t

[Reprinted from THE AERONAUTICAL QUARTERLY Vol. XIX, August 1968]

Published by THE ROYAL AERONAUTICAL SoCIETY

Finite Plane Strain of IncompressibleElastic Solids by the Finite

Element MethodJ. TINSLEY ODEN

(University 01 Alabama in Huntsville)

P"nlld by Thl LlW" Pr... Wlghlman & Co. LId .. Llw". Su ....

Finite Plane Strain of IncompressibleElastic Solids by the Finite

Element MethodJ. TINSLEY ODEN

(University 01 Alabama In Huntsville)

Summary: The finite element method is extended to the problem of finite planestrain of elastic solids. A highly elastic body subjected to two-dimensionaldeformations is repre<;ented by an assembly of triangular elements of finitedimension. The displacement fields within each element are approximated bylinear functions of the local coordinates. Non-linear stiffness relations involvinggeneralised node forces and displacements are derived from energy considera-tions. For demonstration purposes, the non-linear stiffness equations are appliedto the problems of finite simple shear and generalised shear. For finite simpleshear, it is shown that these relations are in exact agreement with finite elas-ticity theory. Convergence rates of finite element representations of these

problems are briefly examined.

1. Introduction

This paper investigates the application of the finite element method to theproblem of finite plane strain of incompressible elastic bodies. Previous investiga-tions have dealt with applications of the finite element concept to plane stressrroblems involving finite deformutiollsH and with general finite elemcnt formula-tions for the analysis of finitc deformations of three-dimensional elastic and visco-elastic bodies of general shapcu.

The development of finite element fomlUlations for the problem of finite planestrain of elastic solids involves several considerations nOI encountered in plane stressformulalions. Available constitutive equations for highly elastic materials, such asnatural rubber and certain solid propellant fuels. assume that such materials areincompressible. The deformation of incompressible bodies determines the stresstensor only to within a hydrostatic pressure. In the general theory of finite elasticdeformations7•8, this hydrostatic pressure is sometimes determined from equilibriumconditions and static (kinetic) boundary conditions. In plane stress problems, forexample, the hydrostatic pressure is determined from the condition that the normalstress in the direction normal to the plane in which the body deforms is zero. Withisochoric deformations. the hydrostatic pressure is arbitrary.

In the present investigation. the incompressibility condition is regarded as acondition of internal constraint. and the hydrostatic pressure appears as a Lagrangemultiplier in the definition of an energy functional. This procedure follows the

Originally received June 1967.

254 The Aeronautical Quarterly

FINITE PLANE STRAIN

general variational principles for constrained continua presented by Truesdell andToupin9

, suggested for incompressible finite elements of three-dimensional bodiesby Oden5 and used for infinitesimal deformations of incompressible solids by Her-mannlO

• The approach adopted in the present investigation leads to stillness relationswhich are non-linear in the generalised displacements. Solutions to representativefinite plane strain problems are examined; it is shown that the present formulationleads to exact solutions in tbe case of finite simple shear. The rate of convergenceof the procedure is also examined for the case of generalised shear.

Notation

Ao undeformed area of finite element

C1• C3 Mooney constants

Fa components of body force

11.13• la strain invariants

PNa global generalised forccs

Sa components of surface force

UN. global generalised displacements

V, V* energy functionals

W strain energy density

CaN displacement coefficients

kN rigid-body displacement coefficients

p hydrostatic pressure

p local generalised forceN ••

UNa< local generalised displacements

1'1/ strain tensor

811 Kronecker delta

E., permutation symbol

A extension ratio

.p strain invariant

(TtJ/J strcss tensor

fiMN• mapping function

2. Finite Plane Deformations

Consider a continuous body which undergoes finite deformations parallel to aplane. To describe the defomlation of this body. a fixed Cartesian coordinate system

August 1968 255

J, T. aDEN

Xi (j = 1,2,3) is established in the undeformed state so that the plane in which thebody deforms is parallel to the X,X2 plane. In addition to the plane deformations,the body may be subjected to a uniform extension parallel to the x, axis.

Let y/ denote the Cartesian coordinates in the deformed body of a particlewhich has coordinates ."(/in the undeformed state. Then

y,,=X,,+U,,; (I)

where Uu (u - 1.2) are the components of displacement and ,\ is the constant exten-sion ratio normal to the plane of deformation. The Lagrangian strain tensor isdefined by Ref. 8 as

_ (iJY", ~y", -81/) .Y'/- ! i)xj dXI .

(2)

whcre 0'1 is the Kronecker delta. Introducing equations (I) into (2) we find, for theI.:ascof finile plane strain,

...,...(au" I "JU~.+ "~UA i~UA) (3u)YII ! i)x~ (Jx, cJx~ cJXft

y",,=o (3b)

y,,=!('\~-l). (3c)

Three invariants can be formed from the deformation tensor 8ij+ 2y.,. Forfinile plane strain, the principal invariants are

/. ::::2 (J + y,,~)+ ,\~/~= I + 2,\ ~+ 2 (I + A') y ••+ tjJ/,=,\2(1 +2y,j.,+tjJ).

l1

(4)

In which (5)

where E"ft and E,\,,, are the two-dimensional permutation symbols.

3. Finite Element Representation

The continuous body is now represented by an assembly of a finitc number01' flat triangular elements as shown in Fig. I. The dimensions of these elemcnts arcassumed to be sulliciently small in comparison with charactcristic dimensions ofthe body that the displacement field within the elcment is adequately approximatedby linear functions of the coordinates x, of the form

(6)

in which au and b", are undetermined constants. By evaluating equation (6) at eachof the three node points of an element, six equations are obtained which can be

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FINITE PLANE STRAIN

Figure 1. Finite elementof un deformed body,

XI

//

-~'--solved to obtain au and ba~ in terms of the node displacements. If these solutionsare introduced into equation (6). it is found that the displacement field correspond-ing to element e is" I.S

where

and

(7)

(8a)

(8b)

In these equations, U~"l<' are the displacement components of node N of element e.E:OIK is the three-dimensional permutation symbol, Ao is the area of undefofmedtriangle. ·\',VA are the Cartesian coordinates of node M. Note that the sum is to betaken on all repeated indices in equations (7) and (8a), including N. M, and K.Ranges of these indices are: a, {3, A, µ= 1.2; M. N, K= 1.2.3; e= 1.2, .. " £,where E is the total number of elements. Unless noted olherwise. subsequentdevelopments pertain to a typical finite element which. for the present. is consideredto be independent of other elements in the discrete model. Thus, the element identi-fication index e is temporarily dropped for simplicity.

Substituting equation (7) into equation (3a) gives. for the strain componentsin a typical finite element.

(9)

with, of course, ')',13 = 0

and ')'33= ~ (A2-I).,-constant.August 1968 257

J. T. ODEN

4. Non-Linear Stiffness Equations

It is now assumed that the body under consideration is perfectly elastic and thatthere exists a potential function W which represents the strain energy per unit ofundeformed volume. If the body is isotropic. W can be expressed as a function ofthe three invariants defined in equations (4). The total potential energy of a typicalfini te element is given by

v = J W d'llo- f F.u. dv - J S,lI, ds,ClO 00

(10)

where '110 is the volume of the undeformed clement, F. are components of bodyforce per unit volume. and S. are components of the surface tractions per unitsurface area (arc) ds.

Although different finite elements in the system may have different materialproperties. individual elements are assumed to be homogeneous. Thus, W is not afunction of the initial coordinates x, and can be factored outside the integral inequation (10). Taking this into account. and substituting equation (7) into equation(10). the potential energy becomes

in which

(11)

(12)

The quantities PH. (N= 1. 2. 3; a= 1,2) are the components of "consistent" general-ised forcc. Physically, PNu is the generalised force at node N in direction a. It isimportant to note that the forces Sa arc, in general. functions of the displacements.since thcir direction and the area on which they act may change with the dcforma-tion.

According to the principle of minimum potential energy

Thus

(13)

(14)

Subsequent considerations are restricted to incompressible isotropic materials.For such materials. W is a function of only invariants 11and 12 and tlle incompressi-bility condition

(15)

must be satisfied, In particular. we consider highly elastic materials with strainenergy functions of the Mooney form

where C1 and C2 are material constants.

258

(16)

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FINITE PLANE STRAIN

Introducing equation (15) into equations (4). it is found that

(17a)

(l7b)

(17c)

Substituting equations (5), (16), and (17) into equation (14) leads to a non-linearstiffness relation for finite plane strain of a finite element of an isotropic. incom-pressible, elastic body; but the incompressibility condition is satisfied only in anaverage sense over the element. Moreover. an arbitrary hydrostatic pressure p doesnot affect Wand hence is not accounled for in the stiffness relation. To obtain anon-linear stilfness relation which includes hydrostatic pressure, we introduce a newenergy functional V* defined by

v*= V + vup (/3 - I). (18)

Here equation (15) is regarded as a condition of intemal constraint and the hydro-static pressurc plays the role of the Lagrange multiplier. This procedure followsthat suggested by Truesdell and Toupin' and Hermann10 and used by Oden· in thcformulation of three-dimensional problems, To be consistent with the linear dis-placement approximation, the hydrostatic pressurc is regarded as a constant foreach finite element.

Minimising v* with respect to UN. without introducing equation (15) leads tothe alternative form of the non-linear stiffness relation which involves the hydro-static pressure:

Equation (19) represents six simultaneous non-linear equations for each finiteclemcnt in thc seven unknowns Unu and p. The seventh equation needed is obtainedby introducing the incompressibility condition in equation (15).

It is interesting to note that. for certain types of deformation. equation (15) isautomatically satisfied for any type of material (e.g .. isochoric deformations such assimple shear). The incompressibility condition then provides no information andthe hydrostatic pressure is indeterminate. In such ca~s. additional information.usually in the form of conditions on the stress components. must be introducedif p is to be evaluated.

The stress tensor, measured per unit area in the deformed body and referredto convected coordinates XI, is given by'

1 (aw aw)(J"'P- - -+-- 2"; (/3) 19Y./I 19Y/l. 'August 1968

o-d= 0, (20)

259

J. T. ODEN

If W is of the Mooney form,

,,..11 = 2C1o'U) + 2C2 [0.11 (A2 + ~,) + 2E"AEII"YAI'] + P (0011~ 2eA

Eo"'YA',)

(ra3= 0

ffl3 = 2y2C1 + 4A2 (I + C""U"a. + 1 C",~C/llU,"'a.U.\f') C2 + p.

tI

(21)

The stiffness relations in equations (19) apply to a single finite element. Toappropriately connect all of the elements together so as to obtain the completediscrete model of a given elastic body. group transformations of the foml proposedhy Wissmannll, Oden" and Oden and Sato2 are used. Briefly, let PH" and UNo denote(he generalised node forces and displacements at node N of the assembled systemof finite elements. In this case. N ranges from I to k. where k is the total numberof nodes in the assembled system. Then

in which HIIN.· = I if node M of clement e is identical to node N of the assembledsystem and OMN'=O if otherwise. fn equations (22), N~I,2, ... ,k; M=1. 2. 3;ex'" I, 2; and e = 1,2, ' ... /:', where E is the total number of finite elements. and allrepeated indices are to be summed throughout their admissible ranges.

Substituting equation (19) into equation (22) leads to global stiffness relationsfor the entire assembly of the elements of the form

(23)

The functions K~'Q(UN") are non-linear in UN". After appropriate boundary condi-tions are applied, equations (23) reduce to a system of independent non-linearalgebraic equations in the unknown node displacements. These are solved numeri·cally. Stresses and strains in individual elements are then obtained from equations(9). (20). and (21).

S. Sample ProblemsConsideration is now given 10 the application of the non-linear stiffness rela-

lions derived previously to some relatively simple problems of finite plane strain

B'

Figure 2.(a) Simple shear deformation and (b) finite element representation of square specimen.

260 The Aeronautical Quarterly

FINITE PLANE STRAIN

for which the exact solutions are known. For convenience, the material is assumedto have a strain energy function of the Mooney form.

5.1. SIMPLE SHEAR

The classical problem of simple shear involves finite plane deformations of arectangular section of a material in which straight line elements parallel to, say.the x, axis are displaced relative to one another in the YI direction, but remainstraight and parallel in the deformed body. A cubic specimen in simple shear isshown in Fig. 2(a).

Let A. B. C. and D denote corner points in the specimen, and let ~ denotethe total horizontal displacement of A and B relative to C and D. Assuming thatA ~ I and employing the finite element network shown in Fig. 2(b). it is found that.for the kth element on the line AC (shaded in the figure),

Il

[0 -I I][c"~]=ll I -1 0

k~ (k -I) ~Ullk= -, U:tJk=U3Ik=

Il

Introducing these values into equations (9). we find, for the strains.

(24)

(25)

Thus. the strain components are independent of 11 (and k) and the same result isobtained for all clements in the network. regardless of the number of elementsused. This follows immediately. of course, from the fact that in simple shear allelements experience pure homogeneous strain. which is precisely the mode ofdeformation assumed in deriving stiffness relations for finite elements. Thus, thetillite elemellt solutioll is exact for the case of simple shear,

For a Mooney material. we find from equation (21) that

(T1l=2C1 + 2C2 (2+~~+ p (1 +~') }U12= -(2C2+ p) ~uZ= 2C1+4C2+p.

(26)

The incompressibility condition is automatically satisfied (isochoric deformation)and equations (26) are in exact agreement with the solutions given by Green andAdkins'.

5.2. GENERALISED SHEAR

In the case of generalised shear. sides AC and BD of the specimen are allowedto neform in a general manner (Fig. 3(a»). For an incompressible. isotropic solid

August 1968 261

e'

J. T. ODEN

Figure 3.(a) Generalised shear deformations and (b) deformed finite element network,

with strain energy of the Moonev form. Green and Adkins (Ref. 8, p. 122) haveshown that the curve AC (and BD) in the deformed state takes the shape of aquadric' parabola. Interesting results are obtained in a finite element analysis of thisproblem in which the displacements of node points along lines parallel to AC areprescribed to fit a simple parabola. The deformed finite element network for 2ftelements (n divisions along each side) is shown in Fig. 3(b).

For the kLh element,

'Yu= ~ (2k _ 1.)2 n n

'Ya= 6.' [(2k)' _ 2k +~]2 n rr'rr"

(27)

the remaining components being zero. Stresses are calculated as before. For example,we find. for the component ull in element k.

(28)

Comparing this result with the exact solution, we find that the difference E isgiven by

E=(p+2C.)~· (2y.-2+ ~r (29)

5.3. INFINITE INCOMPRESSIBLE LOGAs a final example. we consider the problem of finite deformation of an infinite,

incompressible rubber log under the symmetric linc loading indicated in Fig. 4(0).The finite element representation indicated in Fig. 4(b) is used, and it is assumedthat the material is of the Mooney type. In this case. displacements arc not prescribed

262 The Aeronautical Quarterly

p

FINITE PLANE STRAIN

31N

p p

Figure 4. Incompressiblelog, under line loading.

August 1968

200

100

a

(b)

FIgure 5. De/ormedshape.

263

J. T. aDEN

and. taking advantage of symmetry. applicatioll of ClIualiolls (19) and (22) leads toa system of 22 simultaneous. non-linear algebraic equations in the 22 unknownnodal displacements and hydrostatic pressures. For illustration purposes, the lineloading P was taken to be 200 Ib/in' (1379 kN/m~ and for the Mooney constantsC1=43'75 lb/in' (302 kN/m~ and C~=6'25 lb/in' (91 kN/m~) were used.

The non-linear equations were solved with the aid of a digital computer in sixiterations. using the Newton-Raphson method. Results in the form of the deformedprofile are indicated in Fig. 5. In addition to the model in Fig. 4(b), several coarserfinite element representations were used. and results indicate that the nodal displace-ments are not significantly in error. Hydrostatic pressures, of course, converge moreslowly since they are assumed to be constant over each element. For the presentexample. the values of p obtained can, at best. represent only rough averages of thetrue hydrostatic pressures. Thus, whereas a rather crude network is often adequateto predict nodal displacements, a significantly finer network is needed to depictvariations in stress and hydrostatic pressure.

AcknowledgmentThe assistance of Mr. John Key in solving the problem or finite deformation of

the infinile incompressible log is gratefully acknowledged,

ReferencesI. ODEN. J. T. Analysis of large deformalions of clastic membranes by the finite clement

method. ProceetlillRs of thl' lASS lnft'rnational Congress on Large-Span Shells. Leningrad.USSR. September 1966.

2. ODEN, J. T. and SATO.T. Finite deformations of elastic membranes by the finite dementmethod. International Journal of Solids alld Structures, May 1967.

3. BECKER.E. B. A numerical solution of a class of problems of finite elastic dcformations.Thesis presentcd to the University of California at Berkeley, in 1966, in partial fulfilmentof the requircmcnts for the degrce of Doctor of Philosophy.

4, OOEN. J. T. and KUBITZ.... W. K. Numerical analysis of nonlincar pneumatic structures.Proceedings of the IntC!rnational Col/oquium 011 Pneumatic StfllCtllrC's, Stuttgart, May 1967.

5. ODEN,J. T. Numerical formulation of nonlinear elasticity problems. Journal of the Struc-tural Division. American Society of Civil Engineers. Vol. 93. No. SO, pp. 235-255, June1967.

(,. ODEN, J. T, Numcrical formula lion of a class of problems in non-linear viscoelasticity,AdvaTlc('s ill the Astronalltical Scit'TlC('s,Vol. 24, 1967.

7. GREEN, A. E. anti Zl!RN.... W. Tlzl'oreticalelusticity. Clarendon Press, Oxford, 1954.

ll. GREEN,A, E. and ADKINS.J. E. LarJiC!e1l1sticdeformatioll.\'. Clarendon Press, Oxford, 1960.

I). TRUESDEll, C. and. TOUI·IN. R. Thc classical field theories. J::ncydop('dia of Physics.ediled by S. FIUgge, Vol. 111/1. pp. 6(10-<>02,Springer. Berlin, 1%0.

10. HERMANN. L.. R. Elasticity equatiolls for incomprl)Ssible anti nearly incompressiblematerials by a variational theorem. AIAA )ollfllul. Vol. 3, Nu. 10, 1965.

11. WISSM...NN, J. W. Numerical analysis of nonlinear elasti\; boJies. GDI AstronauticsERR-AN-157. San Diego. California, 1963.

264 The Aeronautical Quarterly