [11]. Motivated byexperiments performed by Fleck et al. [12] on thin copper wires Fleck and Hutchinson [13]proposed an alternative theory which takes into account the gradient of the continuum rotation.Di�erent numerical strategies for the treatment of gradient plasticity were investigated by Sluys

et al. [14] and Pamin [15]. Huerta and Pijaudier-Cabot [16] studied the in uence of the discretiza-tion on the regularization performance of di�erent localization limiters. Recent contributions to thenumerics of gradient plasticity at small and large strains are treated by Svedberg and Runesson[17].Non-local integral formulations of continuum damage were proposed by Pijaudier-Cabot and

Bazant [18] and Bazant and Pijaudier-Cabot [19]. The transition to a gradient formulation wasoutlined in the works by Lasry and Belytschko [8] and M�uhlhaus and Aifantis [9] in terms of aTaylor series expansion.In the small strain regime the recent proposal of de Borst et al. [1], Peerlings et al. [2] and

Peerlings et al. [3] to enhance an isotropic damage formulation by spatial gradients has gainedmuch attention for its conceptional beauty and convincing operational performance. A related butalternative formulation has been investigated by de Borst et al. [20]. The extension to anisotropicdamage is proposed by Kuhl et al. [21], the identi�cation of the additional gradient parameter isconsidered by Mahnken and Kuhl [22].The essential ingredient of the gradient enhanced damage is an additional equation for the

determination of the non-local so-called damage equivalent strain which coincides with the non-local stored energy in this work. Thereby, this averaging equation is derived from an approximationto the integral de�nition of the truly non-local damage equivalent strain and takes the format of adi�usion–adsorption–reaction equation. A noteworthy feature from the numerical point of view isthe treatment of the non-local damage equivalent strain as an independent variable.Therefore, the question arises how this proposal with its desirable features may be extended to

the geometrically non-linear case? Here, failure processes of rubber constructions like, e.g. bearingsor tires at large strains are considered as typical target problems. From the continuum mechanicalpoint of view the set up of the additional averaging equation is not trivial. In the large strainsetting several options with di�erent interpretations or motivations are possible. It is therefore theaim of this contribution to propose and to examine the most obvious extensions of the gradientenhanced damage framework to the geometrically non-linear regime.An outline of this contribution is as follows:In the �rst part the strong form of the coupled problem is discussed. To this end the boundary

value problem of geometrically non-linear solid mechanics is brie y reiterated. Then a frame-work of local isotropic damage at large strains is reviewed and the extension to non-localdamage is highlighted. Next, the additional balance equation corresponding to theintroduction of the Non-local Stored Energy (NSE) as an independent variable is advocated. Inparticular, four di�erent options for the constitutive law relating the gradients of the NSE to theNSE- ux entering the balance equation are proposed and are related to the well-known small strainformulation.

Copyright ? 1999 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 46, 757–779 (1999)

FORMULATION OF NON-LINEAR GRADIENT DAMAGE 759

The next part is dedicated to the weak form of the coupled problem. After reiterating thestandard virtual power statement the weak form for the additional balance equation de�ning thenon-local NSE is given with respect to the reference and the current con�guration.In the subsequent part, issues pertaining to the linearization of the weak form are addressed.

Thereby, we compare, in particular, the linearizations of the four options for the NSE- uxconstitutive law.Finally, after selecting a model problem the last part concentrates on a computational comparison

of the four di�erent averaging formulations proposed above. Here the in uence of the mesh spacingand the gradient parameter are investigated computationally and are discussed.

2. STRONG FORM OF THE COUPLED PROBLEM

To set the stage we brie y reiterate the boundary value problem of geometrically non-linear con-tinuum mechanics in strong form.

2.1. Balance of linear momentum

Let B0 and B denote the reference and the current con�guration occupied by a solid at timet0 and t, respectively. The current placements x∈B of the solid are described by the non-lineardeformation map x=e(X) in terms of the reference placements X∈B0. The boundary @B0 toB0 with outward normal N is subdivided into disjoint parts @B0 = @Bx

0 ∪ @Bt0 with @B

x0 ∩ @Bt

0 = ∅where either Neumann or Dirichlet boundary conditions are prescribed. The corresponding bound-ary @B to B with outward normal n is consequently subdivided into disjoint parts @B= @Bx ∪ @Bt

with @Bx ∩ @Bt = ∅. Next, the deformation gradient F, its inverse F−1 together with the Jacobideterminant J are de�ned in terms of the convected covariant and contravariant spatial and refer-ence base vectors gI = @�Ix; gI =∇x�I and GI = @�IX;GI =∇X �I and the corresponding covariantmetric coe�cients gIJ and GIJ

F=∇Xe= gIJgI ⊗GJ with J = det F and F−1 =GIJGI ⊗ gJ (1)

Here, the convected base vectors are merely introduced in order to clearly highlight the relationbetween the material and the spatial formulation in the sequel. Then the covariant Cauchy–Greenand Finger strain tensors C and c are introduced over B0 and B

C=Ft · F= gIJGI ⊗GJ and c= [F · Ft]−1 =GIJgI ⊗ gJ (2)

and are related to the covariant spatial and material metric tensors g and G by standard pull-backand push-forward operations

G=Ft · c · F=GIJGI ⊗GJ and g= [F · C−1 · Ft]−1 = gIJgI ⊗ gJ (3)

Obviously, the metric tensors g and G denote the same second-order unit tensor expanded alongthe di�erent base vectors gI and GI . Again this redundant notation is introduced in order todistinct a material from a spatial description. Please note that the spatial Finger tensor c transformsobjectively under a superposed rigid body motion with e∗(X)=Q · e(X) + k∗ as

c∗=Q · c ·Qt with Q∈SO(3) and k∗ ∈R3 (4)


760 P. STEINMANN

Figure 1. Kinematical quantities related to con�gurations B0 and B

Figure 2. Kinetic quantities related to con�gurations B0 and B

The kinematical quantities related to the undeformed and deformed con�gurations B0 and B aresummarized for convenience in Figure 1.Then, the spatial Cauchy and Kirchho� stress �elds b and c= Jb together with the two-point 1.

Piola–Kirchho� stress �eld � are introduced as

b= �IJgI ⊗ gJ and c= �IJgI ⊗ gJ ⇒ �= �IJGI ⊗ gJ (5)

Thereby, spatial stress tensors inherit the following symmetry properties

b= bt and c= ct ⇒ F · �=�t · Ft (6)

Then, neglecting inertia, the balance of linear momentum manifests itself in B and B0 as

div bt + b= 0 and Div�t + B= 0 (7)

and is supplemented by Dirichlet and Neumann boundary conditions for the deformation e andthe traction vector t on @Bt or T on @Bt

0, respectively

e=ep on @Bx and bt · n= tp on @Bt or �t ·N=Tp on @Bt0 (8)

Here, distributed body forces B and b per unit volume in B0 and B, respectively, are related byB= Jb. The kinetic quantities related to the undeformed and deformed con�gurations B0 and Bare summarized for convenience in Figure 2.

2.2. Isotropic damage framework

In classical isotropic local damage the free energy is postulated as

(d;F)= [1− d]W (F) (9)



Figure 3. Update algorithm for isotropic local damage

and is assumed to be the product of a reduction factor 1 − d in terms of a damage variable06d61 and the local stored energy W of the undamaged material which is supposed to be anobjective and isotropic function in F

W (Q · F)=W (F)=W (F ·Qt) ∀Q∈SO(3) (10)

Then, exploiting the Clausius Duhem inequality �t : l − ¿0 with l= F · F−1 the spatial velocitygradient renders the constitutive relations for � and � in terms of e�ective Kirchho� and 1. Piola–Kirchho� stresses � and � whereby W serves as a potential

ct = [1− d]� t ; � t = @FW · Ft in B and �t = [1− d]�t ; �t= @FW in B0 (11)

Moreover, W turns out to be conjugated to d, thus a damage condition together with an associateddamage evolution law is motivated as

�(W ;d)=�(W )− d60 with d= �@W� (12)

with �(•) a monotonic function of its argument, see e.g. [23] for the small strain case. The damageevolution law together with the set of Kuhn–Tucker equations

�(W ;d)60 and �¿0 and ��(W ;d)= 0 (13)

follow from the postulate of maximum dissipation. The damage consistency condition in the case ofloading characterized by �=0 and �¿0 allows the closed form update for the damage parameter

��(W ;d)= 0 ⇒ W = �¿0 ⇒ d=�(�) (14)

whereby � is computed with �0 the initial damage threshold from

�= max−∞¡s¡t

(W (s); �0) (15)

Within a numerical step by step solution strategy the corresponding update algorithm for isotropiclocal damage is summarized in Figure 3.Non-local or higher gradient continuum theories are motivated by micro defect interaction, see

e.g. [24]. In the case of higher gradient damage the local stored energy W is simply substitutedby its non-local counterpart �W for the computation of the internal variable �

�= max−∞¡s¡t

( �W (s); �0) (16)


762 P. STEINMANN

Figure 4. Update algorithm for isotropic non-local damage

Thus the corresponding update for isotropic non-local or higher gradient damage is simpli�ed tothe algorithm in Figure 4. The central remaining question is now how to compute the non-localstored energy �W which is the essential input data for the algorithm in Figure 4 in the geometricallynon-linear case?As a reminder the line of arguments leading to the proposal of Peerlings et al. [2] and Peerlings

et al. [3] for the small strain case is brie y outlined. A truly non-local integral de�nition of �W inthe spirit of Pijaudier-Cabot and Ba�zant [18] and Ba�zant and Pijaudier-Cabot [19] is given in termsof a weight function g(^) with ^ denoting the distance from point x to its neighbourhood by

�W (x)=

∫V

g(^)W (x + ^) dV∫V

g(^) dV(17)

Following Lasry and Belytschko [8] and M�uhlhaus and Aifantis [9] a Taylor series expansion forW (x + �) around x

W (x + ^)=W (x) +∇xW (x) · ^+ 12!∇x∇xW (x) : [^⊗ ^] + · · · (18)

renders under the assumption of an isotropic weight function g(^)= g(|^|) the following di�erentialequation for the higher gradient approximation of the non-local stored energy

�W =W + div(c∇xW ) + · · · (19)

Finally, an alternative approximation of the same order which circumvents higher continuityrequirements within a �nite element setting by introducing �W as an independent variable wasproposed by Peerlings et al. [2; 3] as

W = �W − div(c∇x �W )− · · · (20)

The central objective of this contribution is to investigate the possibilities for the extension of thisaveraging di�erential equation to the large strain case.

2.3. Balance of micro momentum

First, we recall that the local and the non-local stored energy W and �W are de�ned per unitvolume of the undeformed con�guration B0, thus we may alternatively introduce the local and the



Figure 5. Material and spatial NSE and their gradients related to con�gurations B0 and B

non-local stored energy w and �w per unit volume in the deformed con�guration B from∫B0

W dV =∫B

w dv;∫B0

�W dV =∫B

�w dv ⇒ Jw=W; J �w= �W (21)

Then we de�ne for the material NSE �W the covariant material gradient H in B0 and its corre-sponding covariant push-forward h in B as

H=∇X �W = hIG I ⇒ h=∇x �W =F−t ·H= hIgI (22)

Likewise, for the spatial NSE �w we may introduce the alternative covariant spatial gradient h inB and its corresponding covariant pull-back H in B0 as

h=∇x �w= hIgI ⇒ H=∇X �w=Ft · h= hIGI (23)

Please note that the spatial NSE gradients h and h transform objectively under a superposed rigidbody motion with e∗(X)=Q · e(X) + k∗ as

h∗=Q · h and h∗=Q · h with Q∈SO(3) and k∗ ∈R3 (24)

The di�erent options for the NSE together with their gradients are summarized in Figure 5.Next, in order to set up a balance equation for the computation of the non-local stored energy �W,

we �rst introduce a contravariant vector �eld ` over the deformed con�guration B which weshall call the spatial Cauchy-type NSE ux. Accordingly, we may introduce the weighted spatialKirchho�-type NSE ux p in B and, as a contravariant pull-back, the material Piola–Kirchho�-typeNSE ux � in B0 as

`= �IgI and p= J`=pIgI ⇒ �=F−1 · p=pIGI (25)

With these prerequisites at hand we may propose the following spatial and material balance equa-tions for the non-local stored energy consisting of a di�usion, an adsorption and a reaction term

div `+ �w=w in B ⇒ Div� + �W =W in B0 (26)

which are complemented by Neumann boundary conditions for the NSE ux

` · n= �p on @B� ⇒ � ·N=�p on @B�0 (27)

and possibly Dirichlet boundary conditions for �W and �w

�W = �Wp on @B �w0 and �w= �wp on @B �w (28)

The di�erent NSE uxes and boundary conditions are summarized in Figure 6.


764 P. STEINMANN

Figure 6. Material and spatial NSE ux and boundary conditions in B0 and B

Figure 7. Option 1: Lagrange averaging

The remaining task would be to relate the spatial and material NSE ux to the spatial andmaterial NSE gradients, respectively, whereby an isotropic relation in terms of a scalar so-calledgradient parameter c0 is certainly the simplest choice. Here we have essentially four di�erentoptions involving the alternative parameters c= J−1c0 and C = Jc0:

Option 1: Lagrange averaging. Option 1 is consistent with a Taylor series approximation toan integral non-local model for �W whereby the integral is evaluated entirely in the referencecon�guration

�W (X)=

∫V0

G(�)W (X + �) dV∫V0

G(�) dV⇒ �W − Div(c0∇X �W )=W (29)

Thus the same material particle in the relative distance � to the material point X is weighed bythe same factor G(�) throughout the deformation process. In other words, the domain of in uenceis convected with the deformation, see Figure 7. By comparing the averaging equation with thematerial balance of micro momentum we conclude that the material NSE ux � is directly relatedto the material gradient H of the material NSE �W. Consequently, in the current con�guration thecontravariant metric c−1 enters the relation between the spatial NSE ux ` or p and the spatialNSE gradient h

`=−cc−1 · h and p=−c0c−1 · h ⇒ �=−c0G−1 ·H (30)

With HI =GIJ hJ evaluating p and � explicitly with respect to the covariant base vectors gI andGI renders the representation

`=−cHIgI and p=−c0HIgI ⇒ �=−c0HIGI (31)

Note the introduction of the deformation-dependent spatial gradient parameter c= J−1c0.



Figure 8. Option 2: Euler averaging

Option 2: Euler averaging. Option 2 is consistent with a Taylor series approximation to anintegral non-local model for �w whereby the integral is evaluated entirely in the current con�guration

�w(x)=

∫V

g(^)w(x + ^) dv∫V

g(^) dv⇒ �w − div(c0∇x �w)=w (32)

Thus throughout the deformation process di�erent material particles in the relative distance ^ tothe material particle currently at the spatial point x are weighed by the same factor g(^). Inother words, the domain of in uence is spatially �xed, see Figure 8. By Comparing the averagingequation with the spatial balance of micro momentum we conclude that the spatial NSE ux ` or pis related to the spatial gradient h of the spatial NSE �w. Therefore, in the reference con�gurationthe contravariant metric C−1 enters the relation between the material NSE ux � and the materialgradient H of the spatial NSE

`=−c0 g−1· h and p=−C g−1· h ⇒ �=−C C−1· H (33)

With hI = gIJ hJ evaluating p and � explicitly with respect to the covariant base vectors gI andGI renders the representation

`=−c0 hIgI and p=−ChIgI ⇒ �=−ChIGI (34)

Note the introduction of the deformation-dependent material gradient parameter C = Jc0.

Option 3: Mixed averaging I. Option 3 is inspired by similar relations for the heat ux within athermomechanical coupled setting at large strain where the Kirchho� heat ux, say q, is typicallyexpressed as a negative constant fraction, the conductivity, of the spatial temperature gradient ∇x�.As a formal motivation the metric tensors c−1 and G−1 in option 1 are simply exchanged to themetric tensors g−1 and C−1. Consequently, we may linearily relate the spatial NSE ux ` or p tothe spatial gradient h of the material NSE �W. Correspondingly, in the reference con�guration thecontravariant metric C−1 enters the relation between the material NSE ux � and the materialgradient H of the material NSE

`=−cg−1 · h and p=−c0g−1 · h ⇒ �=−c0C−1 ·H (35)

With hI = gIJ hJ evaluating p and � explicitly with respect to the covariant base vectors gI andGI renders the representation

`=−chIgI and p=−c0hIgI ⇒ �=−c0hIGI (36)


766 P. STEINMANN

Option 4: Mixed averaging II. Option 4 is inspired by similar relations for the Darcy seepagevelocity within a poromechanical coupled setting at large strain where the material Darcy seepagevelocity, say W, is typically expressed as a negative fraction, the permeability, of the materialgradient of the excess pore pressure ∇X �e. As a formal motivation the metric tensors g−1 andC−1 in option 2 are simply exchanged to the metric tensors c−1 and G−1. Consequently, wemay linearly relate the material NSE ux � to the material gradient H of the spatial NSE�w. Correspondingly, in the spatial con�guration the contravariant metric c−1 enters the relationbetween the spatial NSE ux ` or p and the spatial gradient h of the spatial NSE

`=−c0c−1 · h and p=−Cc−1 · h ⇒ �=−CG−1 · H (37)

With H I =GIJ hJ evaluating p and � explicitly with respect to the covariant base vectors gI andGI renders the representation

`=−c0H IgI and p=−CH IgI ⇒ �=−CH IGI (38)

Remark. It is easy to verify that the spatial NSE ux ` transforms objectively under a super-posed rigid body motion with e∗(X)=Q · e(X) + k∗ for all options 1–4

`∗=Q · ` with Q∈SO(3) and k∗ ∈R3 (39)

3. WEAK FORM OF THE COUPLED PROBLEM

As a prerequisite for a �nite element discretization the coupled non-linear boundary value problemhas to be reformulated in a weak form.


The balance of the linear momentum and the corresponding Neumann boundary conditions aretested by a virtual velocity �v to render the virtual work expression

Gextx − Gintx =0 ∀�v (40)

Here, the internal contribution Gintx to the virtual work expands into the standard alternative rep-resentations in terms of spatial or material quantities as

Gintx =∫B0

[1− d]�F : �t dV =∫B0

[1− d]l� : �t dV =∫B

[1− d]l� : �t dv (41)

whereby we expressed the stress explicitly in terms of the e�ective stress and the reduction factor[1−d]. Moreover, �F and l� denote the material and spatial gradients ∇X �v and ∇x�v, respectively,of the virtual velocity �v. Finally, the external contribution Gextx to the virtual work is given as

Gextx =∫B0

�v · B dV +∫@Bt

0

�v · Tp dA=∫B

�v · b dv+∫@Bt�v · tp da (42)

For the spatial discretization we resort to the Bubnov–Galerkin �nite element method with typicalpolynomial expansions �vh; xh;Xh ∈P2 ⊗ P2 for planar quadrilateral elements.




The balance of micro momentum and the corresponding Neumann boundary conditions are testedby a virtual NSE � �! to render the weak form

Gext�w + Grea�w − Gint�w =0 ∀� �! (43)

Thereby, � �! might be interpreted as either the virtual material NSE � �W per unit volume in thereference con�guration (options 1 and 3) or as virtual spatial NSE � �w per unit volume in thespatial con�guration (options 2 and 4). Then, the internal contribution G �w

int to the weak formexpands alternatively in terms of spatial or material quantities as

Gint�w =∫B0

�H ·� dV =∫B0

h� · p dV =∫B

h� · ` dv (44)

whereby �H and h� denote the material and the spatial gradients ∇X � �! and ∇x� �!, respectively,of the virtual NSE � �!. Finally, the reaction contribution Grea�w to the weak form takes the format

Grea�w =∫B0

� �![ �W −W ] dV =∫B

� �![ �w − w] dv (45)

Moreover, the external contribution G �wext to the weak form is given as

Gext�w =∫@B�

0

� �!�p dA=∫@B�� !�p da (46)

For the spatial discretization we resort to the Bubnov–Galerkin �nite element method with typicalpolynomial expansions �!h; �wh; �Wh ∈P1 ⊗ P1 for planar quadrilateral elements.

4. LINEARIZATION


The linearization of the e�ective stresses is given in terms of the fourth-order Lagrange andEuler e�ective tangent operators

��t= L : �F and � t� = E : l� (47)

whereby �� =�� − l� · � denotes the nominal linearization increment of the e�ective Kirchho�stress, i.e. the push-forward of ��=F−1 · ��, and �F and l� represent the material and the spatialgradients ∇X�e and ∇x�e, respectively, of the linearization increment �e. Here, the e�ectivetangent operators contain the geometric sti�ness contributions and expand as

L= @2FFW = EIJKLgI ⊗GJ ⊗ gK ⊗GL and E= EIJKLgI ⊗ gJ ⊗ gK ⊗ gL (48)

Next, the linearization of the damage parameter d is expressed with H (•) the Heaviside function as�d=D� �W with D=H (�)@�� (49)


768 P. STEINMANN

Finally, the linearization of the internal contribution Gintx to the virtual work expands into thefollowing format:

�Gintx =∫B0

�F : [L : �F−D �t� �W ] dV =

∫B0

l� : [E : l� −D�t� �W ] dV (50)

Here, we abbreviated [1− d]L and [1− d]E into L and E, respectively.

Remark. Please note that we have to take into account the relation �W = J �w for options 2 and 4which are parametrized in �w, i.e.

�W = J� �w + �WF−t : �F= J� �w + �Wg−1 : lsym� (51)

thus an additional contribution to L and E has to be considered.


Concerning the linearization of the internal and reaction contributions Gint�w and Grea�w to the weakform we have to distinguish between the four options for the constitutive laws for the NSE ux:

Option 1: Lagrange averaging. For the �rst option the linearization of the NSE ux results in

��=−G−1 · [c0�H] and p� =−h� · c−1 · [c0 h�] (52)

thus the linearization of G �wint results in the simple expression

�G �wint =−

∫B0

�H ·G−1 · [c0 �H] dV =−∫B

h� · c−1 · [c h�] dv (53)

whereby �H and h� denote the material and spatial gradients ∇X� �W and ∇x� �W, respectively, ofthe linearization increment � �W . Note, that there are no additional geometric terms in contrast tothe following options. Moreover, the linearization of the reaction contributions G �w

rea to the weakform takes the format

�G �wrea =

∫B0

� �![� �W − �t : �F] dV =∫B

� �![J−1� �W − �t : lsym� ] dv (54)

Option 2: Euler averaging. For the second option the linearization of the NSE ux results in

��= − C−1 · [C� H +�C ·�] and p� = − g−1 · [Ch� + 2 lsym� · p] (55)

thus the linearization of Gint�w expands into a constitutive and a geometric part

�Gint�w = −∫B0

�H · C−1 · [C� H +�C ·�] dV = −∫B

h� · g−1 · [c0h� + 2 lsym� · `] dv (56)

whereby � H and h� denote the material and spatial gradients ∇X� �w and ∇x� �w, repectively, ofthe linearization increment � �w. Likewise, we introduced the abbreviations �C=�C− 1=2[�C :C−1]C and l

sym� = lsym� − 1=2[lsym� : g−1]g. Moreover, the linearization of the reaction contributions

Grea�w to the weak form takes the format

�Grea�w =∫B0

� �![J� �w + Mt: �F] dV =

∫B

� �![� �w + mt : lsym� ] dv (57)



Figure 9. Model problem: bar under tension

Here, it is interesting that M= �WF−1 − � and m= �wg−1 − b might be interpreted as particularversions of the e�ective Eshelby energy–momentum tensor whereby the original contribution ofthe stored energy is substituted by its NSE counterpart.

Option 3: Mixed Averaging I. For the third option the linearization of the NSE ux results in

��= − C−1 · [c0�H +�C ·�] and p� = − g−1 · [c0 h� + 2lsym� · p] (58)

therefore the linearization of Gint�w expands as well into a constitutive and a geometric part


�H · C−1 · [c0�H +�C ·�] dV = −∫B

h� · g−1 · [c h� + 2lsym� · `] dv (59)

Here, the linearization of the reaction contributions Grea�w to the weak form obviously takes thesame format as for option 1.

Option 4: Mixed Averaging II. For the last option the linearization of the NSE ux results in

��= −G−1 · [C� H +� �C ·�] and p� = − c−1 · [Ch� + 2�l sym� · p] (60)

therefore the linearization of Gint�w expands as well into a constitutive and a geometric part


� H ·G−1 · [C� H +� �C ·�] dV = −∫B

h� · c−1 · [c0h� + 2 l sym� · `] dv (61)

whereby we introduced the abbreviations � �C= − 1=2[�C :C−1]G and �lsym� = − 1=2[lsym� : g−1]c.

Moreover, the linearization of the reaction contributions Grea�w to the weak form takes obviouslythe same format as for option 2.

5. MODEL PROBLEM

As a model problem we examine the bar in Figure 9 under uniaxial tension. The problem statement,which includes a slight geometric imperfection in the middle of the bar, is taken from Peerlingset al. [2] whereby homogeneous Neumann boundary conditions for the NSE ux were prescribedat the boundary. The material is modelled based on the local stored energy function W of theBlatz and Ko expansion for carbon �lled rubbers in terms of the principal values �A of C

W =��2

[[3∑A=1

�A − 3 + 1k

3∏A=1�−kA − 1

k

]+1− ��

[3∑A=1

�−1A − 3 + 1k

3∏A=1�kA −

1k

]](62)


770 P. STEINMANN

Figure 10. Damage evolution versus internal variable

with principal Kirchho� stresses �A=2@�AW�A given by

�A= �[�[�A −

3∏B=1�−kB

]+ [1− �]

[3∏B=1�kB − �−1A

]](63)

For the damage evolution law we specify the function �(�) with �0 the initial damage thresholdand �; h material parameter as

�(�)= 1− exp(h[�0 − �]) (64)

The corresponding damage evolution with an increasing internal variable �¿�0 is depicted inFigure 10. The material parameters are set to �=1·8N=mm2; k ≈ 0; �=0·5; �0 = 1·0N=mm2and h=0·1mm2=N. In the following we examine the in uence of the type of averaging, thediscretization density and the gradient parameter c0. Thereby, due to symmetry, only one half ofthe bar is considered.First the load versus displacement results for the case of the local damage are displayed in

Figure 11 for computations with discretizations of the total bar into 40, 80, 160, and 320 elementswith biquadratic P2 ⊗ P2 expansions for the displacement and bilinear expansions P1 ⊗ P1 for theNSE. Thereby, the load is applied with arclength control. Please observe that we impose truly�nite strains with a maximum stretch of over 2·5. The curves re ect the typical behaviour of localsoftening computations with a non-converging post-peak branch upon mesh re�nement. The kinkin the pre-peak regime of the curves denote the onset of the damage accumulation at an elasticstretch of ≈2. The corresponding distributions of the damage variable with a maximum value of≈1 at the center of the bar are plotted over B0 in Figure 12 with basically a concentration ofdamage in only one element.The corresponding results for the case of gradient enhanced damage based on Lagrange averaging

with a varying gradient parameter c0 = 1; 5 and 10mm are given in Figures 13–18. The load versusdisplacement curves in Figures 13, 15 and 17 demonstrate clearly mesh convergence for all thevalues of the gradient parameter c0. Moreover, higher values of the gradient parameter render



Figure 11. Load versus displacement curves for local damage computations

Figure 12. Damage distribution for local damage computations


772 P. STEINMANN

Figure 13. Load versus displacement curves for non-local damage computations: Lagrange averaging c0 = 1

Figure 14. Damage distribution for non-local damage computations: Lagrange averaging c0 = 1


774 P. STEINMANN





Figure 19. Load versus displacement curves for non-local damage computations: Euler averaging c0 = 5

Figure 20. Damage distribution for non-local damage computations: Euler averaging c0 = 5


776 P. STEINMANN

Figure 21. Load versus displacement curves for non-local damage computations: Mixed averaging I c0 = 5

Figure 22. Damage distribution for non-local damage computations: Mixed averaging I c0 = 5



Figure 23. Load versus displacement curves for non-local damage computations: Mixed averaging II c0 = 5

Figure 24. Damage distribution for non-local damage computations: Mixed averaging II c0 = 5


778 P. STEINMANN

a somewhat more ductile post-peak behaviour after the onset of damage accumulation with amaximum stretch of ≈3, see Figures 13, 15 and 17. The distribution of the damage variable withmaximum value of ≈1 in Figures 14, 16 and 18 display a broadening of the localized zone withincreasing c0. Thereby, the localized zone is resolved with a number of elements and possesses asmooth pro�le of the damage variable.Next, Figures 19 and 20 present the results for the case of gradient enhanced damage based

on Euler averaging with gradient parameter set to c0 = 5mm. Thereby, mesh convergence in theload versus displacement curves is hardly visible in Figure 19. On the other hand, the non-smoothdistribution of the damage variable with maximum value of ≈1 in Figure 20 demonstrates a distinctfeature with respect to the local computations since the localized zone is resolved by more thanone element.Gradient enhanced damage based on mixed averaging I with c0 = 5mm renders again no

observable mesh convergence in Figure 21. Nevertheless, the smooth damage pro�le in Figure22 resembles the distribution obtained for the case of Lagrange averaging. In particular the local-ized zone is resolved by a number of elements.Finally, Figures 23 and 24 for the case of mixed averaging II with c0 = 5mm display a similar

tendency as was observed for Euler averaging.

6. SUMMARY AND CONCLUSIONS

The objective of this work was the extension of a small strain and small deformation formulationof gradient enhanced damage to the geometrically non-linear case. Therefore, material and spatialnon-local stored energy densities were treated as primary variables. For the computation of thecorresponding uxes, material and spatial balance laws were proposed. Thereby the motivation wasprovided by the well documented averaging equation of the small strain limit. Clearly, it was thenthe principal task to construct appropriate constitutive laws for the NSE- uxes. To this end, thesame averaging argument as in the small strain case was pursued, whereby the averaging domainwas speci�ed either as material, termed Lagrange averaging, or spatial, termed Euler averaging.Moreover, by changing the metric tensors two additional formulations termed mixed averaging Iand II were obtained. The associated FE-formulation and solution strategy were discussed in detail,whereby aspects due to the geometrical non-linearities were emphasized.The comparison of these four di�erent formulations for the example of a bar in tension at

truly �nite strains revealed that only the Lagrange averaging procedure inherits all the desireablefeatures encountered in the small strain case: post-peak load versus displacement responses withoutpathological mesh dependence and discretization independent width of localization zones which areresolved by more than only one element row. All other formulations lack the mesh independentpost-peak response, although localized zones are resolved by more than one element. A partialexplanation is that the underlying in uence domain of integration consists of the same materialpoints throughout the deformation history only for the Lagrange averaging case. It is worthwhileto note that the Lagrange averaging option not only performs optimally but it is also the easiestoption for implementation since the linearization of the NSE- ux does not contribute additionalgeometrical sti�nesses.In summary it is believed that this contribution has clari�ed issues on how to formulate and

implement gradient enhanced damage at large strains whereby the motivation was typically pro-vided by the failure processes of rubber constructions at large strains.



REFERENCES

1. De Borst R, Pamin J, Peerlings RHJ, Sluys LJ. On gradient-enhanced damage and plasticity models for failure inquasi-brittle and frictional materials. Computational Mechanics 1995; 17:130–141.

2. Peerlings RHJ, De Borst R, Brekelmans WAM, De Vree JHP. Gradient enhanced damage for quasi-brittle materials.International Journal for Numerical Methods in Engineering 1996; 39:3391–3403.

3. Peerlings RHJ, De Borst R, Brekelmans WAM, De Vree JHP, Spee I. Some observations on localization in non-localand gradient damage models. European Journal of Mechanics A=Solids 1996; 15:937–953.

4. Aifantis EC. On the microstructural origin of certain inelastic models. Journal of Engineering Materials andTechnology 1984; 106:326–330.

5. Aifantis EC. On the role of gradients in the localization of deformation and fracture. International Journal ofEngineering Science 1992; 30:1279–1299.

6. Steinmann P. Views on multiplicative elastoplasticity and the continuum theory of dislocations. International Journalof Engineering Science 1996; 34:1717–1735.

7. Schreyer HL, Chen Z. One-dimensional softening with localization. Journal of Applied Mechanics 1987; 53:791–979.8. Lasry D, Belytschko T. Localization limiters in transient problems for strain-softening in statics and dynamics.International Journal of Solids and Structures 1998; 24:581–597.

9. M�uhlhaus HB, Aifantis EC. A variational principle for gradient plasticity. International Journal of Solids andStructures 1991; 28:845–857.

10. De Borst R, M�uhlhaus HB. Gradient-dependent plasticity: formulation and algorithmic aspects. International Journalfor Numerical Methods in Engineering 1992; 35:521–539.

11. Vardoulakis I, Aifantis EC. A gradient ow theory of plasticity for granular materials. Acta Mechanica 1991; 87:197–217.

12. Fleck NA, Muller GM, Ashby MF, Hutchinson JW. Strain gradient plasticity: theory and experiment. Acta MetallurgicaMaterials 1994; 42:475–487.

13. Fleck NA, Hutchinson JW. A phenomenological theory for strain gradient e�ects in plasticity. Journal of the Mechanicsand Physics of Solids 1993; 41:1825–1857.

14. Sluys LJ, De Borst R, M�uhlhaus HB, Wave propagation, localization and dispersion in a gradient-dependent medium.International Journal of Solids and Structures 1993; 30:1153–1171.

15. Pamin J. Gradient-dependent plasticity in numerical simulation of localization phenomena, Dissertation, DelftUniversity of Technology, Delft, 1994.

16. Huerta A, Pijaudier-Cabot G. Discretization in uence on regularization by two localization limiters. ASCE Journal ofEngineering Mechanics 1994; 120:1198–1218.

17. Svedberg T, Runesson K. An algorithm for gradient-regularized plasticity coupled to damage based on a dual mixedFE-formulation. Computer Methods in Applied Mechanics and Engineering 1998; 161:49–65.

18. Pijaudier-Cabot G, Bazant ZP. Nonlocal damage theory. Journal of Engineering Mechanics ASCE 1987; 113:1512–1533.

19. Bazant ZP, Pijaudier-Cabot G. Nonlocal continuum damage, localization instability and convergence. Journal ofApplied Mechanics 1988; 55:287–293.

20. De Borst R, Benallal A, Heeres O. A gradient-enhanced damage approach to fracture. Journal of Physics IV 1996;6:491–502.

21. Kuhl E, Ramm E, De Borst R. An anisotropic gradient damage model for quasi-brittle materials, 1999, ComputerMethods in Applied Mechanics and Engineering.

22. Mahnken R, Kuhl E. A �nite element algorithm for parameter identi�cation of gradient enhanced damage models.European Journal of Mechanics=A:Solids 1999, in press.

23. Simo JC, Ju JW, Strain- and stress-based continuum damage models: I. Formulation. International Journal of Solidsand Structures 1987; 23:821–840.

24. Bazant ZP. Why continuum damage is nonlocal. Journal of Engineering Mechanics 1991; 117:1070–1087.


Formulation and computation of geometrically non-linear gradient damage

Documents

Transcript of Formulation and computation of geometrically non-linear gradient damage