A new volumetric and shear locking-free 3D enhanced strain element

A new volumetric and shearlocking-free 3D enhanced

strain elementRJ Alves de Sousa

Departamento de Engenharia MecaAtilde nica Universidade de AveiroPortugal

IDMEC plusmn Faculdade de Engenharia Universidade do Porto Portugal

RM Natal JorgeIDMEC plusmn Faculdade de Engenharia Universidade do Porto Portugal

RA Fontes ValenteDepartamento de Engenharia MecaAtilde nica Universidade de Aveiro

PortugalIDMEC plusmn Faculdade de Engenharia Universidade do Porto Portugal

JMA CeAcircsar de SaAcircIDMEC plusmn Faculdade de Engenharia Universidade do Porto Portugal

Keywords Shell structures Strain measurement Shear strength

Abstract This paper focuses on the development of a new class of eight-node solid regnite elementssuitable for the treatment of volumetric and transverse shear locking problems Doing so theproposed elements can be used efregciently for 3D and thin shell applications The starting point ofthe work relies on the analysis of the subspace of incompressible deformations associated with thestandard (displacement-based) fully integrated and reduced integrated hexahedral elementsPrediction capabilities for both formulations are deregned related to nearly-incompressible problemsand an enhanced strain approach is developed to improve the performance of the earlierformulation in this case With the insight into volumetric locking gained and beneregting from arecently proposed enhanced transverse shear strain procedure for shell applications a new elementconjugating both the capabilities of efregcient solid and shell formulations is obtained Numericalresults attest the robustness and efregciency of the proposed approach when compared to solid andshell elements well-established in the literature

Introduction

Several problems of important physical meaning such as incompressible

elasticity plasticity of metals or additionally the macrow of certain macruids involve

the inclusion and treatment of the incompressibility phenomenon

The Emerald Research Register for this journal is available at The current issue and full text archive of this journal is available at

httpwwwemeraldinsightcomresearchregister httpwwwemeraldinsightcom0264-4401htm

It is worth noting the funding by MinisteAcircrio da CieAtildencia e Ensino Superior FundacEumlaAumlo para aCieAtildencia e Tecnologia Portugal under grants PRAXIS XXIBD2166299 as well as FEDER undergrant POCTI33640EME2000 These supports are gratefully acknowledged

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Received September2002Revised May 2003Accepted May 2003

Engineering ComputationsVol 20 No 7 2003pp 896-925

q MCB UP Limited0264-4401DOI 10110802644400310502036

Mathematically incompressibility modelling relates to the imposition of aconstraint over the analysed continuum enforcing the volumetric part of thestrain regeld to be zero or very small when compared to its deviatoriccounterpart However the same treatment involving the frame of the regniteelement method (FEM) imposes some difregculties in particular when low-orderelementsrsquo formulations are employed

Low-order elements offer simpler implementation when compared withhigher-order formulations allowing at the same time straightforward meshoperations to be performed Nevertheless problems can arise in the presence ofincompressibility as stated before leading to the so-called volumetric lockingphenomenon In this case bilinear elements (in two-dimensional analysis) andtrilinear solid elements (in three-dimensional analysis) are not able to grant thecomplete nullity of the volumetric strain This leads to an overestimation ofstiffness values related to the volumetric deformation and in consequence tonear zero values of displacements obtained

Several approaches have been proposed in the latest decades to reduce oralleviate volumetric locking occurrence Reduced and selective reducedintegration (SRI) techniques were the regrst successful irreducible form ofsolutions for locking problems although in the beginning not directedspeciregcally to volumetric locking (Hughes et al 1978 Zienkiewicz et al 1971)For the particular case of the eight-node hexahedral regnite element bothtechniques correspond to the use of a lower quadrature rule (one Gauss pointcorresponding to the elementsrsquo center) in opposition to the so-called completequadrature rule (2 pound 2 pound 2 Gauss points) While the (total) reduced integrationappeared to lead to non-physical (spurious) deformation patterns selectivereduced integration proved quite successful being in some extent thepredecessor of the B-bar method (Hughes 1977) In the latter shape functionsderivatives related to the volumetric part of deformation were replaced byapproximations resulting from a mixed formulation without resorting toreduced integrations

All these methods proved to be effective proposals in alleviating speciregcallythe volumetric locking although their performance in bending dominatedsituations revealed some deregciencies For selective reduced integration andB-bar cases in addition their application is in some extent limited to materialswhose strain (stress) tensor can be decomposed into volumetric and deviatoricparts Other formulations succeeded in using an augmented functional whencompared to that obtained from displacement-based approaches incorporatingadditional regelds into the formulation and leading to the onset of general mixedmethods For the up mixed formulation displacements are interpolated withfunctions providing C 0 continuity requirements while the pressure regeld isintroduced via discontinuous functions between elements (Hughes 2000Zienkiewicz and Taylor 2000)

A new enhancedstrain element

897

In the 1990s the work of Simo and Rifai (1990) introduced the enhancedstrain method In this formulation the strain regeld is enlarged with the inclusionof an extra internal regeld of variables resulting therefore in additionaldeformation modes A particular choice of the extra modes of deformation ledto the QM6 element derived via the incompatible modes method of Wilson et al(1973) a formulation designed to improve the performance of quadrilateralelements in 2D bending problems (Taylor et al 1976)

Due to its versatility the enhanced strain method was successfully appliedin 2D 3D and shell formulations achieving good results even with coarsemeshes (Andelregnger and Ramm 1993 Armero and Dvorkin 2000 de Borstand Groen 1999 de Souza Neto et al 1996 Glaser and Armero 1997 Kasperand Taylor 2000 Korelc and Wriggers 1996 Piltner 2000 Rohel and Ramm1996 Simo and Armero 1992 Simo and Rifai 1990 Simo et al 1993) Thepossibility of respecting some conditions ordffreelyordm adding extra variables(deformation modes) represents an important matter being directly related to agiven elementrsquos performance

Concerning near incompressible situations and speciregcally for 2D planestrain problems the number of additional variables to use in reliablevolumetric locking-free elements can vary from two to four as in the proposalsof CeAcircsar de SaAcirc and Natal Jorge (1999) or Simo and Rifai (1990) among others In3D analysis this number is still a matter of discussion varying from 9 to 12 inaccordance to Korelc and Wriggers (1996) 12 according to Simo et al (1993) orstill 6 to 30 according to Andelregnger and Ramm (1993) only to quote somerelevant publications in the regeld

Still dealing with hexahedral regnite elements but departing from the exposedbefore an interesting application relates to the shell structures modelling Infact 3D elements were the starting point of some shell elements as in the onsetof the degenerated approach with the pioneer work of Ahmad et al (1970) Forthe proper reproduction of this particular kinematics either of Mindlin orKirchhoff-Koiter type some simpliregcations were taken leading to a class ofelements incorporating plane-stress assumptions with undesirable losses in thegeneral character provided by full 3D material laws of solid elements

In this sense a correct reproduction of shells (and plates) structuresbehaviour with the use of tridimensional elements is desirable Apart from theeasier formulation and material considerations solid elements provide astraightforward extension to geometrically non-linear problems in particularin the presence of large rotations once the only degrees of freedom involved are(additive type) translations Another difregculty in dealing with shell elements isthe treatment of corner-like zones (for example in reinforced shell structures)with the resulting drilling degrees of freedom and also the treatment ofconjunction between solids and shells in the same model

For a successful modelling of general shells with no limitations on thicknessvalues transverse shear locking effects must be accounted Transverse shear

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locking phenomenon relates to the inability of a given element to reproduce anull transverse shear strain energy state in pure bending (CeAcircsar de SaAcirc et al2002) Backward in the earlier developments shear locking in degenerated shellelements were revealed as thickness values tended to small values The regrstsuccessful procedures to overcome this artiregcial behaviour were the citedselective numerical under-integration of the transverse shear strain termsThe appearance of spurious zero energy deformation modes required thedevelopment of stabilization or projection techniques in order to recover thecorrect rank of the stiffness matrices involved (Belytschko et al 1992) In casethe full quadrature is maintained some procedures proved to be efregcient inattenuating transverse shear locking Among the successful techniques werethe so-called mixed interpolation of tensorial (transverse shear strain)components (MITC) from the initial work of Dvorkin and Bathe (1984) andlater on Bathe and Dvorkin (1986) Bathe et al (2000) Chapelle and Bathe (2000)and Lee and Bathe (2002)

Additionally it is also worth noting the assumed natural strain (ANS)approach regrst introduced by Park (1986) and Park and Stanley (1986) Bothformulations employ a set of additional sampling points over a regnite element inorder to obtain a substitute or complementary strain regeld leading to thefulreglment of the Kirchhoff-Koiter hypothesis in the thin shell limit Relating tothe MITC this procedure is usually cast into a more general frame than theconventional degenerated formulation An example is the ordfsolid-shellordm class ofelements developed by Doll et al (2000) Harnau and Schweizerhof (2002)Hauptmann and Schweizerhof (1998) and Hauptmann et al (2001) In thisformulation mixed interpolation as described was used for the transverseshear locking while the EAS method is adopted in attenuating membranelocking Thickness volumetric and trapezoidal locking appearance is related tothese elements and proper treatments of the last locking effects are needed inorder to reach a successful and general regnite element (Harnau andSchweizerhof 2002)

In this paper based on the framework of subspace of deformations analysisalready successfully applied in 2D plane strain (CeAcircsar de SaAcirc and Natal Jorge1999) and shell problems (CeAcircsar de SaAcirc et al 2002) a new volumetric and shearlocking-free EAS element for 3D analyses is proposed The main idea is theimprovement of the original strain regeld in an additive way leading to elementswith higher performance in the presence of volumetric locking (3Dapplications) and transverse shear locking (shell structures applications)

Volumetric locking plusmn subspace analysisThe incompressibility problem can be formulated as a constrainedminimization of a functional (CeAcircsar de SaAcirc and Natal Jorge 1999) In simpleterms the goal is to obtain in the linear space of admissible solutions U a regeldof displacement u that minimizes the total energy of the system located in the

A new enhancedstrain element

899

subspace of the incompressible deformations I and simultaneously containedin the space of all the solutions (I U) This statement can be posed in theform

I = u [ U div(u) = 0 (1)

In an approach done by the FEM the linear space of the admissible solutions Uand the respective subspace I previously deregned are approximated by thespaces U h and I h respectively

A two regeld regnite element solution can be expressed for linear elasticityaccording to CeAcircsar de SaAcirc and Natal Jorge (1999) by

K Q

2 Q 0

uh

ph

=fhext

0

(2)

where fext is the vector of applied external forces p is the hydrostatic pressureand K is the stiffness matrix The superscript (h) means that the variable incause is a regnite element approximation

The incompressibility condition is given by the second group of equationsderegned in equation (2)

Quh = 0 (3)

which will deregne the subspace of the incompressible deformations I h as

I h = uh [ U h Quh = 0 (4)

In order to avoid the trivial solution (uh= 0) in equation (3) the regeld ofdisplacements uh should belong to the nullspace of Q that is to the subspaceof the incompressible deformations I h The approximated displacements uh

are contained in I h being therefore a linear combination of a given basis of I h

elementsIf the subspace I h is an approach to the original subspace I it is plausible to

admit that it cannot reproduce all the possible solutions contained in I In factdifferent formulations lead to better or worse approximations for the I subspaceThe volumetric locking phenomenon occurs when for a certain group ofboundary conditions and external forces applied in a near incompressiblesituation the expected solution or some of its components do not appearproperly represented in the subspace I h Different types of elements can offerdistinct subspaces of incompressible deformations originating thereforedifferent solutions (CeAcircsar de SaAcirc and Owen 1986)

To characterize the already deregned subspace I h consider a standardisoparametric eight-node hexahedral element with domain Oe (Figure 1)For small deformations the incompressibility constraint (e ii = 0) can takethe form

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Z

Oe

div(u) dOe =Z

Oe

shy u

shy x+

shy v

shy Z+

shy w

shy z

sup3 acutedOe = 0 (5)

At the element level in any point of the domain the displacement regeld can beinterpolated using the standard linear shape functions (Hughes 2000Zienkiewicz and Taylor 2000)

u lt uhe = N(x Z z )de (6)

One way to guarantee the incompressibility condition is to assure that theintegrand function in equation (5) is zero Substituting equation (6) inequation (5) results

shy u

shy x+

shy v

shy Z+

shy w

shy z= [N ix(x Z z ) N iZ(x Z z ) N iz(x Z z )] de = 0 (7)

for i = 1 nenodes Making use of a complete integration (2 pound 2 pound 2 eight Gauss

points Figure 1) the application of equation (7) leads to

Figure 1Standard eight-node

isoparametrichexahedral element with

2 pound 2 pound 2 Gauss pointsnumerical integration

rule

A new enhancedstrain element

901

2 a 2 a 2 a a 2 c 2 c c c 2 b 2 c a 2 c 2 c 2 c a c 2 b c b b b 2 b c c

2 a 2 c 2 c a 2 a 2 a c a 2 c 2 c c 2 b 2 c 2 b c c 2 c a b c c 2 b b b

2 c 2 c 2 b c 2 a 2 c a a 2 a 2 a c 2 c 2 b 2 b b b 2 c c c c a 2 c b c

2 c 2 a 2 c c 2 c 2 b a c 2 c 2 a a 2 a 2 b 2 c c b 2 b b c b c 2 c c a

2 c 2 c 2 a c 2 b 2 c b b 2 b 2 b c 2 c 2 a 2 a a a 2 c c c c b 2 c a c

2 c 2 b 2 c c 2 c 2 a b c 2 c 2 b b 2 b 2 a 2 c c a 2 a a c a c 2 c c b

2 b 2 b 2 b b 2 c 2 c c c 2 a 2 c b 2 c 2 c 2 c b c 2 a c a a a 2 a c c

2 b 2 c 2 c b 2 b 2 b c b 2 c 2 c c 2 a 2 c 2 a c c 2 c b a c c 2 a a a

2

666666666666666666666664

3

777777777777777777777775

pound de= 0 (8)

where the results for each Gauss point are grouped by rows and

a =1

8(1 + f )(1 + f ) b =

1

8(1 2 f )(1 2 f ) c =

1

8(1 + f )(1 2 f )

and f =

3

p

3

(9)

Equation (8) contains a matrix (Q) of rank seven Being 24 the total number ofdegrees of freedom ie the dimension of the subspace of admissible solutionsU h there will be a dimension 242 7 = 17 for the incompressible deformationssubspace

rank(Q) = 7 ^ nullity(Q) = 17 (10)

If for the same element a reduced numerical integration scheme with only oneGauss point is used (x = Z = z = 0) the imposition of condition (7) will lead toa subspace of dimension 23 when the maximum dimension of the space of theadmissible solutions is 24 The matrix in equation (8) is rewritten as follows

1

82 1 2 1 2 1 1 2 1 2 1 1 1 2 1 2 1 1 2 1 2 1 2 1 1 1 2 1 1 1 1 1 2 1 1 1

pound curren

pound de = 0(11)

and

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rank(Q) = 1 ^ nullity(Q)= 23 (12)

Analysing these two possible bases for the subspace of the incompressibledeformations it can be inferred clearly that the use of the reduced integrationallows the reproduction of more six incompressible displacement modes thanthe case of complete integration Since the admissible incompressible solutionsare nothing more than a linear combination of a given I h basis this can be aquite reasonable explanation for the good performance of the reducedintegration techniques in volumetric locking problems and why the classicalcomplete integration numerical integration locks

For a clear illustration of the last statements a graphical representation ofthe linearly independent elements which form the basis of the incompressibledeformations subspace is shown in Figure 2 The six rigid body motions can beobtained linearly combining these elements The displacement regeld associatedwith each mode is plotted in Table I The modes 1-17 are reproduced both bythe complete and the reduced integration The modes 18-23 are reproduced onlyby the reduced integration being volumetric locking situations for the completeintegration They can be divided into four main groups

(1) simple edges translations in x y and z directions represented by themodes 1-12

(2) expansioncontraction of one face modes 13-17

(3) hourglass modes 18-20

(4) warping modes 21-23

In the following sections the proposed approaches to alleviate locking effects(using the enhanced strain method) are conveniently deregned within thesubspace analysis framework just described

Formulation of volumetric and transverse shear locking-freeelementsAs already referred the reduced integration selective reduced integration andsimilar techniques have proved to be efregcient methods on attenuating thevolumetric locking However the behaviour in bending dominated situations isnot the best and for the case of the SRI techniques the analysis is in somemeasure limited to materials where the stress tensor allows the decompositionin volumetric and deviatoric components

On the other hand the enhanced strain method is a powerful techniquein the sense that permits the inclusion of more or less additional variablesfor the enhanced strain regeld Thus it is possible to construct a formulationwith good behaviour both in near incompressible situation and bendingsituations

However the number of additional variables in the enhanced regeld is acrucial matter If increasing the number of additional variables and

A new enhancedstrain element

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