Development of a non-linear triangular prismsolid-shell element using ANS and EAS techniques

Fernando G. FloresDepartment of Structures, Universidad Nacional de Crdoba, Casilla de Correo 916,

5000 Crdoba-Argentina, and CONICET, fflores@efn.uncor.edu

Abstract

This paper extends a previous triangular prism solid element adequate tomodel shells under large strains to become a solid-shell element, i.e. thatdiscretizations may include just one element across the thickness. A totalLagrangian formulation is used based on a modified right Cauchy-Green de-formation tensor (C). Three are the introduced modifications: a) an assumedmixed strain approach for transverse shear strains b) an assumed strain ap-proach for the in-plane components using a four-element patch that includesthe adjacent elements and c) an enhanced assumed strain approach for thethrough the thickness normal strain (with just one additional degree of free-dom). One integration point is used in the triangle plane and as many asnecessary across the thickness. The intention is to use this element for thesimulation of shells avoiding transverse shear locking, improving the mem-brane behavior of the in-plane triangle, alleviate the Poisson effect lockingand to handle quasi-incompressible materials or materials with isochoric plas-tic flow. Several examples are presented that show the transverse shear andPoisson effect locking free behavior, how the improvement in the membraneapproach alleviates the volumetric locking and the very good performance ofthe introduced element for the analysis of shell structures for both geometricand material non-linear behavior.Keywords: Solid-Shell, Prism, Large strains

1. Introduction

Promoted by the continuous improvement in computer facilities and alsoby the need for improving different aspects of the models to obtain morereliable simulations, the use of solid elements for the simulation of shellshas grown a lot in the last fifteen years. The main advantages when usingsolid-shell elements are: a) general tridimensional constitutive relations can

Preprint submitted to Elsevier June 11, 2013

be used as plane stress state assumption is no needed; b) contact forces arerealistically applied on the outer surfaces which is particularly important forfriction; c) large transverse shear deformations can be considered speciallyif more than one element is used across the thickness; d) special transitionelements between shell elements and solid elements are avoided; e) boundariesnon-parallel to the shell normal or director can be correctly modeled; f)local triads and rotation vectors, that are costly in general and difficult toparameterize and update, are not needed.

For the simulation of strongly non-linear problems due to, for instance,complex constitutive models or contact, the simplex (linear interpolation)elements are preferred and if possible with displacement degrees of freedom(DOFs) only as they are more reliable and robust. Solid-shell elements, i.e.when only one element is used across the thickness, developed until noware hexahedral and particularly the largest developments are devoted to thetri-linear 8-node brick. It is well known that simplex solid elements basedon the standard displacement formulation when used to simulate slenderstructures lock severely. This numerical locking indicates the inability ofthe interpolation functions (and their gradients) to fit the solid behaviorthat many times makes the obtained solutions useless. In bending problemstransverse shear locking appears that increases with the shell slenderness.Besides that a linear interpolation (constant gradient in that direction) doesnot allow to fit a linear through the thickness strain due to the Poisson effect.If the shell is initially curved artificial transverse strains and stresses appearunder pure bending due to curvature thickness locking. Membrane lockingspecially appears on initially curved shells when bending is preponderantwithout middle surface stretching. Finally when dealing with incompressibleor nearly incompressible materials or elastic-plastic materials with isochoricplastic flow (metals typically) volumetric locking appears.

In order to be computationally efficient, it is necessary that solid-shellelements have a different integration scheme on the shell tangent plane (as lowas possible) than across the thickness, where the number of integration pointsmust be arbitrary in order to capture the non-linearities of the constitutivemodel when bending is present. This is particularly important for elastic-plastic models for example springback in sheet metal forming simulations.The use of a single integration point in the plane of the shell requires, in orderto maintain the efficiency, the series expansion of different variables (e.g.the inverse of the Jacobian) and some control of the spurious deformationmodes due to under-integration. The first one brings some limitations onthe allowable distortion of the elements and the latter usually leads to theintroduction of user-defined factors that must be properly tuned. Howeverthe advantages are very important and markedly decrease the storage space

2

for internal variables and the CPU time, particularly in codes with explicitintegration of the momentum equations.

The advances in solid and solid-shell elements aimed to cure the differentlocking problems are numerous. In Reference [21] a detailed state of the artfor this type of elements can be found. To avoid transverse shear locking,the classical mixed assumed strain approximation by Dvorkin and Bathe [6]is the most used when full integration is considered (see for example [14, 29])and a variation of it, modifying and increasing the sampling points to avoidthe occurrence of spurious modes, when reduced integration is preferred (seefor example [4]). In solid elements under large strains the volumetric lock-ing has been mostly resolved using mixed elements with constant pressureas originally proposed by [17], where the pressure degree of freedom can beremoved locally averaging over the element the volumetric strain, leading toan element with constant volumetric strain. This technique is not accept-able for solid-shell elements because it leads to excessively flexible elementsand does not allow an adequate normal strain gradient across the thick-ness. More recently enhanced assumed strain techniques (EAS) [23] havebeen developed consisting of improving the deformation gradient with theinclusion of internal degrees of freedom which are condensed at the elemen-tary level, maintaining displacements as the only global degrees of freedom.EAS techniques not only allows to eliminate the volumetric locking, but alsoto eliminate the problem due to the Poisson effect and can even be usedto improve the performance of the membrane part. It is not free of disad-vantages: instabilities may occur under large compressive strain requiring asignificantly increase of elemental database or the use of an iterative loop ineach element for the determination of the internal degrees of freedom at eachglobal Newton-Raphson iteration. Apparently the solid-shell element withthe best performance developed until now is that presented in Refs. [20, 21]where it is shown that an additional internal degree of freedom together witha reduced integration scheme is sufficient to solve the volumetric locking andthe problem arising from Poisson effect. The element satisfies the membraneand bending patch-tests, shows very good convergence properties and worksproperly under large elastic-plastics strains as shown in [22]. Alternatively[3] have developed an element with an additional local displacement degreeof freedom at the center of the element that improves the interpolation ofthe normal displacement.

In the authors knowledge, the only triangular prism solid element ade-quate to simulate shell is the one proposed in [7]. In this Reference transverseshear locking is cured using an assumed natural strain (ANS) to modify themetric tensor components associated with shell normal direction. Volumetriclocking is alleviated on the one hand by averaging volumetric deformation

3

over the element (restricted to use at least two elements in the thickness)and on the other hand by an assumed strain technique for the in-plane com-ponents that uses pieces of information from adjacent elements ([9]). Thusa simple element is obtained, which does not require stabilization due tounder-integration, formulated in large strains and suitable for contact prob-lems. One obvious and important advantage of a triangular prism elementis that the triangular mesh generators are quite more efficient, and provideelements with a better aspect ratio. Curiously there are not developmentsfor triangular prism solid-shell elements. Probably the reason for this is thefew possibilities given by the interpolation functions of the standard prismelement.

The behavior of the standard (displacement base) 6-node prism and 8-node brick are quite different, thats why the strategies to cure the differentlocking problems may be different. The transverse shear locking of the firstone is quite lower while the latter has a better in-plane behavior. In a planestrain condition a one point quadrature eliminates volumetric locking for aquadrilateral but not for a triangle. Note also that for the same mesh density(measured in the number of nodes) a reduced integration strategy implies thedouble number of integration points for the prism than for the brick.

In this paper we propose to modify the assumed strain 6-node elementprism [7] in order to make it suitable as a solid-shell element, i.e., that justone element can be used in the thickness of the sheet and get correct resultsin thin shells with non-zero Poisson ratios and quasi-incompressible materialsor with isochoric plastic flow.

The next section summarizes the basic formulation of the solid element.Then the improvements in the standard element are presented, starting withthe improvement in the tangent plane of the sheet, followed by the transverseshear formulation and finally the EAS technique used to prevent the Poissoneffect locking. Section 5 presents several examples showing the very goodbehavior of the element and finally some conclusions are summarized.

2. Basic kinematics of the solid finite element

Next the kinematic formulation of the standard 6-node triangular prismelement is presented. The element configurations are described by the stan-dard isoparametric interpolations [31]

X () =6I=1

N I () XI (1)

x () =6I=1

N I () xI =6I=1

N I ()(XI + uI

)(2)

4

where XI , xI , are uI denote the original coordinates, the current coordinatesand the displacements of node I respectively. The shape functions N I ()combine linear polynomials in terms of the area coordinates (, ) on thetriangular base with a linear interpolation () along the prism axis:

N1 = zL1 N4 = zL2

N2 = L1 N5 = L2

N3 = L1 N6 = L2(3)

where the third triangular area coordinate and the linear Lagrangian poly-nomials have been included:

z = 1 L1 =

1

2(1 ) (4)

L2 =1

2(1 + )

In a standard way, defining the Jacobian matrix at each integration point

J =X

(5)

the computation of the Cartesian derivatives of the shape functions can beperformed

N IX = J1 N I (6)

At each element center, coincident with the principal orthotropy direc-tions of the constitutive material, a local Cartesian triad is defined

R = [t1, t2, t3] (7)

that allows to compute the Cartesian derivatives with respect to this localsystem (Y)

N IY = RT N IX (8)

and the deformation gradient F as a function of the present nodal coordinates

Fij =NNI=1

N Iyj xIi (9)

Finally the components of right Cauchy-Green deformation tensor C areobtained

Cij = FkiFkj (10)

5

As most of the constitutive equations are easily dealt with using volu-metric and deviatoric components, at each integration point the deformationtensor is decomposed in a multiplicative form

C = [det (C)]13 CD = J

23CD (11)

defining the volumetric and deviatoric strain components as

= ln J

(12)

eD = ln(C

12D

)the logarithmic strain tensor results

e =

31 + eD (13)

The computation of the strains (12) requires the spectral decompositionof C

C = LT 2 L (14)

where 2 is a diagonal matrix collecting the eigenvalues 2i of C and Lincludes the associated (unit) eigenvectors.

Adopting the hypothesis of additivity of elastic and plastic strain compo-nents, the strain tensor is written as

e = ep + ee (15)

For materials with a yield surface independent of the mean press (Misesor Hills yield functions for instance) it is possible to work exclusively withthe deviatoric strain an stress tensors, making easier and computationallycheaper the integration of the constitutive equations.

As it will be shown in the next section the tensor C is modified usingassumed strain techniques that in one case includes an additional internaldegree of freedom , leading to an improved tensor C. The balance equationsto be solved (variational formulation) are for the large strain case

g1 (u, ) =

Vo

1

2S(C)

: uC dV0 + gext = 0 (16)

g2 (u, ) =

Vo

1

2S(C)

: C dV0 = 0 (17)

6

where S is the second Piola-Kirchhoff stress tensor that can be related toKirchhoff stress tensor T using the following expressions:a) defining the rotated tensor

TL = LT T L (18)

b) the relationship between the rotated tensors is (See for example Reference[5])

[SL] =1

2[TL]

(19)

[SL] =ln (/)12

(2 2

) [TL]c) finally the 2nd Piola-Kirchhoff stress tensor is

S = L SL LT (20)

As an alternative to the logarithmic strain, the spectral decomposition(14) allows to easily deal with large strain hyperelastic materials (elastomers),using models such as Ogden, Mooney-Rivlin, neo-Hookean, etc., that areusually defined in terms of a strain energy written in terms of the principalstretches.

3. Modifications of the standard element

The triangular prism element described above must be substantially im-proved to be used for large strain elastic-plastic analysis of shells includingcontact constraints. Different modifications are introduced over the metrictensor C with that aim.

The discretization of a shell type solid using solid-shell elements impliestwo steps a) a discretization of the shell middle surface with three-node tri-angles in the present case and b) a discretization in the thickness directionusing one (or more) prism elements based on the triangles defined before.Here it will be assumed that the 6-node connectivity associates nodes 1-3and nodes 4-6 with planes nearly parallel to the shell middle surface andthat the latter ones are above the first three nodes along the shell normalat a distance equal to the thickness. Thus middle surface normal direction(local y3) is almost coincident with local natural coordinate .

As shown in previous section the strain tensor is computed from thespectral decomposition of the right Cauchy-Green deformation tensor, thus

7

an interesting possibility is to directly modify the components of C associatedto the behavior intended to improve

C =

Cm11 Cm12 Cs13Cm21 Cm22 Cs23Cs31 C

s32 C33

(21)where the components with the upper index m are those that have the maininfluence on the in-plane (membrane and bending) behavior of the shell andthose denoted with an s are those associated with the transverse shear. Thenthe deformation tensor may be divided into three parts

C = C1 + C2 + C3 (22)

where

C1 = C11t1 t1 + C22t2 t2 + C12

(t1 t2 + t2 t1) (23)

corresponds with the components on the tangent plane,

C2 = C13(t1 t3 + t3 t1)+ C23 (t2 t3 + t3 t2) (24)

are those components mainly associated with the transverse shear strainsand

C3 = C33t3 t3 (25)

is used to compute the through the thickness strain.The changes in each of the parts in which tensor C has been divided

are described next. The proposed approximation to C1 is identical to thatdescribed in [7]. The modification in C2 is similar to that described in [7] butmodifying the position of the sampling points for the assumed mixed strainapproach. As for C3 now an EAS approximation is considered with theinclusion of a single internal degree of freedom. Moreover unlike the originalreference that uses 2 fixed integration points in normal direction, here thenumber of integration points is arbitrary, as usual in solid-shell elements.For optimization reasons the integration scheme leads to a different wayof evaluating the equivalent nodal force vector and the geometric stiffnessmatrix. With regard to the volume integral (performed with respect to thereference configuration) the determinant of the Jacobian of the isoparametricapproach is evaluated at three points, namely the centers of the faces and theelement center, and is quadratically interpolated to the integration points.

8

1 2

3 78

9

. .

.

P1P2

P3

(a) (b)

Figure 1: Patch of elements. (a) spatial view. (b) parametric space view.

3.1. Improvements on the in-plane behavior using the adjacent elementsTo improve the in-plane interpolation the same technique used for rotation-

free shell elements ([9]) is applied here. A four-element patch, involving 12nodes, is defined by the element and its three adjacent ones (see Figure 1.a).This allows to define an in-plane quadratic interpolation at both the upperand lower surfaces, that is used to compute the in-plane deformation gradi-ent and the associated components of the metric tensor. Here we follow theprocedure used in ([9]) that averages at the element center the metric tensorcomponents computed at each mid-side points. For the solid-shell elementexactly the same computations can be performed at both upper and lowerfaces (see Figure 1.b with the notation of the lower face), and a subsequentinterpolation to the integration points. For the lower face the associatedquadratic shape functions are simply:

N1 = (z + ) N7 = z2

(z 1)

N2 = ( + z) N8 = 2

( 1)

N3 = ( + z) N9 = 2

( 1)

(26)

Thus, at each face defined by three nodes of the element and anotherthree from the adjacent elements:

1. A local system (t1, t2) on the shell tangent plane is computed (t1 andt2 are chosen according to the local system R defined at the elementcenter), with t3 normal to the face .

9

2. At each mid-side point (PK) the in-plane Jacobian (X,X) is evaluatedand projected over the Cartesian directions (t1, t2)

J =

[X t1 X t1X t2 X t2

](27)

3. Note that at each mid-side point PK only four of the six derivatives ofthe shape functions defined in (26)

1 2 3 7 8 9N I 1 + 1 z 12 z 12 0N I 1 + z 1 12 z 0 12

are non zero. For instance for mid-side point P1((, ) =

(12, 12

))the

shape function derivatives of nodes 8 and 9 that do not belong to any ofthe two adjacent triangles are null. The Cartesian derivatives of thesefour functions are computed as[

N I1N I2

]K= J1K

[N IN I

]K(28)

4. That allows to compute the in-plane deformation gradient (fK1 , fK2 ) andwith it CKij (i, j = 1, 2). These components are averaged over each faceCf

ij (f = 1, 2 for lower and upper face respectively).5. When and adjacent element is missing (boundary), as originally pro-

posed for rotation-free shells, the values of the components of Cij com-puted from the 3-node central triangle are included for the averaging.

For the prism element nG integration points are used along the normal direc-tion (). At these points the in-plane components of the Cauchy-Green tensorare interpolated using (remind that the modified components are identifiedby an over bar)

Cij () = L1C1ij + L

2C2ij (29)

while their variations (for future use) are

12C1112C22C12

= 12C1111

2C122C112

L1 + 12C2111

2C222C212

L2 = E11E22

2E12

(30)At each face a modified tangent matrix Bf relating the incremental tensor

10

components with the incremental displacements u can be written as

12Cf1112Cf22Cf12

= 13

3K=1

12CK1112CK22CK12

=1

3

3K=1

4J=1

fK1 N

J(K)1

fK2 NJ(K)2(

fK1 NJ(K)2 + f

K2 N

J(K)1

) uJ(K)

=(Bfm)318 u

f (31)

where array uf include only the nodes on each face f (lower or upper).Then it is possible to write[

Bm]336 =

[L1B1m, L

2B2m]

(32)

Note that each matrix is associated with a different set of nodes, becausematrix B1m is associated with the nodes on the lower face and B2m with thenodes on the upper face.

The equivalent nodal force vector stems from the integral

rT1 u =

11

S11S22S12

T [ L1B1m, L2B2m ] Jd u=

11

S11S22S12

T L1JdB1m, 11

S11S22S12

T L2JdB2m u

=

S111S122S112

T B1m, S211S222S212

T B2m u (33)

where

uT =[uT1 u

T2 u

T3 u

T7 u

T8 u

T9 u

T4 u

T5 u

T6 u

T10u

T11u

T12

](34)

3.1.1. Geometric stiffness matrixThe geometric stiffness matrix results from

uTKgu =

V

u

12C111

2C22C12

T S11S22S12

udV (35)=

nGG=1

V G3

2f=1

Lf3

K=1

4I,J=1

{uI

[N I1N

I2

] [ S11 S12S21 S22

] [NJ1NJ2

]}K11

where the sum on G is the numerical integration with nG points along direc-tion .

It is possible to take advantage of the values S computed above (33)leading to

uTKgu =

2f=1

3K=1

4I,J=1

{uI

[N I1N

I2

]K [ nGG=1

V G3Lf[S11 S12S21 S22

]] [NJ1NJ2

]KuJ

}

=

2f=1

3

K=1

4I,J=1

{uI

[N I1N

I2

]K [ Sf11 Sf12Sf21 S

f22

] [NJ1NJ2

]KuJ

}f

(36)

where the contributions from each face may be seen independently one fromthe other. In fact the integrated values Sfij depends on the strains at bothfaces.

3.2. Transverse shear formulationTo cure transverse shear locking, following most of the literature, an in-

terpolation in natural coordinates of mixed tensorial components is used.In Reference [19] a general methodology for Reissner-Mindlin shell elementsis presented, that is particularized for the quadratic 6-node triangular ele-ment, leading to a linear variation of the transverse shear strain tangent tothe side. Assuming a constant value of the shear strain at each side, thetechnique was also used for a linear 3-node element ([30]). For the presentelement the latter case is helpful to write the relevant (mixed) componentsof the right Cauchy-Green tensor as

[C3C3

]=

[ 1 1

] 2C1t3C23C33

= P (, ) c (37)with

c =

2C1t3C23C33

= 2f1t f13f2 f23

f3 f33

(38)where the most relevant components to transverse shear (C3 , C3) has beenwritten with respect to a mixed coordinate system that includes the in-planenatural coordinates (, ) and the spatial local coordinate in the transversedirection (y3). These components are written in terms of the transverse shearstrain tangent to the side computed at each mid-side point (1

( = = 1

2

),

2( = 0, = 1

2

)and 3

( = 1

2, = 0

), see Figure 2.a). Similarly to hexahe-

dral solid-shell elements with reduced integration (one point in the shellplane)[4] the sampling points chosen are those located at lower and upper

12

faces ( = 1). As a result six sampling points located at mid-side of eachtriangular face are necessary as shown in Figure 2.b. Besides that, the nu-merical integration is performed along the prism axis ( = = 1

3), whereby[

C3C3

]= P

( =

1

3, =

1

3

)c = Pac =

2C1t3 C23 + C33

3

[ 1+1

]+

[C33C23

](39)

1 2

3

ft2

ft3

ft1

P2

P3

P1

f31

ft3

ft1

ft2 f3

1

f32

312

(a) (b)

Figure 2: Points for the computation of the transverse shear strains.

Replacing (38) into (39) allows to compose

C2 = C3(t t3 + t3 t)+ C3 (t t3 + t3 t) (40)

where the dual base vectors[

t t t3]computed from the local base[

t t t3]

=[X

X

Xy3

]have been used. The modified Cartesian com-

ponents (again denoted by an over bar) are computed projecting over thelocal Cartesian base

C13 = t1 [C3

(t t3 + t3 t)+ C3 (t t3 + t3 t)] t3

= C3

(a1a

33 + a

31a3

)+ C3

(a1a

33 + a

31a3

)(41)

where the symbol aji = ti tj (with i = 1, 2, 3 and j = , , 3) has beenintroduced. Noting that aji =

ji it simplifies to

C13 = C3a1 + C3a

1 (42)

Proceeding similarly for component C23 we can write in a single expression[C13C23

]=

[a1 a

1

a2 a2

] [C3C3

]= J1p

[C3C3

](43)

13

where J1p is the inverse of the in-plane Jacobian of the isoparametric map-ping. Note that due to the way in which the local system has been definedthe components in (43) are null in the reference configuration.

At the sampling points the necessary deformation gradient componentsare ft (natural coordinate derivative) and f3 (local Cartesian coordinate deriva-tive). At sampling points ft are for the lower (1) and upper (2) faces respec-tively 2f1tf2

f3

1 = x3 x2x1 x3

x2 x1

2f1tf2f3

2 = x6 x5x4 x6

x5 x4

(44)while f3 can be expressed as

f3 =6I=1

N I3xI =

[f f f

] y3y3y3

= (x) jT3 (45)where jT3 are the components in direction y3 of the inverse of the Jacobianof the isoparametric mapping (third column j13 ). Finally the modified trans-verse shear Cartesian components emerge from replacing equations (38) into(39) and these into (43)

[C13C23

]= J1p Pa

f1t f13f2 f23f3 f33

(46)while the right Cauchy-Green tensor components at integration points areobtained interpolating the values at each face[

C13C23

]() =

[C13C23

]1L1 +

[C13C23

]2L2 (47)

Note that the Green-Lagrange strain components associated to the trans-verse shear at each face are directly the interpolated values[

2E132E23

]() =

[C13C23

]()

The tangent matrix Bs, relating displacement increments with strain in-crements, is also obtained by interpolating from both faces

Bs () = B1sL

1 + B2sL2 (48)

14

that requires first computing at the sampling points (for each face)

Bsue =

f1t f13 + f1t f13f2 f23 f2 f23f3 f33 + f3 f33

(49)where (with xe = ue)

2f1tf2f3

1 = x3 x2x1 x3x2 x1

2f1tf2

f3

2 = x6 x5x4 x6x5 x4

(50)and

[

f13 f23 f

33

]=

[u1 u2 u3 u4 u5 u6

]

N1(1)3 N

1(2)3 N

1(3)3

N2(1)3 N

2(2)3 N

2(3)3

N3(1)3 N

3(2)3 N

3(3)3

N4(1)3 N

4(2)3 N

4(3)3

N5(1)3 N

5(2)3 N

5(3)3

N6(1)3 N

6(2)3 N

6(3)3

(51)

then interpolate to the element axis as in (39) and finally convert to theCartesian system

Bfs =(Jfp)1

PaBfs (52)

The associated nodal equivalent forces are computed as:

rT2 =

V

[S13S23

]T [Bs]218 dV

=

11

QT[B1sL

1 + B2sL2]218 Jd

=

11

QTL1JdB1s +

11

QTL2JdB2s

= QT41

[B1sB2s

]418

(53)

where the generalized shear forces Q at each Gauss point are defined as

Q41 = 11

[S13S23

]L1[

S13S23

]L2

Jd (54)

15

3.2.1. Geometric stiffness matrix using QThe above expressions allow to advance in obtaining the geometric stiff-

ness matrix

uTKsGu =

{[B1sB2s

]u

}TQ (55)

where at each face we have

Bfs u =J1p

3

[2f1t f13

2f1t f13 + f2 f23 + f2 f23 + 2f3 f33 + 2f3 f33

2f1t f13 +

2f1t f13 + 2f2 f23 + 2f2 f23 + f3 f33 + f3 f33

]

= J1p

{1

3

[ B1s B2s + 2B3sB1s 2B2s + B3s

]218

}u (56)

considering for instance increments of strain components in the lower face[Q1, Q2

](B1su

)results

[Q1, Q2

] J1p3

[ 2f1t f13 2f1t f13 + f2 f23 + f2 f23 + 2f3 f33 + 2f3 f332f1t f13 +

2f1t f13 + 2f2 f23 + 2f2 f23 + f3 f33 + f3 f33

]denoting by [

Q1, Q2

]=[Q1, Q2

]J1p (57)

we have

[Q1, Q2

](B1su

)=

1

3

(Q1 + Q2) (2f1t f13 +2f1t f13 )+(Q1 2Q2) (f2 f23 f2 f23 )+(2Q1 + Q

2

) (f3 f33 + f3 f33

)(58)

In the numerical implementation the derivatives NJ(K)3 related to f3 arekept in an array but not those related to ft because its values are: 0, 1, and-1 at each side.

3.3. Improvement of the transverse behaviorTo avoid locking due to the Poisson effect when bending is important and

also to help in alleviating the volumetric locking (in quasi-incompressibleproblems) an enhanced assumed strain formulation for the component C3 isused.

3.3.1. Enhanced assumed strain (EAS) techniqueThe standard EAS method interpolates natural (convective) strain com-

ponents. In this case we intend to improve directly the Cartesian componentC33 so a slightly different approach will be used as explained next.

16

At the element center( = = 1

3, = 0

)the Cartesian deformation gra-

dient component along y3 can be computed from the isoparametric interpo-lation

fC3 =6I=1

N IC3 xI (59)

The enhanced gradient in direction y3 is defined as

f3 = fC3 e

(60)

thus the interesting component of the right Cauchy-Green tensor results in

C33 = fC3 fC3 e2 = CC33e2 (61)

Note that for the standard element the normal strain in transverse direc-tions results in a first approximation

3 =C33 =

f3 f3 (62)

e33 = ln (3) =1

2ln (C33) (63)

that is practically constant in y3 direction for the linear element, whereas ifwe now use the enhanced version

e33 =1

2ln(C33)

=1

2ln(CC33e

2)

=1

2ln(CC33)

+

= eC33 + (64)

In this EAS approach the changes that will be produced by the enhancedf3 on the other components C13 and C23 are disregarded, as they are computedas explained in previous sections. Thus here only the influence on componentC33 is considered.

3.3.2. Balance equation and implicit solutionThe variation of the Green strain now involves the internal DOF and

can be written

E33 =1

2C33 = f

C3 fC3 e2 + C33

=

(6I=1

N IC3 uI

) fC3 e2 + C33

= e2BC3 ue + C33 (65)

17

In the last expression the first term replaces the corresponding part of thestandard displacement approach (the difference is the factor e2 on the com-ponents associated to E33).

The balance equation (17) associated to variable is

11S33C33Jd = 0 (66)

thus must be solved iteratively. Denoting the residue by 11S33C33Jd = r (67)

the Newton-Raphson technique allows to approximate the value of nullify-ing the residue (67)

11

[S33C33u

u +S33C33

]Jd + r = 0 (68)

with

S33C33u

= C33D3B + 2S33e2BC3 = C33D3B + 2S33B3 (69)

S33C33

= C33D33C33 + 2S33C33 = C33(D33C33 + 2S33

)(70)

where D is the tangent constitutive matrix (Voigts notation ), D3 is itsthird row (1 6) and D33 its diagonal component. B is the matrix relatingincremental Green strains with incremental displacement when the assumedstrain approximations are used for tensor C and B3 is the first term of (65).

When implicit techniques based on Newton-Raphson method are used toobtain the equilibrium path, the DOF is locally condensed at element levelfrom equation (68) 11

[(C33D3B + 2S33B3

)u + C33

(D33C33 + 2S33

)]Jd + r = 0

(71)H118u + k + r = 0

then = r

k 1k

Hu (72)

18

that is replaced in the corresponding balance equation associated to displace-ment variation (16)

uTV

BTSdV uTGext = uT r (u, ) (73)

once linearized 11

[BT(S

uu +

S

)+

(BT

uu +

BT

)S

]Jd +r (u) = 0

11

[BT(DBu + DT3 C33

)+ SGu + 2S33B

T3

]Jd +r (u) = 0

11

[(BTDB + SG

)u +

(BTDT3 C33 + 2S33B

T3

)

]Jd +r (u) = 0(

KTu + HT

)+r (u) = 0

KTuHT(rk

+1

kHu

)+r (u) = 0(

KT HT 1k

H

)uHT r

k+r (u) = 0

(74)

The contributions of the last expression to the Newton-Raphson schemeare the modified elemental stiffness matrix and equivalent nodal force vector:

KT = KT HT 1k

H (75)

r = rHT rk

(76)

Note that above

k =

11C33

(D33C33 + 2S33

)2Jd (77)

KT = KM + KG =

11

BTDBJd +

11

SGJd (78)

H =

11

(C33D3B + 2S33B3

)Jd (79)

19

3.3.3. Geometric Stiffness MatrixThe contribution of the normal transverse component S33 to equilibrium

is

rT3 =

V

B3S33dV

=

11e2BC3 S33Jd

= BC3 S33 (80)

where

S33 =

11e2S33Jd (81)

so the contribution to the geometric stiffness matrix is:

uTKG3u = fC3 fC3 S33

=6I=1

(uI)T 6

J=1

N IC3 NJC3 S33

1 11

uJ (82)3.3.4. Explicit solution

A straightforward way to address the problem when using explicit inte-gration of the momentum equations is to use the condition (72). The possiblesteps are:

1. Modify the equivalent nodal forces

r = r + HTrk

(83)

and compute a first update of the EAS parameter

n+1 = n rk

(84)

2. Use the standard central difference scheme to compute the incrementaldisplacements u

3. Then compute a second update of the internal DOF

n+1 = n+1 Huk

(85)

Such a scheme requires (besides keeping that is indispensable):

20

Keep k between steps, although it can be recomputed at the beginningof the step the increment in the elemental data base is very low so it isnot worthwhile

Store row vector H to avoid its re-computation, this implies an impor-tant increase in the elemental data base

To know D3 (elastic-plastic relating increment in S33 (second Piola-Kirchhoff) with increments in Green-Lagrange strains), that imply anotably increase of floating point operations at each integration point.This is because D3 is not available in explicit integrators (the sameproblem appears for the stabilization of under-integrated elements)

To avoid the increase of the elemental data base [23] suggest (in implicitcodes) for models with a large number of internal DOFs (12) not to store thecorresponding submatrices (that here have been reduced to vector H and thescalar k) and to use an iterative scheme to update the internal DOFs in theroutine that compute stresses and equivalent nodal forces. These authors saythat with a few iterations (they suggest using just 2) accurate enough resultsare obtained.

On the one hand, if the iterative process is reasonable accurate in a finiteelement code with implicit integration of the momentum equations, it will beeven more accurate if the code is explicit. On the other hand each iterationin the computation of the stresses is proportionally very costly in an explicitcode, so it will be assumed that just one iteration is enough to obtain accurateresults. The numerical experiments on elastic-plastic models indicate thatthis assumption is correct even using the elastic component D33 (the iterativeprocess just requires the diagonal component not the entire row D3).

4. Numerical examples

In the set of examples shown below we denote by SPr the solid-shell ele-ment described above. A suffix Q indicates that the improvement in the mem-brane behavior has been activated. The original solid element on which thesolid-shell element is based is denoted by Prism (developed in [7]). With com-parative purposes results obtained with other elements are includes: Q1SPsis a reduced integration solid-shell hexahedral element with an excellent be-havior [20, 21, 22] while LBST and BBST are rotation-free thin shell triangularelements, the former [8] uses the standard constant strain triangle for themembrane part, while the later includes an assumed strain approach for themembrane part [10] almost identical to the formulation presented above.These rotation-free shell elements compute bending strains resorting to the

21

3X

Y1

2

4

Coordinates [mm]1: (0.04, 0.02)2: (0.18, 0.03)3: (0.16, 0.08)4: (0.08, 0.08)

Figure 3: Patch test

geometrical configuration of a four-element patch, the element and the threeadjacent ones, leading to a non-conforming approach (see [18] for a compre-hensive treatment).

4.1. Patch testThe patch test is understood as a necessary condition for the convergence

of the element. In the case of solid elements it is expected that when nodaldisplacements corresponding to a constant strain gradient (membrane patchtest) are imposed, constant efforts are obtained in all the elements. In thecase of standard shell elements displacements and rotations are imposed atnodes corresponding to a constant curvature tensor (bending patch test) anda constant moment tensor is expected to in all elements. In the case of asolid-shell element clearly this must satisfy at least the membrane patch testand, although it may not be necessary, it is highly desirable that the elementsatisfies the bending patch test as this will lead to a more robust and reliableelement.

The Figure 3 shows a patch of elements that has been widely used toaccess quadrilateral shell elements and hexahedral solid shell elements. Hereeach hexahedral has been replaced by two prismatic elements. The size of thelargest sides is es a = 0.24mm and the size of the shortest side is b = 0.12mm,while the thickness considered is t = 0.001mm. The lower surface has beenlocated at coordinate z = t/2. The mechanical properties of the materialare: Youngs modulus E = 106MPa and Poisson ratio = 0.25. Because theproblem considered is linear just 2 integration points across the thickness areused located in the usual Gauss quadrature positions ( = 1/3).

4.1.1. Membrane patch testThe prescribed nodal displacements (on the boundary nodes) are defined

by the linear functions

ux = (x+y

2) 103 uy = (y + x

2) 103 (86)

22

and uz = 0 only on the nodes in the lower face to allow contraction due toPoisson effect. Using present element SPr the correct results are obtainedfor both the displacements of the interior nodes according to (86) and theelement stresses (xx = yy = 1333, 3 MPa y xy = 400 Mpa). The internalDOF is null at all the elements. The same results are obtained with theversion SPrQ as the deformation gradient is constant.

4.1.2. Bending patch testIn this case the displacement field associated with a constant bending

stress state is given by (103):

ux =(x+

y

2

) z2

uy =(y +

x

2

) z2

uz =(x2 + xy + y2

) 12

(87)

that is prescribed on the exterior nodes of both shell faces. Again the resultsobtained with both element versions are correct. The bending stresses at theintegration points are xx = yy = 0.3849 MPa y xy = 0.1155 MPa whilethe displacements at the interior nodes correspond exactly with expression(87). The internal DOF results = 0.3333 108. Thus the element satisfythe bending patch test also.

4.2. Cooks membrane problemThis example (see Figure 4) involves a large amount of shear energy

and is commonly used to assess in-plane bending performance. Plane straincondition will be considered here with two different material behavior: a) aquasi-incompressible elastic material with G = 80.1938GPa and K = 40.1104GPa corresponding with a Poisson ratio = 0.4999 and b) an elastic-plastic material with elastic properties G = 80.1938GPa andK = 164.21GPaimplying a Poisson ratio = 0.29 and J2 plasticity with isotropic hardeningas a function of the effective plastic strain ep defined by

y = 0.45 + 0.12924ep + (0.715 0.45)(1 e16.93ep) [GPa].

The applied load is 100kN for the elastic case and 5kN for the elastic-plasticmaterial.

The plane strain condition implies coefficient C33 = 1 at all points ( =0), thus the version without ANS for the in-plane components locks due tothe almost incompressibility constraint in the same way that a constant straintriangle does. Because of that this example is intended to assess how the im-provement in the membrane field collaborates to cure the volumetric locking.The Figure 5 shows a convergence analysis as the mesh is refined. The verti-cal displacement of point C has been plotted versus the number of divisions

23

F48

4416

C

Figure 4: Cooks membrane problem. Geometry, units in mm.

per side. For comparison results obtained with three two-dimensional solidelements have been included, a linear triangle with a F formulation ([26])that averages the volumetric component over two adjacent elements (onlyresults for the elastic case are available) and two 4-node quadrilaterals: Q1P0entirely formulated in displacement with 22 integration for the shear com-ponents and the volumetric strain averaged over the element and Q1EA ([13])based on the EAS technique including four internal degrees of freedom. Inthe Figure 5.a the plots identified by 0.4999 and 0.499 indicates the Poissonratio used to obtained those results with element SPrQ. The first value is theproposed Poisson ratio for the benchmark and the second a value slightlylower that allows to assess the element sensitivity to the incompressibilityconstraint. On one hand it can be seen that for the proposed Poisson ratioboth quadrilaterals show a clearly better behavior and that a large meshdensity is required to reach convergence with present element. On the otherhand the curve = 0.499 shows that in that case convergence is good enoughand similar to the triangle by [26]. For the elastic-plastic model (Figure 5.b)where although plastic flow is isochoric the elastic behavior is compressible,the results plotted indicate that present element has a better convergenceproperties than both quadrilaterals..

4.3. Cantilever beam with a point loadThis problem has been analyzed by a numerous of authors (see for exam-

ple [25, 12, 21]). A cantilever plate strip of length L = 10mm width B = 1mmand thickness t = 0.1mm is subjected to a transverse load F = 40N. For the

24

Elements per side

Vert.

Dis

pl.C

[mm

]

0 5 10 15 20 25 30 352

3

4

5

6

7

8

SouzaNetoQ1POQ1E40.49990.499

Elements per sideVe

rt.D

ispl

.C

[mm

]0 5 10 15 20 25 30 352

3

4

5

6

7

8

Q1POQ1E4SPrQ

(a) (b)

Figure 5: Cooks membrane problem. a) elastic quasi-incompressible. b) J2 plasticity

selected Youngs modulus E = 106MPa the behavior is one with large dis-placements but small strains. Using different values of Poisson ratio ( = 0.0, = 0.3 , = 0.4999) it can be assessed if the proposed assumed strain tech-niques allow to avoid respectively the transverse shear locking, the Poissoneffect locking and the volumetric locking.

The final deformed configurations (vertical displacement is 70% of thelength) is achieved in ten equal load steps. The discretization includes 16divisions in length, one in the width and one across the thickness with twointegration points. The Figure 6.a shows the vertical displacement of the tipversus the load factor [0:1] for 5 different values of the Poisson ratio. Thecase = 0 allows to compare with the reference value (uz = 7.08 mm) andto see if the approach used to cure transverse shear locking is adequate. Theresult obtained uz = 7.06 mm indicates that effectively the element is freeof transverse shear locking. The second value of Poisson ratio ( = 0.30)is used to assess if the EAS technique avoids the appearance of locking dueto Poissons effect. In this case the computed displacement is uz = 7.01mmthat although is not exactly the same value obtained for = 0 shows thatthe proposed method avoids the Poissons effect locking allowing a propergradation of the transverse normal strain. Finally the last three values ofPoisson ratio (0.49, 0.499 y 0.4999) allow to observe if the performance of theelement deteriorates significantly in the quasi-incompressible range. It can beseen that although differences grow with Poisson ratio, this are below 4% forthe higher value considered. Besides, Figure 6.b plots the tip displacement as

25

Tip Displacement [mm]

Loa

dFa

cto

r

0 2 4 6 80

0.2

0.4

0.6

0.8

1

0.00.30.490.4990.4999

Number of elementsTi

pdi

spla

cem

en

t[mm

]0 8 16 24 326

6.25

6.5

6.75

7

0.00.30.4990.4999BBST 0.0

(a) (b)

Figure 6: Bending of a cantilever plate strip

a function of the mesh density (number of divisions along the length) for fourdifferent Poissons ratio. Results obtained with element BBST and = 0, thatfor this example converges quite rapidly, are also plotted for comparison. Forthe present element it can be seen that convergence deteriorates for Poissonsratio larger than 0.499.

4.4. Cylindrical roofThis third example is the linear analysis of a cylindrical shell under self

weight. The roof is free along the straight sides and supported by rigiddiaphragms at the curved sides. Using symmetry considerations just one-quarter of the roof is modeled. Five structured meshes were used to assessconvergence with the same number of elements along each side. Of the twopossible mesh orientations, the one shown in Figure 7.a was considered. Asthe problem is linear just one element with two integration points is usedacross the thickness.

As this is a membrane dominated problem, the membrane approxima-tion is crucial for a fast convergence, then this example allows to assess theimportance of the ANS for in-plane components in non-isochoric problems.The results for the vertical displacement at the mid-side point of the free side(reference value is wA = 3.610.) are plotted in Figure 7.b as a function ofthe number of elements per side. The figure includes two curves correspond-ing to both formulations of the membrane part of the solid-shell element.Also the results obtained with shell elements LBST and BBST, which main

26

XY

Z

diaphragm

300

symm

etry

A

symmetry

free

300=40

Elements per side

Dis

pla

cem

en

tofA

0 5 10 15 20 25 30 350

0.5

1

1.5

2

2.5

3

3.5

4

LBSTBBSTSPrSPrQ

(a) (b)

Figure 7: Scordelis cylindrical roof. (a) geometry (b) displacement of point A

difference is the membrane approach, are included for comparison as theirbehavior should be similar to the solid-shell element presented here. The re-sults of present element SPr are almost identical to those obtained with thethin shell element LBST except for the coarse mesh where the shell element ismore flexible because its bending approach is non-conforming. When usingthe version with improved membrane behavior results quickly converge tothe reference value in a manner similar to the shell element with improvedmembrane behavior BBST .

4.5. Semi-spherical shell with a 18o holeThe pinched hemisphere is considered in order to introduce initially dou-

ble curved geometry. This is an extensively analyzed shell problem in thecontext of large elastic displacements. The Figure 8.a shows the geometryconsidered once symmetry conditions are introduced and the loads applied.This is mainly an inextensional bending problem where the Poisson effect isimportant and the membrane behavior is not. Because of the double curva-ture the curvature-thickness locking and the membrane locking may appear.Two meshes have been considered that include 16 and 24 elements per side.The coarse one is usually used to determine if the element suffers any lockingdue to the initial curvature, i.e. if the results differs by more than 5% of thetarget values some degree of locking exists. The middle surface radius is esR = 10mm and the thickness is t = 0.04mm (R/t = 250). The mechanicalproperties adopted are E = 6.825 104GPa and = 0.3.

27

XY

Z

(a) (b)

Figure 8: Semi-spherical shell with a hole. Original and deformed geometry.

The Figure 8.b shows the deformed configuration for an inward displace-ment of the loaded point equal to 60% of the shell radius. Whilst the Figure9 plots the displacement (absolute values) of the loaded points where thelargest displacement corresponds to the inward load. The results presentedin Reference [24] using a shear deformable shell element (mesh 16 16) andconverged results obtained with element BBST using a 32 32 mesh are alsoincluded. Results obtained with both formulations of present elements withboth meshes are plotted. Those obtained with the coarsest mesh (16 ele-ments per side) are 8% lower than the target values, in contrast with thosepresented in Reference [21] where an excellent approximation is obtainedwith the same mesh using element Q1SPs. The results corresponding to thefiner mesh (24 elements per side) are in good agreement with the convergedresults.

4.6. Slit annular plateAn annular plate with a radial cut is clamped at one of the slit edges

and subjected to a transverse load in the other one. The Figure 10.a showsthe original geometry and indicates the size and mechanical properties of thematerial. The Figure 10.b shows the deformed configuration for the maxi-mum load factor considered. This is a popular benchmark to assess shellelements under large rotations. Initially proposed by [2] it has been consid-ered by many authors and in Reference [27] converged results are presentedin tabular form that have been used here for comparison. The latter re-sults have been obtained using the element SR4 present in commercial codeAbaqus[1] employing a mesh with 10 80 elements.

Two discretizations have been considered here, the first with 5 36 di-visions and the second with 10 72 divisions. The vertical displacements

28

Load

Dis

pla

cem

en

ts

0 20 40 60 80 1000

1

2

3

4

5

6

BBSTSIMOSPrQ_16SPrQ_24SPR_16SPR_24

(a) (b)

Figure 9: Semi-spherical shell with a hole. Displacements of the loaded points.

X Y

Z

A

B

Ri

Re

Re=10

=0

Pmax=0.8

Ri=6

E=21x106

thick. = 0.03

(a) (b)

Figure 10: Slit Annular plate. (a) Initial geometry. (b) deformed geometry

29

Displ. points A & B

P/P m

ax

0 5 10 150

0.2

0.4

0.6

0.8

1

SzeSPr-5x36SPrQ-5x36SPrQ-10x72

Figure 11: Slit annular plate. Displacements of the free slit.

of two points on the loaded edge, denoted in Figure 10.b as A and B areused for comparison. The Figure 11 plots the evolution of displacementsmentioned above in terms of load factor. It includes converged results fromReference [27], results for both meshes considered when using SPrQ versionand results for the coarsest mesh for element SPr. A comparison of the resultsfor the coarse mesh allows to observe the influence of the ANS for membranepart that for the maximum load factor indicate a difference in displacementslarger than 4%. Comparing the results obtained with the SPrQ version forboth meshes with the converged ones it can be seen that for low values of theload factor the three curves almost coincide but for higher load factor thedisplacements with the coarse mesh separate from the reference values untilalmost a 4% for the maximum load factor. The results for the fine mesh arein excellent agreement with those provided in [27]

4.7. Hinged cylindrical panel under point loadThis example considers a rectangular cylindrical panel simple supported

along the straight sides and free along the curved sides, that is subjectedto a vertical point load in its center (see Figure 12). The middle surfacegeometry is defined by the length of the panel L = 508mm, the radius of thecylinder R = 2540mm and the half angle = 0.1rad. The behavior of thepanel presents a limit point, followed by a strong loss of strength and a finalstiffening once the curvature is inverted. Two different thicknesses for thesame mid-surface geometry have been considered t = 12.7 and t = 6.35 that

30

LR

Simple

supp

orted

FreeY

X

Z

Figure 12: Cylindrical Panel under point load.

for the thin case leads to a snap back of the loaded point. This example hasbeen widely used to assess the performance of shell elements and non-linearpath-following techniques.

For this problem three meshes have been considered with 4, 6 and 8 ele-ments per side. In this case 2 elements in the thickness direction have beenused that allows to introduce the hinge in the middle surface and then tocompare with solutions obtained with shell elements. The vertical displace-ment of the loaded point A (for both thicknesses) and the mid point of thefree side B (only for the thin case) are used to study convergence and forcomparison with other results. Two sets of converged results are included,those obtained from reference [27] using a shear deformable shell element(meshes with 16 16 and 2424 elements for the thick and thin case respec-tively) and our own results obtained with the thin shell element BBST usinga mesh with 16 16 elements. Also to compare with another solid-shell ele-ment results obtained for the coarsest mesh (4 4 2) with element Q1STs[21] are plotted. For the thick case (Figure 13.a) there are some discrepan-cies when comparing the displacements obtained using present element withshell elements but they are almost identical for the three meshes considered.Also they are very similar to those obtained with the hexahedral solid-shellelement on the postcritical path. For the thin case (Figure 13.b) the resultsobtained with the three meshes seem to converge to those obtained with shellelements. The coarsest mesh is clearly inadequate to obtain reliable resultsfor the entire path, particularly for the descending post-critical path wherethe snap back of the loaded point occurs. This can also be seem in the resultsobtained with element Q1STs [21].

4.8. Square thin film under in-plane shearThis example has been previously analyzed in [28] using the commercial

code Abaqus[1]. Experimental data is also available [15]. The problem con-

31

Center Displacement

Loa

d

0 5 10 15 20 25 300

500

1000

1500

2000

2500

30004el/s6el/s8el/sSzeQ1STsBBST

Point B

Point A

Displacement

Loa

d

0 5 10 15 20 25 30-600

-400

-200

0

200

400

600

800

4el6el8elSzeQ1STsBBST

(a) (b)

Figure 13: Cylindrical panel. a) t = 12.7 b)t = 6.35.

a

a

=1

Figure 14: Square thin film under in-plane shear

sists of a square membrane (see Figure 14) with side a = 229mm made ofa thin film of Mylar with thickness t = 0.0762mm. The Mylar mechanicalproperties are E = 3790MPa and = 0.38. The top and bottom edgesare clamped and the lateral edges are free. The top edge is subjected to auniform horizontal displacement = 1mm along the edge. The Figure 14also show a perspective view of the deformed membrane (scaled 5X in thetransverse direction).

Three uniform structured mesh with 2626, 5151 and 101101 nodes,with 1250, 5000 and 20000 elements respectively have been considered. Inthe sequel these meshes will be referenced as mesh 25, 50 and 100, associatedto the number of subdivision along each side

Figure 15 plots two out-of-plane displacement profiles along the center ofthe square in both Cartesian directions. The plot on the left corresponds to

32

Coord. x

Dis

pl.z

0 50 100 150 200

-1

0

1

2

3

2550100BBST

Coord. y

Dis

pl.z

0 50 100 150 200

-1

0

1

2550100BBST

(a) (b)

Figure 15: Square membrane under in-plane shear. Transverse displacement profiles alongthe center of the square. a) y = a/2. b) x = a/2.

y = 114.5 mm and the plot on the right to x = 114.5 mm. The results for thethree meshed defined above are included. These deformed configurations aresimilar to the experimental evidence[15] and also coincide with the numericalresults presented in [28] obtained with program Abaqus [1] using the quadri-lateral shell element S4R5. In that work mesh 100 was used and reported asthe minimum acceptable mesh according to their convergence studies. Herethe profiles for mesh 100 obtained with shell element BBST [11] have alsobeen included. For present element (SPrQ) the mesh 25 does not capturethe correct number of waves, but meshes 50 and 100 are almost coincidenton the wrinkled zones. It can be seen that the results are coincident withthose obtained with shell element BBST but on the free boundaries where themembrane is in slack state. Note also that for mesh 50, that leads to verygood results, the element aspect ratio is x/t = 60.

4.9. Elastic-plastic conical shellThis a second example that allows to study the performance of present

element under large displacements and large elastic-plastic strains. The ge-ometrical details are shown in Figure 16. The elastic mechanical propertiesare E = 206.9MPa, = 0.29 whilst the plastic behavior obeys von Misesyield function with isotropic hardening ruled by function

y (ep) = 0.45 + 0.12924ep + 0.265

(1 e16.93ep) [MPa].

The reference load is P = 0.01N/mm that is scaled by the load factor .

33

PH

R2

R1

t

R1= 1mm

R2= 2mm

H = 1mm

t = 0.1mm

Figure 16: Conical shell geometry

Due to the axisymmetric nature of the problem only a sector of = 0.1rad is included in the discretization applying adequate kinematic constraints.Four uniform discretization along the meridian direction has been consideredwith 4, 8, 16 and 32 divisions. In the hoop direction only one division isused and the number of integration points across the thickness has been setto 5. The Figure 17 plots the load factor versus the vertical displacement ofthe loaded line. The converged results obtained by [21] that discretize onequarter of the geometry using 32 32 elements have been included. Theresults obtained with element SPrQ for the four meshes are plotted but onlythose for the coarsest mesh using element SPr, again to assess the influence ofthe membrane improvement. From the plot it can be said that the mesh with8 elements along the meridian gives results that are practically converged.Element SPr is clearly stiffer than its counterpart SPrQ.

4.10. Deep drawing and elastic springbackThis is one of the Numisheet93 benchmarks ([16]) where a U-shaped

(one axis bending) deep drawing is performed and the elastic springback ismeasured. The Figure 18 shows the forming tools geometrical details. Theblank is made of an aluminum alloy with elastic properties E = 71GPa and = 0.33. The plastic behavior is defined by the Lankford ratios r0 = 0.71,r45 = 0.58 and r90 = 0.7 with isotropic hardening given by

y (ep) = 576.79 (0.01658 + ep)0.3593 .

For the friction between blank and forming tools a coefficient = 0.162 hasbeen adopted while the blank folder force is 2.45 kN. The punch is first moved70mm downwards and all the tools are then removed to allow the springback.

34

Displacement of the loaded line

Loa

dfa

cto

r

0 0.25 0.5 0.75 1 1.25 1.5 1.75 20

1

2

3

4

5

SPr-4SPrQ-4SPrQ-8SPrQ-16SPrQ-32Q1STs

Figure 17: Conical shell convergence study

F/2F/2

5055

R5

R5 R56

BLANK HOLDER

Blank size: 350

Z

X

STR

OKE

70 1 DIE

PUNCH

55

52

6

Figure 18: Deep drawing of a strip

35

AB

15

C

35

2

1

Figure 19: Elastic springback

This example assess the behavior of the element under moderate to largeplastic strains (less than 20%) and the bending behavior after such defor-mation. From symmetry consideration just one quarter of the problem hasbeen discretized using 10 uniform divisions in the half-width and 75 divi-sions in the half-length with mesh size of 5mm in the zones that are neverin contact with the the tools shoulder and mesh size of 1.5mm on the zonesthat are plastically bent. The mesh size of 1.5 mm is the maximum that canbe used to correctly capture the contact due to the low radius of the tools(5mm), as a higher mesh size leads on the one hand to erroneous contactforces and on the other to a wrong prediction of the springback as discussedin [3]. The Figure 19 shows a profile of the blank after the springback stage.The measured parameters used for comparison with experimental values arealso shown in this figure. Using 5 integration point the following values havebeen obtained (the values in brackets are the reported experimental range) = 87mm [81 99], 1 = 111o[110 116] and 2 = 69o[68 76]. It can beseen that all the parameters computed are within the experimental values.

4.11. Deep drawing of a square sheetThe last example considered is the deep drawing of a thin sheet cor-

responding to other of the benchmarks proposed in NUMISHEET93[16].The Figure 20 shows the geometry of the tools. The undeformed sheet issquare with side length 150 mm. The elastic mechanical properties of themild steel considered are: elastic modulus E = 206GPa and Poisson ratio

36

Punch

Stro

keM

in.50

BlankHolder

35 50437

482

XR5

Z

R10

X

Y

R12

8535 2

352

85

Punch

Figure 20: Geometry of the tools (dimensions in mm) for the Numisheet 93 benchmark

= 0.3. For the plastic behavior the classical Hills yield function withconstant coefficients F = 0.283 , G = 0.358 , H = 0.642 , L = 1.065 ,M = 1.179 , N = 1.289 was assumed. These coefficients were computedfrom the Lankford ratios R0 = 1.79, R90 = 2.27 and R45 = 1.51. Isotropichardening is defined by the yield stress along rolling direction (X direction)0 = 567.3 (0.00713 + ep)

0.264 .The symmetry conditions shown in the figure have been considered, then

just one quarter of the geometry has been included in the model. The dis-cretization of the sheet includes 30 elements on each side (1800 elements inthe plane) and 7 integration points in the thickness direction. The blankholder force used is 19,6 kN and the adopted friction coefficient is = 0.144.The simulation considered a punch stroke of 40 mm.

The Figure 21 plots the punch force versus the punch travel. This figureincludes the results obtained with the 3-node triangular shell element BBSTwith 7 through the thickness integration points and with the solid elementPrism with four layers, in both cases with the same in-plane mesh, for com-parison. Results of element version SPr are not reported as in constrainedproblem of this type it suffers from a severe volumetric locking due to theisochoric plastic flow. It can be seen that the differences between the differentmodels are very small.

The Figure 22 shows the contour fills of the effective plastic strain for thepresent solid-shell model (center), and those obtained with the solid (Prism)and shell (BBST) models used for comparison. For the shell model the zonewith the largest plastic strain is less widespread and at the point with thelargest thickness increase (mid-side points) the equivalent plastic strain islower. The differences between the solid model and the solid-shell model are

37

Punch Travel [mm]

Pun

chFo

rce

[kN]

0 10 20 30 400

20

40

60

80

SPrQNBSTPrism

Figure 21: Punch force versus punch travel.

quite small.

5. Conclusions

In this paper we have developed a triangular prism solid-shell elementsuitable for nonlinear analysis with elastic-plastic large strains. In the for-mulation assumed strain techniques have been used to prevent transverseshear locking and to improve the membrane behavior. To avoid the lock-ing due to the Poisson effect an enhanced assumed strain method with justone internal degree of freedom has been proposed. The volumetric locking is

Prism-4 layers

XY

Z

0.80.70.60.50.40.30.20.10

SPrQ BBST

Figure 22: Equivalent plastic strain for the final punch travel

38

alleviated as the combination of the effects of the membrane and the trans-verse approaches. The formulation is simple and can effectively achieve theobjectives. The main conclusions are:

Transverse shear locking disappears completely in all cases analyzed

The improvement in the membrane field is not only important in mem-brane dominated problems (e.g. deep drawing simulations), it resultscrucial to cure the volumetric locking.

The EAS approach for the transverse normal strain effectively avoidsthe locking due the Poisson effect in bending and collaborates in themitigation of the volumetric locking in quasi-incompressible problems.

For explicit time integration, one local (elemental) iteration for theupdate of the internal degree of freedom seems to be enough to obtainreliable results. Even for elastic-plastic problems the use of the elasticmechanical parameters leads to the correct results.

For double curvature surfaced, the element converges to the correct re-sults but the not with the same speed as reduced integration hexahedralsolid-shell elements.

The element did not show any problem in large elastic-plastic straincases.

Acknowledgments

The author acknowledges the financial support from CONICET (Ar-gentina) and SeCyT-UNC.

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