An enhanced strain 3D element for large deformation elastoplasticthin-shell applications

R. A. Fontes Valente, R. J. Alves de Sousa, R. M. Natal Jorge

Abstract In this work a previously proposed solid-shellfinite element, entirely based on the Enhanced AssumedStrain (EAS) formulation, is extended in order to accountfor large deformation elastoplastic thin-shell problems. Anoptimal number of 12 enhanced (internal) variables isemployed, leading to a computationally efficient perfor-mance when compared to other 3D or solid-shell enhancedelements. This low number of enhanced variables is suf-ficient to (directly) eliminate either volumetric andtransverse shear lockings, the first one arising, for in-stance, in the fully plastic range, whilst the last appears forsmall thickness’ values. The enhanced formulation com-prises an additive split of the Green-Lagrange materialstrain tensor, turning the inclusion of nonlinear kine-matics a straightforward task. Finally, some shell-typenumerical benchmarks are carried out with the presentformulation, and good results are obtained, compared towell-established formulations in the literature.

Keywords Solid-shell elements, Enhanced strains,Volumetric and transverse shear lockings, Geometric andmaterial nonlinearities, Thin shells

1IntroductionFinite element analysis of shell structures goes back intime until the onset of the so-called degenerated approachin works of Ahmad et al. [1] and Zienkiewicz et al. [101],as well as in early papers of Ramm [78], and afterwardswith Hughes and Liu [55] and Hughes and Carnoy [57],among others. Soon it was verified that brick elementswere prone to the appearance of volumetric and transverse

shear locking effects. The first one is characteristic ofcommon metal plasticity models, where plastic deforma-tion is taken to be isochoric or, in other words, incom-pressible [19]. The second one comes from the analysis ofthin shells, were the limit between ‘‘thick’’ and ‘‘thin’’geometries is somewhat difficult to establish, with theoccurrence of locking not only strictly relying on thick-ness/length ratios, as demonstrated by Chapelle, Bathe andco-workers [11, 12, 29, 30, 31].

In order to circumvent these parasitic phenomena,selective reduced integration (or, equivalently, u/p for-mulation, mean-dilatation technique and B-bar methods)– for volumetric locking – and the ‘‘mixed interpolation oftensorial components’’/assumed strain method – fortransverse shear locking - had arisen as possible andsuccessful techniques. In the literature see, for instance,references [4, 45, 53, 54, 61, 68, 69, 74, 85, 86, 96] for thegrounds of computational treatment of incompressibility,in elastic and elastoplastic finite element cases, and also [9,38, 56, 67] for earlier works dealing with transverse shearlocking.

In the specific case of shell elements, original plane-stress assumptions were enough to avoid or postponeincompressibility issues in the nonlinear material range[5, 47, 55, 78, 88], although at the expense of a rotationtensor inclusion. As more generality was needed, higherorder theories including thickness change via extensibledirector fields and/or ‘‘layerwise’’ approaches weredeveloped, including (or not) rotational variables, as in[6, 7, 14–18, 22–25, 39, 40, 41, 52, 84, 89, 93], to name buta few.

Despite the good results obtained by these formulationsin thick and thin shell problems, interest in trilinearbrick-type elements, resting just on translation-type de-grees-of-freedom, has been increasing over the last decade.A relative advantage gained with this kind of formulationwould then be the avoidance of a specific treatment forrotation variables. On the other side, for this kind of ele-ments, locking pathologies must be appropriately treatedwhile keeping its scope of application independent ofthickness values. Such an hexahedral solid element shouldalso naturally incorporate kinematical formulations typicalof shell approaches with, at the same time, the automaticaccount for thickness variations.

According to Wriggers et al. [99], reliable three-dimensional elements for shell-type applications withfinite strains can be obtained using the Enhanced AssumedStrain (EAS) method of Simo and co-workers [87, 90, 92].Representative lines of research in this field are, for

Computational Mechanics 34 (2004) 38–52 � Springer-Verlag 2004

DOI 10.1007/s00466-004-0551-7

Received: 25 August 2003 / Accepted: 13 January 2004Published online: 27 February 2004

R. A. F. Valente (&), R. J. A. de SousaDepartment of Mechanical Engineering,University of Aveiro Campus de Santiago,3810-193 Aveiro, PortugalE-mail: robertt@mec.ua.pt

R. A. F. Valente, R. M. N. JorgeIDMEC, Faculty of Engineering,University of Porto, Porto, Portugal

Funding by Ministerio da Ciencia e do Ensino Superior (FCTand FSE) (Portugal) under grant PRAXIS XXI/ BD/21662/99; aswell as the funding by FEDER, under grant POCTI/EME/47289/2002, are gratefully acknowledged.

38

example, the intensive work of Schweizerhof et al. [37]Freischlager and Schweizerhof [44], Harnau andSchweizerhof [48], Hauptmann and Schweizerhof [49],Hauptmann et al. [50], Klinkel and Wagner [62], Klinkelet al. [63], Wagner et al. [98], Miehe [73] and recently Vu-Quoc and Tan [97] and Legay and Combescure [65]. Allthese works have the common feature that enhancedassumed strain, assumed strain method and/or selectiveintegration procedures have been combined in order toobtain a wide class of solid-shell elements with goodperformances. For typical shell problems, solid-shell ele-ments can then represent an alternative with, as statedbefore, a simpler formulation when compared to shellelements, although more advantages can be specified. Inmetal forming simulations involving two-sided contactalong the thickness direction (presence of blank-holder)and in composites delamination problems (with a moreaccurate evaluation of interlaminar shear and normalstresses), numerical simulations can be effectively carriedout with this class of finite elements.

The grounds of the present work rely on the recentpaper of Alves de Sousa et al. [2], where a new class ofthree-dimensional EAS elements for incompressible caseswas introduced. Starting with a sound analysis of thedeformation subspace granting the incompressibilitycondition ðdiv u ¼ 0Þ, an enhanced strain field wasdeveloped and introduced into the functional of the clas-sical displacement-based solid element. It was then shownthat the inclusion of 6 enhanced variables, acting on thevolumetric components of the strain field, was sufficient toavoid the volumetric locking phenomenon. A first pro-posal for a new 3D element, characterized by a total of 18internal variables has proved to be effective in solvinggeneral three-dimensional problems (HCiS18 solid ele-ment). The adopted EAS approach avoids the direct use ofclassical selective reduced integration, which is consistentonly for material models with decoupled isochoric andvolumetric behavior. Another important feature was thatthe element has proved to be reliable in thin shell prob-lems. However, for the specific case of shell structures, theauthors have verified, also in the last reference, that the useof only 12 enhanced parameters (leading to more com-putational efficiency) was enough for the obtention ofsound results. In this case, 6 enhanced variables areresponsible for the elimination of transverse shear lockingeffects, without resorting to assumed strain methods, andfollowing previous works of the authors [28, 43]. This lastelement, then coined HCiS12 solid-shell element, is nowextended and applied in large deformation elastoplasticshell problems.

The distinguishing characteristic of the presentformulation can be summarized by the fact that only theenhanced assumed strain method is used to simulta-neously treat volumetric and transverse shear locking inclassical thin-shell problems. This point contrast withthe generalized use of the assumed natural strainapproach (for the transverse shear locking) and/or theselective reduced integration technique (for near-incompressibility constraints) in well-established solid-shell formulations in the literature. As a first step in theformulation, linear benchmarks were provided in

reference [2], with the extension of the methodology toaccount for nonlinear geometric as well as elastoplasticproblems being carried out in the present work. Besidesleading to an unified and ‘‘neat’’ formulation for thesolid-shell element as a whole, the present proposal(coming from a subspace analysis detailed in [28] and[2]) relies upon an enhanced strain field based on thederivatives of a three-dimensional ‘‘bubble-function’’.This specific choice of functions is grounded on im-proved results obtained for distorted meshes in incom-pressibility conditions, and also in bending-dominatedsituations for two-dimensional problems, as can beinferred from previous works of the authors (references[26, 27], respectively).

This paper is organized as follows. In Sect. 2, kinematicaspects of the HCiS12 solid-shell element are revisited,along with expressions for stress and strain tensors in theconvective frame. Section 3 deals with the enhanced strainvariational formulation, establishing the grounds for thefinite element implementation described in Sect. 4. In thelatter, an overview of the algorithmic aspects related tononlinear material and geometric behaviors is performed,focusing on the advantages of both the adoption of aconvective frame for tensorial representation and theadditive strain enhancement employed. Finally, results aregiven in Sect. 5 for a class of benchmarks in shell andsolid-shell elements evaluation, and comparisons arecarried out with distinct formulations well-established inthe literature.

2Kinematics of the solid-shell elementDuring deformation, it is theoretically useful to establishsome configurations related to whom each particle in theanalyzed body can be referred to. In this sense, it is pos-sible to consider a reference (or material) and current (orspatial) configurations M � R3 and S � R3, respectively.A third configuration employed is the parametricconfiguration P � R3, defined by a set of curvilinear(convective) coordinates

n ¼ ðn1; n2; n3Þ 2( � ½�1; 1� � ½�1; 1� � ½�1; 1� ð1ÞWithout loss of generality, and within the incremental-iterative context, the reference configuration can be relatedto a converged state ðnÞ (last increment) whereas thecurrent configuration points to the unknown configurationðnþ 1Þ (corresponding to the next load increment). Forthe solid-shell topology treated in this work, any point inthe reference configuration can be defined by a positionvector as

nx nð Þ ¼ 1

21þ n3� �n

xu n1; n2� �

þ 1

21� n3� �n

xl n1; n2� �

ð2ÞThe corresponding position after deformation (currentconfiguration) can be defined by an analogous expression,now referred to state ðnþ 1Þ. In Eq. (2) it is worth notingthe use of position vectors xuð�; �Þ and xlð�; �Þ, of auxiliarypoints corresponding to upper ðn3 ¼ þ1Þ and lowersurfaces ðn3 ¼ �1Þ, respectively, and referred to a fixedorthonormal global frame ðe1; e2; e3Þ. From expression (2)

39

it is straightforward to arrive at different formulationscommonly used in the literature ([73], [48], [97], amongothers)

nxðnÞ ¼ 1

2nxu n1; n2� �

þn xl n1; n2� �� �

þ 1

2n3 nxu n1; n2

� ��n xl n1; n2

� �� �ð3aÞ

¼ n xm n1; n2� �

þ 1

2n3 na n1; n2

� �nv n1; n2� �� �

ð3bÞ

The right-hand side of Eq. (3b) is characteristic of shellformulations, with the introduction of the mid-surfaceposition vector xm, the shell thickness a ¼ a n1; n2

� �and

the unit director v n1; n2� �

. In all the equations before, it isimplicit the definition of a preferred through-thicknessorientation, common in solid-shell approaches (see, forinstance, references [37, 48, 49, 50, 62, 63, 73, 97, 98]). Inthis context, ðn1Þ and ðn2Þ represent inplane curvilinearcoordinate axes while ðn3Þ denotes the through-thicknessorientation.

The displacement field for any point from a convergedstate until the current position can be given by the relation

nþ1nuðnÞðiÞ ¼ nþ1 xðnÞðiÞ �n xðnÞ ð4Þ

for a given iteration ðiÞ. Also from the position vectors it ispossible to define the covariant base vectors from thepartial derivatives (dropping the iteration index for brevityreasons)

nþ1gkðnÞ ¼onþ1xðnÞ

onkð5aÞ

ngkðnÞ ¼onxðnÞonk

ð5bÞ

each one directly related to its contravariant counterpartsnþ1gk and ngk, respectively, with ðk ¼ 1; 2; 3Þ [8].

The relative deformation gradient between configura-tions ðnÞ and ðnþ 1Þ in the above defined convective basisthen takes the usual form

nþ1nF ¼ onþ1x

onx¼ nþ1 gk �n gk ð6Þ

It is now possible to define the (displacement-based)Green-Lagrange strain tensor Eu (as well as itscomponents), between states ðnÞ and ðnþ 1Þ as

nþ1nEu ¼ 1

2nþ1

nFT nþ1nF� I2

� �¼ Eu

klngk �n gl ð7aÞ

nþ1nEu

kl ¼1

2ngk �

onþ1nu

onlþ onþ1

nu

onk�n gl

�

þ onþ1nu

onk� o

nþ1nu

onl

�ð7bÞ

and including the second-order identity tensor I2. Thestrain tensor in Eqs. (7) is a material entity work-conju-gated to the 2nd Piola-Kirchhoff stress tensor S

S ¼ Skl ngk �n gl ð8Þ

The material Green-Lagrange strain tensor will be addi-tively enhanced with incompatible (element-wise defined)strain terms and, in conjunction with the stress tensor S,will form the variational structure of the proposed ele-ment, as shown in the next section. In a departure from themajority of works in the literature, only the enhancedassumed strain (EAS) approach will be employed, in anunified way, to solve either the transverse shear and vol-umetric lockings, with a minimum number of variablesand following previous works of the authors ([2, 26–28,43).

3EAS variational formulationThe starting point of the present formulation is theEnhanced Assumed Strain method, in its linear version asoriginally presented by Simo and Rifai [87]. Even dealingwith nonlinearities, the proposed approach keeps theoriginal frame of additive enhancement over the dis-placement-based Green-Lagrange strain tensor, in a waysuccessfully advocated at first by Andelfinger and Ramm[3] (linear cases), Bischoff and Ramm [18] (nonlinearcases), after that by Klinkel and Wagner [62] and Klinkelet al. [63] and, more recently, by Fontes Valente et al. [43]and Vu-Quoc and Tan [97]. As showed in these references,this approach is indeed computationally simpler (andleading to virtually the same results) than the oneadvocated by Simo and Armero [90] and subsequentlysuccessfully used, for example, by Miehe [73].

In general terms, the starting point is the Hu-Washizu-de Veubeke 3-field functional for static cases [18]

PHWVðu;E; SÞ ¼Z

V

WsðEÞdV

þZ

V

S :1

2FTF� I2

� �� E

� �dV�Pext ð9aÞ

Pext ¼Z

V

u � �bqdVþZ

Sr

u ��tdS ð9bÞ

where the displacement ðuÞ, the Green-Lagrange strain ðEÞand the 2nd Piola-Kirchhoff stress ðSÞ are independentvariables. In the previous expressions, it is worth notingthe displacement-driven strain energy ðWsÞ, the tractionand volume force vectors ðtÞ and ðqbÞ, respectively, alto-gether with the corresponding prescribed fields ð�tÞ andð�bÞ, over control area Sr and volume V. It is worth notingthat all variables are referred to the reference configura-tion, while the boundary conditions for the displacementfield were omitted in Eq. (9).

The total strain field is then assumed to be composed ofa compatible (displacement-based) (7) and incompatible(element-wise) parts, in the form [87]

E ¼ Eu þ Ea ð10Þwhere the left indexes relating to the configuration wereomitted for the sake of simplicity and the right indexesreports to the driven field of strain (displacement – u orenhanced variables – a). The substitution of Eq. (10) into

40

(9a), together with (9b), and the orthogonality condition[87, 90, 92]Z

V

S : Ead V ¼ 0 ð11Þ

reduces the number of independent variables in the ori-ginal functional to just two. The weak form of this modi-fied functional is obtained with the Gateaux or directionalderivative, leading to the total variation [97]

dP u; Eað Þ ¼ dPint � dPext ð12aÞ

dPint ¼Z

V

dEu þ dEað Þ :oWs Eu þ Eað Þo Eu þ Eað Þ dV ð12bÞ

dPext ¼Z

V

du � �b qdVþZ

Sr

du ��t dS ð12cÞ

The weak form can be expanded via a truncated Taylorseries about the solution (fixed point) at the kth state

ujk;Eajk� �

[18]

dP ujkþ1;Eajkþ1

� ��dP ujk;Eajk

� �

þD dP½ � ujk; Eajk� �

� Du;DEað Þð13Þ

where, in the present context, the ðDÞ operator relates to afinite variation between ðkÞ and ðkþ 1Þ states. The finiteelement interpolation for the displacement-based andenhanced strain fields is described next, along with theexplicit expression for the D dP½ � operator and the mainadvantages of including the additive approach as in (10).

4Finite element implementation including nonlinearities

4.1General aspectsIn each element’s domain, the displacement field (with thecorresponding variation and increment) is interpolated inthe form

u � uh ¼ NðnÞd; du � duh ¼ NðnÞdd;

Du � Duh ¼ NðnÞDdð14Þ

where matrix N encopreses the usual isoparametric com-patible shape functions for a 8-node 3D element, relatingthe continuum displacement field and the correspondingvector of elemental degrees-of-freedom ðdÞ (in the presentcase with a total of 24 translational components) [10].

The displacement-based Green-Lagrange strain tensordefined in Eqs. (7) can then be related (as well as itsvariations) to vector ðdÞ at the element level, i.e.,

Eu ¼MuðnÞd; dEu ¼MuðnÞdd; DEu ¼MuðnÞDd ð15Þwhere Mu is the conventional strain-displacement matrix,for the present case with 6� 24 components, and thespecific arrangement of the convective strain tensorcomponents is according to

Eu ¼ Eun1n1 Eu

n2n2 Eun3n3 Eu

n1n2 Eun1n3 Eu

n2n3

n oTð16Þ

Once the major application field of the present formu-lation relies on shell structures, in each of theð2� 2� 2Þ Gauss points within an element, local framesare constructed, based on the inherent thickness direc-tion and following the guidelines often taken fordegenerated shell elements [10]. This local frame is up-dated, from the last converged configuration, using theincremental rotation tensor ðnþ1

nRÞ, extracted via a polardecomposition from its respective incremental deforma-tion gradient (6). The strain-displacement operator (15)is defined in this orthonormal frame and, as a conse-quence, the strain tensor assumes automatically a coro-tational character, as well as its work-conjugated stresstensor. This approach is responsible to provide astraightforward constitutive update in the nonlinearrange (for further details, see references [71, 72] and,particularly, [43]).

4.2Interpolation of the enhanced strain tensorAlong with the strain tensor ðEuÞ, it is necessary todefine its enhanced counterpart ðEaÞ, responsible for theattenuation of volumetric and transverse shear lockinginherent to a conventional displacement formulation. Informal terms, the enhanced strain field is analogous tothe displacement-based one, being possible defined therelations

Ea ¼ MaðnÞa; dEa ¼ MaðnÞda; DEa ¼ MaðnÞDa

ð17Þwhere the enhanced operator ðMaÞ involves a set ofinternal variables ðaÞ, discontinuous between elements andeliminated by a static condensation procedure at eachelement level. The enhanced strain field, in componentsform, is equivalent to its displacement-based counterpart,as presented in Eq. (16).

The grounds of the present formulation points to therecent work of Alves de Sousa et al. [2]. In this work, theoriginal (displacement-based) trilinear 3D element wasanalyzed in detail, with the description of its subspacebasis components able to enforce the incompressibilitycondition ðdivðuÞ ¼ 0Þ. These linearly independent com-ponents are responsible for the range of possible defor-mation modes within an element while, on the other hand,the absence of specific modes leads invariably to the onsetof volumetric locking.

In order to suppress this parasitic phenomenon, themissing deformation modes are directly introduced intothe formulation. The enhanced assumed strain approachwas then chosen to achieve this goal, and a first mixed-formulated element with 18 internal variables was intro-duced in reference [2]. The HCiS18 element representedthen a general approach, with reliable results either inthree-dimensional and thin-shell cases.

However, for shell applications, an approach involvinglesser variables then the latter was also introduced byAlves de Sousa et al. [2] for linear applications. From theuse of the enhanced assumed strain method to treattransverse shear locking in thin shells, adopted initially forlinear cases by Cesar de Sa et al. [28] and further

41

developed for nonlinear cases by Fontes Valente et al. [43],it was possible to devise a solid-shell element with arelative low computational cost (due to the use of 12enhanced variables), with no rotations and good perfor-mances irrespective of thickness/length ratios. It is worthemphasize that the referred 12 internal variables areresponsible for the treatment of volumetric and transverseshear locking altogether, with no further enhancement orassumed strain method being necessary. This HCiS12solid-shell element can be characterized by an interpola-tion matrix as follows

and resorting to the three-dimensional bubble function

NaðnÞ ¼1

2ð1� n1n1Þð1� n2n2Þð1� n3n3Þ ð19Þ

About the specific form of the enhanced operator (18), it isnoticeable the distinctions between the present formula-tion and classical works in the enhanced strain field forthree-dimensional elements [3, 24], namely in the last 9columns of ðMa

(jHCiS12Þ matrix, and specifically in the lownumber of internal variables involved in the present case.Another point of interest in the present formulation is thatan enhancement on the in-plane displacement-basedstrain terms is not strictly necessary. In fact, no substantialgain were obtained if – instead of using the presentapproach – a full 3D enhanced element (including 18enhanced variables to improve each strain componentcoming from the displacement formulation) is adopted formembrane-dominated linear shell problems (as showed bythe authors in reference [2]).

After the adoption of the enhanced interpolation matrix(18), defined in the convective frame ðn1; n2; n3Þ, itstransformation onto the local corotational frame is carriedout with the mapping

MajHCiS12 ¼detJð0ÞdetJðnÞT0Ma

(jHCiS12 ð20Þ

involving the Jacobian operators directly relating theconvective and local frames, and evaluated at the centerof the element and in each Gauss point (Jð0Þ and JðnÞrespectively). Furthermore, transformation matrix ðT0Þinvolves the components of the Jacobian inverse, beingalso evaluated at each element’s center. This scalingprocedure is common in literature dealing with theenhanced assumed strain approach, and the formalstructure for the operator ðT0Þ, can be found, for

instance, in the works of Andelfinger and Ramm [3] andKlinkel et al. [63].

4.3Algorithmic aspect of the linearized discrete weakform and the constitutive update procedureAfter the description of the interpolation functions andvariables for both the displacement-based and enhancedstrain fields, the second member of the right-hand side ofthe linearized weak form (13) can be stated (dropping thekth iteration indices) as [18, 62, 63, 97]

D½dP�ðd; aÞ � ðDd;DaÞ ¼ oðdPint � dPextÞoðd; aÞ � ðDd;DaÞ

ð21ÞAfter including the interpolation functions (14), (15) and(17), the variations in (21) take the matrix form

dPintðd; aÞ ¼ ddðTÞZ

V

ðMuÞðTÞS dV

þ daðTÞZ

V

ðMaÞðTÞS dV ð22aÞ

dPextðdÞ ¼ ddðTÞZ

V

NðTÞ�b qdV

þ ddðTÞZ

Sr

NðTÞ�t dS ð22bÞ

where the ðMaÞ matrix refers to the enhanced strainoperator in the local frame and presented in (20).

Focusing on the variation of the internal part (22a) ofthe whole potential, it is possible to state that

D½dPint� � ðDd;DaÞ ¼ oðdPintÞod

� Ddþ oðdPintÞoa

� Da

¼ ddT ðKlguu þ Knlg

uu ÞDdþ KuaDa� �

þ daT½KauDdþ KaaDa�ð8dd; 8daÞ ð23Þ

The linear and nonlinear geometric (initial stress) stiffnessmatrices (K

lguu and K

nlguu , respectively) are defined as in a

fully displacement-based formulation [10]. The main

Ma(HCiS12

¼

oNa

on1 0 0 o2Na

on1on2o2Na

on1on3o2Na

on2on3 0 0 0 0 0 0

0 oNa

on2 0 o2Na

on1on2o2Na

on1on3o2Na

on2on3 0 0 0 0 0 0

0 0 oNa

on3o2Na

on1on2o2Na

on1on3o2Na

on2on3 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 oNa

on1oNa

on2o2Na

on1on2 0 0 0

0 0 0 0 0 0 0 0 0 oNa

on1oNa

on2o2Na

on1on2

2

6666666666664

3

7777777777775

ð18Þ

42

result of the inclusion of the enhanced parameters into thevariational formulation is the appearance of the couplingstiffness matrices Kau and Kua, as well as the introductionof the fully-enhanced stiffness operator Kaa, all of thempossessing the same structure as in the linear formulationof Simo and Rifai [87]. In fact, the adopted additive ap-proach (10) leads to a straightforward algorithmic exten-sion from the linear case, with no inclusion of nonlineargeometric stiffness matrices associated with the enhancedvariables, as in [90] and [73], and generating a final systemof equations, on matrix form, with the structure [18, 62,63, 97]

Klguu þ K

nlguu

�Kua

Kau Kaa

" #Dd

Da

�

¼

R

V

NT�bqdVþR

Sr

NTtdS�R

V

Muð ÞTS dV

�R

V

Mað ÞTS dV

8><

>:

9>=

>;

ð24ÞThe internal force vectors related to displacement andenhanced fields, ð

R

V

ðMuÞTS dV andR

V

ðMaÞTS dV, respec-

tively) come from the discrete form of Eq. (12b).Besides these nonlinear geometric aspects, the constit-

utive update of the stress tensor needs a specific treatmentdue to nonlinearities coming from the account ofplasticity.

The spatial incremental Cauchy stress tensor (in thelocal frame) can be directly related to the (corotational)2nd Piola-Kirchhoff stress tensor [13]. This approach leadsto the adoption of a hypoelastic constitutive model rep-resentative of a Green-Naghdi objective stress rate, whichwill derive in an additive constitutive update of the stresstensor also in the nonlinear material range. Doing so, andwith the last converged increment representing the refer-ence configuration (updated Lagrangian approach) it ispossible to define an evolution equation for the spatialstress field in terms of the material (mixed) strain field, inthe form

nþ1r ¼ nrþ nþ1nr

¼ nrþ nþ1nS

¼ nrþ C nþ1nE ð25Þ

In order to obtain a second order accurate procedure, themid-point configuration between states ðnÞ and ðnþ 1Þ isused for the evaluation of the displacement-based andenhanced strain components

nþ1nE ¼ nþ1

2nMu nþ1

ndþ nþ12nMa nþ1

na ð26Þ

Doing so, the continuous update of the local frame withresort to the relative rotation tensor is theoreticallyequivalent to successive push-forward and pull-backoperations over the spatial Cauchy stress tensor [36].

In algorithmic terms, a conventional backward-Eulerprocedure is employed in the update of the stress tensor.In this sense, the incremental Green-Lagrange straintensor is consider to be additively composed of an elastic

and elastoplastic counterpart. The implementation stepsfollowed, to some extent, those detailed for the specificcase of shell elements in the classical work of Brank et al.[21].

5Numerical examplesIn the following, numerical benchmark problems areconsidered in order to evaluate the performance of theHCiS12 solid-shell element, as well as the validity of thepresented kinematical and constitutive formulation. Rela-tive convergence tolerances for forces and displacements’norms in Newton-Raphson procedure were set to1:0� 10�5. Both convergence indicators were treatedsimultaneously in each analysis. Problems including non-smooth (unstable) load-deflection paths were solvedresorting to the ‘‘cylindrical’’ standard arc-length methodof Crisfield [33, 34] with the refinements introduced by deSouza Neto and Feng [35], and following the algorithmicguidelines detailed by the authors in [43].

5.1Thick-wall sphere problem with geometric nonlinearityEnhanced strain methods are known to provide non-stable response in the nonlinear range for large homo-geneous compressive strain states. This pathology hasbeen identified by a number of authors, with a soundanalysis being carried out initially by Wriggers and Reese[100] and, subsequently, in references [64, 79, 80–82], toname but a few. The common point in all these works isthe focusing on plane-strain and full three-dimensionalproblems.

As the formulation for the present case, on the otherside, is devoted to simulation of typical shell problems, theanalysis of a (free) thick-wall sphere subjected to aninternal pressure field was chosen in order to attest thelevel of occurrence of numerical instabilities in compres-sion loading cases.

The problem is accounted for following the guidelinesof Kasper and Taylor [60], who considered a geometricallylinear, nearly incompressible, problem. The authors havealso previously analysed this fully-linear case in reference[2], with encouraging results. In the present work, mate-rial, boundary conditions and geometric data are kept thesame (elastic modulus E ¼ 250, inner radius Ri ¼ 7:5,external radius Re ¼ 10:0 and a varying Poisson’s ratio m).However, a nonlinear geometric behavior is now alsotaken into account, altogether with a higher pressure load(internal pressure p ¼ 2:0). Mesh topology also follows theone presented in reference [60], being reproduced inFig. 1, where a symmetric (undeformed) one-eight of thetotal volume is depicted.

The evolution of the radial displacements with thePoisson’s ratio, for node points located at both the internaland external radius, is shown in Table 1. Apart from thevariation of the results as incompressibility condition isprogressively achieved, it is worth reporting that, for val-ues of m 0:49999999, convergence is completely lost andhourglass patterns appears (even for the relatively lowdisplacement values involved).

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5.2Elastic large deflections (membrane) bending problemThis example relates to the analysis of a clamped beamwhich is loaded by a transverse force F ¼ 1000 on its freeedge (see, for instance, references [88, 89]). The resultantin-plane bending deformation is reproduced using a meshof 10 solid-shell elements, in both regular and distortedpatterns [15, 73]. The elastic properties refer to a bulkmodulus of j ¼ 83:33� 105 and a Lame’s parameter ofl ¼ 38:46� 105, whilst the geometry is characterized bythe relationship height/width/length of h=w=l ¼0:1=0:1=1:0 consistent units.

In order to infer the effect of mesh distortion in thenonlinear geometric range, three meshes are considered asrepresented in Fig. 2. The first two meshes are definedfollowing the previous references, the skewed pattern ofthe second mesh obtained, in the present work, with thetranslation of nodes (0.05 unit) along the beam axis. Athird mesh is taken into account (coming from the 90

rotation of the second one) in order to evaluate a realtridimensional mesh distortion level. The load-deflectioncurves for the displacement of nodes A and B is presentedin Fig. 3 and compared to the solution presented by Simo,Fox and Rifai [88, 89]. For the shown meshes the resultsare almost the same, being in good agreement with thosepresented in the last references. It is worth noting thatresults with HCiS12 solid-shell element also correspond to

those obtained by Miehe [73] (with a solid-shell en-hancedþassumed strain formulation) and by Betsch et al.[15] (using a bilinear shell formulation incorporatingextensibility of the director field and also enhancedstrains).

5.3Elastic large deflections (out-of-plane) bending problemConsider now an elastic cantilever beam with lengthL ¼ 10 and rectangular section with constant width w ¼ 1,clamped in one end and subjected to (out-of-plane) pointloads on the opposite (free) end. This example has beentreated by a number of authors, either with extensible-director shell or solid-shell formulations, such as in [24],[41], [73], [76] and [89], among others.

Following the last two references, the thickness of thebeam is taken as a ¼ 0:1, and the material properties aredefined as E ¼ 107 and m ¼ 0:3. The external load con-sidered has a constant value of F ¼ 40� k, where for the

Table 1. Nonlinear geometric thick-wall sphere – Evolution ofradial displacements for an internal pressure level p ¼ 2:0

Poisson’sratio (m)

radial displacement ð�10�2Þ

Ri ¼ 7:5 Re ¼ 10:0

0.3 8.750 6.3050.49 8.049 4.5890.499 8.014 4.5090.4999 8.000 4.5100.49999 8.000 4.5400.499999 8.000 4.5830.4999999 7.989 4.598

Fig. 1. Nonlinear geometric thick-wall sphere – Finite elements’mesh adopted, with a total of 2100 solid-shell elements

Fig. 2. Membrane (in-plane) bending benchmark – Initial con-figurations for 3 different meshes

Fig. 3. Membrane (in-plane) bending benchmark – Evolution ofdisplacements (in the load direction) with load-level for points Aand B

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present case k is a geometrical factor, function of thethickness, and defined as ðk ¼ 103 � a3Þ. In the presentwork three mapped meshes were considered, with 10, 16and 20 HCiS12 elements along the length direction.

For the small deformation theory, a solution of 16:0consistent unities for the tip displacement can be advanced[41], according to linear beam theory. In this case, HCiS12element gives results of 15:820 (10 elements’ mesh), 15:880(16 elements’ mesh) and 15:890 (20 elements’ mesh).

In case of large rotations and displacements, the sameload level as before is now applied in 10 equal steps, asproposed by el-Abbasi and Meguid [41]. Following thisreference, solutions are compared to a theoretical onerelying on inextensional elastica, and coming from thework of Frisch-Fay [46]. The analysis of the present resultsfor the three meshes and the theoretical one is presented inFig. 4. For the three meshes, it is noticeable the perfor-mance of the proposed element, with solutions in goodagreement with the reference one. Still referring to thework of el-Abbasi and Meguid [41], no Poisson’s lockingappears with HCiS12 element in this test case.

Focusing on the 16� 1� 1 mesh, and based on theproposal of Hauptmann et al. [50], a set of numericalanalysis with different Poisson’s ratio is carried out. Theload level is the same as before, being likewise applied in10 equal steps. The results presented in Fig. 5 clearly showa virtually insensitivity of the load-deflection curve for thevarious Poisson’s coefficients presented.

5.4Nonlinear geometric pinching of a clamped cylinderIn this test problem an elastic cylindrical shell, fullyclamped at one end, is subjected to a pair of concentratedloads at its free end. Following references dealing withshell elements [20, 43, 58, 94] a regular mesh of 16� 16elements is employed (Fig. 6). An additional mesh topol-ogy with 20� 20 solid-shell elements over the midsurfaceof the cylinder is also considered. In both cases, only 1element along the radial direction is employed.

Elastic constitutive parameters are Young modulusE ¼ 2:0685� 107 and Poisson coefficient m ¼ 0:3. Thelength of the cylinder is L ¼ 3:048, with mean radius ofR ¼ 1:016 and thickness a ¼ 0:03. Nominal load in Fig. 6is Ftot ¼ 1600� k, where the load factor employed issupposed to vary between k ¼ 0:0 to k ¼ 1:0. Due tosymmetry, only 1=4 of the structure needs to be meshed.

The results obtained with HCiS12 element for thedeflection of the point under the concentrated load isrepresented in Fig. 7, and compared to those of Brank et al.[20]. The value of the cylinder radius is highlighted in thepicture, establishing the physical limit of the deformation.From the last picture it is clear that the refined mesh of20� 20� 1 is necessary in order the results can be in goodagreement with those from the reference. However, the

Fig. 4. Out-of-plane bending benchmark – Evolution of dis-placements with load level for different meshes and m ¼ 0:3

Fig. 5. Out-of-plane bending benchmark – Influence of Poisson’scoefficient on the deflection of 16� 1� 1 mesh

Fig. 6. Clamped cylinder problem – Mesh, loading and boundaryconditions

45

overall predictive capability of the proposed formulationin an example traditionally analyzed with shell elements isworth noting. In Fig. 8 successive deformed equilibriumconfigurations at different load stages are shown for thecoarser mesh over one half of the cylinder, until a value ofdisplacement near its radius.

5.5Unstable behavior of a shallow roof structureIn this classical shell example, the snap-through and snap-back load-displacement path of a cylindrical shell is ana-lyzed (see references [32, 33, 42, 43, 51, 70, 75, 83, 95], toname but a few) . The structure, schematically represented

in Fig. 9, is mapped with 5� 5 HCiS12 elements over one-quarter of its surface, along with 2 elements in the thick-ness direction. The imposition of these two elements isrelated to the proper reproduction of the hinged supportover the straight edges. About the input data for thisproblem, linear dimensions are L1 ¼ 508:0 andL2 ¼ 507:15, with a nominal radius R ¼ 2540:0 andthickness value of a ¼ 6:35. Material parameters areE ¼ 3102:75 and m ¼ 0:3. The maximum load level attainedis equal to Ftot ¼ 1000.

The displacement along OZ direction for points A andB is reproduced in Fig. 10, plotted against the referenceload level and compared to the solution advanced byHorrigmoe and Bergan [51], coming from a shell for-mulation. It is noticeable the agreement between bothsolutions, with the proposed approach spanning thewhole nonlinear range in a total of 39 arc-length con-trolled steps.

Fig. 7. Clamped cylinder problem – Deflection curve for loadedpoints

Fig. 8. Clamped cylinder problem – Sequence of deformed con-figurations for displacements of: a 0.26 R, b 0.58 R, c 0.72 R, d0.97 R

Fig. 9. Shallow roof problem – Geometric model with load andboundary conditions

Fig. 10. Shallow roof problem – Results for points A and B

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5.6Geometric- and material-nonlinear analysisof a pinched hemispherical shellThe well-known nonlinear geometric hemispherical shelltest case, introduced by Simo et al. [88], is now consideredwith the inclusion of elastoplastic effects. This combinednonlinear behavior has been previously investigated byMasud and Tham [72], based on a class of reducedintegrated solid elements [66, 71].

According to reference [72], geometric and elasticparameters, as well as restraint conditions, are kept thesame as in the work of Simo et al. [88], while a new set ofplastic properties in coherent unities (initial yield stressr0 ¼ 6:825� 105; isotropic linear hardening factorHiso ¼ 6:825� 106 ) are now introduced.

In [72], a maximum load level of F ¼ 400:0 is proposed(in opposition to the value of 200:0 in [88]), along withmapped meshes of 16� 16� 2 and 18� 18� 2 elements.For the sake of comparison, the results using the lasttopology are included in this work.

In the present simulation, a coarser mesh of 16� 16� 1HCiS12 solid-shell elements was adopted. The resultsobtained for the displacement along the OX and OY direc-tions (traction and compression external loads, respec-tively) are represented in Fig. 11. It is noticeable the goodcorrespondence between the present and reference results,even with the lower number of elements in the earlier case. Itis also worth noting that the complete deformation path isobtained in 20 steps, 5 times less than the number ofincrements adopted by [72]. The deformed configurationfor the maximum load level is shown in Fig. 12.

5.7Elastic and elastoplastic stretchingof a short cylinder with free endsA cylindrical shell is submitted to a pair of concentratedforces, inducing large displacements and rotations to theelements. A schematic representation is presented inFig. 13 where, due to symmetry reasons, only one octant of

the hole shell is showed. Previous publications dealingwith this example include, among others, the works ofPeng and Crisfield [77], Sansour and Bufler [83], Jiang andChernuka [59], Brank et al. [20], Masud et al. [71], Ibra-himbegovic et al. [58] and Fontes Valente et al. [43]. In allthese cases, only geometric nonlinearities were considered,whereas the nonlinear material analysis have previouslybeen considered by Masud and Tham [72].

The starting point relies on an initially cylindricalgeometry characterized by a length of L ¼ 10:35, radius ofR ¼ 4:953 and a constant thickness value a ¼ 0:094.Material properties are: Young modulus E ¼ 10:5� 106,and Poisson’s ratio m ¼ 0:3125. No boundary conditionsare applied to the free ends of the shell, being the load pairresponsible for the equilibrium of the cylinder. Plasticparameters are the initial yield stress r0 ¼ 1:05� 105, anda linear isotropic hardening coefficient ofHiso ¼ 10:5� 105, as adopted in reference [72].

Mesh topologies are analogous to those employed in[43], that is: 12� 8� 1 HCiS12 elements, regular anddistorted, and also 16� 8� 1 elements (regular only). Thenumbers relate to elements along the periphery, the semi-length and radial directions, respectively. In Fig. 13, thedistorted pattern with 12� 8� 1 elements is schematically

Fig. 11. Elastoplastic pinched hemispherical shell – Load-deflection diagram

Fig. 12. Elastoplastic pinched hemispherical shell – Orthogonalviews of deformed geometry

Fig. 13. Stretching of a cylinder – Schematic view (one octant)with 12� 8� 1 elements in a distorted mesh

47

represented. The main goal is to evaluate the effects ofmesh distortions in the final solution of the proposedsolid-shell element. For this distorted mesh, the elementmost far away from the load point is 10 times larger thanthe smaller one.

The maximum load level, both for the elastic and elas-toplastic cases, is Ftot ¼ 40000� k, where k ranges from0:0 until 1:0. Results for the deflection of points A and B,as a function of the k parameter, are given respectively inFigs. 14 and 15 (for the purely nonlinear geometric case)and in Figs. 16 and 17 (for the elastoplastic nonlineargeometric case). For the present HCiS12 element, all curveswere obtained with the arc-length procedure, as statedbefore, in a total of 26 automatic load steps.

In both the elastic and elastoplastic cases, the presentsolution tends to follow the one obtained in [71] and [72],but with less elements. For the coarser meshes, in theelastic case, there is a deviation of results when compared

to the earlier reference. It is also interesting to note, forthis example, the increase in displacements when distortedelements are employed, in opposition to what happenedwith the class of fully enhanced shell elements, recentlypresented by the authors [43]. For the refined meshemployed, the results are acceptable either in the elasticand elastoplastic regimens, still keeping a lower number ofincrements to achieve the full deformation path.

5.8Elastoplastic analysis of a simply supported platewith pressure loadsIn this test case the inflation of a square plate is analyzed.This example has been treated before in a number ofreferences, including Miehe [73], Betsch and Stein [17],Eberlein and Wriggers [40], Doll et al. [37], Hauptmannet al. [50] and Harnau and Schweizerhof [48]. Along thesereferences, a variety of mesh topologies is employed, while

Fig. 14. Stretching of a cylinder – Results for point A, elastic case

Fig. 15. Stretching of a cylinder – Results for point B, elasticcases

Fig. 16. Stretching of a cylinder – Results for point A, elasto-plastic case

Fig. 17. Stretching of a cylinder – Results for point B, elasto-plastic case

48

in references [37, 48] a higher number of Gauss points (6)is employed for the integration in thickness direction.

The geometric properties for this test are defined by therelations length/width/thickness of 508=508=2:54 consis-tent unities [17]. The plate is submitted to an uniformlydistributed load of p ¼ 60 p0, where p0 ¼ 10�2. Materialproperties are given as E ¼ 6:9� 104, with a perfectlyplastic law characterized by an initial yield stress ofr0 ¼ 248 (Hiso ¼ 0). Boundary conditions only restrain thedisplacements in the direction normal to the plate, beingapplied to just the lower nodes of the mesh (defined overone quarter of the plate). Do to this fact, it is valid theoccurrence of a sort of ‘‘edge-rotations’’ and, as the pressurevalue increases, the plate assumes a ‘‘pillow-type’’ defor-mation mode, changing from a bending dominated defor-mation (in the beginning) to a membrane dominated one.

From the meshes available for comparison, a15� 15� 1 topology, refined in the corners (inspired byEberlein and Wriggers [40]) and a second one with24� 24� 1 mapped elements (following Betsch and Stein[17]) are adopted. The so-called ‘‘6-parameter’’ formula-tion on reference [40] is the one chosen for comparisonpurposes. For both meshes, the maximum load level isattained in 45 load steps, and the resulting out-of-planedisplacement curves are shown in Fig. 18, where the cen-tral node of the plate was monitored. It can be seen thatthe present results are in agreement with both referencesolutions, although the predictive capability of few HCiS12elements approaches the values obtained with the refinedmesh, even with a two-point integration rule acrossthickness direction. The deformed configurations for thefull load and both meshes are represented in Fig. 19.

5.9Elastoplastic nonlinear geometric responseof a pinched cylinderThis example deals with the elastoplastic deformation of athin-walled cylinder, submitted to a pair of concentratedforces. It is a classical test to analyze the behavior of finiteelement in problems involving localized plasticity and

strong shape modifications. Earlier works dealing with thisproblem includes the contributions of Simo and Kennedy[91], Wriggers et al. [99], Hauptmann and Schweizerhof[49], Miehe [73], Eberlein and Wriggers [40] and Wagneret al. [98], among others.

For the present case, comparisons will be carried outwith the results presented in references [73] and [99].Following these guidelines, the cylinder geometry is char-acterized by a relation radius/thickness/length of300=3=300 consistent unities, respectively. The boundaryconditions are such that the circular shape of the cylinder’send is preserved although free deformation in longitudinaldirection is allowed. Once each element has upper andlower nodes, a ‘‘hard support’’, in the sense of Wagner et al.[98], is considered. A von Mises yield criterion is assumed,with yield stress r0 ¼ 24:3 and a linear isotropic hardeningparameter Hiso ¼ 300. Elastic parameters are j ¼ 2500 andl ¼ 1154 [73]. Two discretizations were accounted for: acoarse mesh of 16� 16� 1 elements [98, 99] and a finerone with 32� 32� 1 elements [40, 73, 99], applied overone eighth of the cylinder, benefiting from symmetricgeometry. The simulation is performed in a load-controlledway and the whole path is covered, with the arc-lengthmethod, in 65 steps for both meshes. The obtained resultsfor the displacement value in the external load direction(against this same load) are presented in Fig. 20. A goodagreement between the present and reference results isobtained, with the coarser mesh leading to a load-deflec-tion path similar to the obtained with the refined mesh. Forthis last case, the HCiS12 element agrees quite well with theresults presented in the work of Miehe [73]. It is also worthnoting that the apparent ‘‘snap-through’’ behavior notice-able in the 16� 16� 1 mesh almost disappear with theadoption of the 32� 32� 1 mesh, reproducing a mesh-dependent behavior already pointed out by Hauptmannand Schweizerhof [49]. In Fig. 21 a sequence of deformedconfigurations for the adopted meshes is presented, start-ing from the undeformed point and ranging until thephysically acceptable displacement value of 300 consistentunities (equal to the cylinder radius).

Fig. 18. Simply supported plate – Results for the out of planedeflection of the center node

Fig. 19. Simply supported plate – Deformed configurations forthe analyzed meshes at full load

49

6ConclusionsIn this work a recently proposed formulation for a newsolid-shell element, entirely based on the EnhancedAssumed Strain approach, was extended and tested inlarge deformation elastoplastic and nonlinear geometric

problems typically solved by shell elements. The adoptedadditive enhanced strain field, involving a minimumnumber of internal variables, is responsible for the treat-ment of transverse shear locking (for low limiting thick-ness values) and volumetric locking (arising on the fullplastic range) in an unified way, without resort to othertechniques, such as the reduced integration approach orthe assumed natural strain method. The presentednumerical simulations show high predictability charac-teristics when compared to other well-established (shelland solid-shell) formulations in the literature.

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