Computers and Structures 140 (2014) 14–22

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Computers and Structures

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Spurious modes in geometrically nonlinear small displacement finiteelements with incompatible modes

http://dx.doi.org/10.1016/j.compstruc.2014.04.0040045-7949/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +1 6179265199.E-mail address: kjb@mit.edu (K.J. Bathe).

Theodore Sussman a, Klaus-Jürgen Bathe b,⇑a ADINA R & D, Inc., Watertown, MA 02472, United Statesb Massachusetts Institute of Technology, Cambridge, MA 02139, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 15 March 2014Accepted 16 April 2014

Keywords:Incompatible finite elementsSpurious modesSmall displacementsSmall strainsGeometrically nonlinear analysisCritical strain value

We demonstrate the existence of spurious modes in finite elements with incompatible modes when ageometrically nonlinear displacement analysis with small displacements and strains is performed. Thespurious modes are a direct consequence of the incompatibility of the elements with displacementboundary conditions. We derive a critical compressive strain condition analytically, and show that thecritical strain can be quite small, with small displacements, if the geometric aspect ratio of the elementsis large but still practical. In numerical examples we give further insight and results in correspondencewith the analytical theory, and demonstrate that spurious modes can be triggered in practical small strainanalyses when using elements with incompatible modes.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The two-dimensional 4-node and three-dimensional 8-nodeincompatible modes finite elements (here now referred to as ICMelements) have been proposed in Ref. [1], improved for elementgeometric distortions in Ref. [2] and extensively used for manyyears. As the formulation of the ICM elements is quite well known,we do not give the formulation details here; for a general descrip-tion of the elements, see for example Ref. [3]. The key advantage ofthese elements, as compared with the elements not containingincompatible modes, is that the ICM elements do not exhibit shearlocking when subjected to pure bending. The undistorted ICMelements pass the linear analysis patch tests, and, with propercorrections, the distorted elements also pass these tests [2,3].

The elements are in fact a special case of the ‘enhanced strainelements’ (referred to as EAS elements) proposed in Ref. [4] andextensively studied, see for example Refs. [5–11] and thereferences therein. The concept of enhancing the strain assump-tions in finite elements is quite appealing to avoid lockingphenomena but it was also shown that the EAS elements, andhence ICM elements, are in fact unstable in large strain conditions[8–10]. To remedy the behavior of the elements, a stabilizationtechnique was proposed [11] with the use of deformationdependent stabilization parameters. While perhaps useful,

however, in our developments of finite element procedures weprefer to not use such factors [3,12].

Although the ICM elements are perhaps best suited to modelpure bending, in practice the elements are used in very complexnonlinear analyses in which the elements are subjected to a widevariety of stresses and boundary conditions. Thus it is valuable tounderstand the behavior of these elements in as many situationsas possible. In fact, the present paper is motivated by an attemptto explain the unexpected and unphysical results obtained in theanalysis of a large industrial problem involving only small strains.

The purpose of the present paper is to demonstrate theexistence of spurious modes (displacement patterns with zero ornegative energy) in meshes of geometrically nonlinear ICMelements subjected to only small displacement and small strain con-ditions. The spurious modes are associated with the incompatibilityof the incompatible modes with the boundary conditions, and thespurious modes are triggered when the strains reach a certaincritical value. The critical value is highly dependent on the elementgeometric aspect ratios, with elements having a large aspect ratiogiving a small critical value.

In Section 2, we derive the tangent bending modulus of a two-dimensional undistorted rectangular 4-node ICM element, assum-ing geometric nonlinear conditions, and a linear elastic materialwith zero Poisson’s ratio. The theoretical results show that evenin small displacements only, the element can become unstable.

In Section 3, we present some numerical examples that illus-trate and give insight into the derived theoretical results, and that

r

x1

x2

L1

Unitthickness

L2

s

12

3 4

(a) Initial configuration

Homogeneous deformationUndeformed

(b) Homogeneous deformation applied

Homogeneousdeformation

Incrementalcompatibledisplacements

Undeformed

u11

u13

u12

u14

(c) Incremental bending applied,compatible displacements shown

Homogeneousdeformation

Incrementalincompatibledisplacements

Undeformed

(d) Incompatible displacements due toincremental bending

Fig. 1. 2D 4-node ICM element under homogeneous deformation and incrementalbending.

T. Sussman, K.J. Bathe / Computers and Structures 140 (2014) 14–22 15

show the spurious modes for various element assemblages in two-and three-dimensional analyses. We also include the results of apractical analysis case.

Finally we give some closing remarks in Section 4.

2. Derivation of the element critical strain criterion

In the derivation of the critical strain condition, we makevarious simplifying assumptions in order to have a clear derivationand obtain insight into the reasons for the instability to occur.These assumptions will be introduced during the derivation.

The principle of virtual work for the total Lagrangian formula-tion is [3]

dW ¼Z

Sijdeij dV ð1Þ

where dW is the virtual work, Sij is the 2nd Piola–Kirchhoff stresstensor, referred to the original configuration, eij is the Green–Lagrange strain tensor, referred to the original configuration andthe integral is performed over the original volume. For brevity, wehave dropped the left super- and subscripts commonly used toindicate the time and reference configuration [3].

The increment in the principle of virtual work can be written

DdW ¼Z

@Sij

@eklDekldeij þ SijDdeij

� �dV ð2Þ

where the D denotes an increment.In the following we assume small strain conditions with linear

isotropic elasticity, Poisson’s ratio equal to zero and planardeformations in the x1–x2 plane. Eq. (2) becomes

DdW ¼ZðEðDe11de11 þ De22de22Þ þ 4GDe12de12 þ S11Dde11

þ S22Dde22 þ 2S12Dde12Þ dV ð3Þ

in which S11 ¼ Ee11; S22 ¼ Ee22; S12 ¼ 2Ge12 and the symmetries ofSij and eij are employed.

Using the definition eij ¼ 12 ðui;j þ uj;i þ uk;iuk;jÞ where ui;j ¼ @ui

@xj, ui

are the displacement components, xi are the (original) coordinatecomponents, and k is an index used in conjunction with theEinstein summation convention, we obtain for small strains

e11 � u1;1; e22 � u2;2; e12 �12ðu1;2 þ u2;1Þ ð4a;b; cÞ

De11 � Du1;1; De22 � Du2;2; De12 �12ðDu1;2 þ Du2;1Þ ð5a;b; cÞ

Dde11 ¼ Du1;1du1;1 þ Du2;1du2;1; Dde22 ¼ Du1;2du1;2 þ Du2;2du2;2;

Dde12 ¼ 12 ðDu1;1du1;2 þ Du2;1du2;2 þ Du1;2du1;1 þ Du2;2du2;1Þ

ð6a;b; cÞ

so that Eq. (3) becomes

DdW ¼Z

EðDu1;1du1;1 þ Du2;2du2;2Þ dV þZ

GðDu1;2du1;2

þ Du1;2du2;1 þ Du2;1du1;2 þ Du2;1du2;1Þ dV

þZ

S11ðDu1;1du1;1 þ Du2;1du2;1Þ dV þZ

S22ðDu1;2du1;2

þ Du2;2du2;2Þ dV þZ

S12ðDu1;1du1;2 þ Du2;1du2;2

þ Du1;2du1;1 þ Du2;2du2;1ÞdV ð7Þ

We now consider the single rectangular two-dimensional 4-node ICM element shown in Fig. 1. This element is first subjectedto a homogeneous deformation that produces the stresses S11,

S22, S12. Since this deformation is homogeneous, the incompatiblemodes in the element are not triggered.

Now consider the following incremental nodal displacements:Du1

1 ¼ Du31 ¼

L24 Dh, Du2

1 ¼ Du41 ¼ �

L24 Dh in which the superscript

denotes the node number and in which Dh controls the magnitudeof the displacement. Here Dh represents an incremental rotation asshown in the figure (assuming small strains, the original value of L2

is used instead of the deformed value). The incremental nodaldisplacements cause an internal incremental displacement withinthe element of

Undeformed

MM

Incrementaldeformation

Homogeneousdeformation

Fig. 2. Total displacements due to incremental bending.

16 T. Sussman, K.J. Bathe / Computers and Structures 140 (2014) 14–22

Du1 ¼L2

4rs Dh ð8Þ

where r, s are the element isoparametric coordinates.Simultaneously, an incremental incompatible displacement of

Du2 ¼ ð1� r2ÞDa ð9Þ

is introduced, where Da is the magnitude of the incremental incom-patible mode (Da represents a length rather than an angle; a is cus-tomarily used to represent the magnitude of an incompatiblemode). Thus there are two independent incremental degrees offreedom, Dh and Da, and two corresponding variations, dh and da.

Since the element is rectangular, we do not need to include thecorrections to the incompatible modes shape functions requiredfor the element to pass the patch test.

Clearly

Du1;1 ¼L2

2L1s Dh; Du1;2 ¼

12

r Dh;

Du2;1 ¼ �4L1

r Da; Du2;2 ¼ 0 ð10a;b; c;dÞ

Using Eqs. (5), (6) and (10), and performing the indicated integrals,Eq. (7) becomes in matrix form

DdW ¼ ½ dh da �k11 k12

k12 k22

� �Dh

Da

� �ð11Þ

where

k11 ¼ ðEþ S11Þ L32

12L1þ ðGþ S22Þ L1L2

12

k12 ¼ � 2L23 G

k22 ¼ ðGþ S11Þ 16L23L1

ð12a;b; cÞ

The incremental forces conjugate to the incremental displacementsare thus

DFh

DFa

� �¼

k11 k12

k12 k22

� �Dh

Da

� �ð13Þ

Using the condition DFa = 0 we obtain

DFh ¼ kDh ð14Þ

where

k ¼ k11 �k2

12

k22¼ ðEþ S11Þ

L32

12L1þ Gþ S22 �

G2

Gþ S11

!L1L2

12ð15Þ

is the tangent stiffness associated with Dh. The incremental forceDFh can be interpreted as an incremental bending moment DM.

We now assume that the strains are small, then the magnitudesof S11, S22 are much smaller than E, G. The terms in parentheses inEq. (15) can be approximated by

Eþ S11 � E ð16aÞ

Gþ S22 �G2

Gþ S11¼ Gþ S22 � G

11þ S11=G

� �� Gþ S22 � Gð1� S11=GÞ ¼ S11 þ S22 ð16bÞ

so that Eq. (15) becomes

k ¼ Eþ ðS11 þ S22ÞL1

L2

� �2 !

L32

12L1ð17Þ

Then using S11 ¼ Ee11; S22 ¼ Ee22, where e11; e22 are the strainscorresponding to the homogeneous deformations, Eq. (17) becomes

k ¼ EL32

12L11þ ðe11 þ e22Þ

L1

L2

� �2 !

ð18Þ

Using the bending moment of inertia I ¼ L32

12 (for unit thickness), andthe incremental curvature Dv ¼ Dh

L1(again assuming small strains

since the original value of L1 is used), we see that the tangentbending modulus is

ðEIÞjtangent ¼DFh

Dv ¼ EI 1þ ðe11 þ e22ÞL1

L2

� �2 !

ð19Þ

When both the displacements and strains are small, the Green–Lagrange strains are very close to the infinitesimal strains, andthe 2nd Piola–Kirchhoff stresses are very close to the Cauchy stres-ses, so under small displacement / small strain conditions, Eqs. (17)and (19) reduce to

k ¼ Eþ ðs11 þ s22ÞL1

L2

� �2 !

L32

12L1ð20Þ

ðEIÞjtangent ¼ EI 1þ ðe11 þ e22ÞL1

L2

� �2 !

ð21Þ

where eij is the infinitesimal strain tensor and sij is the Cauchystress tensor.

The incremental deformation corresponding to the imposedbending is shown in Fig. 2, and we can make the followingobservations:

4-node element without incompatible modes: The bendingmodulus of this element can be obtained by setting Da = 0, thenthe tangent bending modulus is given by Eq. (12a):

ðEIÞjtangent ¼ L1k11 ¼ ðEþ S11Þ þ ðGþ S22Þ L1L2

� �2� �

L32

12

� 1þ GE

L1

L2

!20@

1AEI

ð22Þ

(using small strain assumptions). The tangent bending modulus isseen to be too stiff due to shear locking, and the shear locking effectis worsened as the ratio L1

L2increases. This shear locking is the

primary motivation for introducing the incompatible modes.4-node ICM element

(1) The absence of the shear modulus in Eq. (17) shows thatshear locking is not present in the ICM element.

(2) For zero strains, the tangent bending modulus from Eq. (19)is EI. Of course this result is expected since the purpose ofincompatible modes is to represent the case of pure bending.

(3) There is a critical value of the sum e11 + e22, corresponding toa zero tangent bending modulus:

ðe11 þ e22Þjcrit ¼ �L2

L1

� �2

ð23Þ

When L1� L2, the critical value of e11 + e22 is very small, thus thesmall strain assumptions made throughout this derivation are valid

AB

CL2=0.5

L1=5

D

Not drawn to scaleE=1E6

T. Sussman, K.J. Bathe / Computers and Structures 140 (2014) 14–22 17

when L1� L2. We will assume that L1� L2 throughout our furtherdiscussion.

(4) The situation of having only the strain e11 nonzero is shownin Fig. 3. The compressive axial force associated with thiscritical strain is

=0

(a) Initial geometry, showing dimensions and nodes

Pjcrit ¼ ðEL2ÞL2

L1

� �2

¼ 12EI

L21

ð24Þ

u/2u/2

Prescribed traction

Prescribed traction

(b) Transverse traction boundary conditions

u/2

v/2

v/2

u/2

(c) Transverse displacement boundary conditions

M M

(d) Incremental bending

Fig. 4. One element bending tests.

which compares quite well with the Euler buckling loadPjEuler ¼ p2EI

L21

for a beam with both ends pinned. Clearly only anupper bound to the Euler buckling load can be expected, sincethe bending shape used in the element (Eq. (9)) is only an approx-imation to the Euler buckling mode shape.

Three-dimensional 8-node ICM element: The correspondingthree-dimensional 8-node ICM element contains the same criticalstrain as the two-dimensional 4-node ICM element (since thetwo-dimensional incompatible mode shapes are used in thethree-dimensional element). For example, considering an undis-torted brick element with side lengths L1, L2, L3, subjected to a

compressive strain e22, the critical strain is e22jcrit ¼ �L2

maxðL1 ;L3Þ

� �2.

3. Numerical examples

We present in this section the results of various example solu-tions. These finite element solutions illustrate the results of oursimple theory and also show that the instabilities can easily arisein practice. When we refer to strain values, we always meanGreen–Lagrange strains, even though these strains are numericallysmall.

3.1. One element bending tests

The purpose of these tests is to illustrate the results given inSection 2, and to show some of the fundamental characteristicsof the responses of different element types.

A single two-dimensional rectangular element is subjected tohomogeneous deformation, then incremental bending, as shownin Fig. 4. The element dimensions are shown in Fig. 4a and the ele-ment length/width ratio is 10/1. The Poisson ratio is assumed to bezero.

The homogeneous deformation in the longitudinal (horizontal)direction is always applied as a prescribed displacement. Thehomogeneous deformation in the transverse (vertical) direction isapplied in two ways: as a transverse prescribed traction (Fig. 4b),and as a transverse prescribed displacement (Fig. 4c). Theincremental rotation is applied as an additional prescribed rotationof Dh = 0.001 radians (Fig. 4d). Combining all of the above, theboundary conditions are

All models:

uA1 ¼

u2þL2

4Dh; uB

1¼�u2�L2

4Dh; uC

1 ¼�u2þL2

4Dh; uD

1 ¼u2�L2

4Dh

UndeformedBuckling mode

Deformation, =11 crit

Fig. 3. Buckling mode due to incremental bending.

Transverse prescribed traction:Traction along line AB = Traction along line CD

Transverse prescribed displacements:

u2 along line AB ¼ v2; u2 along line CD ¼� v

2

In the finite element solution, the general total Lagrangian formula-tion is used [3], containing as a special case the theory in Section 2.In the first solution step, the homogeneous deformation is applied,and in the second solution step, the incremental rotation isimposed. The conjugate reaction force DM obtained from thesecond solution step is then used to calculate the bending modulusEI ¼ DM L1

Dh.Four homogeneous deformation cases are run:

(1) No deformation: e11 = e22 = 0(2) e11 ¼ �0:01; e22 ¼ 0(3) e11 ¼ 0; e22 ¼ �0:01(4) e11 ¼ e22 ¼ �0:005

In cases 2–4, the homogeneous deformations correspond to thecritical strain condition given by Eq. (23).

For each homogeneous deformation case, the two transverseboundary conditions of prescribed traction and prescribeddisplacement are run.

18 T. Sussman, K.J. Bathe / Computers and Structures 140 (2014) 14–22

Finally three different element types are used: 4-node elements(without incompatible modes), 4-node ICM elements andquadratic 8-node elements (without incompatible modes).

The results are given in Table 1. In this table, the bendingmodulus is normalized by the analytical bending modulusE L3

212 ¼ 1:04167E5. We observe the following:

(1) Physically the transverse traction boundary conditionshould result in a much softer bending modulus than thetransverse prescribed displacement boundary condition,since pure bending is permitted with the transverseprescribed traction boundary condition, and pure bendingis suppressed by the transverse prescribed displacementboundary condition. We are not trying to accuratelycompute the bending modulus for the transverse prescribeddisplacement boundary condition, but we do expect that thisbending modulus be much stiffer than the bending moduluscorresponding to the transverse traction boundarycondition.

(2) For the 4-node element, the same results are obtained withboth transverse boundary conditions, and these results aremuch stiffer than the analytical beam bending modulus. Thisresult is expected, since the 4-node element does not includea bending deformation shape.

(3) For the 4-node ICM element, the same results are obtainedwith both transverse boundary conditions, and these resultsare in agreement with the theory of Section 2. However theresults are physically unrealistic for the transverseprescribed displacement boundary condition. This must beexpected, since the displacements corresponding to theincompatible mode do not satisfy the transverse prescribeddisplacement boundary condition.

(4) For the 8-node element, the response for the transversetraction boundary condition matches the results from the4-node ICM element, and the response for the transverse

Table 1Results for the 1-element bending tests.

Elementtype

Transverse boundarycondition

Normalized EI, fromcalculated solution

Com

(a) e11 = e22 = 04-node Traction 51.00 514-node Displacement 51.00 Sam4-node ICM Traction 1.00 Sam4-node ICM Displacement 1.00 Mu8-node Traction 1.00 Sam8-node Displacement 51.00 Sam

(b) e11 = �0.01, e22 = 04-node Traction 49.97 Ne4-node Displacement 49.97 Sam4-node ICM Traction �0.03 Ne4-node ICM Displacement �0.03 Tra8-node Traction �0.03 Sam8-node Displacement 49.97 Sam

(c) e11 = 0, e22 = �0.014-node Traction 50.00 Ne4-node Displacement 50.00 Sam4-node ICM Traction 0.00 Zer4-node ICM Displacement 0.00 Tra8-node Traction 0.00 Sam8-node Displacement 50.00 Sam

(d) e11 = e22 = �0.0054-node Traction 49.98 Ne4-node Displacement 49.98 Sam4-node ICM Traction �0.02 Ne4-node ICM Displacement �0.02 Tra8-node Traction �0.02 Sam8-node Displacement 49.98 Sam

displacement boundary condition matches the results fromthe 4-node element. This is all physically realistic.

3.2. Bending of an assemblage of 3 elements, transverse displacementboundary conditions

The assemblage of three elements shown in Fig. 5a is subjectedto transverse displacement boundary conditions and incrementalrotations, similar to the tests in Section 3.1. The problem dimen-sions are chosen so that the critical strain is�0.01, as in Section 3.1.The general total Lagrangian formulation is again employed andthe Poisson ratio is assumed to be zero.

The longitudinal strain is always 0. The transverse strain isvaried from 0 to �0.01.

The results are given in Table 2.4-node element assemblage: For zero transverse strain, the

bending modulus is much larger than the analytical beam bendingmodulus. When the critical strain is imposed in the transversedirection, the bending modulus decreases only slightly.

4-node ICM element assemblage: For zero transverse strain, thebending modulus is about nine times larger than the analyticalbeam bending modulus. When the critical strain is imposed inthe transverse direction, the bending modulus decreases to zero.

The incremental deformations of the 4-node ICM elementassemblage are shown, magnified by a factor of 500, in Fig. 5band c. Fig. 5b shows the actual incremental deformations, includ-ing the contribution from the incompatible modes, and Fig. 5cshows the incremental deformations not including the incompati-ble modes (which is how deformed ICM elements are typicallyplotted in practice). The corresponding total longitudinal stressesare also shown in Fig. 5c.

Thus we see that the critical strain criterion is applicable to anassemblage of ICM elements, when the adjacent elements can forman interlocking (hour-glass) pattern.

ments

times the analytical value; result as predicted by equation 22.e result as for transverse traction boundary conditione as analytical solution

ch too softe result as for 4-node ICM elemente result as for 4-node element

arly the same result as with e11 = e22 = 0e result as for transverse traction boundary condition

arly zero, as predicted by equation 19. This corresponds to Euler buckling.nsverse displacement boundary conditions not matched by incompatible mode.e result as for 4-node ICM elemente result as for 4-node element

arly the same result as with e11 = e22 = 0e result as for transverse traction boundary condition

o, as predicted by equation 19.nsverse displacement boundary conditions not matched by incompatible mode.e result as for 4-node ICM elemente result as for 4-node element

arly the same result as with e11 = e22 = 0e result as for transverse traction boundary condition

arly zero, as predicted by equation 19.nsverse displacement boundary conditions not matched by incompatible mode.e result as for 4-node ICM elemente result as for 4-node element

0.5

15E=1E6 =0

(a) Initial mesh

(b) Incremental deformations, including fromincompatible modes

(c) Incremental deformations from compatibledisplacements only, and total longitudinal stresses

Fig. 5. Bending of an assemblage of elements, transverse displacement boundaryconditions.

E=1E6

Planestrain

80

20

=0, 0.3 Presc.verticaldisp.

=5E-9

Fig. 6. Frequency analysis of a plane strain compressed block, unconfined in thelongitudinal direction.

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Displacement

Eige

nval

ue (×

1E1

0)

Mode 1, ICM, =0

Mode 2, ICM, =0

Critical displacement

Mode 2, ICM, =0.3

Mode 1, ICM, =0.3

Mode 1, no ICM, =0

Fig. 7. Eigenvalues vs. compressive displacement for frequency analysis of a 2Dcompressed block, unconfined in longitudinal direction.

Fig. 8. Spurious mode and modal longitudinal stresses for frequency analysis of a2D compressed block, unconfined in longitudinal direction.

T. Sussman, K.J. Bathe / Computers and Structures 140 (2014) 14–22 19

3.3. Frequency analysis of a two-dimensional compressed block,unconfined in longitudinal direction

The block shown in Fig. 6 is compressed and a frequencyanalysis is performed in the compressed configuration. A uniformmesh of 4-node ICM elements with the total Lagrangian formula-tion is employed. A consistent mass matrix is used in the frequencyanalysis.

As the length to width ratio of the elements is 8/1, the criticalstrain is 1/64, corresponding to a critical compressive displace-ment of 0.315.

The lowest two eigenvalues are graphed as a function of thecompressive displacement in Fig. 7. At the critical compressivedisplacement, the lowest eigenvalue becomes negative, and for aslightly larger compressive displacement, the second lowesteigenvalue also becomes negative. The results are in excellentagreement with the theory in Section 2.

The results are affected only slightly by the choice of Poisson’sratio.

Also shown for comparison is the lowest eigenvalue when4-node elements (without incompatible modes) are used. The eigen-value decreases only slightly over the range of compressive displace-ments shown. Again this result is in accordance with theory.

Fig. 8 shows the mode shape and modal longitudinal stresses ofthe model with ICM elements and zero Poisson’s ratio, calculatedat the critical compressive displacement. The spurious interlocking(hour-glass) pattern is clearly visible.

Table 2Results for bending of an assemblage of 3 elements, transverse displacement boundary co

Element type e22 Normalized EI, from calcula

4-node 0 2534-node �0.01 2484-node ICM 0 8.844-node ICM �0.01 0.00

3.4. Frequency analysis of a compressed block, confined in longitudinaldirection

The same problem as in Section 3.3 is solved, except that theblock is confined in the longitudinal direction and Poisson’s ratiois equal to zero (Fig. 9). Again, a uniform mesh of 4-node ICMelements with the total Lagrangian formulation is employed.

Different mesh refinements, all with the same element length towidth ratio of 8/1, are considered. The critical compressivedisplacement (compressive displacement at which the lowestmode has a zero eigenvalue) is plotted as a function of the numberof elements in the mesh (Fig. 10).

The longitudinal boundary condition tends to suppress theinterlocking spurious modes, so that the critical compressive dis-placement is higher than the theoretical value; however, as themesh is refined, the critical compressive displacement approachesthe theoretical value. A typical spurious mode shape and the modallongitudinal stresses are shown in Fig. 11.

3.5. Limit load analysis of a thread in a power screw

Fig. 12 shows a schematic of a power screw, and the threadgeometry of the screw. The material of the power screw isidealized as an elastic–plastic material with bilinear isotropic

nditions.

ted solution Comments

Virtually no stiffness. nonphysical spurious mode

E=1E6

80

20=0

Presc.verticaldisp.

=5E-9

Fig. 9. Frequency analysis of a compressed block, confined in the longitudinaldirection.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

10 100 1000 10000 100000

Number of elements

Theoretical value

5 10

10 20

20 4040 80 80 160C

ritic

al c

ompr

essi

ve d

ispl

acem

ent

Fig. 10. Results for frequency analysis of the compressed block, confined in thelongitudinal direction.

(a) Schematic drawing of power screw

All fillet radii 0.1All dimensions in mm

Material properties:

E=2.07 105 N/mm2

ET=2.07 103 N/mm2=0.3y=700 N/mm2

8

2 1

7

0.5

29o

(b) Thread geometry and material properties

Nut

Not drawn to scale

Lead is equal to twice the pitch.

Screw

Fig. 12. Power screw geometry.

20 T. Sussman, K.J. Bathe / Computers and Structures 140 (2014) 14–22

hardening. The nut is idealized as rigid, and frictionless contact isassumed between the screw and the nut.

Fig. 13 shows a three-dimensional model of 180 degrees of thethread. A mesh of 8-node ICM elements is used, along with a few 6-node prism elements, and the total Lagrangian formulation isemployed. Notice that the pitch of the thread is included in themesh.

The aspect ratio of the elements in the thread is quite large,with the elements much longer in the tangential direction thanin the other directions. However, it would lead to a very inefficientsolution if a mesh with good aspect ratio elements were used forthis geometry and in practice meshes with long slender elementsare used.

An axial prescribed displacement is applied to the screw, andthe thread of the screw contacts the thread groove of the nut.

When the threads come into contact, a plastic zone forms at thefillet between the screw thread and the screw. On further loading,a spurious mode is triggered. This mode can be visualized byplotting (non-smoothed) bands of the effective stresses as shownin Fig. 14.

Fig. 11. Spurious mode shape and modal longitudinal stresses for frequen

The effect of the spurious mode is also evident in the force–displacement curve (Fig. 15), in which we see that the force dropssuddenly when the spurious mode is triggered. For comparison, inFig 15, we also give the results obtained when the ordinary8-node element is used (that is, when incompatible modes are notincluded), and when a mixed displacement/pressure formulationis used with the 8/1 element. Here, the 8 corner nodes are usedfor the displacement interpolation and 1 pressure degree offreedom is employed [3]. Notice that the ordinary 8-node elementmesh is too stiff, which we expect since plasticity is occurring,plastic strains are incompressible and therefore volumetric lockingoccurs in the ordinary 8-node element. The 8/1 element isformulated to alleviate volumetric locking, and the mesh givesresults almost identical to the 8-node ICM element mesh untilthe spurious mode in the ICM element assemblage is triggered.

Clearly the contact surfaces and surrounding elements areacting as displacement boundary conditions on the ICM elements,and the incompatible displacements in the ICM elements are notcompatible with the displacement boundary conditions.

This problem demonstrates that a spurious mode can occur inpractical finite element analysis involving only small displace-ments when using the ICM elements.

cy analysis of a compressed block, confined in longitudinal direction.

X

Z

Symmetric boundary conditions

Prescribed vertical displacements

(a) Isometric view

(b) Front view, showing thread pitch

Rigid target surface

(c) Detail of front view, showing rigid target surface for nut

Y

Fig. 13. 3D finite element mesh of power screw.

(b) After spurious mode is triggered

(a) Before spurious mode is triggered

Fig. 14. Effective stresses in power screw mesh.

0

2000

4000

6000

8000

10000

12000

14000

16000

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Vertical displacement (mm)

Rea

ctio

n fo

rce

(N)

8-node ICM element

Spurious mode triggered

Ordinary 8-node element8/1 element

Fig. 15. Force–deflection curves for power screw model.

T. Sussman, K.J. Bathe / Computers and Structures 140 (2014) 14–22 21

4. Concluding remarks

While the difficulties of using elements with incompatiblemodes in large strain analyses are well known, we considered inthis paper geometrically nonlinear elements subjected to onlysmall strains and small displacements. This is clearly a very largefield of analysis.

In Section 2, we derived the tangent bending modulus of thegeometrically nonlinear 4-node ICM element with the followingsimplifying assumptions: undistorted rectangular element, linearelastic material, zero Poisson’s ratio, and small strains. Thesesimplifications were made to obtain basic insight into the elementbehavior. The same derivation can also be performed with a non-zero Poisson ratio, but the effect of Poisson’s ratio is small (as longas no incompressible confined material condition is approached),the resulting formulas are more complex and little further insightis gained.

When the tangent bending modulus is zero in the presence ofonly transverse traction boundary conditions, Euler bucking isseen, the associated mode shape is a physical response and henceis not spurious. But when the tangent bending modulus is zero in

the presence of transverse displacement boundary conditions, anonphysical response is seen: the zero tangent bending modulusthen corresponds to a spurious mode.

It is clear from the derivation that the spurious modes areinseparable from the desirable characteristics of the ICM elements,namely the absence of shear locking, and that the spurious modes

22 T. Sussman, K.J. Bathe / Computers and Structures 140 (2014) 14–22

are a direct consequence of the incompatibility of the incompatiblemodes with prescribed displacement boundary conditions.

In Section 3, we illustrated the nonphysical response obtainedwith the ICM elements. All these results are in correspondencewith the developed simple theory, and show that the element defi-ciency might be widely seen in actual finite element solutions (ifelement aspect ratios are large). The fundamental reason is thatICM elements are in practice embedded within finite elementmeshes where for a typical element, the neighboring elementsand contact conditions provide in essence displacement boundaryconditions.

We did not investigate in this paper whether there are addi-tional deficiencies of the ICM elements due to element geometricdistortions, but focused on the simplest case for a theoreticalanalysis.

Based on the experiences presented in this paper, we recom-mend that the ICM elements be used with caution in geometricnonlinear analyses, even involving only small strains, and thatwhen the ICM elements are used, these elements do not have alarge geometric aspect ratio.

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