Incompatible mode method for finite deformationquasi-incompressible elasticity

F. Gharzeddine, A. Ibrahimbegovic

Abstract In this work we develop a geometrically non-linear version of the method of incompatible modes,suitable for quasi-incompressible ®nite deformationhyperelasticity. The proposed method is featuring theprincipal axis representation of the theory, facilitating thechoice of the strain energy function (in terms of theprincipal stretches) and simplifying the stress computa-tion. The choice of the spatial Cauchy-Green strain mea-sure, leading to a very sparse structure of the strain-displacement operators, and the operator split solution ofequilibrium equations, leading to reduced secondarystorage requirements, further increase the computationalef®ciency. A set of numerical examples is used to illustratea robust performance of the constructed plane strain ele-ment with a single incompatible mode in quasi-incom-pressible deformation patterns.

1IntroductionThe method of incompatible modes was initially proposedas an interesting possibility to enhance the performance oflow-order (typically 4-node) element in bending domi-nated deformation patterns (e.g., see Wilson et al. 1973) inlinear elasticity. Keeping such a motivation in mind, theoriginal method of incompatible modes has been extendedto other choices of interpolations (e.g., 9-node element, seeIbrahimbegovic and Wilson 1991) and fully nonlinearproblems (e.g., see Ibrahimbegovic and Frey 1993).

It was soon noted that the incompatible mode methodalso performs quite well and can be used as a successfulalternative to mixed methods (e.g., see Atluri and Reissner1989) in linear analysis of quasi-incompressible elasticmaterials, either in its original version (see Ibrahimbego-vic and Wilson 1991) or in its modi®ed version, currentlyreferred to as enhanced strain method (e.g., see Simo andRifai 1990). The extension of the enhanced strain methodto fully nonlinear analysis (e.g., Simo and Armero 1992)was also shown to inherit a superior performance (withrespect to the standard isoparametric interpolations) inquasi-incompressible large deformation patterns.

However, several works have soon followed drawingattention to a lack of robustness of the enhanced strainmethod in the large strain problems. Namely, the analysispresented in Wriggers and Reese (1996) identi®ed theappearance of a non-physical zero eigenvalue at a certainstage of the compression of a single quadrilateral elementin a homogeneous state of strain. Several authors (Cris®eldet al. 1995; De Souza et al. 1995, among others) have alsoreported similar performances. Although the standarddisplacement model does not present this de®ciency it iswell known that the presence of incompressibility of thematerial (e.g. rubber), leads to the so-called locking phe-nomenon with such a discretization. A couple of possi-bilities to eliminate hourglass instability effect incompression have been presented by Korelc and Wriggers(1996) or Wriggers and Korelec (1996) and Glaser andArmero (1997).

In this paper, we present a geometrically nonlinearversion of the original method of incompatible modes,geared towards quasi-incompressible large deformationproblems. The low-order (4-node) ®nite element inter-polations are still used for constructing the basis of theelement, but the main difference from our previous workon geometrically nonlinear method of incompatible modes(see Ibrahimbegovic and Frey 1993) is in the choice ofincompatible modes. Namely, it is demonstrated by theresults to follow that the incompatible modes which arethe most appropriate for bending are less so for quasi-incompressible deformation patterns and that the latterrequires somewhat different choice of the incompatiblemodes.

The outline of the paper is as follows. In the nextsection, we brie¯y recall the fundamental ingredients ofthe ®nite deformation elasticity in principal axes. In Sects.3, 4 and 5 we discuss the numerical implementationdetails. Section 6 includes some illustrative numericalsimulations that assess the performance of this elementin compression. Finally, the conclusions are drawn inSect. 7.

2Formulation of finite deformation elasticityAmong different possibilities to formulate the ®nite de-formation elasticity theory, we choose the one employingthe spatial strain measure, i.e. the strain measure de®nedin the current con®guration. More precisely, our choiceis the Cauchy-Green deformation tensor, b, which repre-sents the corresponding transformation of the metrictensor in the initial con®guration, G

Computational Mechanics 24 (2000) 419±425 Ó Springer-Verlag 2000

Received 14 December 1998

F. Gharzeddine (&), A. IbrahimbegovicUniversite de Technologie de CompieÁgne,Dept. GSM, Lab. G2MS, CNRS UPRES A 6066,BP 529, 60206 CompieÁgne, France

This work was supported by MENERT and CNRS (LG2mS). Thissupport is gratefully acknowledged.

419

b � FGÿ1FT �1�In (1) above, F is the deformation gradient. We assumehyperelastic material, which implies the existence of astrain energy function w���. Invariance requirements im-pose that the strain energy function be de®ned in terms ofinvariants of the tensor b, or in a more convenient formwhich is obtained by using the principal stretches ki, i.e.

w � w�ki� �2�The principal stretches are the solution to the followingeigenvalue problem

�bÿ �ki�2gÿ1�mi � 0 �3�where g is the metric tensor in the current con®guration.

Standard thermodynamics considerations directly leadto the constitutive equation for energy-conjugate stresstensor given in terms of the Kirchhoff stress, s. Namely,in context of purely mechanical theory the latter can bede®ned as

D :� s � dÿ owog

Lv�g� �4�

where d is the rate-of-deformation tensor and Lv�g� is theLie derivative of g. Considering that Lv�g� � 2d, from (4)above we get

s � 2owog

�5�

Isotropy implies that the Kirchhoff stress tensor s sharesthe same eigenvectors mi found in the eigenvalue problemin (3), so that we can write

�sÿ sigÿ1�mi � 0 �) s �

Xi

sigÿ1mi gÿ1mi ; �6�

where si are the principal values of the Kirchhoff stress.One can readily compute an explicit form of the con-

stitutive equation in the principal axis representation. Tothat end, we ®rst compute the directional derivative of theeigenvalue problem statement in (3) to get (e.g., seeIbrahimbegovic 1994)

oki

og� ki

2gÿ1mi gÿ1mi �7�

Next, by using the chain rule and the last result, we canrewrite (5) in the principal axis representation as

s �X

i

kiowoki

gÿ1mi gÿ1mi �8�

Comparing the last result with the one in (6), we concludethat

si � kiowoki

�9�

Therefore, the principal axis methodology leads to a veryef®cient computational framework, since the tensor cal-culus can be reduced to manipulation of scalars, i.e. thecorresponding principal values.

3Variational basis of the incompatible mode methodIn this work, we apply the method of incompatible modesto the ®nite strain quasi-incompressible elasticity, fol-lowing in the footsteps of the previous works on thesubject (e.g., see Simo and Armero 1992 or Ibrahimb-egovic and Frey 1993). To that end, the key assumption tobe used is the one on the multiplicative decomposition ofthe deformation gradient in which one abandons thestandard deformation gradient, F, in favor of an enhancedgradient, ~F,

~F � �I� a�F �10�In (10) above, I is a unit, two-point tensor, a is the spatialdisplacement gradient, so that I� a represents a spatialdeformation gradient superposed on the compatible de-formation gradient F. The graphical interpretation of themultiplicative decomposition is given in Fig. 1.

The spatial deformation gradient a can formally beconstructed by making use of the material gradient A, i.e.

a � AFÿ1 �11�Hence, in the subsequent consideration we use A as one ofthe state variables.

The weak form of the balance equations can now berewritten as

G�u;A;w� :�ZB

s :1

2��r~uw�Tg� gr~uw�dBÿ Gext � 0

�12�where the spatial gradient of the virtual displacement ®eldw is recomputed with the enhanced deformation gradient~F as

r~uw � rw~Fÿ1 �13�However, in contrast with the standard displacement-based interpolation case, we get an additional equationassociated with the variation of the incompatible modeparameters f, which can be written as

G�u;A; f� :�ZB

s :1

2��r~uf�Tg� gr~uf�dB � 0 �14�

where r~uf is the spatial form of the variation of theenhanced displacement gradient given as

r~uf � rf~Fÿ1 �15�

Fig. 1. Multiplicative decomposition of the deformation gradientinto compatible and an enhanced deformation gradient

420

4Discrete approximation of the incompatible mode methodThe crucial simpli®cation introduced in this section is thechoice of Euclidean coordinates, which leads to a simplecoordinate representation of the metric tensor in terms ofthe unit matrix, i.e.

g 7! I �16�Moreover, in the Euclidean framework the total displace-ment vector, d, is well de®ned, and so that the positionvectors in the current con®guration can be constructed as

u � x� d �17�where x is the corresponding position vector in thereference con®guration.

In each element, Be, the displacement vector is inter-polated in the standard isoparametric manner (e.g., seeHughes 1987) as

djBe �Xnen

a�1

Na�x�da �18�

where da are the nodal displacements and Na�x� are theelement shape functions for a particular choice of elementwith nen nodes. The deformation gradient can then beconstructed as

F :� rujBe �Xnen

a�1

�xa � da� rNa;

rTNa � oNa

ox1;oNa

ox2;oNa

ox3

� � �19�

where `' denotes the tensor product.As already noted by Ibrahimbegovic and Frey (1993),

the enhanced displacement gradient can formally be con-structed as the gradient of an incompatible displacement®eld, so that we can write

A �Xnim

b�1

ab rM̂b; rTM̂b � oM̂b

ox1;oM̂b

ox2;oM̂b

ox3

� ��20�

where Mb�x� are incompatible mode shape functions3

and nim is the chosen number of incompatible modes.As noted by Ibrahimbegovic and Frey (1993), the in-compatible mode shape functions must be modi®ed tomake them orthogonal in the energy norm to any con-stant stress ®eld, and thus to ensure the patch testsatisfaction. This can be carried out according to (seeIbrahimbegovic and Wilson 1991)

M̂b � Mb ÿ 1

Be

ZBe

Mb dB �21�

Using the isoparametric interpolations for the virtualdisplacement ®eld and the variation of the enhanced dis-placement gradient, results with

r~uwjBe �Xnen

a�1

wa r~uNa �22�

r~ufjBe �Xnim

b�1

fb r~uM̂b �23�

where wa and fb are, respectively, the virtual displace-ments and virtual enhanced gradient interpolationparameters, and

r~uM̂b � ~FÿTrM̂b �24�The symmetric part of the last two expressions can bewritten in a matrix notation as

sym�r~uw� 7! �jBe �Xnen

a�1

Ba�d;A�wa �25�

and

sym�r~uf� 7! cjBe �Xnim

b�1

Gb�d;A�fb �26�

where B and G can be written, respectively, as

Ba�d;A� �

eT1 �eT

1r~uNa�eT

2 �eT2r~uNa�

eT3 �eT

3r~uNa�eT

1 �eT2r~uNa� � eT

2 �eT1r~uNa�

eT2 �eT

3r~uNa� � eT3 �eT

2r~uNa�eT

3 �eT1r~uNa� � eT

1 �eT3r~uNa�

26666664

37777775 �27�

and

Gb�d;A� �

eT1 �eT

1r~uM̂b�eT

2 �eT2r~uM̂b�

eT3 �eT

3r~uM̂b�eT

1 �eT2r~uM̂b� � eT

2 �eT1r~uM̂b�

eT2 �eT

3r~uM̂b� � eT3 �eT

2r~uM̂b�eT

3 �eT1r~uM̂b� � eT

1 �eT3r~uM̂b�

26666664

37777775 �28�

with ei as the unit base vectors which constitute the unitmatrix I � �e1; e2; e3�. It is apparent from (27) and (28)above that the main advantage of choosing the spatialstrain measures is resulting sparse structure of the strain-displacement operators B and G, which is essentially thesame as the one from the linear theory.

With this matrix notation in hand, the weak form of thegoverning equations in (12) can be restated as

G�d;A;w� :� Anel

e�1

Xnen

a�1

wTa

ZBe

BTa �d;A��s�d;A�dBÿ fa

� �( )

:� Anel

e�1

Xnen

a�1

wTa�raÿ fa�

" #� 0 �29�

where fa is the external load vector,sT � hs11; s22; s33; s12; s23; s31i is the vector containing allthe components of the Kirchhoff stress tensor in (8), andAnel

e�1 is the ®nite element assembly operator (e.g., seeHughes 1987) over all nel elements in the mesh.

3 In choosing the compatible and incompatible displacementshape functions, one must ensure that Na

TMb = ;; Typically,

functions Mb are polynomials of one degree higher that those inNa.

421

Similarly, incompatible mode based residual in (14) canbe rewritten as

G�d;A; f� :�Xnim

b�1

fTb

ZBe

GTb �d;A��s�d;A� dB

:�Xnim

b�1

fTb hb � 0; 8 e 2 �1; nel� �30�

The crucial difference between the last two equationsconcerns the continuity requirements and the resulting®nite element discretization. The equilibrium equation in(29) is associated with the variation of displacement ®eldwith the nodal values of the interpolation parameters, wa,which are shared by all the elements in the neighborhoodof node a. On the other hand, the equilibrium equation in(30) is associated with the variation of the incompatiblemode ®eld, constructed for each element independently.Therefore, the former leads to a set of global, whereas thelatter to a set of local nonlinear algebraic equations. Asdescribed in the following section, a special solution pro-cedure is proposed in order to take advantage of thisparticular structure of the nonlinear equations on hand.

5Operator split in the method of incompatible modesThe main idea of the operator split (e.g., see Chorin et al.1978) is to separate the solution to the second group ofdiscrete equilibrium equation in (30) (for ®xed values ofthe nodal displacements, the best iterative guess), from thesubsequent computation of a new iterative value of thenodal displacement vector from the global equilibriumequations in (29). A more detailed discussion is providednext.

To that end, we assume that d�i� is the best (iterative)value provided for the nodal displacement vector. Keepingthis value ®xed, we proceed to compute the correspondingincompatible mode parameters which will satisfy theequilibrium equation in (30). Since the latter is a set ofnonlinear equations, we need to use an iterative procedure;At a typical iterative sweep, �j�, of such a procedure, wemake use of the linearized form of the equation in (30)

Lin�G�d�i�;A�j�; f��

:� G�d�i�;A�j�; f�� �Xnim

a�1

Xnim

b�1

fTa�He

1;ab � He2;ab�gb

�31�where

He1;ab �

ZBe

GTa�d;A�DGb�d;A� dB �32�

and

He2;ab �

ZBe

sabI3 dB; sab �X3

i;j�1

sij�eTi r~uM̂a��eT

j r~uM̂b�

�33�A typical iterative step of this element-based procedurereduces to

g�j� � ÿH�j�ÿ1h�j�

a�j�1� � a�j� � g�j� �34�The converged values of incompatible mode parameters,satisfying the equilibrium equation in (30), are used toconstruct an element-wise approximation of the incom-patible mode ®eld, �A. Having obtained the incompatiblemode values, we proceed to the next sweep of the globaliterative procedure in order to provide an improved valueof the total displacement ®eld. In that respect we make useof the following linearized forms of the discrete equilib-rium equations

Lin�G�d�i�; �A;w��

:� G�d�i�; �A;w� �Xnen

a�1

Xnen

b�1

wTa�Ke

1;ab � Ke2;ab�ub

�Xnen

a�1

Xnim

b�1

wTa�Fe

1;ab � Fe2;ab�gb �35�

and

Lin�G�d�i�; �A; f�

:� G�d�i�; �A; f� �Xnim

a�1

Xnen

b�1

fTa�Fe

1;ab � Fe2;ab�ub

�Xnim

a�1

Xnim

b�1

fTa�He

1;ab � He2;ab�gb �36�

where

Ke1;ab �

ZBe

BTa�d;A�DBb�d;A�dB �37�

Ke2;ab �

ZBe

pabI3dB;

pab �X3

i;j�1

�sij�eTi ruNa��eT

j ruNb��38�

and

Fe1;ab �

ZBe

GTa�d;A�DBb�d;A�dB �39�

Fe2;ab �

ZBe

rabI3dB; rab �X3

i;j�1

�sij�eTi r~uM̂a��eT

j r~uNb�

�40�By de®nition of incompatible mode approximation �A itfollows that all element-based residuals vanish, i.e.

G�d�i�; �A; f� � 0; 8 e 2 �1; nel� �41�Hence, (36) can be used to compute the increments inincompatible mode parameters in terms of incrementaldisplacement.

ga � ÿXnen

b�1

�Hÿ1F�abub �42�

422

The result in (42) can then be replaced into the linearizedform of the global balance equations in (35), in order toeliminate the incompatible mode parameters

Anel

e�1K̂d � rÿ f ; Kÿ FTHÿ1F �43�

6Numerical examplesTo test the developed plane strain element formulation for®nite elastic deformations a simple model is considered interms of a compressible neo-Hookean material, with thestrain energy of the form

W � 1

2K�ln J�2 � l

2�k2

1 � k22 ÿ 2� ÿ l ln J; k3 � 1

where J � det F, K and l are the Lame material parame-ters.

We denote by Q1=I4 the resulting ®nite element with 2

incompatible modes M1�n� � 1ÿ n2 and M2�g� � 1ÿ g2,and by Q1=I2 the one with a single incompatible modeM�n; g� � n2 � g2. It is interesting to note that the latterchoice also inherits a very robust performance in bending-dominated problems (e.g., see Arunakirinathar and Reddy1995).

6.1Checkerboard pressure modeWe consider the 4 element mesh shown in Fig. 2. We as-sume that all the nodes on the boundary of the mesh are®xed (i.e., the displacements are set to zero), apart themiddle node on the right-hand side, which can movehorizontally. A compressive load is imposed at that noderesulting with highly constrained compressive deforma-tion. The simulation carried out with Q1=I4 produces apathological result where the sign of the Jacobian alter-nates between negative and positive values (see Fig. 3)which is referred to as the checkerboard pressure mode(see Hughes 1987). The same analysis repeated with Q1=I2shows no presence of this checkerboard mode.

6.2Homogeneous compression testWe investigate here the example discussed previouslyby Wriggers and Reese (1996). The right half of a rectan-gular block is discretized by using 10� 10 elements.Symmetry boundary conditions are imposed along the

vertical center line. The bottom of the block is restrainedin the vertical direction only, with an imposed verticaldisplacement at the top. The selected material parametersare K � 40000 and l � 80:2. The block is compressed to ahomogeneous state of deformation. As depicted in Fig. 4,the load de¯ection curves, computed with Q1I2 and Q1I4,practically coincide. When using Q1I4, with up to 45%compression all eigenvalues remain positive. At this stageone eigenvalue turns negative. Figure 5 represents theeigenmode at this ®rst buckling point, exhibiting certainamount of hour-glassing, already noted by Wriggers andReese (1996). A second negative eigenvalue appears at adeformation state of 46:2%. The next two bifurcationpoints are 46:5% and 47:2% of compression. We note thatall four bifurcation points are detected at about the samecompression state. The ®rst bifurcation point detected byQ1=I2 is at a stage of 86%. The associated eigenvector,plotted in Fig. 6, exhibits no hour-glassing. It is interestingto point out that all bifurcation points always appear atabout the same stage of compression even when we re®nethe mesh. The latter is the consequence of the homoge-neous deformation state.

6.3Non-homogeneous compression testIn this example we perform a simulation of non-homoge-neous compression test, as very recently proposed by Reeseet al. (1998). The geometry and the loading of the structure

Fig. 2. Model used for checkerboard pressure mode

Fig. 3. Checkerboard pressure mode ± sign of computedpressure in each element

Fig. 4. Load-de¯ection curve computed with Q1=I2 and Q1=I4elements

423

are plotted in Fig. 7. The material parameters are chosenthe same as in the previous example. The boundary con-ditions at the top of the structure are chosen in such a waythat the nodes cannot move in the horizontal direction. Theanalysis is performed by imposing a pressure of 600. Theresult convergence with mesh re®nement is depicted in Fig.8. The percentage of compression quoted in Fig. 8 appliesto the upper middle node. We can see that by increasing thenumber of elements over the height, we converge to thesolution obtained by Reese et al. (1998), i.e. 50% of com-pression of the structure for the applied load level.

7ConclusionThis works provides an extension of the classical methodof incompatible modes (e.g., see Wilson et al. 1993) to

large strain quasi-incompressible elasticity problems. Themain ®nding is that the original choice of Wilson-type ofincompatible modes, initially proposed for capturing thebending dominated deformation patterns, does not nec-essarily remain optimal for capturing the quasi-incom-pressible deformation patterns, and it suffers from thesame kind of sensitivity already noticed for the enhancedstrain method.

However, reducing the number of incompatible modesto a single one, M�n; g� � n2 � g2, eliminates the fore-mentioned sensitivity, and leads to a robust element per-formance. The latter has been con®rmed by a con®nedpressure (checkerboard provoking) test and a couple ofother examples recently proposed in the literature.

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Fig. 5. Eigenvector associated with the ®rst bifurcationdetected by Q1=I4

Fig. 6. Eigenvector associated with the ®rst bifurcationdetected by Q1=I2

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Fig. 8. Non-homogeneous compression test result: convergencewith mesh re®nement for Q1=I2

Fig. 9. Non-homogeneous compression test: Deformed con®gu-ration at 48:5%

424

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