Compl/tns & Str""lI"n. Vol. 3. PI'. 175-194. PeI1!amon Press 1973. Prinlcd in Great Britain

SOME RESULTS OF FINITE ELEMENT APPLICATIONS INFINITE ELASTICITYt

J. T. ODEN

Depallment of Engineering Mechanics, The University of Alabama in Huntsville,P.O. Box 1247, Huntsville, Alabama 35807, U.S.A.

Abstract-This paper examines a number of problems connected with the finite element analysis of finiteelastic deformations. A brief review of fomlUlation of equations governing finite deformations of highlyclastic clements is given. The convergence of finite element approximations for static problcms in elasticityis studied. Incn:mental stilTness equations are derived in general form and various types of incrementalloading techniques are examined. A number of representative solved problems in I1nite elasticity are given.

1. INTRODUCTION

TilE DEVELOPMENT of finite-element models of the behaviour of highly elastic bodies at finitestrain has made it possible to study quantitatively rather general problems in finite elasticityfor the I1rst time since the foundations of the subject were established over a century ago.A detailed discussion of the formulation of such finite-element models. along with a numberof representative problems and references, is given in a recent book [I]; we do not intendto reproduce these details here (see also Refs. [2, 3]). However, while it can be said thatencouraging results have been obtained. there are still numerous mathematical problemsconnected with the numerical analysis of tinite elastic deformations that have not yet beencompletely resolved. These include the problem of convergence and accuracy of finite-ele-ment approximations of the highly nonlinear equations of finite elasticity, problems con-nected with the nonuniqueness of solutions common in finite elasticity, questions of generalinstability, branching of solutions, the conditioning of nonlinear systems. methods of solu-tions of large nonlinear systems. and, of course, the vast realm of problems in dynamicfinite elasticity which have only recently begun to be explored.

The present paper contains preliminary investigations into a number of the abovementioned problem areas along with investigations of numerical schemes for the solutionof large systems of nonlinear equations and numerical results obtained from solutions of anumber of representative problems. In the following section. we review brielly the basicformulation of the equations governing finite deformation of highly elastic elements.In Section 3 of the paper, we look briefly at the question of convergence of the finite-element approximation for static problems in finite elasticity. We discover that under reason-able assumptions concerning bounded ness of the strain energy function. the convergencecriteria are essentially the same as those commonly used in linear elliptic problems. Forstatic phenomena, we adopt the premise that the nonlinear equations describing the modelare best treated incrementally. Accordingly, in Section 4 of the paper we develop andinterpret incremental stiffness equations for finite elasticity. We give here a rather general

t Presented as an Invited Paper at the National Symposium on Computerized Structural Analysis andDesign at the School of Engineering and Applied Science, George Washington University, Washington,D.C .. 27-29 March (1972).

175

176 J. T. ODEN

development of the incremental equations of finite elasticity and the incremental stiffnessequations for the finite-clement model, which uses a notion of equilibrium paths in 'state'space and the idea of Frechet-differentiation offunctionals on this space. Then, followingarguments that are now well established in the literature, we view the incremental loadingprocess as an invariant imbedding technique that transforms the system of nonlinear alge-braic equilibrium equations into a system of first order nonlinear differential equations.We then investigate a number of methods for solving these equations for the important casesin which multiple solutions and instabilities exist. In particular, we demonstrate the effec-tiveness of several of the methods by means of simple examples. Finally, we list a number ofrepresentative test problems that have been solved recently.

2. FINITE ELEl\'lENTS OF ELASTIC BODIES

PreliminariesThe procedure through which the equations of motion of a typical element in a finite-

element model of an elastic body are derived is discussed elsewhere [1-3]; consequently, weneed only review the essential features here.

First of all, we consider the motion of a material body f!4 from a reference configurationCo c.E 3 to a current configuration C of euclidean 3-space. We establish in Co a fixed frameof reference Xl and we label the material particles of f!4 Xl so that the numbers Xl and Xl coin-cide in Co. For simplicity, we shall take Xl to be cartesian. The motion of fA is the one-parameter family of mappings Xl =xl(X 1, X2

, X3, t) which gives the spatial position (place)Xl of the particle X=(X1, X2, X3) at time t. The three functions UI=Xi_XI arc the com-ponents of displacement, and Y/j=(u/, j+uJ. I+Uk, I"k. j)/2 are the components of theGreen-Saint Venant strain tensor (commas denote differentiation with rcspect to materialcoordinates; i.e. (JuJoXj=u/. j)' If dAo is an element of material arca of a surface in [JIwhile in Co and dA is the same area in C, then net 'external' force at a particle in dA canbe written adA or tdAo, where a and tare Piola-Kirchhoff stress vectors. We prefer touse t since

(1)

where tlj = tjl are the components of the second Piola-Kirchhoff stress tensor, 0111 are com-ponents of a unit vector normal to dAo, and Gj=(Jij+u/. )i/ (i/ being the orthonormalbasis of XI); that is to say, the unit normal to dAo can be used in defining t. However, t/joll/is obviously a contravariant vector.

Linear momentum is conserved at a particle X of [JI if

(2)

wherein Po is the mass density of fA while in Co, Fm are the cartesian components of bodyforce and iim=o2um(x, t)/ot2 are the components of acceleration.

In the case of an elastic (hyperelastic) body, there exists a strain energy function W(Yij)which has the property

(3)

Some Results of Finite Element Applications in Finite Elasticity

Thus, linear momentum is balanced at each particle X in an elastic body if

177

(4)

Equation (4) is the general equation of motion of hyperelastic bodies. To apply it to specificsituations, W must be furnished as a function of "Iij' F", must be specified, we must haveconditions on IIi and til =a WjOyij on the boundaries afA of fA, and appropriate initial con-ditions on II",(X, 0) and ti",(x, 0) must be supplied.

Finite-element modelsThe body ~, or some approximation ii of it, is now decomposed into E finite elements

connected together at appropriate nodes. A typical element fJ4e is isolated from the modeland the local displacement field u}e)(x, t) is approximated according to (e.g. Refs. [1-3])

(5)

where the repeated indices N are summed from I to Ne, Ne being the total number of nodalpoints of the element, lj!N(X) are the usual local interpolation functions and uf(t) are thecomponents of displacement at node N of the element at time t. The displacement com-ponents uf(t) are related to the components ut at node /). of the connected model §jaccording to

(6)

(e) (e)

wherein n~is the boolean matrix (n~= 1 if node N of element e coincides with node /).(e)

of ifj and n~=0 if otherwise) and the repeated /). is summed over all global nodes, /).= 1,2, ... , G. Thus the global model of the displacement field is

(7)

where, assuming local and global material reference frames coincide,

(8)

The exact displacement field "i that exists in an elastic body at time t is considcred as anclement of a Hilbert space :ff; its finite-clement approximation Iii of (7) is its closest approxi-mation, in a sense to be described subsequently, in a finite-dimensional subspace l' of :ff

178 J. T. aDEN

spanned by the functions qJd(X) of (8). Let u(X) and v(X) be two elements in :If, theirdomains being the body fll. Then an inner product of u and 0 shall be defined by

(u. v) ==f,;, II(X)v(X)d,~

where dfll = do is the Lebesq ue measure of X.

(9)

The approximate displacement field iii does not, in general, satisfy the linear momentumequations (2): indeed, we refer to the vector

(10)

as the residual of the linear momentum at particle X at time t. The Ritz-Galerkin methodamounts to choosing Rill to be orthogonal. in the sense of (9), to the subspace "Y. That is.if v= VdqJd(X)E"Y, we set

(II)

Since v is an arbitrary element of "Y, this implies that (Rill' qJd) is zero. Now we observethat

f E f /, (I)

(Rm, qJI,.) =. ~ RmqJddfll = Jt ~.R~~l).)1 nft/J Afl(X)d@ (J 2)

where R~~) is the restriction of Rm to IfJe and we have made use of (27). However, the inter-polation function t/JJ.:l have almost disjoint (compact) support (i.e. t/J~)(X)=O, X$@e)'Thus, kn~t/JN=Un~t/JN almost everywhere, and we can write

E <el(Rm, qJd)= L nf(R~e), t/JN)=0.

e= 1(13)

However, we again note that the sets oflocal basis functions {t/J~)(X)} have disjoint support.Therefore, terms in the series (9) vanish independently. and we have locally

(R<el .I,(e» = 0m ' "PN . (14)

Equations (13) and (14) establish the essence of the finite element method: The vanishingof the local residual independently over each element insures its vanishing globally over theconnected model gg provided the relation (6) holds. Consequently. we need study only thelocal approximations (\4); global approximations can then be generated via (6) and (13).

Equations of motion of a finite elementIntroducing (10) into (\4) and performing the indicated integrations. we obtain for the

general local equations of motion of an element

(\ 5)

Some Results of Finite Element Applications in Finite Elasticity 179

where N, M = 1. 2, ... , Ne; i. j, k = I, 2, 3; vo(e) is the initial volume of PJ(e), IIINM is the massmatrix and PNk are the components of generalized force at node N:

In )16h, Sk are the cartesian components of prescribed surface traction and are generallyfunctions of II/, j'

In other work [I]. we have chosen to interpret the derivation of (15) differently. Suppose(15) holds and let VI= Ijf1/1N be the local velocity field. Then, multiplying (15) by liZ, we get

It is easily verified that this result can be rewritten in the form

We recognize the first term as the rate-of-change of kinetic energy K in the element whilethe integrand of the second term is simply tijYij. The term PNktjf is the mechanical powern developed in the element. Consequently, (15) implies that

However, llJyij=e- V.q-Poh, where E is the internal energy, q is the heart flux, and his the heat supply per unit mass. Thus, we obtain the first law of thermodynamics,

where

u=f Edvo·Vo

In other words, approximations iii which satisfy (15) insure that linear momentum andenergy are conserved globally over a finite element. In the present investigation, of course,we take Q=O.

Since (/j is given by (3) for elastic bodies, we may rewrite (15) in the form

(17)

where we have used the identity

180 J. T. ODEN

Isotropic bodiesFor isotropic bodies, we chose to write W as a function of the principal invariants

It> 12 and 13 of the deformation tensor GIj=O/j+211j· Then the equations of motion of anelement take the form

(18)

If the material is incompressible W= W(Il' 12), 13= 1, and (18) is replaced by the pair ofequations

(19)

Here hMµM(X) is the local approximation of the element hydrostatic pressure, µM(X) beingappropriate local interpolation functions, and (19)2 is the finite element analogue of theincompressibility condition [1].

3. CONVERGENCE OF FINITE-ELEMENT APPROXIMATIONS INNONLINEAR ELASTICITY

We shall now consider briefly the question of convergence of finite-element approxi-mations ofthe form (5) with the specific aim of showing that, under reasonable assumptions,the criteria commonly used for selecting the local interpolation functions tfrN(X) of (5) forlinear problems [I] are still adequate for many nonlinear problems in elasticity. We con-sider here only the relatively simple case of static behavior of hyperelastic bodies charac-terized by strain energy functions which are smooth functions of the strains. The followingdefinitions and lemmas are necessary:

Let

denote a finite-element model of a body !!J and let Xe and Ye denote two particles in element{!Je. If IIXeI12=Xie+X~e+Xie' the diameter he of {!Je is defined as the largest dimensionof the element:

he= max IIXe- Yell·x, Yd'.

(20)

The mesh h of a finite-element model ij is the maximum diameter of all of the elementsin ij;

(21)

Let z~denote the space of ordered n-tuples of positive integers, a=(cxl, a2' ... , cxn)eZn+.Using the notation lal=cxl+cx2+ ... +CXn, tX!=CXl!CX2! ... cxn!, and D",!l=al"'11l/aX~'ax~2

Some Results of finite Element Applications in Finite Elasticity 181

... ax~"(x=(X l' x2, ..• X,,)ei1n, the Taylor expansion of a function u(X) whose deriva-tives of order p + 1 are continuous on an n-dimensional domain giri" can be written

where

y'"u(X+Y)= > -D",lI(X) + Rp+t

I"'~por!

R_ 1 op+tu(X+OY)

p+1- - (Y(p+I)! ax. ax iJX it-Xi,)'" (Y, -x, )'I 12'" ip+J p+l ,,+1

(22)

O~O~I. (23)

Using this notation, a finite-element approximation ii(X) of I/(X) is said to be a repre-sentation of order q (see Ref. [1]) if

wherein

ii(X)= E A~q>:(X)1"'1 Sq-t

(24)

E (e)

cp:(X)= U n"iljlN(X);e-l

l1,r=1,2, ... ,G

a, lleZ~; lal, IplS; q-I

N=1,2, ... ,Ne

(25)

Here G is the total number of nodal points in the model and Ne the number of nodesbelonging to element f!I e'

The function I/(X) of (22) is a member of the space CP(Bl") of functions with all deriva-tives up to order p continuous on Bl". If q<p, the function ii(X) is an element of a finitedimensional subspace d/t of CP(Bl"). The completion of CP(9l") is the Sobolev spaceW~(9in) of functions whose derivatives of all orders up to p are square-integrable in aLebesque sense on [)In. The norm of an element in W~([)ln) is

(26)

(27)

where

ilII IIL(91")=f r 1I2dge.

Let u(X)eCP(ERn), IOP+IU(X)/OX'tOX"", aX'H II~K, K being a constant, and9l" - "jtn= rp. Then a function u(X)eifl c CP(El") is a finite-element q - I-order interpolantof u(X) if

(28)

IH2 J. T. ODEN

for every nodal point X~E9t", ~= 1,2, ... , G; and 1(XI~q-l.Lemma I (cf. Ref. [I)). Suppose u(X) is an elcment of a spaccllt of finite-elemcnt

reprcsentations of a function l/(X)ecm(~)m~p + I, which contains a complete polynomialof degree p. Let.)71contain a pth-order interpolant a(X) of l/(X). Then

(29)

wherc II is the mesh of the finite-element model ~n of ~"cEn, 1~I~p,and K is a constantindependent of II.

Alternately. we can use the stronger estimate.Lemma 2 (cf. Ref. [4]). Tfthe functions q>1 of(25) belong to W~(9i''') and if the conditions

of Lcmma 1 are satisfied. then

(30)

where 5=0. I, .... q and O~q~p.With the error estimates provided by (29) and (30). we can proceed dircctly to the proof

of convergencc or finite-element approximations in linite elasticity. To fix idcas, considcra hyperelastic body 91 with material surface aPA = !f. If we write the strain energy per unitundeformcd material volume Vo as a function <l> of displacement gradients "r. s rather thanstrains, then

and thc equations of equilibrium [(2) with um=O] become

[ v<l> ] +PoF r =0 .all r. s '.'

(31)

(32)

If IVr are arbitrary functions in wj(91) satisfying the kinematic boundary conditions on !f,

then the generalized soillrion of (32) is the displacemcnt field IIFWj(91) satisfying

(33)

wherc (u. \') is defined in (9) and o~" is the portion of the boundary over which stresscsSs=ofl,Oq}/our• ,are prcscribed.

Now if 91 is in stable cquilibrium. it is a well known fact that the potential energyfunctional

n[lIr]=f. [<1}(lIr.s)-PoFrllr]d9l-f. Ssllsd!f~ c~~

(34)

is a relative minimum at the displaccment field II: corresponding to a stable equilibriumcontiguration. Let;j denote a finite elemcnt model of 91 ([Ji-9I=</» and 111be a finitedimensional subspace spanned by the (inite clement interpolation functions cp~(X). Theelement urE1JI which minimizes fI among all elements in °Il is. of course, the finite element

Some Results or Finite Element Applications in Finite Elasticity 18:\

approximation of the 'exact' displacement field II~. Then, if Ii,EUU is an intcrpolant of u~ ofat least first ordcr, it is clear that

(35)

Hence, if lir = II~ + IVr•

(36)

where we have used (33) and <D = <1>[11:., + 011',. s]. 0 ~ 0 ~ I.Now the strain energy function $(lIr •• ) for most known materials. has bounded second

and third derivatives with respect to IIr. s except possibly at certain isolated points. Thus,

(37)

Since wr = lir - u:. we can invoke (30) [or (29)] with s = I, P = 1, and II the mesh of aJ. Then.for Co and C1 positive constants.

(38)

so that from (35)

(39)

These results show that as the model is refined (i.e. 11-0), then n[i1,J-n[u:J. which com-pletes the proof. In other words. the finite element approximations of the solutions of (32)converge in energy to the solution of (33).

4. INCREl\lENTAL EQUATIONS AND ANALYSIS

Among the most effective methods for the solution of the large systcms of nonlinearequilibrium equations encountered in the finite element analysis of nonlinear elasticityproblems are the so-called incremental loading. continuation. or imbcdding methods.Summaries of the basic ideas are given clsewhcre (e.g. Refs. [I. 3]). In thc present section.we shall examine general incremcntal forms of the equilibrium equations. and we proposea number of variants of the incremental loading method which appear to be attractivecandidates for treating problems involving bifurcations and multivalued solutions. We alsodemonstrate certain aspects of these methods by means of simple examples.

Incremental stiffness relatiolls. While our rcsults are essentially the same. we followa somewhat diffcrent plan here than that of Oden and Key [5].

Considcr a IS-dimensional vector space f/. the elements of which are ordercd quad-ruples of displacement ficlds, stress tensors. body forces and surface tractions over the samematerial body fA: i.e. if A is a typical element of Y'.

(40)

184 J. T. ODEN

The algebraic system [1/ is indeed a vector space. since we can define vector addition andscalar multiplication in the obvious way:

The equilibrium equations may be considered as a mapping..¥ of a subset t! of [1/ intothe null vector of a 3-dimensional vector space "1/3:

(41)

if XEt14(we define ..¥(A)=tiiOll,(<5Jm+um. j)-S", if XEO:Jls and ..¥(A)=lIj-gj if Xeo:JlJ.The subset rf of [1/ is also a vector space, since it is precisely the null space of the operator..¥.

Let se[O, co) be a real valued parameter and consider the elements A(s)erf x [0, co).The one-parameter function A = A(s) defines an equilibrium path of the body !!4 if (i) A(s) isdifferentiable with respect to s (almost) everywhere in 8 x [0, co) and (ii) dA/dserf x [0, co).In other words, the set of states A which. for each s, satisfy

.¥(A(s» = 0 (42)

define points on an equilibrium path.We now introduce a norm topology in [1/ which allows us to speak of Fretchet differen-

tiability of the operator..¥. Indeed, if r is an arbitrary element of [1/. the Fretchet differen-tial £5..¥(A. n is the operator linear in r, satisfying

lim 1I..¥(A+o-.¥(A)-<5.¥(A, 011 =0IIrll-+o

We are interested in the case Ilvll == sup Ivl. Since A(s) is assumed to be differentiable withrespect to s, it is meaningful to consider

r=A(s+&) -A(s)=A&+O(~S2)

where, of course, A == dA/ds. Then. if II rII-+0 as &-+0. we may write

In view of (40). we have on every equilibrium path

(44)

Here b" denoles the Fretchet differential operator with respect to A; in more specific terms,we may write (44) in the form

(45)

where

Some Results of Finite Element Applications in Finite Elasticity 185

(46)

etc., anddul,

Uj= ds'iii =dtij.

- ds' (47)

Equation (44) [or (45)] is the general incremental equilibrium equation for nonlinearcontinua. It holds for quasi-static deformations of virtually any type of material, so thatwe can obtain forms appropriate for clastic bodies as a special case. Introducing (3) into(41), incorporating the result into (45), and assuming compatibility of strains, we obtainthe general equilibrium equations for hyperelastic bodies:

(48)

By setting

(49)

in (48) and retaining only linear terms in C" $ we obtain the general equations of equilibriumfor the problem of infinitesimal deformations superimposed on finite deformations ofhyperelastic bodies (cf. Ref. [6D:

Here C, are the increments ill the displacements and !:J.Fm are the increments in body force.Similar equations must, of course, be written for the boundary conditions.

Discrete models. We would now like to apply the ideas discussed previously to finite-element models of the equilibrium equations of hyperelastic bodies. The procedure isessentially the same as before, but now we work in a finite dimensional subspace !J of f/.We shall describe the process locally.

Consider the finite-element approximation (5) of the displacement field over a finite-element !JI~. We seek, as usual, a Galerkin approximation over Ell e of the generalizedproblem

(.#'(A), V)=O. (51)

In the present case, this is equivalent to seeking elements in the nulI space C of the discreteoperator equation

.#'A(A)=f tiirJIN, k5jk+ rJIM, kutl)duO - PNk =0.00(_)

(52)

186 J. T. ODEI'

We may now speak of generalized equilibrillm path in IS; that is, a set of A's dependingon a parametcr s which satisfy (51). Clearly, every solution of (42) is also a solution of (51),but the converse is not true. By seeking finite-clement approximations of (51) dependingon a parameter s we cffcctively seek approximations to a generalized equilibrium path forwhich. at each s.

(53)

The discrete incremental equations are then

(54)

To vary A, howevcr. we now need only vary the applied forces Fm and Sm and the nodaldisplacements u:(s); i.c.

(55)

For a hyperelastic body. this equation leads to

(56)

where M. N. R= l. 2, .... Ne: i.j. k. p, q. s= 1,2,3

(57)

and

(58)

Here we have taken into account the fact that thc tractions Sk depend upon the displace-ments IIf as well as prescribed tractions Tm'

For simplicity in notation. we designate the first integral in (56) as A:~Rl(U), the secondas Bhl( u). and we denote

(59)

Then we arrive at the system of first-order nonlinear differentia I equations.

(60)

Some Results of Finite Element Applications in Finite Elasticity 187

The matrix AN~k contains the familiar stiffness lUatrix of the linear theory plus the so-called initial displacement matrix; BNRk is the 'geometric' or 'initial stress' stiffness matrixand CNRk is a 'load correction' matrix. By replacing Ii:and 4Nk by increments ou: and oqNk,we get the general incremental stiffness relations for finite deformations of elastic finiteelements.

Incremel/falloading. Aside from the physical interpretation of the rcsults. we can applythe procedure described previously formally to reduce virtually any system of nonlinearequations to a system of first-order differential equations. Such techniques are referred to asimbedding or continuation methods and have been in use in one form or another for overtwo decades. The method appears to havc been first applied formally to nonlinear problemsin structural mechanics by Goldberg and Richard [7] in 1963. Further details and referencesare given in Ref. [I]. To arrive at such imbeddings, wc now prcfer to view the equilibriumequations of a finite-element model of an elastic body to be of the form

nU. p)=O (61 )

where f is an n-vector of nonlinear equilibrium equations in the unknown nodal displace-ments (and possibly hydrostatic pressures). as represented by the II-vector U, and a loadparameter p. If, as before. we assume U and p are functions of a parameter s. then we maywritc

where

. .f=J(U. p)U+g(U. p)P=O

J(U. p)=[ofdiJUj): g(U. p)= {iJfdiJp}.

(62)

(63)

Suppose that we wish to solve (61) for a specil1c value of the load. p = Q. We then setp=p(s), where s is a real parameter e(O. I] such that p(I)= Q. The problem then becomesone of integrating the system of nonlinear diffcrential equations

(64)

from s=O to s= 1. Often. it is convenient to simply set /;= I.The standard incremcntal loading method is based on the integration of (64) using

Euler's method: i.e. we use the algorithm

(65)

where J,=J(ur. pr), etc. However, (65) is often unstable and fails at points at which Jrbecomes singular. Alternately, more sophisticated numerical integration schemes suchas the Runge-Kutta method [1,3] or corrector-predictor techniques [1,8] can be employed(see also Refs. [9, 10)). However, these also may prove to be unstable or to be incapableof producing multivalued solutions unless gradie/lt tests (3] can be employed. These testsare merely checks as to the definiteness of the Jacobian matrix J(U, p) at the end of cachload increment; if J changes from positive-definite to negative definite (or vice versa), anegative load increment (or positive) must be applied if the process is to be continued success-fully.

To illustrate several of these variants of the ineremental loading method and to point

188 J. T. ODEN

out important limitations and pitfalls, we shall consider the response of the simple two-bar frame shown in Fig. 1. Though simple in appearance, the response of this structurepossesses many features inherent in the most complex nonlinear structural behavior:multi-valued solutions, bifurcations, etc. Figure I indicates the variation of the verticaland horizontal displacements of the center node with the applied load P, all nondimen-sionalized; i.e. y=vjb and 1i=lljb are the nondimensional vertical and horizontal displace-ments and P =Pd3jaoE is the nondimensional applied load, ao being the initial cross-sectional area of the bar and E the modulus.

40

30'-

-20'--

I I, II II I

I I. II I

15/ 117. JI ., I

I II

..L.....24 / 3cj

~v/\J

FIG. 1. Nonlinear response of a simple elastic structure.

It is clear that the response of the structure is sensitive to the aspect ratio µ=blc. Ascan be seen in Fig. 2, for low flat frames, say 0~ II ~ I, Ii=0 and y increases rapidly withsmall increases in load. Bifurcations are, for such geometries, impossible and only theslightest resemblance to the snap-through phenomenon can be experienced, with P cr~O+.In fact, if 11=0 the initial stiffness of the structure to transverse loads is zero. At highervalues of II Vt~ 1.0), greater stiffness at small loads is experienced and, for each µ in thisrange, there is a definite load P~for which snap-through takes place. Then, for loads P~P 4'

multiple equilibrium configurations are possible. meaning that any numerical techniquedesigned to solve the nonlinear equilibrium equations must reckon with the possibility ofnonunique solutions for a single load. For tall thin frames (µ~.J2), the possibility ofnonzero horizontal displacements arises. Then points of bifurcation are reached at loadsp<p •. In such cases, even for a simple two-degree-of-freedom structure such as this, asmany as six different equilibrium configurations can exist for a single value of the load P.Not all of these, of course, are stable.

Sume Results of Finite Element Applications in Finite Elasticity 189

60'-

50

µ '0-0 050·810 1·3 1'51-7 1·9

FIG. 2. Load-deflection curves for various aspect ratios ~t=blc.

The multiplicity of equilibrium configurations and the desirability of knowing the res-ponse of the structure for a number of values of P suggests. once again, an incrementalmethod of the type described previously. As a first choice. we attempt to analyze the frame-work for the case 11 = 1.0 using nothing more than the standard incremental loading method(Euler's method) without modification. The results are shown in Fig. 3 compared withthe exact solution. We observe that as long as the Jacobian J(U. p) of (63) is nonsingularand the load increment is sufficiently small, the standard Euler-incremental-loading pro-cedure gives reasonable results. Some error does accumulate, however. and near the criticalpoint at which det J ~O, the scheme becomes unstable and fails.

The use of more sophisticated numerical integration schemes does not seem to helpmatters. As is also indicated in Fig. 3. a fourth-order Runge-Kulla integration scheme.the powerful Adams-Moulton corrector-predictor scheme, and Hamming's integrationscheme, also a corrector-predictor method. are either incapable of 'shooting beyond' thefirst singular point for J(U, p) or they overshoot the snap-through phase of the response.

It is obvious that the difficulties indicated are a result of the fact that the integrationschemes mentioned are either incapable of anticipating the need for changing the sign of theload increment at (or beyond) critical points or they only paritally anticipate changes inslope of the tangent plane to an equilibrium path. To overcome this difficulty, gradient testsare required [3]: i.e. it is necessary to test the definiteness of J(U. p) at the end of each incre-ment in the neighborhood of critical points. For example, if J changes from positive tonegative definite during an increment. a negative load increment should be applied(i.e. the load should be decreased): if J changes from negative to positive definite duringan increment, a positive load increment should be applied (i.e. the load should be increased);

190 J. T. QDEN

FIG. 3. Comparison of incremental loading methods.

if J changes from either negative or positive definite to negative or positive semidefinite, thetest fails and the process is repeated using a larger load increment. Several variations ofsuch tests are possible. but their principal aim is to determine the proper sign of the succeed-ing load increment. Figure 4 indicates numerical results obtained using gradient tests atcritical points in J. A substantial improvement in the prediction of the response is seen;however, numerical instabilities are still possible as considerable round-off error mayaccumulate when the solution is continued far into the post buckling range.

One way of reducing round-ofT error is to jntroduce a 'corrector' at the end of each loadincrement. Figure 5 indicates the results obtained by applying Euler's method, the Runge-Kutta method, and Hamming's method, with gradient tests, to the analysis or the frameconsidered in Fig. 1. In the Euler and Runge-Kulla solutions, error E>O.OOOI in v waseliminated at the end of each load increment using the modified Newton-Raphson method.That is, the displacements computed at the end of each load increment were used as startingvalues for the modified Newton-Raphson iterations and these iterations were continueduntil the difference in successive iterates was less than a prescribed E (in this casc. E=O.OOOI).The modified Newton-Raphson was used in these calculations; it is clearly a more econo-mical choicc than standard Newton-Raphson iterations since it involves the inversion ofonly the Jacobian matrix calculated at the end of the load increment. Solutions obtainedusing Hamming's corrector-predictor method with gradient tests are also shown in thefigure. While rcsults obtained using Hamming's method do depend upon the increment size,the Hamming corrector has the advantage of not requiring the solution of a large system ofequations at the end of each increment. However, while the corrector algorithm can be usedrepeatedly in the spirit of successive-approximations to further improve the results, successof the method is strongly dependent on increment size. In the present form of the method,

Some Results of Finite Element Applications in Finite Elasticity 191

the gradient tests are not compatible with the corrector algorithm and, when combined,an unstable scheme results.

1-2

10

08

06

P 04

02

o

-02'-

-004

......-.. Hommings method

_. Euler or runge-kuHo- _:.-0. modified Newton-Rophson corrections

-Exact

s~/ .'\

..( \\\

FIG. 4. Incremental loading solutions with gradient tests.

JJ

FlO. S. Results obtained using gradient tests and correctors.

192 J. T. ODEN

5. TEST CASES

20

We shall conclude this investigation by citing briefly the results of applying the theoryand methods discussed earlier to a number of representative test cases.

Consider first the problem of a thick hyperelastic spherical shell constructed of a Mooneymaterial, W= C1(l\ - 3)+C2(I2 -3), with C1 =80 lbfin2

, C2 =20 Ibfin2 and subjectedto a uniform external pressure p on its exterior surface. The geometry of the shell. and aI1nite-element model consisting of triangular elements of revolution, are shown in Fig. 6.The shell was restrained to deform axisymmetrically, and the general incremental loadingprocess described previously, Euler's integration scheme with gradient tests and Newton-Raphson corrections. was used to solve the nonlinear equations. Results in the form ofcomputed deformed shapes of the shell at various values of external pressure are shown inFig. 6 (see also Ref. [II». Here we can trace the response of the shell from small prebucklingdeformations, to snap-through, post-buckling behavior of a combined bending profileseen in thc Figure 7. Figure 7 contains representative load-deflection curves of various noda Ipoints computed in this analysis.

Pressure, Ib/in2

--- 0,0now .. -2'50-'- +25-- -0,5-.-.- -12·0_.-- -20,0

FIG. 6. Computed deformed shapes-bending, buckling, and inflation of a highly clasticspherical shell.

Thc results of an earlier analysis of a thin circular membrane are reproduced in Figs.8 and 9. The membrane is initially 0.05 in. thick and is composed of a Mooney materialwith C\ =24.0 psi, C2 =3.0 psi. Since no linearized solution exists (i.e. J(O, 0):=0), it isnecessary in such problems to determine accurately a slightly deformed equilibrium con-figuration of the membrane in order to initiate the incremental loading process. This isaccomplished by applying a small pressure (p =0.05 Ibfin2), estimating nodal displacc-ments for a slightly dcflected shape, and performing Newton-Raphson iterations untilthe equilibrium equations are satisfied. The incremental process is then initiatcd relativeto this deformed position. We observe that between p=O.1 and 0.5 Ibfin2

• the response(for this particular node displacement) is virtually linear, but at p'l':::, 1.3 Ib/in2 a smallincrease in pressure produces a disproportionally large increase in displacement. We alsoobserve that thc responsc is extremely sensitive to changes in the second Mooney constant

Some Results of Finite Element Applications in Finite Elasticity 193

20

N.S"-.0 16

18

~ 14::l<II<II

~ 12c.

'0CI>

0.c.<t

N,-0'3 -0,2 -0-1 0 0-1 0-2 0-3 0-4

Nodal displacment. in

Scale for B

3<>

2<>

N.c:,

1-5~

e1·0::l....•It

FIG_ 7. Representative computed load-displacement curves.

Apex displacement. in.

FIG. 8. Variation in the apex displacement of circular Mooney membranes with appliedpressure.

c2· As indicated in the figure, a change from CtlC2 =0.125 to CtlC2 =0.0625 correspondsto a significant decrease in the stiffness in the membrane. The numbers indicated at variouspoints along the curve indicate the polar extension ratio A at the crown of the membraneand the number of Newton-Raphson iterations required to reduce the residaul error in eachequilibrium equation to /0-4. Figure 9 indicates the computed deformed profiles forvarious values of interval pressure for a material of the Hart-Smith type [12].

194 J. T. OOEN

A.=7{)(pluS clomp slippage)

o----<l Finite element---- Experimental-------- Path of nodes ",-

./"/

//

//

II

IIIII\\\\\ I \\~,'....... ~ ........

6 6

FIG. 9. Computed and experimentally determined profiles.

Acknowledgement-The support of this work by the U.S. Air Force Office of Scientific Research underContract F44620-69-C-0124 is gratefully acknowledged. It is also a pleasure to acknowledge the assistanceof Messrs. J. N. Reddy and J. E. Key in performing some of the calculations reported herein.

REFERENCES[I] J. T. OOEN,Finite Elements of Nonlinear Cominua. McGraw-Hili, New York (1972).[2] J. T. OOENand J. E. KEy, Analysis of finite deformations of elastic solids by the finite element method.

Proc. IUTAM Colloq. High Speed Computing Elastic Stmct., Liege (1971).[3] J. T. OOEN, Finite element applications in nonlinear elasticity. NATO Advanced Study Institute 011

Finite Element Methods ill Comilluum Mechanics, Lisbon (1971).[4] G. STRANGand G. FIX, Fourier analysis of the finite element method in Ritz-Galerkin theory. Studies

appl. Math. 48, 265-273 (1969).[5] J. T. OOEN and J. E. KEY, On some generalizations of the incrementai stiffness relations for finite

deformations of compressible and incompressIble finite elements. Nuc. Eng. Design 15. 121-134 (1971).[6] A. E. GREENand J. E. AOKINS,Large Elastic Deformatiol/s. Oxford University Press (1960).[7] J. E. GOLDBERGand R. M. RICHARO,Analysis of nonlinear structures. J. stmct. Div., ASCE 89, No.

ST4, 333-351 (1963).[8] T. H. PtANand P. TONG,Variational formulation of finite-displacement analysis. IUTAM Symposium

0/1 High Speed Computillg of Elastic Struclures, Liege (1970).[9] J. A. STRICKUN,W. E. HAIsLER and W. A. VONRresI!MANN,Self correcting initial value formulations

in nonlinear structural mechanics. AIAA J0lU'1U119,2066-2067 (1971).[10] D. A. MASSETrand J. A. STRICKLIN,Self-correcting incremental approach in nonlinear structural

mechanics. AIAA JOllrnol4, 2464-2465 (1971).[11] J. T. OOENand J. E. KEY, A note on the finite deformations of a thick elastic shell of revolution. P/'oc.

II/Iematiollal COlifere1lce011 Slructural Mechanics in Nuclear Reactor Technology, Berlin (1971).(12) L. J. HART-SMITHand J. D. C. CRISP, Large elastic deformations of thin rubber membranes. II/t. J.

Engng Sci. 5, 1-24 (1967).

(Received 14 February 1972)