a new 8-node element for analysis of three-dimensional...

Computers and Structures 202 (2018) 85–104

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Computers and Structures

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A new 8-node element for analysis of three-dimensional solids

https://doi.org/10.1016/j.compstruc.2018.02.0150045-7949/� 2018 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: [email protected] (K.J. Bathe).

Yeongbin Ko a, Klaus-Jürgen Bathe b,⇑a 227 Dongil-ro 86, Nowon-gu, Seoul 01618, Republic of KoreabDepartment of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

a r t i c l e i n f o

Article history:Received 10 January 2018Accepted 26 February 2018

Keywords:3D finite element for solids8-node elementMITC methodEAS methodIncompatible modesShear and volumetric locking

a b s t r a c t

We propose a new 8-node hexahedral element, the 3D-MITC8 element, for the analysis of three-dimensional solids. We use the MITC method and find the assumed strain field from a thought experi-ment using a truss idealization. For geometric nonlinear analysis, when needed to suppress hour-glassdeformations, the formulation also uses automatically displacement-based contributions to the shearstrains. The element shows a much better predictive capability than the displacement-based element.It is computationally more effective than the 8-node element with incompatible modes, and consideringaccuracy, in linear analysis performs almost as well, and in nonlinear analyses we do not observe spuri-ous instabilities. We show that the new 3D solid element passes all basic tests (the isotropy, zero energymode and patch tests) and present the finite element solutions of various benchmark problems to illus-trate the solution accuracy reached with the new element.

� 2018 Elsevier Ltd. All rights reserved.

1. Introduction

A three-dimensional 8-node hexahedral solid finite element isfrequently employed for the finite element analysis of solids inengineering practice. The element can be used to model manythree-dimensional (3D) solids and performs considerably betterthan the 4-node tetrahedral element. However, the standard 8-node 3D solid element does not satisfy the inf-sup conditions,hence the solution accuracy can severely deteriorate due to shearand volumetric locking [1,2].

To improve the behavior of the standard displacement-basedelement, additional ‘‘incompatible modes” are frequently used[3]. The incompatible modes technique is a special case of theenhanced assumed strain (EAS) method, and the resulting 8-node3D solid element requires, compared to the standard puredisplacement-based element, an additional 9 internal degrees offreedom to represent the conditions of pure bending [3–6]. The ele-ment is quite powerful since it alleviates both shear and volumet-ric locking, but it uses the additional degrees of freedom and canshow a non-physical instability in the analyses of nonlinear prob-lems [5–8].

The instability of the EAS elements has been observed to occurin both small and large strain nonlinear analyses [7–14]. If an ele-ment mesh is subjected to compression, a spurious hour-glassbending mode may occur in elements eventually resulting into

an indefinite stiffness matrix at a certain critical compressive strain[8]. Initially, the hour-glass deformations are small but as theygrow, the incremental analysis leads to a spurious collapse of themodel.

To treat the spurious instability special solution methods andvarious element formulations have been developed. The varia-tional principle for nonlinear analysis has been modified, stabiliza-tion parameters have been proposed, and mixed-enhancedelements have been developed [9–14], see these references andthe references therein. However, further developments are ofmuch interest and, based on our success of developing reliableand efficient shell elements based on the MITC technique [1], webelieve that we can also obtain an effective 3D eight-node MITCelement.

In this paper we propose a new 8-node hexahedral elementbased on the standard displacement interpolations and the MITC(Mixed Interpolation of Tensorial Components) approach[1,2,10,15–20]. To obtain a stable element, we choose the tyingpositions and strain interpolations based on the physical behaviorof a simple truss structure that idealizes the 8-node solid element.For geometric nonlinear analysis to suppress hour-glass deforma-tions, the formulation also uses automatically when needed a sta-bilization scheme based on displacement-based contributions inshear strains. Using these key ideas for the assumed strain field,we find the 3D-MITC8 element to give solution stability, good solu-tion accuracy, and to be computationally efficient. We also extendthis element to obtain the 3D-MITC8/1 element based on a mixeddisplacement-pressure formulation.

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86 Y. Ko, K.J. Bathe / Computers and Structures 202 (2018) 85–104

In the next section we present the concepts we use for the tyingand interpolation of the strains in the MITC procedure. Then, inSection 3, we propose the new 3D-MITC8 and 3D-MITC8/1 ele-ments using the total Lagrangian (T.L.) formulation. Of course,the linear behavior corresponds to the first step in the T.L. formu-lation. Further, in Section 4, the stability and accuracy of the ele-ment are assessed using basic tests (the isotropy, zero energymode and patch tests) and the solutions of various benchmarkproblems. Finally, in Section 5, we present our conclusions.

2. Tying and interpolation of strains in the MITC procedure

In this section, we present the concepts we employ in the MITCprocedure for the new element. The tying positions and interpola-tions of the assumed strain components are developed consideringstability in linear and nonlinear analyses.

The geometry of an 8-node hexahedral solid element is shownin Fig. 1. The element domain is given and the strain componentsare defined corresponding to the three natural coordinates, r, sand t. For the 3D solid element, there are three normal (in thedirections of r, s and t) and three shear (on the planes of rs, stand tr) strain components. The six assumed strain componentsare denoted by 0~err , 0~ess, 0~ett , 0~ers, 0~est and 0~etr .

The choice of assumed strain interpolation must be such thatthe solid element is stable corresponding to each strain compo-nent. To obtain insight, we idealize the hexahedral domain as atruss structure with 8 joints that correspond to the nodes, seeFig. 2(a), of the 8-node 3D element. The selected truss structureis shown in Fig. 2(b). This structure is stable and consists of theminimum number of 2-node truss elements. We next considerthe location and direction of each truss element to correspond toan assumed strain component. The location of tying is given bythe truss element but to obtain better accuracy using the 3D ele-ment in analyses we can move these locations to correspondingGauss integration points.

If the truss structure we use is stable with the minimum num-ber of truss elements, we can expect that in linear analysis the 3DMITC element will also be stable and will not lock, because a min-imum number of truss elements is used. The use of the truss struc-ture to idealize the solid element is similar to how the classicaltransverse shear assumption was developed by Dvorkin and Bathefor 4-node shell elements, notably for the MITC4 shell element[15]. Here the 4-node shell element transverse shear behaviorwas idealized by the behavior of four 2-node isoparametric beamelements located along the edges of the shell element, with eachbeam assuming a constant transverse shear strain [1,15].

Fig. 1. A standard 8-node hexahedral 3D solid element.

We place a tying location at the center of each truss, and inter-polate the assumed strain components according to these loca-tions, see Fig. 2(c). The normal strains (0~err , 0~ess and 0~ett) areinterpolated bilinearly over the planes defined by their respectivetying locations and the shear strains (0~ers, 0~est and 0~etr) are interpo-lated linearly between the tying points.

While the resulting assumed strain field yields stability in linearanalysis, there is an instability that can arise in nonlinear analysis.The phenomenon has been widely observed for enhanced assumedstrain elements when initially regular meshes undergo compres-sion [7–14]. Indeed, for the incompatible modes elements, spuri-ous bending deformations or hour-glass modes are seen at acritical state even in small strains [8]. For an 8-node hexahedralelement, possible 2D and 3D hour-glass modes are depicted inFig. 3(a) and (b), respectively. This behavior occurs if an 8-nodeelement has the ability to express pure bending deformationsand the surrounding elements cause mixed behavior of bendingand compression. The behavior is possible for the incompatiblemodes element.

The mechanism of this nonlinear instability was studied bySussman and Bathe [8], where it was found that a spurious bendingdeformation occurs when a critical compressive strain state isreached. For the two different kinds of hour-glass modes we treatthe potential instabilities separately. We suppress the accumula-tion of a 3D hour-glass mode by using the incremental displace-ments to calculate the constant compressive strain. Further, wesuppress the 2D hour-glass modes and their coupling to the 3Dhour-glass mode by interpolating an additional stabilizing shearstrain term bilinearly on the respective planes defined by themid-points on the edges, see Fig. 3(c).

Incorporating these ideas, the assumed strain field is proposedas

0~err ¼ A0rr þ A1

rrsþ A2rrt þ A3

rrst;

0~ess ¼ A0ss þ A1

sst þ A2ssr þ A3

sstr;

0~ett ¼ A0tt þ A1

ttr þ A2ttsþ A3

ttrs;

0~ers ¼ A0rs þ A1

rst þ S1rsr þ S2rssþ S3rsrs;

0~est ¼ A0st þ A1

str þ S1stsþ S2stt þ S3stst;

0~etr ¼ A0tr þ A1

trsþ S1trt þ S2trr þ S3trtr;

ð1Þ

in which the Akij and Skij are the unknown strain coefficients. The con-

stants Akij (k = 0, 1, 2, 3) and corresponding interpolations allow

overall stability in linear and nonlinear analyses. The constants Skij(k = 1, 2, 3) are designed to automatically suppress 2D hour-glassdeformations in nonlinear solutions and are significant only whencompressive strain has been accumulated.

3. Formulation of 3D-MITC8 element

We use the left-superscript t to denote the current configura-tion (or ‘time’) of the element. We employ the total Lagrangian for-mulation with the reference configuration at time 0 indicated bythe left subscript 0.

The geometry and displacement of the standard 8-node hexahe-dral 3D solid element is interpolated by [1]

tx ¼X8i¼1

hiðr; s; tÞtxi

¼ txix þ tyiy þ tziz

¼ tx ty tz½ �T ;

with txi ¼ txi tyitzi½ �T ,

Fig. 2. Identification of tying strains and positions. (a) 8 joints forming a hexahedron. (b) An ideal (minimally stable) truss structure. (c) Tying positions for each assumedstrain component.

Y. Ko, K.J. Bathe / Computers and Structures 202 (2018) 85–104 87

hiðr; s; tÞ ¼ 18ð1þ nirÞð1þ gisÞð1þ fitÞ;

n1 n2 n3 n4 n5 n6 n7 n8½ �¼ �1 1 1 �1 �1 1 1 �1½ �;

g1 g2 g3 g4 g5 g6 g7 g8½ �¼ �1 �1 1 1 �1 �1 1 1½ �;

f1 f2 f3 f4 f5 f6 f7 f8½ �¼ �1 �1 �1 �1 1 1 1 1½ �; ð2Þ

in which hiðr; s; tÞ is the standard 8-node interpolation function forthe isoparametric procedure, and ix, iy, iz are base vectors in the glo-bal Cartesian coordinate system, see Fig. 1. The incremental dis-placement vector u from the configuration at time t to theconfiguration at time t þ Dt is

uðr; s; tÞ ¼ tþDtxðr; s; tÞ � txðr; s; tÞ;

u ¼X8i¼1

hiðr; s; tÞui

¼ uix þ viy þwiz¼ ui v i wi½ �T

with ui ¼ ui v i wi½ �T : ð3ÞThe covariant base vectors and displacement derivatives for the 3Dsolid element are given as

tgr ¼@ tx@r

; tgs ¼@ tx@s

; tgt ¼@ tx@t

; u;r ¼ @u@r

;

u;s ¼ @u@s

; u;t ¼ @u@t

; ð4Þ

and we define the following ‘distortion vectors’

txrs ¼ 18

X8i¼1

nigitxi;

txst ¼ 18

X8i¼1

gifitxi;

txtr ¼ 18

X8i¼1

finitxi;

txrst ¼ 18

X8i¼1

nigifitxi; ð5Þ

Fig. 3. Identification of tying strains and positions. (a) Two-dimensional hour-glass modes and corresponding distortion vectors. (b) Three-dimensional hour-glass mode andcorresponding distortion vector. (c) Tying positions for each assumed strain component.

Fig. 4. Characteristic geometry vectors. (a) Three covariant base vectors at thecenter of the element. (b) The three covariant base vectors characterizing a regularmesh.


which measure hour-glass deformations. The three ‘2D distortionvectors’ txrs, txst and txtr correspond to 2D deformations, and the‘3D distortion vector’ txrst corresponds to a 3D deformation, seeFig. 3(a) and (b).

We define the following covariant base vectors at the elementcenter,

tg_

i ¼ tgið0;0;0Þ; with tg_

1 ¼ tg_

r ;tg_

2 ¼ tg_

s;tg_

3 ¼ tg_

t: ð6Þ

These three vectors characterize the regular geometry (tV_

e) of theelement domain (tVe), see Fig. 4.

The seven vectors tg_

r , tg_

s, tg_

t , txrs, txst , txtr and txrst establish thegeometry of the 8-node hexahedral element and we define the cor-responding incremental displacements from time t to t þ Dt using

tXC ¼ tg_

rtg_

stg_

ttxrs

txsttxtr

txrst

h i;

UC ¼ tþDtXC � tXC ¼ ur us ut urs ust utr urst½ �; ð7Þ

where we note that the displacement vectors ur , us, ut , urs, ust , utr

and urst are independent of each other.The contravariant base vectors are obtained from the covariant

base vectors using the standard relationship [1]

tgi � tg j ¼ d ji ; with tg1 ¼ tgr ;

tg2 ¼ tgs;tg3 ¼ tgt ;

tg1 ¼ tgr; tg2 ¼ tgs; tg3 ¼ tgt : ð8Þ

The covariant Green-Lagrange strain components in the configura-tion at time t with respect to the reference configuration at time0 are


t0eijðr; s; tÞ ¼

12ðtgi � tgj � 0gi � 0gjÞ: ð9Þ

Using Eq. (3) in Eq. (9) applied at time t and t þ Dt, the incrementalcovariant strain components are

0eijðr; s; tÞ ¼ tþDt0eijðr; s; tÞ � t

0eijðr; s; tÞ

¼ 12ðtgi � u;j þ u;i � tgj þ u;i � u;jÞ; ð10Þ

which can be separated into a linear part (0eij) and a nonlinear part(0gij)

0eij ¼ 12ðtgi � u;j þ u;i � tgjÞ; 0gij ¼

12ðu;i � u;jÞ; ð11Þ

in which tgi ¼ @tx@ri

and u;i ¼ @u@ri

with r1 ¼ r, r2 ¼ s, r3 ¼ t. Note that

0e11 ¼ 0err , 0e22 ¼ 0ess, 0e33 ¼ 0ett , 0e12 ¼ 0ers, 0e23 ¼ 0est and0e31 ¼ 0etr , and similarly for 0gij.

We use the strain components transformed to a fixed covariantcoordinate system defined at the element center in the referenceconfiguration

0 e_

ij ¼ 0eklgki g

lj; 0g

_

ij ¼ 0gklgki g

lj with g j

i ¼ 0g_

i � 0g j; ð12Þand

0kðr; s; tÞ ¼0jð0;0;0Þ0jðr; s; tÞ with 0jðr; s; tÞ ¼ det 0gr

0gs0gt

� �; ð13Þ

as done in previous works [4–6,10,14,16].We also use the following geometric measure in geometric non-

linear analysis

tc ¼ �X8i¼1

nitxi

!�X8i¼1

nitxi

!�

X8i¼1

gitxi

!�X8i¼1

gitxi

!

�X8i¼1

fitxi

!�X8i¼1

fitxi

!;

t0c ¼ tc � 0c; ð14Þwhere tc measures the normal compressive deformation of the hex-ahedral element and t

0c accumulates the history of the measure.In order to correctly account for the element domain (0Ve), we

employ the volume-averaged constant incremental strains [16]

0 e_conij ¼ 1

0V

Z0Ve

0 e_

ijd0Ve; 0g_con

ij ¼ 10V

Z0Ve

0g_

ijd0Ve;

0V ¼Z

0Ve

d0Ve; ð15Þ

which results in the same constant strain part as given by the stan-dard 8-node displacement-based element.

We express the 3D distortion vector as

txrst ¼ tdrtg_

r þ tdstg_

s þ tdttg_

t ; ð16Þwith

tdr ¼ txrst � tg_r ; tds ¼ txrst � tg

_s; tdt ¼ txrst � tg_t with tg

_

i � tg_ j ¼ d j

i ;

and obtain likewise the corresponding displacement vector

urst ¼ tdrur þ tdsus þ tdtut : ð17ÞIn Eq. (17) the 3D distortions are expressed using the regular defor-mations. To obtain overall stability in nonlinear analysis, the 3Dhour-glass deformations need to be suppressed using the constantstrain parts. Hence, we use Eq. (17) in Eq. (15) and denote the result-ing ‘constant strain’ as 0~econij and 0~gcon

ij , see Appendix A for details.

Based on the above considerations, we thus use for the 3D-MITC8 element the assumed incremental linear strains

0~err ¼ 0~econrr

þffiffiffi3

p

4 0 e_

�0; 1ffiffi

3p ; 1ffiffi

3p�

rr0kðr; s; tÞ sþ t þ

ffiffiffi3

pst

� �

þffiffiffi3

p

4 0 e_

�0;� 1ffiffi

3p ;� 1ffiffi

3p�

rr0kðr; s; tÞ �s� t þ

ffiffiffi3

pst

� �

þffiffiffi3

p

4 0 e_

�0; 1ffiffi

3p ;� 1ffiffi

3p�

rr0kðr; s; tÞ s� t �

ffiffiffi3

pst

� �

þffiffiffi3

p

4 0 e_

�0;� 1ffiffi

3p ; 1ffiffi

3p�

rr0kðr; s; tÞ �sþ t �

ffiffiffi3

pst

� �;

0~ess ¼ 0~econss

þffiffiffi3

p

4 0 e_

�1ffiffi3

p ;0; 1ffiffi3

p�

ss0kðr; s; tÞ t þ r þ

ffiffiffi3

ptr

� �

þffiffiffi3

p

4 0 e_

�� 1ffiffi

3p ;0;� 1ffiffi

3p�

ss0kðr; s; tÞ �t � r þ

ffiffiffi3

ptr

� �

þffiffiffi3

p

4 0 e_

�1ffiffi3

p ;0;� 1ffiffi3

p�

ss0kðr; s; tÞ �t þ r �

ffiffiffi3

ptr

� �

þffiffiffi3

p

4 0 e_

�� 1ffiffi

3p ;0; 1ffiffi

3p�

ss0kðr; s; tÞ t � r �

ffiffiffi3

ptr

� �;

0~ett ¼ 0~econtt

þffiffiffi3

p

4 0 e_

�1ffiffi3

p ; 1ffiffi3

p ;0�

tt0kðr; s; tÞ r þ sþ

ffiffiffi3

prs

� �

þffiffiffi3

p

4 0 e_

�� 1ffiffi

3p ;� 1ffiffi

3p ;0�

tt0kðr; s; tÞ �r � sþ

ffiffiffi3

prs

� �

þffiffiffi3

p

4 0 e_

�1ffiffi3

p ;� 1ffiffi3

p ;0�

tt0kðr; s; tÞ r � s�

ffiffiffi3

prs

� �

þffiffiffi3

p

4 0 e_

�� 1ffiffi

3p ; 1ffiffi

3p ;0�

tt0kðr; s; tÞ �r þ s�

ffiffiffi3

prs

� �;

0~ers ¼ 0~econrs þ 12 0 e

_ð0;0;1Þrs � 0 e

_ð0;0;�1Þrs

� �0kðr; s; tÞt

þ tanhðt0cÞ0 e_stabrs ðr; sÞ0kðr; s; tÞ;

0~est ¼ 0~econst þ 12 0 e

_ð1;0;0Þst � 0 e

_ð�1;0;0Þst

� �0kðr; s; tÞr

þ tanhðt0cÞ0 e_stabst ðs; tÞ0kðr; s; tÞ;

0~etr ¼ 0~econtr þ 12ð0 e

_ð0;1;0Þtr � 0 e

_ð0;�1;0Þtr Þ0kðr; s; tÞs

þ tanhðt0cÞ0 e_stabtr ðt; rÞ0kðr; s; tÞ; ð18aÞ

0 e_stabrs ¼

ffiffiffi3

p

8

�0 e_

�1ffiffi3

p ; 1ffiffi3

p ;1�

rs þ 0 e_

�1ffiffi3

p ; 1ffiffi3

p ;�1�

rs

�r þ sþ

ffiffiffi3

prs

� �

þffiffiffi3

p

8

�0 e_

�� 1ffiffi

3p ;� 1ffiffi

3p ;1�

rs þ 0 e_

�� 1ffiffi

3p ;� 1ffiffi

3p ;�1

�rs

��r � sþ

ffiffiffi3

prs

� �

þffiffiffi3

p

8

�0 e_

�1ffiffi3

p ;� 1ffiffi3

p ;1�

rs þ 0 e_

�1ffiffi3

p ;� 1ffiffi3

p ;�1�

rs

�r � s�

ffiffiffi3

prs

� �

þffiffiffi3

p

8

�0 e_

�� 1ffiffi

3p ; 1ffiffi

3p ;1�

rs þ 0 e_

�� 1ffiffi

3p ; 1ffiffi

3p ;�1

�rs

��r þ s�

ffiffiffi3

prs

� �;


0 e_stabst ¼

ffiffiffi3

p

8

�0 e_

�1; 1ffiffi

3p ; 1ffiffi

3p�

st þ 0 e_

��1; 1ffiffi

3p ; 1ffiffi

3p�

st

�sþ t þ

ffiffiffi3

pst

� �

þffiffiffi3

p

8

�0 e_

�1;� 1ffiffi

3p ;� 1ffiffi

3p�

st þ 0 e_

��1;� 1ffiffi

3p ;� 1ffiffi

3p�

st

��s� t þ

ffiffiffi3

pst

� �

þffiffiffi3

p

8

�0 e_

�1; 1ffiffi

3p ;� 1ffiffi

3p�

st þ 0 e_

��1; 1ffiffi

3p ;� 1ffiffi

3p�

st

�s� t �

ffiffiffi3

pst

� �

þffiffiffi3

p

8

�0 e_

�1;� 1ffiffi

3p ; 1ffiffi

3p�

st þ 0 e_

��1;� 1ffiffi

3p ; 1ffiffi

3p�

st

��sþ t �

ffiffiffi3

pst

� �;

Fig. 6. Cantilever problem (1 � 4 mesh, E ¼ 3:0� 104 and m ¼ 0:0). (a) Regularmesh. (b) Distorted mesh.

Fig. 5. Mesh used for the patch test.

Table 1Predicted y-displacement at the tip (point A) in the cantilever problem.

Elements Regular mesh Distorted mesh

H8 0.235611 0.204659H8I9 0.347833 0.3453073D-MITC8 0.347833 0.304211Exact solution 0.347833

0 e_stabtr ¼

ffiffiffi3

p

8

�0 e_

�1ffiffi3

p ;1; 1ffiffi3

p�

tr þ 0 e_

�1ffiffi3

p ;�1; 1ffiffi3

p�

tr

�t þ r þ

ffiffiffi3

ptr

� �

þffiffiffi3

p

8

�0 e_

�� 1ffiffi

3p ;1;� 1ffiffi

3p�

tr þ 0 e_

�� 1ffiffi

3p ;�1;� 1ffiffi

3p�

tr

��t � r þ

ffiffiffi3

ptr

� �

þffiffiffi3

p

8

�0 e_

�1ffiffi3

p ;1;� 1ffiffi3

p�

tr þ 0 e_

�1ffiffi3

p ;�1;� 1ffiffi3

p�

tr

��t þ r �

ffiffiffi3

ptr

� �

þffiffiffi3

p

8

�0 e_

�� 1ffiffi

3p ;1; 1ffiffi

3p�

tr þ 0 e_

�� 1ffiffi

3p ;�1; 1ffiffi

3p�

tr

�t � r �

ffiffiffi3

ptr

� �;

ð18bÞ

in which the superscript ‘stab’ denotes the strain terms obtaineddirectly from the displacements for stabilizing the spurious hour-glass modes. These strain terms are multiplied by the function tanh(t0c) and hence the effect of stabilization is maximum at large com-pressive deformations. In linear analysis there is no accumulated

Fig. 7. Cook’s problem (4 � 4 mesh, E ¼ 1:0, plane stress conditions with m ¼ 1=3;plane strain conditions with m ¼ 0:3, 0.4 or 0.499).

Table 3Predicted y-displacement at point A in Cook’s problem with plane stress conditions.

Elements Mesh

2 � 2 4 � 4 8 � 8 16 � 16 32 � 32

H8 10.9771 17.3332 21.5484 23.2055 23.7209H8I9 20.3889 22.6184 23.4766 23.7740 23.88403D-MITC8 19.0821 22.2906 23.4092 23.7695 23.8893Reference solution 23.9642

Table 2Predicted stress-xx (�102) at the support (point B) in the cantilever problem.

Elements Regular mesh Distorted mesh

H8 0.466667 0.351417H8I9 0.700000 0.6182223D-MITC8 0.700000 0.580764Exact solution 0.700000

Fig. 8. Convergence curves for Cook’s problem with plane strain conditions. The bold line represents the optimal convergence rate.


deformation, t0c in Eq. (14) is zero, and thus the strain terms in Eq.

(18b) are not activated.We use as tying positions the Gauss numerical integration

points 12 a ¼ 1ffiffi

3p

� �for the in-plane bilinear interpolations of the nor-

mal and shear strains in Figs. 2 and 3 and the mid-points of thefaces 1

2 a ¼ 1� �

for the linear interpolations of the shear strains inFig. 2. These positions of tying the normal and shear strainsdecrease the solution errors.

We employ the same assumed strain field for the incremental

nonlinear strain 0g_

ij, and hence the incremental strain componentsin the global Cartesian coordinate system are

0easij ¼ 0~eklðii � 0g_kÞðij � 0g

_lÞ; 0edilii ¼ 0~econkl ðii � 0g_kÞðii � 0g

_lÞ;

d0gasij ¼ d0~gklðii � 0g

_kÞðij � 0g_lÞ; d0gdil

ii ¼ d0~gconkl ðii � 0g

_kÞðii � 0g_lÞ; ð19Þ

in which superscripts ‘as’ and ‘dil’ denote assumed and dilata-tional strain parts. The dilatational strains are constructed onlyfrom the constant part of the assumed strains to alleviate volu-metric locking.

The constant volumetric strain is obtained as [1,21]

0evol ¼ 0edilxx þ 0edilyy þ 0edilzz ¼ Bvolue;

d0gvol ¼ d0gdilxx þ d0gdil

yy þ d0gdilzz ¼ duT

eNvolue; ð20Þ

and the deviatoric strain components are obtained from

0edevij ¼ 0easij � 13 0evoldij ¼ Bdev

ij ue;

Table 4CPU time used for evaluating the element stiffness matrices of Cook’s problem (320 �320 element mesh).

Elements CPU time (s) used Normalized CPU time

H8 16.1 1.0H8I9 24.2 1.503D-MITC8 18.3 1.143D-MITC8/1 19.0 1.17A laptop with dual core Intel 3.50 GHz CPU and 8.0 GB RAM is used

d0gdevij ¼ d0gas

ij � 13d0gvoldij ¼ duT

eNdevij ue; ð21Þ

with the nodal displacement vector ue ¼ uT1 . . . uT

8

� �T . The effi-cient computation of the strain-displacement matrices given inEqs. (20) and (21) is presented in Appendix B.

Fig. 9. Clamped square plate subjected to in-plane moment (plane strain condi-tions, L ¼ 1:0, E ¼ 1:0, M ¼ 1:0, and m ¼ 0:3, 0.4 or 0.499). (a) Problem descriptionwith regular mesh (4 � 4 mesh). (b) Distorted mesh (4 � 4 mesh).


The stiffness matrix (tK) and internal force vector (t0F) are [1]

tK ¼Z

0Ve

jBvolTBvold0Ve þ

Z0Ve

BdevT

ij Cdevij Bdev

ij d0Ve

þZ

0Ve

13ð0t Sxx þ 0

t Syy þ 0t SzzÞNvold0Ve þ

Z0Ve

0t SijNdev

ij d0Ve;

t0F ¼

Z0Ve

BvolT 13

t0Sxx þ t

0Syy þ t0Szz

� �d0Ve þ

Z0Ve

BdevT

ijt0Sijd

0Ve; ð22Þ

in which t0Sij denotes the second Piola-Kirchhoff stress measured in

the global Cartesian coordinate system, and j ¼ E3ð1�2mÞ is the bulk

modulus with Young’s modulus E and Poisson’s ratio m. Forthe linear isotropic material considered in this study,

Cdevxx ¼ Cdev

yy ¼ Cdevzz ¼ E

1þm and Cdevxy ¼ Cdev

yz ¼ Cdevzx ¼ E

2ð1þmÞ.

The incremental Green-Lagrange strains in Eq. (10) areexpressed using the linear and nonlinear strains in Eqs. (20) and(21), and the global incremental strain values are then obtained as

0evol ¼ Bvolue þ 12uTeN

volue; 0edevij ¼ Bdevij ue þ 1

2uTeN

devij ue: ð23Þ

The second Piola-Kirchhoff stress is updated as

tþDt0Sij ¼ t

0Sij þ 0Sij; 0Sij ¼ j0evoldij þ Cdevij 0edevij ; ð24Þ

and the nodal geometry is updated using Eq. (3).

Fig. 10. Convergence curves for the clamped square plate subjected to in-plane momenoptimal convergence rate.

Alternatively, we can construct an 8-node element using theassumed strain field in Eq. (18) to obtain a mixed displacement-pressure (u/p) formulation for alleviating volumetric locking.

The stiffness matrix (tK) and internal force vector (t0F) of the3D-MITC8/1 element are

tK ¼ tKuu � ð1=KppÞtKuptKT

up;

tKuu ¼Z

0Ve

BdevT

ij Cdevij Bdev

ij d0Ve þZ

0Ve

13

t0Sxx þ t

0Syy þ t0Szz

� �Nvold0Ve

þZ

0Ve

t0SijN

devij d0Ve;

tKup ¼ �Z

0Ve

BvolTd0Ve; Kpp ¼ �

Z0Ve

1jd0Ve ¼ �

0Vj

;

t0F ¼

Z0Ve

BvolT 13

t0Sxx þ t

0Syy þ t0Szz

� �d0Ve þ

Z0Ve

BdevT

ijt0Sijd

0Ve: ð25Þ

The single pressure variable (pe) is

pe ¼ �ð1=KppÞKTupue; ð26Þ

and the second Piola-Kirchhoff stress is updated as

tþDt0Sij ¼ t

0Sij þ 0Sij; 0Sij ¼ �pedij þ Cdevij 0edevij : ð27Þ

t using (a) uniform meshes and (b) distorted meshes. The bold line represents the


When compared with the 3D-MITC8 element, the 3D-MITC8/1element shows the same performance in linear analyses and cangive slightly different solutions in nonlinear analyses. For thenumerical solutions below, the results are the same as for the3D-MITC8 element unless otherwise noted.

In the finite element solutions, we use 2 � 2 � 2 Gauss integra-tion over the element volume considered.

Fig. 11. Pressurized cavity problem (plane strain conditions, L ¼ 1:0, E ¼ 1:0,p ¼ 1:0 for uniform and p0 ¼ 1:0 for smoothly varying loads, and m ¼ 0:3, 0.4 or0.499). (a) Problem description with uniform pressure (4 � 8 mesh). (b) Smoothlyvarying pressure.

4. Illustrative solutions

In this section, we present the performance of the new 3D solidelement through the solutions of various 3D examples. Compar-isons are made with the standard displacement based 8-node ele-ment referred to as ‘‘H8” as well as the standard 8-nodeincompatible modes element referred to as ‘‘H8I9”, see Refs.[1,3–5].

First, the new 3D solid element is assessed considering the basicnumerical tests. Then the performance of the new 3D solid elementis investigated in linear plane stress and plane strain analyses,namely, a cantilever beam problem, Cook’s problem, a clampedsquare plate problem, a pressurized cavity problem, and a clampedplate subjected to a uniform pressure. The geometric nonlinearsolutions are included to test the occurrence of instability: a can-tilever subjected to tip forces, a rubber block problem, and a panelproblem. We use ADINA to obtain the solutions of the geometricnonlinear problems with the incompatible modes element [22].In all analyses we assume that the material response is describedby Young’s modulus E and Poisson’s ratio m, where we recognizethat there is a strain limit that such material can be subjected tobefore a material model instability occurs [1].

Using the derivations in Appendix B, we have found that the cal-culation of the 3D-MITC8 element stiffness matrix is not muchmore expensive than the calculation of the H8 element stiffnessmatrix commonly used (see Section 4.3 for an example).

4.1. Basic numerical tests

We consider the isotropy, zero energy mode and patch tests inlinear analysis.

The spatially isotropic behavior and invariance to the nodalnumbering is important for any element [1,17,19]. The 3D-MITC8element passes the isotropy test.

In the spurious zero energy mode test, the stiffness matrix of asingle unsupported element should show only zero eigenvaluescorresponding to the correct rigid body modes. Six rigid bodymodes corresponding to the 3 translations and 3 rotations shouldbe present. The 3D-MITC8 element passes this test.

In the patch tests [1,15,23], the patch of elements shown inFig. 5 is subjected to the minimum number of constraints to pre-vent rigid body motions, and forces are applied at the boundarycorresponding to the constant stress states. The predicted normaland shearing stresses should be the analytically correct values atany location within the patch of elements. The 3D-MITC8 elementpasses the patch tests.

4.2. Cantilever problem

We solve the cantilever problem shown in Fig. 6. The structureis subjected to a shearing force at its tip. The cantilever is modeledusing regular and distorted meshes with four elements as in Refs.[20,24,25].

Tables 1 and 2 give the tip vertical displacement at point A andthe xx-component of stress at point B with reference to the refer-ence solutions. As when using the H8I9 element, the use of the3D-MITC8 element gives the exact solution with a regular mesh.

For the distorted mesh case, the solution using the 3D-MITC8 ele-ment is much more accurate than when using the H8 element, andnot much less accurate than obtained using the H8I9 element.

4.3. Cook’s cantilever problem

We consider Cook’s problem shown in Fig. 7 [4,20,24,26]. Thecantilever is clamped at one end and is subjected to a distributedshearing force of total magnitude P at its tip. We consider the solu-tions using plane stress conditions with Poisson’s ratio m ¼ 1=3,plane strain conditions with m ¼ 0:3, 0.4 and 0.499, and meshesof N � N elements with N ¼ 2, 4, 8, 16 and 32. The plane strain con-dition is modeled by imposing w ¼ 0 for the entire mesh.

Table 3 gives the vertical displacement at point A for the planestress case. The performance of the 3D-MITC8 element is similar tothat of the H8I9 element. For the plane strain case, we measure thesolution error using the s-norm [27,28] with reference solutionsobtained using a 72 � 72 mesh of the standard 27-nodedisplacement-based element for the cases m ¼ 0:3, 0.4 and the27/4 element for the case m ¼ 0:499 [1].

The s-norm used is given by

kuref �uhk2s ¼ZXref

DeTDsdXref ; with De¼ eref � eh; Ds¼ sref �sh;

ð28Þand then the relative error is


Eh ¼ kuref � uhk2skuref k2s

; ð29Þ

where uref is the reference solution, uh is the solution of the finiteelement discretization, the subscript h corresponds to the elementsize used, e and s are the strain and stress vectors, and Eh is the rel-ative error.

The behavior of an element is optimal if the convergence for theelement in a sequence of meshes is given by

Eh ffi Ch2; ð30Þ

in which C is a constant independent of the material properties andh is the element size.

Fig. 8 shows the convergence of the relative error as a functionof the element size h ¼ 1=N. For m ¼ 0:499, the convergence isseverely degraded when using the H8 element. The 3D-MITC8and H8I9 elements behave nearly optimally and show comparableperformance.

To indicate the computational effectiveness of the new element,Table 4 gives the CPU time used to calculate the element stiffnessmatrices for a fine mesh with N ¼ 320, that is, 102,400 elements.We employ the standard formulations for the H8 and H8I9 ele-ments, with the internal degrees of freedom of the H8I9 elementcondensed out [1,3–5]. The time used for the 3D-MITC8 elementis larger than for the standard H8 element, but not by much.

Fig. 12. Convergence curves for the pressurized cavity problem using (a) the uniform and

4.4. Clamped square plate subjected to an in-plane moment

We solve the clamped square plate problem shown in Fig. 9 [20]and assume plane strain conditions with Poisson’s ratios m ¼ 0:3,0.4 and 0.499. For the solutions, we use regular and distortedmeshes with N � N elements and N ¼ 2, 4, 8, 16 and 32, seeFig. 9(a) and (b), where the element edges are discretized in theratio L1:L2:L3:. . .:LN = 1:2:3:. . .:N. The following boundaryconditions are imposed: u ¼ v ¼ w ¼ 0 along AD, and w ¼ 0 forthe entire mesh.

The relative error is measured using the s-norm with thereference solutions obtained using a 72 � 72 mesh of the standard27-node displacement-based element for the cases m ¼ 0:3, 0.4 andthe 27/4 element for the case m ¼ 0:499. Fig. 10 shows a compara-ble performance of the 3D-MITC8 and H8I9 elements, where theconvergence is nearly optimal for the uniform and distortedmeshes.

4.5. Pressurized cavity problem

We consider a square structural domain with an internally pres-surized circular cavity as shown in Fig. 11. Plane strain conditionsare assumed with Poisson’s ratios m ¼ 0:3, 0.4 and 0.499. Because ofsymmetry, only one-quarter of the structure is modeled usingN � 2N elements with N ¼ 2, 4, 8, 16 and 32. The following bound-

(b) smoothly varying loads. The bold line represents the optimal convergence rate.

Fig. 13. Clamped square plate with uniform pressure (L ¼ 1:0, E ¼ 1:7472� 107,p ¼ 1:0 and m ¼ 0:3). (a) Problem description with regular mesh (4 � 4 mesh). (b)Distorted mesh (4 � 4 mesh).


ary conditions are imposed: u ¼ 0 along BC, v ¼ 0 along DE, andw ¼ 0 for the entire mesh.

The structure undergoes in-plane compression or bendingaccording to the type of internal pressure p used. We test the casesof uniform internal pressure, see Fig. 11(a), and smoothly varyingpressure, see Fig. 11(b).

Fig. 12 shows the convergence of the solutions measured usingthe s-norm, for which the reference solution is obtained with a 36� 72 mesh of the standard 27-node displacement-based elementfor the cases m ¼ 0:3, 0.4 and the 27/4 element for the casem ¼ 0:499. Unlike the H8 element which severely locks near theincompressible limit, the 3D-MITC8 and H8I9 elements behaveoptimally.

4.6. Clamped square plate subjected to a uniform pressure

We consider a clamped square plate transversely loaded by auniform pressure, see Fig. 13(a) [17,19,20]. Utilizing symmetry,only one quarter of the plate is modeled using N � N elements withN ¼ 4, 8, 16, 32 and 64. We vary the ratio of the shell thickness tothe overall dimension of the structure, t=L ¼ 1=10, 1/100 and1/1000. The element size is h ¼ L=N. The boundary conditions areu ¼ hy ¼ 0 along BC, v ¼ hx ¼ 0 along DC andu ¼ v ¼ w ¼ hx ¼ hy ¼ 0 along AB and AD. We use only one layerof elements in the thickness direction.

We also perform the convergence studies with the distortedmesh pattern shown in Fig. 13(b), where each edge is discretizedby the elements in the following ratio: L1:L2:L3:. . .:LN =1:2:3:. . .:N for the N � N element mesh.

Fig. 14 shows the convergence measured using the s-norm; thereference solution is obtained using a 72 � 72 element mesh of theMITC9 shell element [28,29]. The element behavior is optimal if theconvergence curves have the optimal slope and the curves areindependent of the plate thickness and material properties. Forthe 3D solid elements, we can hardly expect to see as good a per-formance as when using shell or plate elements. However, the 3D-MITC8 element shows the optimal convergence behavior whenusing the regular mesh. The convergence behavior deteriorateswhen a distorted mesh is used, but the performance of the 3D-MITC8 element is better than the performance of the H8I9 element.

4.7. Cantilever in large displacements

We solve the cantilever problem shown in Fig. 15; the structureis subjected to tip forces of magnitude F. To test the mixed effect ofcompression and bending, two tip forces are applied at corners ofthe structure. The increment in load per step is 2.0. The fullyclamped boundary condition of u ¼ v ¼ w ¼ 0 is applied at thebottom.

Figs. 16 and 17 present the load-displacement curves and thedeformed shapes, respectively. The reference solution is obtainedusing the standard 27-node displacement-based element. The3D-MITC8 and incompatible modes elements follow the referencebehavior, however, using the H8I9 element, the cantilever is pre-dicted to already collapse at F ¼ 68:0 due to the propagation ofan element instability near the bottom of the model.

4.8. Rubber blocks in large displacements and large strains

We solve the rubber block problems shown in Fig. 18 [12,13,20].The structures are supported at the bottom, loaded with the pres-sure p in plane strain conditions with m ¼ 0:49. The increment inload per step is 1.2.

We analyze the rectangular and trapezoidal structures usingcoarse meshes and, of course, do not claim to have reached solu-

tion convergence of the continuum problems. The purpose of thesolutions is to investigate whether an instability arises when usingthe 3D-MITC8 element.

Fig. 19 shows the calculated load-displacement curves. SincePoisson’s ratio is not very close to 0.5, we obtain the reference solu-tions using the standard 27-node displacement-based element.The standard H8 element shows a too stiff behavior. Using theH8I9 element, the analysis stops at the spurious collapse loadsp ¼ 80:4 and p ¼ 85:2 for the rectangular and trapezoidal struc-tures, respectively. The solutions using the 3D-MITC8 elementare as stable as when using the 2D-MITC4/1 or the standard 27-node elements. Fig. 20 shows deformed shapes obtained usingthe 3D-MITC8 element; these shapes are acceptable in contrastto those obtained when using the H8I9 element which show spu-rious hour-glass patterns.

4.9. Rubber panel in large displacements and large strains

We consider the panel shown in Fig. 21. The pressure load p isapplied asymmetrically on the top in order to also have significantbending of the structure. The increment in load per step is 1.6.Since the mesh is coarse, we do not claim to have reached conver-gence in the solution of the continuum problem. The purpose ofthis problem is to investigate whether a spurious instability arises.

Fig. 22 shows the load-displacement curves. The reference solu-tion is obtained using the standard 27-node displacement-basedelement. While the analysis using the H8I9 element predicts a spu-rious path of solution, the prediction using the 3D-MITC8 elementclosely follows the reference load-displacement curve. Thedeformed shapes are compared in Fig. 23. Unlike the H8I9 element

Fig. 14. Convergence curves for the clamped square plate problem using (a) regular and (b) distorted meshes. The bold line represents the optimal convergence rate.


mesh which shows strong hour-glass shapes, the 3D-MITC8 ele-ment mesh deforms smoothly as in the reference solution.

5. Concluding remarks

We proposed a new 3D 8-node solid element based on the MITCapproach for linear and nonlinear analyses. We established theassumed strain field from physical considerations using a trussstructure to idealize the solid element and previously obtainedknowledge regarding MITC element formulations, specifically the2D-MITC4, the 2D-MITC4/1 and the MITC4 shell elements[15,20]. However, unlike in previous MITC element formulations,we needed to include stabilization terms of displacement-basedshear strains to suppress spurious modes in geometrically nonlin-ear solutions. These stabilization strain terms are only, and thenautomatically, activated in geometrically nonlinear analyses whenneeded.

The new 3D-MITC8 passes all basic element tests and we pre-sented various problem solutions to illustrate the predictive capa-bilities of the element. In linear analyses, the element compareswell in the accuracy of solutions reached with the H8I9 elementbut is computationally more efficient because no incompatiblemodes are used. In geometrically nonlinear analyses, the elementdid not show instabilities as observed when using the H8I9 ele-ment. However, it would be valuable to further study the element

and in particular to obtain deeper insight into the element formu-lation through a mathematical analysis.

Appendix A. Representation of assumed strain field usingcharacteristic vectors

Here we decompose the covariant strain components using thecharacteristic vectors and use the decomposition to represent theconstant assumed strain in Eq. (18).

Note that the incremental linear and nonlinear strain parts inEq. (11) can be expressed as

0eijðr; s; tÞ ¼ ðtxIuJ þ tyIv J þ tzIwJÞnijIJðr; s; tÞ;

0gijðr; s; tÞ ¼ ðuIuJ þ v Iv J þwIwJÞnijIJðr; s; tÞ; ðA:1Þin which I; J ¼ 1; . . . ;8 are node numbers. Note that any part ofincremental strains can be expressed using the corresponding nodalconstants (nijIJ with nijIJ ¼ njiIJ ¼ nijJI ¼ njiJI). The geometry isdescribed and corresponding incremental displacements are

tXC ¼ tx½1� tx½2� tx½3� tx½4� tx½5� tx½6� tx½7�� ¼ tg

_

rtg_

stg_

ttxrs

txsttxtr

txrst

h i;

UC ¼ tþDtXC � tXC ¼ u½1� u½2� u½3� u½4� u½5� u½6� u½7�� ;

Fig. 17. Deformed shapes for the cantilever.

Fig. 15. Cantilever subjected to tip forces (2 � 1�30 mesh, E ¼ 1:0� 103 andm ¼ 0:0).

Fig. 16. Load-displacement curves for the cantilever.


u½K� ¼ u½K� v ½K� w½K�� T with K ¼ 1; . . . ;7: ðA:2ÞUsing Eq. (A.2), we define following representative incremental

strains (e½KjL� and g½KjL�) and the corresponding nodal constants (n½KjL�IJ ),

e½KjL� ¼ 12

tx½K� � u½L� þ u½K� � tx½L�� ¼ txIuJ þ tyIv J þ tzIwJ� �

n½KjL�IJ ;

g½KjL� ¼ 12

u½K� � u½L� þ u½K� � u½L�� ¼ ðuIuJ þ v Iv J þwIwJÞn½KjL�IJ ;

n½KjL�IJ ¼ 1

128ð1½K�I 1½L�J þ 1½K�J 1½L�I Þ with I; J ¼ 1; . . . ;8 and K; L¼ 1; . . . ;7;

1½1�I 1½2�I 1½3�I 1½4�I 1½5�I 1½6�I 1½7�I

h i¼ nI gI fI nIgI gIfI fInI nIgIfI½ �:

ðA:3ÞUsing Eq. (A.2) in Eq. (4),

tgr ¼@ tx@r

¼ tx½1� þ stx½4� þ ttx½6� þ sttx½7�;

u;r ¼ @u@r

¼ u½1� þ su½4� þ tu½6� þ stu½7�;

tgs ¼@ tx@s

¼ tx½2� þ ttx½5� þ rtx½4� þ trtx½7�;

u;s ¼ @u@s

¼ u½2� þ tu½5� þ ru½4� þ tru½7�;

tgt ¼@ tx@t

¼ tx½3� þ rtx½6� þ stx½5� þ rstx½7�;

u;t ¼ @u@t

¼ u½3� þ ru½6� þ su½5� þ rsu½7�: ðA:4Þ

Using Eqs. (A.1), (A.3) and (A.4) with Eq. (11) we obtain the nodalconstants for the covariant strain

nrrIJðr; s; tÞ ¼ n½1j1�IJ þ 2sn½1j4�

IJ þ 2tn½1j6�IJ þ 2stðn½1j7�

IJ þ n½4j6�IJ Þ

þ s2n½4j4�IJ þ 2s2tn½4j7�

IJ þ t2n½6j6�IJ þ 2st2n½6j7�

IJ þ s2t2n½7j7�IJ ;

Fig. 18. Rubber blocks subjected to pressure (6 � 1 � 15 mesh, E ¼ 1:0� 103 andm ¼ 0:49). (a) Rectangular block. (b) Trapezoidal block.

Fig. 19. Load-displacement curves for the rubber blocks. (a) Rectangular block. (b)Trapezoidal block.


nssIJðr; s; tÞ ¼ n½2j2�IJ þ 2tn½2j5�

IJ þ 2rn½2j4�IJ þ 2trðn½2j7�


þ t2n½5j5�IJ þ 2t2rn½5j7�

IJ þ r2n½4j4�IJ þ 2tr2n½4j7�

IJ þ t2r2n½7j7�IJ ;

nttIJðr; s; tÞ ¼ n½3j3�IJ þ 2rn½3j6�

IJ þ 2sn½3j5�IJ þ 2rsðn½3j7�


þ r2n½6j6�IJ þ 2r2sn½6j7�

IJ þ s2n½5j5�IJ þ 2rs2n½5j7�

IJ þ r2s2n½7j7�IJ ;

nrsIJðr; s; tÞ ¼ n½1j2�IJ þ tðn½1j5�

IJ þ n½2j6�IJ Þ þ rn½1j4�

IJ þ trðn½4j6�IJ þ n½1j7�

IJ Þþ sn½2j4�

IJ þ stðn½4j5�IJ þ n½2j7�

IJ Þ þ rsn½4j4�IJ þ 2rstn½4j7�

IJ

þ t2n½5j6�IJ þ t2rn½6j7�

IJ þ st2n½5j7�IJ þ rst2n½7j7�

IJ ;

nstIJðr; s; tÞ ¼ n½2j3�IJ þ rðn½2j6�

IJ þ n½3j4�IJ Þ þ sn½2j5�

IJ þ rsðn½4j5�IJ þ n½2j7�

IJ Þþ tn½3j5�

IJ þ trðn½5j6�IJ þ n½3j7�

IJ Þ þ stn½5j5�IJ þ 2rstn½5j7�

IJ

þ r2n½4j6�IJ þ r2sn½4j7�

IJ þ tr2n½6j7�IJ þ r2stn½7j7�

IJ ;

ntrIJðr; s; tÞ ¼ n½1j3�IJ þ sðn½3j4�

IJ þ n½1j5�IJ Þ þ tn½3j6�

IJ þ stðn½5j6�IJ þ n½3j7�

IJ Þþ rn½1j6�

IJ þ rsðn½4j6�IJ þ n½1j7�

IJ Þ þ trn½6j6�IJ þ 2rstn½6j7�

IJ

þ s2n½4j5�IJ þ s2tn½5j7�

IJ þ rs2n½4j7�IJ þ rs2tn½7j7�

IJ ; ðA:5Þwith nrrIJ ¼ n11IJ , nssIJ ¼ n22IJ , nttIJ ¼ n33IJ , nrsIJ ¼ n12IJ , nstIJ ¼ n23IJ andntrIJ ¼ n31IJ . Using Eq. (17) with Eq. (A.3),

~n½Kj7�IJ ¼ tdrn

½Kj1�IJ þ tdsn

½Kj2�IJ þ tdtn

½Kj3�IJ with K ¼ 1; . . . ;6;

~n½7j7�IJ ¼ td2

r n½1j1�IJ þ td2

s n½2j2�IJ þ td2

t n½3j3�IJ þ 2tdr

tdsn½1j2�IJ

þ 2tdstdtn

½2j3�IJ þ 2tdt

tdrn½1j3�IJ : ðA:6Þ

Substituting ~n½Kj7�IJ for n½Kj7�

IJ (K = 1, . . . , 7) in Eq. (A.5), we obtain thenodal constants for the assumed covariant strain~nIJðr; s; tÞ ¼ ~n11IJ ~n22IJ ~n33IJ ~n12IJ ~n23IJ ~n31IJ

� �.

Appendix B. Calculation of strain-displacement matrices

We aim to obtain good computational efficiency in the evalua-tion of the element matrices.

Rearranging the strain transformation in Eq. (12),

n_

IJðr; s; tÞT ¼ G_

nIJðr; s; tÞT ;

n_

IJðr; s; tÞ ¼ n_

11IJ n_

22IJ n_

33IJ n_

12IJ n_

23IJ n_

31IJ

h i;

nIJðr; s; tÞ ¼ n11IJ n22IJ n33IJ n12IJ n23IJ n31IJ½ �;

Fig. 20. Deformed shapes for the rubber blocks. (a) Rectangular block. (b)Trapezoidal block.

Fig. 22. Load-displacement curves for the rubber panel.

Fig. 21. A rubber panel subjected to surface loading (3 � 3 � 9 mesh, E ¼ 1:0� 103

and m ¼ 0:49).


G_

¼

G_

1

G_

2

G_

3

G_

4

G_

5

G_

6

0BBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCA

¼

g11g

11 g2

1g21 g3

1g31 2g1

1g21 2g2

1g31 2g3

1g11

g12g

12 g2

2g22 g3

2g32 2g1

2g22 2g2

2g32 2g3

2g12

g13g

13 g2

3g23 g3

3g33 2g1

3g23 2g2

3g33 2g3

3g13

g11g

12 g2

1g22 g3

1g32 g1

1g22 þ g2

1g12 g2

1g32 þ g3

1g22 g3

1g12 þ g1

1g32

g12g

13 g2

2g23 g3

2g33 g1

2g23 þ g2

2g13 g2

2g33 þ g3

2g23 g3

2g13 þ g1

2g33

g13g

11 g2

3g21 g3

3g31 g1

3g21 þ g2

3g11 g2

3g31 þ g3

3g21 g3

3g11 þ g1

3g31

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA;

Fig. 23. Deformed shapes for the rubber panel.

0jG_ ¼ C

_

1 C_

2 C_

3 C_

4 C_

5 C_

6

� �: ðB:1Þ

We define the following representative constant vectors evaluatedat the tying positions of the assumed strain components in Eq. (18)


C10

C20

C30

C40

C50

C60

C70

C80

C90

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA

¼ 10V

X8L¼1

1 gL fL gLfL½ �TC_�nL

1ffiffi3

p ;gL 1ffiffi3

p ;fL1ffiffi3

p�

1

1 fL nL fLnL½ �TC_�nL

1ffiffi3

p ;gL 1ffiffi3

p ;fL1ffiffi3

p�

2

1 nL gL nLgL½ �TC_�nL

1ffiffi3

p ;gL 1ffiffi3

p ;fL1ffiffi3

p�

3

1 nL gL nLgL½ �TC_�nL

1ffiffi3

p ;gL 1ffiffi3

p ;fL1ffiffi3

p�

4

1 gL fL gLfL½ �TC_�nL

1ffiffi3

p ;gL 1ffiffi3

p ;fL1ffiffi3

p�

5

1 fL nL fLnL½ �TC_�nL

1ffiffi3

p ;gL 1ffiffi3

p ;fL1ffiffi3

p�

6

1 nL gL nLgL½ �TfLC_�nL

1ffiffi3

p ;gL 1ffiffi3

p ;fL1ffiffi3

p�

4

1 gL fL gLfL½ �TnLC_�nL

1ffiffi3

p ;gL 1ffiffi3

p ;fL1ffiffi3

p�

5

1 fL nL fLnL½ �TgLC_�nL

1ffiffi3

p ;gL 1ffiffi3

p ;fL1ffiffi3

p�

6

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA

;

0V ¼X8L¼1

0j�nL 1ffiffi

3p ;gL 1ffiffi

3p ;fL

1ffiffi3

p�;

G11 G1

2 G13

G21 G2

2 G23

G31 G3

2 G33

G41 G4

2 G43

0BBBB@

1CCCCA ¼

X4L¼1

1nLgL

nLgL

0BBB@

1CCCA G

_�0;nL 1ffiffi

3p ;gL 1ffiffi

3p�

1 G_�gL 1ffiffi

3p ;0;nL 1ffiffi

3p�

2 G_�nL

1ffiffi3

p ;gL 1ffiffi3

p ;0�

3

� �;

n10

��TIJ

n20

��TIJ

n30

��TIJ

0BBBB@

1CCCCA ¼ 1

8

X8L¼1

1 nL gL nLgL½ �~n�0;nL 1ffiffi

3p ;gL 1ffiffi

3p�

rrIJ

1 nL gL nLgL½ �~n�gL 1ffiffi

3p ;0;nL 1ffiffi

3p�

ssIJ

1 nL gL nLgL½ �~n�nL

1ffiffi3

p ;gL 1ffiffi3

p ;0�

ttIJ

0BBBBBBB@

1CCCCCCCA

¼

n½1j1�IJ þ 1

3n½4j4�IJ þ 1

3n½6j6�IJ þ 1

9~n½7j7�IJ

2ffiffi3

p n½1j4�IJ þ 1

3~n½6j7�IJ

� �2ffi3

p

n½2j2�IJ þ 1

3n½5j5�IJ þ 1

3n½4j4�IJ þ 1

9~n½7j7�IJ

2ffiffi3

p n½2j5�IJ þ 1

3~n½4j7�IJ

� �2ffi3

p

n½3j3�IJ þ 1

3n½6j6�IJ þ 1

3n½5j5�IJ þ 1

9~n½7j7�IJ

2ffiffi3

p n½3j6�IJ þ 1

3~n½5j7�IJ

� �2ffi3

p

0BBBBB@

n40

��TIJ

n50

��TIJ

n60

��TIJ

0BBBB@

1CCCCA ¼ 1

8

X8L¼1

1 nL gL nLgL½ �~n�nL

1ffiffi3

p ;gL 1ffiffi3

p ;fL1ffiffi3

p�

rsIJ

1 gL fL gLfL½ �~n�nL

1ffiffi3

p ;gL 1ffiffi3

p ;fL1ffiffi3

p�

stIJ

1 fL nL fLnL½ �~n�nL

1ffiffi3

p ;gL 1ffiffi3

p ;fL1ffiffi3

p�

trIJ

0BBBBBBB@

1CCCCCCCA

¼

n½1j2�IJ þ 1

3n½5j6�IJ

1ffiffi3

p n½1j4�IJ þ 1

3~n½6j7�IJ

� �1ffiffi3

p n½2j4�IJ þ 1

3~n½5j7�IJ

� �n½2j3�IJ þ 1

3n½4j6�IJ

1ffiffi3

p n½2j5�IJ þ 1

3~n½4j7�IJ

� �1ffiffi3

p n½3j5�IJ þ 1

3~n½6j7�IJ

� �n½1j3�IJ þ 1

3n½4j5�IJ

1ffiffi3

p n½3j6�IJ þ 1

3~n½5j7�IJ

� �1ffiffi3

p n½1j6�IJ þ 1

3~n½4j7�IJ

� �

0BBBBB@

G14 G1

5 G16

G24 G2

5 G26

!¼X2L¼1

1fL

G_ð0;0;fLÞ

4 G_ðfL ;0;0Þ

5 G_ð0;fL ;0Þ

6

� �;

H14 H1

5 H16

H24 H2

5 H26

H34 H3

5 H36

H44 H4

5 H46

H54 H5

5 H56

H64 H6

5 H66

H74 H7

5 H76

H84 H8

5 H86

0BBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCA

¼X4L¼1

X2M¼1

1nLgL

nLgL

fMnLfMgLfM

nLgLfM

0BBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCA

G_�nL

1ffiffi3

p ;gL 1ffiffi3

p ;fM

�4 G

_�fM ;nL

1ffiffi3

p ;gL 1ffiffi3

p�

5 G_�gL 1ffiffi

3p ;fM ;nL

1ffiffi3

p�

6

� �;

ðB:2Þin which the signs are f1 f2

� � ¼ 1 �1½ �, n1 n2 n3 n4� � ¼

�1 1 1 �1½ � and g1 g2 g3 g4½ � ¼ �1 �1 1 1½ �. Wenote that the constants in Eq. (B.2) are calculated only once for eachelement.

We define the following representative nodal constants

n00

��IJ ¼ n½1j1�

IJ n½2j2�IJ n½3j3�


IJ n½1j3�IJ

h i;

ffi n½1j6�IJ þ 1

3~n½4j7�IJ

� �23

~n½1j7�IJ þ n½4j6�

IJ

� �ffi n½2j4�

IJ þ 13~n½5j7�IJ

� �23

~n½2j7�IJ þ n½4j5�

IJ

� �ffi n½3j5�


� �23

~n½3j7�IJ þ n½5j6�

IJ

� �

1CCCCCA;

13 n½4j4�


� �13 n½5j5�


� �13 n½6j6�


� �

1CCCCCA;

n70

��TIJ

n80

��TIJ

n90

��TIJ

0BBB@

1CCCA ¼ 1

8

X8L¼1

1 nL gL nLgL½ �fL~n�nL

1ffiffi3

p ;gL 1ffiffi3

p ;fL1ffiffi3

p�

rsIJ

1 gL fL gLfL½ �nL~n�nL

1ffiffi3

p ;gL 1ffiffi3

p ;fL1ffiffi3

p�

stIJ

1 fL nL fLnL½ �gL~n

�nL

1ffiffi3

p ;gL 1ffiffi3

p ;fL1ffiffi3

p�

trIJ

0BBBBB@

1CCCCCA

¼

1ffiffi3

p n½1j5�IJ þ n½2j6�

IJ

� �13 n½4j6�

IJ þ ~n½1j7�IJ

� �13 n½4j5�

IJ þ ~n½2j7�IJ

� �2

3ffiffi3

p ~n½4j7�IJ

1ffiffi3


IJ

� �13 n½4j5�

IJ þ ~n½2j7�IJ

� �13 n½5j6�

IJ þ ~n½3j7�IJ

� �2

3ffiffi3

p ~n½5j7�IJ

1ffiffi3


IJ

� �13 n½5j6�

IJ þ ~n½3j7�IJ

� �13 n½4j6�

IJ þ ~n½1j7�IJ

� �2

3ffiffi3

p ~n½6j7�IJ

0BBBB@

1CCCCA;


E1jIJ E2jIJ E3jIJ� � ¼ 1

4

X4L¼1

1nLgL

nLgL

0BBB@

1CCCA n

�0;nL 1ffiffi

3p ;gL 1ffiffi

3p�T

IJ n

�gL 1ffiffi

3p ;0;nL 1ffiffi

3p�T

IJ n

�nL

1ffiffi3

p ;gL 1ffiffi3

p ;0�T

IJ

;

E1jIJ ¼

n00

��TIJ þ 1

3 n½4j4�IJ þ n½6j6�

IJ þ 13n

½7j7�IJ n½5j5�


IJ 0 n½4j5�IJ

h iT1ffiffi3

p 2n½1j4�IJ þ 2

3n½6j7�IJ 0 2n½3j5�

IJ n½2j4�IJ þ 1

3n½5j7�IJ n½2j5�

IJ n½3j4�IJ þ n½1j5�

IJ

h iT1ffiffi3

p 2n½1j6�IJ þ 2

3n½4j7�IJ 2n½2j5�

IJ 0 n½1j5�IJ þ n½2j6�


IJ þ 13n

½5j7�IJ

h iT13 2n½1j7�

IJ þ 2n½4j6�IJ 0 0 n½4j5�

IJ þ n½2j7�IJ n½5j5�


IJ

h iT

0BBBBBBBBB@

1CCCCCCCCCA;

E2jIJ ¼

n00

��TIJ þ 1

3 n½6j6�IJ n½5j5�

IJ þ n½4j4�IJ þ 1

3n½7j7�IJ n½6j6�


IJ 0h iT

1ffiffi3

p 2n½1j6�IJ 2n½2j5�

IJ þ 23n

½4j7�IJ 0 n½1j5�


IJ þ 13n

½6j7�IJ n½3j6�

IJ

h iT1ffiffi3

p 0 2n½2j4�IJ þ 2

3n½5j7�IJ 2n½3j6�

IJ n½1j4�IJ þ 1

3n½6j7�IJ n½2j6�


IJ

h iT13 0 2n½2j7�

IJ þ 2n½4j5�IJ 0 n½4j6�



IJ

h iT

0BBBBBBBBB@

1CCCCCCCCCA;

E3jIJ ¼

n00

��TIJ þ 1

3 n½4j4�IJ n½4j4�


IJ þ 13n

½7j7�IJ 0 n½4j6�

IJ n½4j5�IJ

h iT1ffiffi3

p 0 2n½2j4�IJ 2n½3j6�

IJ þ 23n

½5j7�IJ n½1j4�


IJ n½1j6�IJ þ 1

3n½4j7�IJ

h iT1ffiffi3

p 2n½1j4�IJ 0 2n½3j5�

IJ þ 23n

½6j7�IJ n½2j4�

IJ n½2j5�IJ þ 1

3n½4j7�IJ n½3j4�

IJ þ n½1j5�IJ

h iT13 0 0 2n½3j7�

IJ þ 2n½5j6�IJ n½4j4�



IJ

h iT

0BBBBBBBBB@

1CCCCCCCCCA;

n14

��IJ

n24

��IJ

!¼ 1

2

X2L¼1

1fL

nð0;0;fLÞ

T

IJ ¼n0

0

��TIJ þ n½6j6�

IJ n½5j5�IJ 0 n½5j6�

IJ 0 0h iT

2n½1j6�IJ 2n½2j5�

IJ 0 n½1j5�IJ þ n½2j6�


IJ

h iT0B@

1CA;

n15

��IJ

n25

��IJ

!¼ 1

2

X2L¼1

1fL

nðfL ;0;0Þ

T

IJ ¼n0

0

��TIJ þ 0 n½4j4�

IJ n½6j6�IJ 0 n½4j6�

IJ 0h iT

0 2n½2j4�IJ 2n½3j6�



IJ

h iT0B@

1CA;

n16

��IJ

n26

��IJ

!¼ 1

2

X2L¼1

1fL

nð0;fL ;0Þ

T

IJ ¼n0

0

��TIJ þ n½4j4�

IJ 0 n½5j5�IJ 0 0 n½4j5�

IJ

h iT2n½1j4�

IJ 0 2n½3j5�IJ n½2j4�


IJ þ n½1j5�IJ

h iT0B@

1CA;

s00��IJ ¼ n0

0

��IJ þ

13

n½4j4�IJ þ n½6j6�




IJ n½4j5�IJ

h i;


S4jIJ S5jIJ S6jIJ� � ¼ 1

8

X4L¼1

X2M¼1

1nLgL

nLgL

fMnLfMgLfMnLgLfM

0BBBBBBBBBBBBB@

1CCCCCCCCCCCCCA

n

�nL

1ffiffi3

p ;gL 1ffiffi3

p ;fM

�TIJ n

�fM ;nL

1ffiffi3

p ;gL 1ffiffi3

p�T

IJ n

�gL 1ffiffi

3p ;fM ;nL

1ffiffi3

p�T

IJ

;

S4jIJ ¼

s00��TIJþ 1

3 2n½6j6�IJ þ n½7j7�

IJ 2n½5j5�IJ þ n½7j7�

IJ13n

½7j7�IJ 2n½5j6�

IJ 0 0h iT

1ffiffi3

p 0 2 n½2j4�IJ þ n½5j7�

IJ

� �2 n½3j6�

IJ þ 13n

½5j7�IJ

� �n½1j4�IJ þ n½6j7�


IJ n½1j6�IJ þ 1

3n½4j7�IJ

h iT1ffiffi3

p 2 n½1j4�IJ þ n½6j7�

IJ

� �0 2 n½3j5�

IJ þ 13n

½6j7�IJ

� �n½2j4�IJ þ n½5j7�

IJ n½2j5�IJ þ 1

3n½4j7�IJ n½3j4�

IJ þ n½1j5�IJ

h iT13 0 0 2 n½3j7�

IJ þ 13n

½5j6�IJ

� �n½4j4�IJ þ n½7j7�



IJ

h iT2 n½1j6�

IJ þ 13n

½4j7�IJ

� �2 n½2j5�

IJ þ 13n

½4j7�IJ

� �0 n½1j5�


IJ þ 13n

½6j7�IJ n½3j6�

IJ þ 13n

½5j7�IJ

h iT1ffiffi3

p 0 2 n½2j7�IJ þ n½4j5�

IJ

� �0 n½4j6�



IJ þ 13n

½7j7�IJ

h iT1ffiffi3

p 2 n½1j7�IJ þ n½4j6�

IJ

� �0 0 n½4j5�


IJ þ 13n

½7j7�IJ n½5j6�

IJ þ n½3j7�IJ

h iT23 0 0 0 n½4j7�


IJ

h iT

0BBBBBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCCCCA

;

S5jIJ ¼

s00��TIJþ 1

313n

½7j7�IJ 2n½4j4�

IJ þ n½7j7�IJ 2n½6j6�

IJ þ n½7j7�IJ 0 2n½4j6�

IJ 0h iT

1ffiffi3

p 2 n½1j4�IJ þ 1

3n½6j7�IJ

� �0 2 n½3j5�

IJ þ n½6j7�IJ

� �n½2j4�IJ þ 1

3n½5j7�IJ n½2j5�


IJ þ n½1j5�IJ

h iT1ffiffi3

p 2 n½1j6�IJ þ 1

3n½4j7�IJ

� �2 n½2j5�

IJ þ n½4j7�IJ

� �0 n½1j5�



IJ þ 13n

½5j7�IJ

h iT13 2

�n½1j7�IJ þ n½4j6�

IJ

�0 0 n½4j5�



IJ þ n½3j7�IJ

h iT0 2 n½2j4�

IJ þ 13n

½5j7�IJ

� �2 n½3j6�

IJ þ 13n

½5j7�IJ

� �n½1j4�IJ þ 1

3n½6j7�IJ n½2j6�


IJ þ 13n

½4j7�IJ

h iT1ffiffi3

p 0 0 2�n½3j7�IJ þ n½5j6�

IJ

�n½4j4�IJ þ 1

3n½7j7�IJ n½4j5�


IJ þ n½1j7�IJ

h iT1ffiffi3

p 0 2�n½2j7�IJ þ n½4j5�

IJ

�0 n½4j6�



IJ þ 13n

½7j7�IJ

h iT23 0 0 0 n½4j7�


IJ

h iT



;

S6jIJ ¼

s00��TIJ þ 1

3 2n½4j4�IJ þ n½7j7�

IJ13n

½7j7�IJ 2n½5j5�

IJ þ n½7j7�IJ 0 0 2n½4j5�

IJ

h iT1ffiffi3

p 2 n½1j6�IJ þ n½4j7�

IJ

� �2 n½2j5�

IJ þ 13n

½4j7�IJ

� �0 n½1j5�


IJ þ 13n

½6j7�IJ n½3j6�

IJ þ n½5j7�IJ

h iT1ffiffi3

p 0 2 n½2j4�IJ þ 1

3n½5j7�IJ

� �2 n½3j6�

IJ þ n½5j7�IJ

� �n½1j4�IJ þ 1

3n½6j7�IJ n½2j6�


IJ þ n½4j7�IJ

h iT13 0 2 n½2j7�

IJ þ n½4j5�IJ

� �0 n½4j6�



IJ þ n½7j7�IJ

h iT2 n½1j4�

IJ þ 13n

½6j7�IJ

� �0 2 n½3j5�

IJ þ 13n

½6j7�IJ

� �n½2j4�IJ þ 1

3n½5j7�IJ n½2j5�

IJ þ 13n

½4j7�IJ n½3j4�

IJ þ n½1j5�IJ

h iT1ffiffi3

p 2 n½1j7�IJ þ n½4j6�

IJ

� �0 0 n½4j5�


IJ þ 13n

½7j7�IJ n½5j6�

IJ þ n½3j7�IJ

h iT1ffiffi3

p 0 0 2 n½3j7�IJ þ n½5j6�

IJ

� �n½4j4�IJ þ 1

3n½7j7�IJ n½4j5�


IJ þ n½1j7�IJ

h iT23 0 0 0 n½4j7�


IJ

h iT



: ðB:3Þ


Here, the part of nM0

��IJ related to ~n½Kj7�

IJ is updated using Eq. (A.6) once

for each element per each increment of nonlinear solution. The

remaining parts of nM0

��IJ , and the constants Ej

��IJ (j ¼ 1,2,3), nk

j

��IJ

and Sj

��IJ(j ¼ 4, 5, 6) are independent of solution increments, loca-

tions within elements and are the same for all elements and henceare only calculated once.

It is possible to represent the assumed strain field in Eq. (18) interms of nodal constants as

~nasrrIJ ¼ ~ncon

rrIJ þffiffiffi3

p

40kðr; s; tÞsn_s

rrIJ þffiffiffi3

p

40kðr; s; tÞtn_t

rrIJ

þ 34

0kðr; s; tÞstn_strrIJ;

~nasssIJ ¼ ~ncon

ssIJ þffiffiffi3

p

40kðr; s; tÞtn_t

ssIJ þffiffiffi3

p

40kðr; s; tÞrn_r

ssIJ

þ 34

0kðr; s; tÞtrn_trssIJ;

~nasttIJ ¼ ~ncon

ttIJ þffiffiffi3

p

40kðr; s; tÞrn_r

ttIJ þffiffiffi3

p

40kðr; s; tÞsn_s

ttIJ

þ 34

0kðr; s; tÞrsn_rsttIJ;

~nasrsIJ ¼ ~ncon

rsIJ þ12

0kðr; s; tÞtn_trsIJ þ

ffiffiffi3

p

8tanhðt0cÞðn

_rrsIJr þ n

_srsIJs

þffiffiffi3

pn_rs

rsIJrsÞ0kðr; s; tÞ;

~nasstIJ ¼ ~ncon

stIJ þ12

0kðr; s; tÞrn_rstIJ þ

ffiffiffi3

p

8tanhðt0cÞðn

_sstIJsþ n

_tstIJt

þffiffiffi3

pn_st

stIJstÞ0kðr; s; tÞ;

~nastrIJ ¼ ~ncon

trIJ þ12

0kðr; s; tÞsn_strIJ þ

ffiffiffi3

p

8tanhðt0cÞðn

_ttrIJ t þ n

_rtrIJr

þffiffiffi3

pn_tr

trIJtrÞ0kðr; s; tÞ; ðB:4Þ

~nconrrIJ

~nconssIJ

~nconttIJ

~nconrsIJ

~nconstIJ

~ncontrIJ

� �T ¼X9M¼1

CMT

0 nM0

��IJ;

n_srrIJ

n_trrIJ

n_strrIJ

26664

37775¼

G21 G1

1 G41 G3

1

G31 G4

1 G11 G2

1

G41 G3

1 G21 G1

1

2664

3775E1jIJ ;

n_t

ssIJ

n_r

ssIJ

n_tr

ssIJ

26664

37775¼

G22 G1

2 G42 G3

2

G32 G4

2 G12 G2

2

G42 G3

2 G22 G1

2

2664

3775E2jIJ ;

n_rttIJ

n_sttIJ

n_rsttIJ

26664

37775 ¼

G23 G1

3 G43 G3

3

G33 G4

3 G13 G2

3

G43 G3

3 G23 G1

3

2664

3775E3jIJ; n

_trsIJ ¼ G2

4n14

��IJþ G1

4n24

��IJ;

n_r

rsIJ

n_s

rsIJ

n_rs

rsIJ

26664

37775¼

H24 H1

4 H44 H3

4 H64 H5

4 H84 H7

4

H34 H4

4 H14 H2

4 H74 H8

4 H54 H6

4

H44 H3

4 H24 H1

4 H84 H7

4 H64 H5

4

264

375S4jIJ ; n

_rstIJ ¼ G2

5n15

��IJ þG1

5n25

��IJ ;

n_s

stIJ

n_t

stIJ

n_st

stIJ

26664

37775¼

H25 H1

5 H45 H3

5 H65 H5

5 H85 H7

5

H35 H4

5 H15 H2

5 H75 H8

5 H55 H6

5

H45 H3

5 H25 H1

5 H85 H7

5 H65 H5

5

264

375S5jIJ ; n_s

trIJ ¼ G26n

16

��IJ þG1

6n26

��IJ ;

n_t

trIJ

n_r

trIJ

n_tr

trIJ

26664

37775 ¼

H26 H1

6 H46 H3

6 H66 H5

6 H86 H7

6

H35 H4

6 H16 H2

6 H76 H8

6 H56 H6

6

H45 H3

6 H26 H1

6 H86 H7

6 H66 H5

6

264

375S6jIJ: ðB:5Þ

In Eq. (B.4), we carry out the computations that depend on the nat-ural coordinates (r, s, t) (for each Gauss point), once for each differ-ent nodal coupling (nodes I and J). Using Eqs. (B.2) and (B.3) to Eq.

(B.5), we calculate (once per each element) the constants n_i

kkIJ , n_ jkkIJ ,

n_ijkkIJ , n

_kijIJ , n

_iijIJ , n

_ jijIJ and n

_ijijIJ , and update ~ncon

ijIJ in each increment.The nodal constants for the global volumetric and deviatoric

strains (nvolIJ and ndevijIJ ) are obtained from

nvolIJ ¼ ndilxxIJ þ ndil

yyIJ þ ndilzzIJ ;n

devijIJ ¼ nas

ijIJ �13nvolIJ dij;

nasijIJ ¼ ~nas

klIJðii � 0g_kÞðij � 0g

_lÞ;ndiliiIJ ¼ ~ncon

klIJ ðii � 0g_kÞðij � 0g

_lÞ: ðB:6Þ

The deviatoric strain-displacement matrices (Bdevij and Ndev

ij ) areobtained by

Bdevij ¼ Bdev

ij1 � � � BdevijI � � � Bdev

ij8

� �;Bdev

ijI ¼X8J¼1

ndevijIJ

txJ 0 0

0 ndevijIJ

tyJ 0

0 0 ndevijIJ

tzJ

0BB@

1CCA;

Ndevij ¼

Ndevij11 � � � Ndev

ij18

..

.Ndev

ijIJ...

Ndevij81 � � � Ndev

ij88

0BBB@

1CCCA;Ndev

ijIJ ¼ ndevijIJ

1 0 00 1 00 0 1

0B@

1CA; ðB:7Þ

where similar relation also holds for volumetric strain-displacement matrices, Bvol and Nvol with nvolIJ .

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