.' .

FOro1ULATION A~D APPLICATION OF CERTAINPRIMAL AND MIXED FINITE ELEMENT MODELS

OF FINITE DEFO~~TIONS OF EJ~STIC BODIES

J. T. Oden

Presented in the Session onNonlinear Problems

-at the

INTERNATIONAL SYMPOSIUM ON COMPUTINGMETHODS IN APPLIED SCIENCE AND ENGINEERING

Held at the I.R.I.A., the Institut de Researched'Informatique et d'Automatique, Racquencourt,

France in December, 1973 under the Sponsorship ofI.F.I.P., the International Federation

for Information Processing

To be Published by

Springer-Verlag, New York

IV,

•

J. T. Oden

Texas Institute for Computational MechanicsThe University of Texas

1. INTRODUCTION

The evolution of automatic computing machinery and the advent of

modern numerical techniques such as the finite-element method have had

a profound effect on finite elasticity. Not many years ago the sub-

ject was largely an academic one, studied. only by purists at a few

institutions and, o~ling to its complexity, scarcely ever applied to

anything of much practical importance. Today finite elasticity theory

has become an important tool in the analysis of a variety of complex

systems, including air cushions and bags, flexible storage tanks bear-

ing pads, shock absorbers, balloons, deceleration systems, membranes,

inflatable structures, as well as the study of veins, arteries, human

organs, and other biological tissue. To be sure, the subject is still

in its infancy, but new developments in computational methods give

strong promise that this infant will grow to maturity.

The numerical analysis of problems of finite elasticity by finite

element methods began around seven years agp with a series of studies

of elastic membranes [1-5]. Since then, a number of papers have

appeared on the subject dealing largely with specific details of the

formulation and with applications to bodies of revolution, plane

strain problems, stretching and inflation of thin sheets, and with

certain computational details such as methods for solving the large

systems of nonlinear equations inherent in such analyses. Summary

accounts of these and related investigations can be found in the mono-

graph [6]. For more recent applications, see [7-9].

The mission of the present paper is four fold: First, we sum-

marize certain features of several formulations of primal and mixed

finite-element models of both quasi-static and dynamic behavior of

highly elastic bodies. Secondly, we present; when possible, certain

error estimates and convergence results, and, thirdly, we discuss a

number of computational methods that have proved to be effective in

recent calculations. Finally, we cite numerical results obtained by

applying these methods to representative problems in finite elasticit~

Following this introduction, we describe basic properties of Galerkin

models of the equations of finite elasticity, while in Section 3 we

~. ~odescribe in some detail properties of a one-dimensional model. Under

appropriate assun~tions, we are able to prove convergence of primal

and a mixed approximation. We also introduce these notations of con-

sistency and stabiloity of the finite element approximation. In Sec-

tion 4 we comment on several computational methods, with emphasis on

incremental loading methods for quasistatic problems. We also intro-

duce here a finite-element-based, Lax-Wendroff type scheme for shock

wave calculations. In the fifth section we cite numerical examples

which include nonlinear elastostatics problems, stability and post-

buckling behavior, large-amplitude transient motions, and shock waves.

2. GALERKIN APPROXIMATIONS IN FINITE ELASTICITY

The motion of a hyperelastic body n is governed by the system of

nonlinear equations (see, e.g. [10,11)

VeS(u(x,t» + pF = pa in onx- - -

~(,!(~,t» . n = T , ~E: anI , t > a- -,!(~,t) = ~(~,t), XE: an2, t > a (2.1)

Here Vx

is the material gradient, ~ = (xl,x2

,x3

) describes labels of a

material point x.cl2,~(~(~,t) is a nonlinear operator on the displace-

ment vector ~(~,t) which represents the first Piola-Kirchhoff stress

tensor, F is the body force, p the initial mass density, and2 - 2" 2~ = a ~(~,t)/at = '! = Dt~ the acceleration. It is understood that

V .S = div S. The boundary an is the sum of two parts, aQI and an2,x - -

on which the stress vector S . n, n being a unit normal to an, is pre-- - -scribed as ~ and the displacement is prescribed as ~, respectively.

For such hyperelastic bodies, the stress S is derivable from a poten-

tial function W, referred to as the strain energy for unit initial

volume, in the sense that

S = a W . Fy

F = I + H : H = V ~ ux(2.2)

where ! = (~+ ~T + ~T~)/2 is the strain tensor and H is the dis-

placement gradient.

Let A(x), B(x) be second-order tensors and a(x), b(x) be vectors- - .- - - -defined over n. We define inner products according to

Here tr denotes the trace of the traction Band Lebesque integra-

<A, B> tr A . Bdn (a,b) = J a

nA •

bdn (2.3 )

tion is implied. The completions of the space of second-order tensors

in the norm induced oby <','> and the space of vectors in the norm

associated with (.,.) are Hilbert spaces, denoted q(n) and ~2(n)

respectively. The completion in the norm (.,.)1/2 of the space of

vector-valued functions which vanish on aQ, is denoted ~~l(n).

Let y(x) be an arbitrary vector in ~~l(n). Then, upon taking the

inner product of (2.1) with Y(~) and using the Green-Gauss Theorem, we

obtain from (2.1) the nonlinear variational problem: find ~e:~~l(n)x

(O,t*] such that

( p a , v) + < S (u) , V Qv> = 1 (v)-- -- x- ~

and t e: [O,t*), wherein - (2.4)

l(Y) = (p~,~) + I t'Y ds

an2

Moreover, u is also subject to the initial conditions,

(2.5)

(u(' ,o)-u(·) ,v) = 0-0 -

(U(·,o)-v(·),v) = 0- -0-

Le~ ~i~)be a,basis of ~2(n); then each y in ~2(n) is of the form1 1 ·~l~(~)=LV (~)~i~), v (~)EL2(n). Let Wm (n) be the Sobolev space of

functions whose derivatives of order m are in L2(n). We now construct

a subspace s~(n) of w~l(n), of finite dimension.G, by identifying

a collection of G linearly independent functions .l(~) '.2(!) ,... f .• G(~)·

The identification of these basis functions defines an L2(n)--

orthogonal projection nh of wOl(n) into Shk(n) such that the image of01 k m

any v(x)e:W (n) under nh is of the form'- m

Gnhv(x) = L (v,.a). (x) -

- 0 a -a=l

where (vl,v2)o ~ In

vlv2dn and .a(x) -

The subspace S~(Q) is assumed to have

V(!) (2.7)

G -1L (9a,.S) 0 .13 (~).athe following properties:

(i) There exists a constant C independent ofoh and v such thato

wherein

index:

IInhvll < C Ilvll V h (2.8)m - 0 m

IIvll; = IIvll2 = I II L Davll dn, ~ being a multi-W; (n ) n I ~ I <m - 0

a = (al,a2,a3); ai = integer::. 0, lal = al,a2,a3

(ii) If p(~) is a polynomial of degree < k,

(2.9)

wherein Ivl~+l.=

(2.11)

(iii) For any r such that a ~ r ~ k, there exists a constant

C > 0 such that

I I (I - IIh) v II s ~ Chk+1- s Iv Ik+1 (2•1a )

JG 2

( E DaV) dn. In all of these relations,n 1~I=k+l -

h is a real number, generally selected so that 0 ~ h ~ 1 (i.e.,

h is a mesh parameter).

Let E be the real numbers and let [O,T]~E be a finite time

interval. Let B[O,T) be a linear space of functions defined on [O,T)r(generally we take B[O,T] = C [O,T], r=O,l,2,or3). The space

s~(n, [O,T]) := S~(n)XB[O,T) is termed the space of semidiscrete

Galerkin functions. Galerkin'smethod amounts to seeking among the

elements in S~(Q, [O,T)) the function(s)

3 G .10

~(~,t) = E E ~i (~,t) A (t)~a(~)i=l a=l

which satisfy (2.4) for Y E s~(n):

(2.12)

Here D~g = ~ and ~f~,t) = E~i(~,t)~m(~'O). Upon introducing (2.8)

into (2.9), we obtain a large system of nonlinear differential equa-

tions in the coefficients Aia(t):

L M~ oAia(t) + <S( E g.Aia(t)~ ) '---a a ~o> = Fjo(t). 1a µ -. _1 Cl X 0 µ µ1,Cl 1,a J

j 'ia j. L (~i<Pa'~ ~a)A (0) = ('!o''! <ps);1,a

(2.13)

Here Mias and F~8 denote the mass matrix and the generalized forces,

(2.14)

with i,j = 1,2,3; a,8 = 1,2 ,...,G .

'Finite Element Formulations. The finiteNelement concept provides

for a direct and systematic a~plication of the Galerkin method to

arbitrary domains n which, in general, leads to well-conditioned

systems of equations. We partition n into E subdomains (finite ele-

ments) so that

E11 = U

e=lQ e 4>, e 'I f 0(2.15)

On each element e we introduce a system of local interpoltaion func-

tions X:~(~)such that

(2.16)

(2.17 )

in n (H,N=l, 2, ... ,N ).e ethen given by

a4>~ (~)

(e) N116 is a Boolean transformation:

XM denotes one of N prescribed nodes- e

"coordinate" functions cp~(~) are

NE e (e~ 0

= U L n X-~ (x)e=l N 6 N

here

Global

where

coincides with X6e:11and is zero if otherwise.

-l'.be 1Dc.a~f,unctions defined in (2.1-2) generally are in S~ (iTe)

with h = dia Q [see (2.6)-(2.8)]. Property (2.12)2 is responsiblee ,for the banded character of <e(A1a), aj4>S> in linear problems, and

makes it possible to formulate a local Galerkln approximation of the

type (2.10) for each independent element. For example, suppose

I~I = 0 in (i~t2) ~i.e., only values are prescribed at nodes) and

~a (~) = U L 11N XN

-(x). Also let g. (x,t) = i, . i, = 0 0 , • Then, fore N a - _1 - -1 -J 1J

each finite element we have the local equations,

(e) (e) (e) 0 e e eL mN~.la.'N (t) + <S(1: L aON (t)!-.xN(·),aoXM('» = f'N(t) (2.18)N l' 1 - i N J J 1 e 1

where <','> is the inner product in (2.2) obtained using restructionse

of the arguments to n ande

m (e).= e Xe) f~N = l(i,X~)(e) (e)a (2.19 )(pXN, ; ; aiN = 1: 11N AoNM M 1 -1 1aa

The global equations are obtained by "connecting the elements

together" using (2.15) and the fact that Fo (t), the global general-. d f ' . 1 E (e) 1a [2] .1ze orce, 1S prec1se y L E 11N f~ (t). See 6,1 for addit10na1

details. e N a 1N

It is ~!ell-kno~n that finite-element interpolants can be easily

constructed so that (2.6) and (2.8) are satisfied, with h defined as

h = max {hl,h2,"',h }, h =diaineter Q. Indeed, Ciarlet and Raviarte e e[13],e in extending work of Fried [14), have shown that for a wide

class of elements (2.8) can be replaced by

(2.20)

where if p is the diameter of the largest sphere that can be in-escribed in an element Q, p=min {p}. Often p=vh for v = constant>O,e e eso that (2.16) reduces to (2.8).

3. ACCURACY AND CONVERGENCE STUDIES

In this article we shall investigate several special forms of

the Galkerin approximation (2.9). In some cases we construct esti-

mates of accuracy and prove convergence. In particular, we consider

incompressible, homogeneous, isotropic, hyperelastic bodies for which

the strain energy function W is of the form

I =13 (3.1)

where I. are the principal invariants of the deformation tensor~G = I + 2y [see (2.2)]; i.e., using the summation convention~

(3.2)rrngwhere yr = grmy , yare the covariant components of y, and

s ms ms -are the covariant components of the metric tensor associated withix at t=O.

Among forms of W(Il,I2) in use in the characterization of natural

and synthetic rubbers, we mention as examples the Mooney form,

W = Ml(Il-3) + M2(I2-3)

the Hart-Smith form

(3. 3) 1

w = c{Jexp

and the B iderman form

(3.3)2

~ = t . F E = a Wr (3.4)

where in oij are components of ~ and the Lagrange multiplier h is

called the hydrostatic pressure (see [6,pp.236-242]).

3.1 Nonlinear Quasistatic Two-Point BVP. As a first simple case, we

consider the stretching of a long, thin cylindrical rod of

incompressible, hypcrelastic material, fix;ed at x1=Oo, subjected to

a static tensile force at its free end of p, and to a force per unit

length of f. Assuming that the transverse normal stresses are

negligible, the hydrostatic pressure can be eliminated ab initio

and the longitudinal Piola-Kirchhoff stress is found to be

S (>.) (3.5)

where>. is the longitudinal extension ratio;o i.e., if u(x) is the

longitudinal displacement and u' = du/dx, then>. = 1 + u'.

Denote D = d/dx. We shall use the notation S(>') and S(Du)

interchangeably, even though S(>.)0= S(l+Du). Likewise, for one-

dimensional problems of this type, <A,B> = (A,B) . The weak form ofothis boundary-value problem is then: find u such that

(S (Du), Dv) = 1 (v) ,o

1V v e: H (O,L) (3.6)

whereas the associated Galerkin problem is find U e: S~ (O,L) such that

(S (DU) , DV) = 1 (V) ,k

V V e: Sh(O,L) (3.7)

Denoting A=l+DU, we easily arrive at the orthogonality condition

(S(A) - S(A), DV) = 0 ,oh

V V e: Sk(O,L) (3.8)

by setting v=V in (3.6) and subtracting (3.7).

Assuming S(>') e: cl, we have

>. = A + a (A- A) = >. - A (3.9)

'. . with 0 < e ~ 1, S' (>..):;ds(I)/dL Hence

(3.10)

For most hyperelastic materials, there are numbers ).Jo').Jl> 0

such that).J < S' (~)o< ).Jlfor every >..> A , where >..is some criticalo - - c cextension ratio> O. Assuming this is the case,

where E = A- IT>..= l+u'=IT I-IT u'=(I=IT )Du=E and e =>..-A=D(u-U)=De .A h h h h u' A uConsequently (Q[12]),

Iloeullo µl I I I< - lEu' 0- ).Jo(3.11)

We summarize these results in the following theorem:

Theorem 3.1 Let the first Piola-Kirchhoff stress S(>..)in a

homogeneous rod of isotropic hyperelastic material be such that

constants ).Jo').Jl> 0 exist so that

dS ().)o <).J < < ).Jo - dA - 1 (3.12)

V >..£ [>..,00]. Let U denote the Galerkin approximation of the solutioncu of (3.6) and eu = u-U be the error. Then (3.11) holds. Moreover,

if U e:oS~(O,L) for which (2.6)-(2.8) hold, then

(3.13)

where k is a constant independent of h.

The results of (3.13) follow from (2.8) and (3.11) together

with the observation that Ileuill ~ cllDeullo and Ileullo ~ hCIllDeullo

for the

order k

vergent

Remark:

simple case under consideration. Thus, if polynomials of

are used in constructing S~(O,L), the approximation is con-

and II De II = 0 (hk) , lie II = 0 (hk+1) .u 0 0 u 0.. 1 .. 1 dS

Observe that u - - DS(u ) = u - - d' u . Hencep x P 1\ xx

C(Ou) -I 1 dSp dA

(3.14)

3.2. Stretching of a Rod of Mooney Material. More precise results

can be obtained if specific forms of W(Il,I2) are identified. To

illustrate, suppose the material is of the Mooney type [see (3.3),].

Then it can be shown that

S(,,) = 2(1 - ,,-3)(Hi1 + H2); h = - A-l[Ml + A-1M2 (1 + 1.3)

so that

(3.15)

(3.16)

For stretching of the rod, " > X = 1; hence, S satisfies the wave- ccondition and, in fact,

Thus

dS< dA (3.17)

IIDe II < 3 (1 H.l1/M2 ) liE ,IIu 0 - u 0(3.18)

Ile",,-2(l-ex/A)-lilo

(3.19)

...convergoe.nce.0£ .t.he.Hydroos.ta.t..icPress-ure. .The ...app.roximation H (x) 0 f

the hydrostatic pressure h(x) also converges, and, in fact from (3.15)

2e 2"e -eII h - H II ~ II Ml + {" 2 (1--+) + 2 X X } M2 II x

o 1\ A ( l-e / X) 0X

Here ex = A - A = Du-DU = Deu' which is bounded above by the interpo-

lation error DE in (3.13).u

3.3 Mixed Finite Element Models. In the linear theory it is known

that [15] improvements in the accuracy of approximations of deriva-

tives can be obtained by using mixed models in which two or more

dependent variables are approximated simultaneously. We shall explore

certain properties of a mixed nonlinear formulation here. In parti-

cular, consider the following canonical form of the basic nonlinear

quasistatic two-point boundary-value problem

v(x) ::S(Du) u(o) = 0(3.20)

Dv(x) = - f ; v(L) = PIA oThese equations corresponds to the longitudinal deformations of a thin

rod of initial cross-sectional area A and length L fixed at x = a andosubj_~ct~d_to a forc::~~~_x_~_ L. _~nd~r:..~nd~nt3Ppro~~ma t:,ions_are _now __, _

areG H

(3.21)IThU = L (U,cpCl) ~ (x) ; P v = L (v,wl'l) w" (x)a=l 0 a 6=1 0 0

H k mand {wt,}6=1 are bases of Sh(O,L) and Ti(O,L)

each generated using appropriate finite-element models.

be the actual solution of (e.20). The finite-element

appropriate solutions are the elements U*·£ S~(O,L) and V* £ T~(O,L)

Gwhere {CPa}a=l

respectively,

Let u* and v*

such that

Pi(V* - S(U*» = a , and ITh(DV* + f) = 0

with V*(L) = PIA ; U*(o) = a (say).o

Lemma 3.1. Let (3.2) hold and denote

(3.22)

E: = V* - p V*V i

e = u* - U* . e = v* ~ V* . E = u*, v ' u

E = v* - P v*v i

IT u*h (3.23)

Then the following hold:

P (S(u*) - S(U*» + e = Ex x v v (3.24)

ITh De = 0.v (3.25)

v* + PiS(u*} in (3.22), whereas

upon replacing ITh f by -IThDV*.

errors associated with u and v;

(3.24) is obtained by adding and substracting Piv* +(3~23) follows from (3.20) and (3.22)

Proof:

In (3.23) eu' ev are approximation

E , E are interpolation errors,u vand £u' £v are termed projection errors, respectively.

Let u U and vV. We shall term (3.22) a consistent Galerkin

approximation of (3.20) if

lim Li(V) = 0 and140

where Lh(V) and Mhi(u,v) are

lim Mh1(u,v) ~ 0h,140

the lack-of-consistency functions

(3.26)

~l(U,V) = (S(Du),v) - (S(DIThu,Plv) (3.27)

A simple calculation reveals that

LR.(v) ~ Ilvllo IIDEvllo and HhR.(u,v) ~·llvllo II·S(Du) - S(DIThU) 110+ I I S (D IThU) II "E 1 I (3 . 2 8 )o v 0

While several different defintions of stability suggest them-

selves at this juncture, we shall choose to refer to the Galerkin

approximation (3.22) as stable whenever constants a,S> a exist such

that

-For every ueU and VEV.

stability.

This may be a rather strong requirement for

Theorem 3.2. Let the Piola-Kirchhoff stress S(Du) satisfy the

wave condition and let the subspaces S~ (O,L) and T~(O,L) be such that

(2.6) - (2.8) are satisfied and I IDE I I and I IDE I I vanish as h, R.o v 0 u 0

~ O. Then the Galerkin approximation is consistent.

Proof: This is obvious. Simply set S(Du) - S(DIThU) = S' (D~)DEu

in (3.28) and use (3.14). It then follows that Lh, MR.h + 0 as h,R.~O

Theorem 3.3. Let the conditions of Theorem 3.2 hold and let the

Galerkin scheme be stable in the sense of (3.29). Then it is con-

vergent in the sense that I IDe I I vanish as h,R.~O.u 0

Proof: According to the triangle inequality,

(3.30)

Since S(DuA)obeys the wave condition, (3.24) yields

(3.31)

Hence,

(3.32)

However, IThD£u = IThDEv' in accordance with (3.25). Since, by hypo-

thesis, (3.29) holds, we have

Combining. this with (3.30) and (3.31) gives

. -~~-.~-------

II De II < (1 +! II IThDP n S I (~) II ) II DE II +! II DE IIU 0- a ~o 0 u 0 a V 0

and

(3.33)

(3.34)

Since S(~) satisfies the wave condition, DPg,S' is bounded. Conse-

quently, both De and e are bounded above by I IDE I I , I IDE I I andu v u 0 v 0

I lEvi 10, which vanish as h,l ~ O.

3.4. Time-Dependent Problems. In the case of time dependent problems

we shall use the combined finite-difference/finite-element approxima-

tion2 .

p (Ot (U1.,V)o 0+ (S(DUi), DV) = l(V)o

(3.35)

. ---- - --VES~(?,L), wherein o~ is a second-order central difference ope~ator

and u1. = U(x,i6t); a = t < t, < ••• < tR = T; t'+l - to = 6t. If the3 0 1. 1.

exact solution UEC [o,T], then

2 . . . .p (Otu1.,v) + (S(DU1.),Dv) = 10(V1.)+ p (w1.6t,v)o 0 0 0

(3.36)

where wi is a bounded function of t. The following lemmas are proved

in [15]:

Lemma 3.2. Let (3.35) and (3.36) hold and let S(u) satisfy the

wave condition. Then

and

(3.38)

wherein a., e· are constants> 0 and independent of 6t and hand. 1. 1.. .e1. = u(x,i6t) - U(x,i6t), E1. = u(x,i6t) - IThu(x,i6t), £1.= U(x,i~t)

- IThu(x,i6t).

Theorem 3.4. Let (3.35) and (3.36) hold and let S(u) satisfy the

wave condition. Then the finite-element/difference approximation error

e(x,t) is such that for sufficiently small h and ~t,

(3.39)

Proof: This result is obtained immediately from (3.37) and (3.38)

by observing that p (o~ £, e:) - P ~~_~~ =~o_~e, e:) in (3.37) and then

-using-T3. 38). Since the d. ,8. arc arbitrary, find constants Kl and 1~2• . . 2 2 ~ J. 2

such that I IDe11I~ ~ klh" + k211t , from which (3.39) follows.

The question of numerical' stability, of course, arises here. In

[15] it is shown that the approximation (3.35) is stable in energyi ~ i ( i / )1/2 . 1 2whenever (h/6t) > v.C /r2, where C = max[S' Ux) Po ' 1 = , ,~ max max

---with--"I = 2/13 and v2 = 2. The case involving vI pertains to the use--------

of the consistent mass matrix of (2.11) in the calculations, whereas

v2 corresponds to the case in which masses are lumped. We describe

some specific numerical results in Section 5 that were obtained using

approximations of the type described here.

3.5 Shock Waves in Elastic Rods. Consider once again the one-dimen-

sional hyperelastic rod problem and again denote by A(x,t) the

longi tudinal extension ratio, :\.= 1 + oDu. Since.c2 (:\.):;!.ao/a A ispo ..

the intrinsic material (acoustical) wave s~eed (_~~~_s~}'~_c:~_DU =_~A, _we have the equation

(3.40)

This alternate form of the wave equation is useful in studying shock

waves in nonlinear materials. Whenever ~ ~ c3, the expansions

2qn+l(x) = qn(x) + 6to2S(:\.n(x» + ~t 02[s' (Xn(x)qn(x)] + O(6t3)

p p

An+l(x) = Xn(x) + 6tqn(x)2

+ 0 (lIt3)+ 6t D2S(Xn(x» (3.41)P 2p

'can be used to develop a finite-element based-Lax-Wendroff type

integration scheme. In (3.41), q(x,t) = X (x,t) and Xn(x) :;X (x,n6t)

where 0<tl<t2<"'<tR=T and llt=to+l-to. We describe such a scheme- - - - 1 ~

in the next section and cite some numerical results obtained using

it in Section 5.

4. COMPUTATIONAL 11ETHODS FOR NONLINEAR

SYSTE~S OF EQUATIONS

In this section we shall describe a number of computational

methods that have proved to be effective in the solution of the large

systems of nonlinear equations associated with the models described

previously. We begin with the description of methods for systems

of nonlinear algebraic equations encountered in static problems.

4.1 Incremental Loading/Newton-Raphson Method. The idea of trans-

forming a system of nonlinear algebraic equations into an equivalent

system of ordinary differential equations and then solving them by

oby numerical integration, \Vas introduced independently by Lahaye [16),

Davidenko [17), and Goldberg and Richard [18], and a brief history

of the method can be found in '[19) and [6]. We shall outline one form

of the method developed in [6) and [20] which has been used to solve

very large systems having multiple roots [21].

For nonlinear problems, the finite-element method described in

§3 leads, for quasi-static problems, to a system of n nonlinear

equations of the form

(4.1)

wherein f is an n-vector of nonlinear equations in the n unknown

generali;ed displacement x = (xl"" ,Xn)T and a parameter p represent-

ing the applied load. By assuming that ~ and p are functions of a

parameter s, we can compute

df.. a =~ = ds =af(x,p) af(x,p)

- * + : p (4.2)

Thus (4.1) is transformed into the system of differential equations

J(x,p)x + g(x,p)p = 0- ~ ~ --where o!(~,p) = a~(~,p)o/a~ is the Jacobian matrix and g(~,p)

is a "load-correction" vector.

(4.3)

- af (x,o)" -

ap

We now proceed to integrate (4.3) numerically. While a number

of sophisticated numerical schemes could be used, numerical experi-

.ments have repeated indicated that, for large systems, none have

any definite advantages over the standard Euler technique, provided

a corrector of some type is introduced to reduce errors at the end of

each increment ~s. Hence, suppose s E [0,1] and introduce the parti-

tion a = s <sl<"'<s =1, s'+l-s.=~s. Denoting xr=x(s ), pr=p(s ),o n 1. 1. - - r r

d r r+l r . hI'an ~p =p -p, we arr1.ve at t e recurrance re at1.on

r r r+l r r r r~ (; , p ) (~ -~) + g (~ ' p ) ~p = 0 (4.4)

Ordinarily we set ~pr=~s and prescribe the load increments ~pr -

hence the term "incremental loading." However, it is sometimes

convenient (in fact, necessary) to treat ~pr as an unknown. Then we

append to (4.4) t~e approximation

of the arc

(~xr-l) 'l' ~xr- -2

length ds

+ ~pr-l~pr = ~s2

T 2. n+l= d~ 4~ + dp 1.nE .

(4.5)

r r+l rHere ~~ =~ -~~.

,'We now propose to reduce the accumulated round-off error at the end of

each load increment using New~on-Rapshom iterations such that

r,m+l _ r,m _ J-l( r,m r)f(,r r)x - x _ ~ ,p _ ~ ,p (4.6)

where the starting value is ~r,o~~r and m=0,1,2, ... ,k. The number k

is determined by prescribing initially an exccptable error E such that

r m T r m 1/2 II r mil .[(~ ' ) (~ ' ») = x' En ~ £ for m > k. Ortega and P~e~n-

boldt [19] prove a version of the following theorem

Theorem 4.1 (Cf[19]). Consider the system of equations (2.1)

where ;-:D ~ En x [0,1] -+ En. Let; be differentiable with respect

to x and let its derivative J(x,p) = af/ax be continuous and non-- - - --singular on Dx[O,l] for all s e: [0,1] and assume a solution ~(s)

[or ~(p(s»)) exists. Then there exists a partition of [0,1] and

integers kl,k2, ... such that the sequence {~r,m}, m=O, ... ,kr-l

remains in D and, after N load increments, lim ~N,m = ~(l) .

In applications, the major problem with this method is its

inability to handle, without major modifications, cases in which

J(x,p) is discontinuous or nonsingular. Such cases are encountered

frequently in nonlinear elasticity in the form of bifurcations and

limi'tpoints on the 'equilibrium path r: ~ = ~ (p) .

4.2. Stability, Bifurcations, and Limit Points. We consider a

modification of the procedure described above which can be used to

o determine limit points and points of bifurcation and to carry the

solution beyond these along stable equilibrium paths.

Ideally, at critical points x such that det J(x ,p ) = 0, we-c - -c cintrocuce a change of variables x = ~¥ such that the matrix

(4.7)

is diagonal and of rank r < n, with zeros in the last n-r entries.

Let ¥ = (Yl'¥2)T, ¥l being the first r rows of y, and let ~o be an

arbitrary r-vector of constants. We set ~ = ~(~0'Y2)'holding p

constant. This moves the solution off an equilibrium path r but

in a direction tangent to r at the critical point ~c' Holding ~

constant, we iterate on p until f(x,p*) = 0 (approximately). If- - -p* > p , the postcritical equilibrium path is stable. If p* < p ,c cit is unstable, whereas if p*~p, the test fails, a new ~l is selected

(I~ll > I~ol) and the process is repeated. Once a postcritical

equilibrium path is reached, the incremental loading process is

continued with prescribed load increments bp such that bp > 0 if

P* < p and ~p < 0 if p* < P •c eTo determine ~e with sufficient accuracy, we employ a procedure

described by Gallagher [22], and evaluate the sign of the determinant

of q(~,p) at each load increment. This is a numerically sensitive

undertaking. Generally J must be appropriately scaled, and x- -cis only estimated by linear or quadratic interpolation. However,

the test is necessary since bifurcation points can be inadvertently

by-passed in the incremental loading process.

The elaborate procedure for post-critical analysis just described

is generally too slow and expensive for the practical study of large

scale nonlinear problems. An alternative that has been used

effectively is described in [9]. In this process, each bifurcation

point (i.e., each critical point through which two or more stable

equilibrium paths cross) is interpreted as a limit point (i.e., a

critical point involving only one equilibrium path) of a perturbed

system obtained by introducing imperfections into the original

system of equations. These imperfections are generally in the form

of perturbations in either the stiffness coefficients, the location

of loads, or both, and are represented by an imperfection parameter

~. The incremental scheme is based on the observation that 'for small

~, the post-critical equilibrium path of the imperfect system

approaches asymptotically that of the "perfect" system and bifurca-

tion points in the perfect system are reduced to limit points in the

imperfect system. Thus, we proceed with the usual incremental loading

Newton-Raphson technique, checking det J as described previously,. 1 ., 1 . r . h d- Th t t r-l .unt1 a cr1t1ca p01nt ~ =~c 1S reac e. e sys em a ~ 1S

perturbed, and the equilibrium path of the perturbed system is traced

beyond x. Newton-Raphson iterations (with ~=O) then return the-csystem to the correct equilibrium path and the incremental loading

process is re-initiated. Examples of postbuckling problems solved

in this way are given in the next section.

4.3 Explicit Integration Procedures. Instead of numerically inte-

grating (4.3), which requires an implicit integration scheme, we can

construct the system of differential equations,

X + Cf(x,p) = a- -- - -(4.8)

where ~ is a damping matrix, and x(O) is usually O. Then the solution

~* of (4.1) is the steady-state solution of (4.8), provided (4.8) is a

stable dynamical system. The damping matrix g can generally be se-

lected so that (4.8) is stable. Taking g to be the diagonal matrix

b!, one choice of an explicit integration scheme for (4.8) is

r+l r (r )x = x + ~s c ; ¥ ,p (4.9)

~s + s +l-s , s E [0,1]. Since p is new held constant, x(s) may notr r ,0 ~

intersect with an equilibrilli~ path at any points other than s=O and

s=l. We cite an example problem solved using this method in the

next section.

4.4 Simple Explicit Scheme for Transient Response. We now turn

to the elastodynamics problem (2.9), which, for the present, can be

thought of as a system of second-order nonlinear differential

equations of the form

Mx + !(~,p)= a (4.10)

x (0) = x-0

x(o) - v-0

Here ~ is the mass matrix described in (2.10). We describe some

results in the next secion obtained using the following scheme:

The matrix l:1 is replaced by a "lumped" mass matrix mI. It can be

shown that this does not deteriorate the accuracy of the approxi-

ornationfor sufficient~y smooth ~(t) and it leads to a better defini-

tion of wave fronts. This step also makes it particularly easy to

solve (4.10) using explicit schemes. Next, we replace (4.10) by the

equivalent system

-1 ) v(O) = :ro(4.11)

v = -f(x,p ,m_ -

x = v , x(O) = ~o

which we approximate

xr+l

the divided central difference scheme

(4.12)

This scheme is easily programmed and has yielded surprisingly good

results for some large problems.

in Elastic

R~(n) with

Suppose

4.4. A Finite-Element Based Lax-Wendroff-Type Scheme for Shock Waves

Materials. Consider two finite-element subspaces S~(Q) and

bases {$a(x)}~=l and {$~(x)}~=l' respectively, (n = [O,L]).

Q(x,t) = IB~(t)~~(x)a

(4.13)

(5.1)

If A and Q are finite-element approximations of A and q of (3.41), a

finite-element/Lax-Wendroff scheme is obtained by introducing (4.13)

into (3.41) and equ~ting the projections of the residuals in s~(n) andm

R1(n) to zero:

2LH Bf (n+l) = In Br (n+l) = L {H - ~(DS I (An)I/J ,DlJ!)} Br (n)r ~r r ~r r lH 2p r ~

lit L(S' (An)D¢ DI/J )Aa(n) + 1np a' II 2a

2LG A~(n+l) = I{G - ~(S' (An)D<j> DIj>)}Aa(n) + ~t L(~ I )B~(n)aB a8 p a' ~ p~' lila.8 8 ~

(4.14)

Here Aa(n) ::Aa(nAt), etc. and 1~, l~ are terms contributed by (kno"m)

generalized forces. In general, we replace Ga~ and H~r in (4.14) by

"lumped" matrices so as to produce an explicit integration scheme.

5. SOME NUMERICAL RESULTS

We conclude this investigation by citing numerical results ob-

tained by applying the methods described previously to a number of

representative problems.

5.1. Large Deformation of an Elastic Frame. As a first example, we

comment on the numerical analysis of large deformations and post-

buckling behaviour of a Hookean two-bar frame shown in Fig. 1, sub-

jected to a vertical load P. If u = u/b and v = v/b are non-dimen-

sionalized horizontal and vertical displacements of the center node

and P = Pd3/a Eb3 is a non-dimensional load, a being the initial baro 0

area and E the modulus, then the system is described by the equations

[20 ]2 2 2 2

(v-µ)(v -2µv + u) = P; u[u - (2µv - 2(2-v »] = 0

where µ = c/b. For µ < 12, limit-point behaviour is encountered

since u = 0 V P. For µ ~ 12, bifurcations in the equilibrium path

exist, as indicated in Fig. 1. Figure 2 shows numerical solutions

obtained using the uncremental-loading/Newton Raphson procedure de-

scribed in Section 4.1, together with a postbuckling analysis of the

type mentioned in Section 4.2, for the cases µ = 1.0 and µ = 1.2.

Good agreement is obtained.

5.2. Biaxial Strip Problem, The stretching of a rectangular strip

has become a standard test problem for computational methods in

nonlinear elasticityo. In Figure 3 we illustrated results obtained

using the explicit time-integration technique described in §4.2

on the finite element model of a strip of Nooney material, initially

8.0 in. square, 0.05 in. thick, Ml = 24.0 and M2

= 1.5 lbs. per

sq. in., for various values of damping: c = .001 to .0010.

Another version of basically the same problem was described in [24J.

Again very rapid solution times were obtained.

5.3. Stability and Postbuckling Behavior. For completeness, we cite

in Figures 4-6 some recent results obtained in [9J on stability and

postbuckling behavior of hyperelastic bodi~s at finite strain. The

bodies considered are neo-Hookean (C2 = 0) 'with Cl = 24 Ibs. per sq.

in., all 1/3 in. thick. In Figure 4, a 2.0 x 8.0 in. body is sub-

jected to an axial load in the x2-direction with the surfaces x2 = afixed and x2 = L fixed against rotation. The finite element model

consists of 64 triangular elements and 72 unknown nodal displacements.

The resulting 72 nonlinear equilibrium equations, which are sixth-

degree polynomials, were solved by the method of incremental loading

with Newton-Raphson corrections. The load was applied in increments

.cfAP = 00..4 lbs.. The determinant of the stability matrix changed

from a positive to negative value at the seventh increment of load

(P = 2.8 lbs.) indicating that a bifurcation point occurred within

the increment. The incremental solution was restarted at a load of

p = 2.4 Ibs, with load increments ~p = 0.05 lbs, for which case the

Newton-Raphson technique failed to converge at a load of 2.65 Ibs.

Hence, the critical load pc was isolated to the range

2,60 < pc < 2.65 Ibs. This agress well with the critical load which

is pc = n2EI/L2 = 2.35 lbs., if the modulus of elasticity is approxi-

mated by E = 6Cl and the moment of inertia I and length L were evalu-

ated in the deformed state near the critical load. Before buckling,

the vertical center line remained vertical, and the vertical dis-

placements were linear. The vertical displacement at buckling was

0.51 in. or 6.4 percent of the original height, with a maximum

change in width of 0.07 in. or 3.5 percent of the original width.

Figure 4a shows the column at state of zero load, the deformed shape

at the critical load, and the postbuckled shape. After the first

critical load was determined, a small horizontal load (p=O.15 Ibs.)

was applied at the top (X2=L) of the column to give the system an

initial deformation. While holding the horizontal load constant,

the ~ertical load was again increased from zero to the critical by

"the incremental technique. For this case the equilibrium path of the

imperfect system was stable to u load of 3.60 lbs. where limit-point

type buckling was indicated. Holding the vertical load constant at

3.0 Ibs., the horizontal load was incremented to zero to project to

the postbuckled path of the perfect system. The motion followed by

the structure in removal of the horizontal load shown by line segment

I-J in Figure 5. With the system now presumably on the postbuckled

path of the perfect system, the vertical load was increased from •

3.0 lbs. to the second critical load of 3.75 Ibs. < pc < 3.80 Ibs.

were limit-point type buckling was experienced. The first two In-

creasing load increments from point J (Figure 5 indicates that the

system has not returned to a point of complete relative minimum of

the total potential energy. This action i5 attributed to the vleak

condition of the system. The postbuckled path of the perfect system

from point K to the second critical point at point L does appear to

be of correct form ir. that it follows nearly parallel to the imperfect

system.

A second example is indicated in Figure 6. Here we see an arch-

type structure modelled with 72 triangular elements and 110 degrees-

of-freedom. A load was applied along the axis of symmetry in incre-

ments of 0.2 Ibs., and the bifurcation point was passed in the in-

crement from 1.4 to 1."61.bs. "This is shmvn graphiocally in Figure 6.

The incremental solution was restarted at a load of P = 1.4 lbs. with

the load increment oP = 0.02 lbs. In this particular case, the

incremental solution did actually pick up ·the postbuckled path at a

load of P = 1.50 lbs. Geometric deformations measured rapidly from this

point (Figure 6), and the critical load pc was isolated to the inter-

val 1.515 Ibs < pc < 1.520 Ibs. Successive plots of the deformed

structure with increasing load (Figure 7) are interesting in that they

show the step-wise transition to a buckled mode. It is noted that or.e

of the members reverses curvature, which is typical of this type

structure. A small couple was applied at the vertex of the structure

and the load was increased to the critical. The critical load of the

imperfect system was found to be 1.00 < pC", 1. as lb. The critical

load of the perturbed system occurs well below that of the perfect

system, which indicates that the perfect system exhibits unstable

symmetric bifurcation. This, of course, is typical of this type

structure.

5.4. Nonlinear Elastodynarnics-Shock Evaluation. We consider a thin

rod of Mooney material (M1 = 24.0 psi, M2 = 1.5 psi) with the follow-

ing vndeforrned characteristics: length = 3.0 in., cross-sectional

.area::: 0.0314 in2, mass density::: 10-1, Ib.sec2/in.4• For the finite

element model, we take 60 evenly spaced elements, so that h :::0.05

in. and NO = 61, andowe consider a concentrated, time-dependent load

which varies sinusoidally is applied at the free end; a complete

loading cycle occurs in 0.002 seconds. It is clear from the computed

response shown in Figure 8 that shocks develop quickly for this kind

of loading. Unlike the response for the tensile step load where the

unloading wave is produced by simply removing the load, the sinusoidal

load actually "pushes" the end of the rod: The instant the load

starts to decrease is the moment when the first wavelet is generated

which propagates faster than the preceding one. Thus, at some time

subsequent to when the compression cycle starts, a compression shock

forms in the rod.

A comparison behleen tvlO integration, velocity-centered central

differences and the finite-elenent/Lax-Wendroff scheme, is also shown

in Figure 9 for the sinusoidal loading. In this case, it is clear

that the internal energy behind the compression shock renders the

central difference scheme unacceptable. It is interesting to note,

hO\'lever,that the tension cycle evidently "absorbs" the large·

oscillations preceding it and again produces a smooth wave front.

The detailed response to this loading is shO\vn in Figure 10. From

.tile.r.esponseoshown, we !lotive sev,eral intoeresting fea tllres of non-

linear wave motion. First, the compressive shock wave is reflected

from the wall as a compressive shock wave by almost doubling the

compressive stress, but the tension part of the stress wave is re-

flected with only a small increase in stress. Secondly, at t = 4.7

milliseconds, two compressive shocks are to collide, with relatively

little deterioration, possibly owing to the fact that mechanical work

of the external forces is continuously supplied to the system. In

addition, where we compare the response at t = 3 milliseconds to

that at t = 5 milliseconds, we find that it approximately repeats

itself, again indicating relatively little deterioration. Finally,

we note that, as in the development of shocks from Lipschitz con-

tinuous data, the shock forms subsequent to initiation of the

compressive cycle. Thus we are led to examine the positive slope

characteristics in the x-t plane to see if they preduct tCR for this

type of loading. Figure 11 shows that if we assume straight

compression characteristics of positive slope, the cusp of the

corresponding envelope in the x-to plane does, in fact, give a good

estimate of the tCR observed in the stress~time plots.

~.'0

·,

,

ACknowlcdgemen~. Portions of this work ~ere sponsored by the U.S.

Air Force Office of Scientific Research under Contract F44620-69-C-

0124 to the University of Alabama in Huntsville. The work reported

on nonlinear elastodynamics \'laS completed with the assistance of

Mr. R.B. Fost under the support of a grant, GK-39071, from the U.S.

National Science Foundation to the University of Texas at Austin.

6 . REFERENCES

1. Oden J.T. Int'l Congress on Large-Span Shells, Leningrad,

1966.

2. Oden, J.T. and Sato, T. Intll. J. of Solids and Structures,

1, 471-488, 1967.

3. Oden, J.T. and Kubitza, \~.K. Proceedings, Int'l. CoIl. on

Pneumatic Structures, Stuttgart, 82-107, 1967.

4. Becker, E.B. "A Numerical Solution of a Class of Problems

of Finite Elastic Deformation", PhD Dissertation, The University of

California, Berkeley, 1961.

5. Oden, J.T. J. St. Div., ASCE, 93, No. ST3, 235-255, 1967.

6. Oden, J.T, Finite Elements of Nonlinear Continua, McGraw-

Hill, New York, 1972.

7. Oden, J.T. J. Compo Structures, 2, No.7, 175-194, 1973.

08. 'Oden, J. T. 0 Nonlinear Elas'1:icity, Edited "by 'W. Dickey,

Academic Press, New York, (To appear) .

9. Sandidge, D. and Oden, J.T. Proceedings, Midwestern Conf.

Appl'd. Mech., Pittsburg, 1973.

10. Truesdell, C. and Noll, W. Encyclopedia of Physics, Edited

by S. Fluggee, III/3, springer-Verlag, New York, 1965.

11. Green, A.E. and Adkins, J.E. Large Elastic Deformations,

Second Edition, Clarendon Press, Oxford Press, 1972.

12. Oden, J.T. Mechanics Today-1973, Edited by S. Nernrnet-Nasse~

pergamoon Press, Oxford, (To appear) .

13. Ciarlet, P.G. and Ciarlet, P.A. Arch. Rat. Mech. and Anal,

46, 3, 177-199, 1972.

14. Fried, I. "Discretization and Round-Off Error in the Finite

Element Method Analysis of Elliptic Boundary-Value Problems and

Eigenvalue Problems," PhD Dissertation, Massachusetts Institute of

Technology, Cambridge, 1971.

15. Oden, J.T. and Fost, R.B. Int'l. J. Num. Meth. Engr'g., 6,

357-365, 1973.

16. Lahaye, E. Acad. Royal Belgian Bull. Cl. Sc., 5, 805-822,

1948.

17...18 .

1963.

Davidenko, D. Dokl. Akad, Nauk. USSR, 88, 601-604, 1953.

Goldberg, J. and Richard, J. Struct. Div. ASCE, 89, 333-351,

19. Ortega, J .M. and Rheinboldt, \~.D. Iterative Solution of

Nonlinear Equations in Several Variables, Academic Press, New York,

235, 1970.20. Oden, J.T. (NATO) Lectures on Finite Element Methods in

Continuum !v1echanics,Edited by J.T. Oden and E. Arantes e Oliveira,

The UAH Press, Huntsville, 1972.,21. Oden, J.T. J. Compo Structures,

22. Gallagher, R.H. Nat' 1. Sym. Comp. Struct' 1 Anal. and Designs,

Washington, D.C., 1972, (See also J. Compo Struct.

23. Oden, J.T. and Key, J.E. Int'l J. Nurn. Meth. in Engr'g.

(To appear) .

• 3 • 0

2 • 0

1 • 0

o

-1 .0

1 • 8

1 • !

1 • '+

1 • 2

1 • 0

• 8

• 6

• It

• 2

c

lJ=1.7

'+ • 0

v

Figure 1. Nonlinear Response of a Two-Bar Frame.

Exact Path0000 Calculated Path

µ= 1 • 5

Figure 2. Analysis of a Two-Bar Frame.

..

'/~~~ >

-- /.::I'/....

~;//--

~t:: 3Q)

S~ 2m.-{

0.1ffl

..-lQ

Static Solution

---.-------~:.=~ -=-== :--=.=-. ==--.~_0- ,OCIO

----- .000$.0003

-x-.ooo2---- .000 I

t (milliseconds)Figure 3. Stretching of a Thin Strip

P = 0

a

P ::::pC

b

p > pc

·c

Figure 4. Deformed and Undeformed Geometry of Buckled Structure.

• 4f •• ( '7') ,

Lx(72) 0=0

L-- -

I 'V;x(7l)

I/K _ K

~~I ~15~~3 j/~ I

"c

.0 p IT<X(71lrl'-"

x(72)

t:'lc:t: 20H

1··

oo 1.0 2.0 3.0

DISPLACEr·1ENT (in.)

4.0

Figure 5. Load-displacement Curves.

0.70.60.50.40.3

,,' ,.- -,,' ,,,'-, ," ,/" /, ,, ,, ,, ,,- ,,

0.20.1o

o

2.0

1.0r--..0rl'-"

t:'lc:t:0H

DISPLACEMENT (in.)

Figure 6. Frame Instability at Finite Strain.

•

p=pc

p>pC

Figure 7. Computed Buckled and Post-Buckled Frame.

t=4.4 I t=4. 6 I II t=4. 7

~ ~~, .

2 3 1 2 3 1 2 31 ~

2 3 2 3 1 2 3 1 2 3

I

Distance Along th~ Rod - inches

Figure 8 . Time history of stress wave response to sinusoidal end load.

6 0

It 0,+=1. am! \ 1 / \ t=2. a I/ L=2.J I L, .J \ 1\t =, ./ \ l =3 ./ \ L./ \t= 3. 7

0'rlCf)

0.- 2 0

o l 1 Vi V III 1 \1 \\.- It 0.. -Cf)

(/J

Q) 60H.µCf.) 4 0

2 0

0

- 20

•

.. ..

LAX-HENDROFF

t= 2. Oms 1. Oms

""\II \

/ \I \

/ \// \

VELOCITY CENTRAL DIFF.

t= 2.0ms 1.Oms/',/ \I \

/1 \/ \

o

4 0

nt= 3.0ms I ~ t= 3.Oms40 1\

~ nd

",' III II IVE . I II i!~ -" 11!111 I

_J IIIII1111

120

----11 2 3 o 2

I3

Distance Along the Rod _ Inches

Figure 9. Comparison of Response Computed Using Two IntegrationSchemes.

~ .. ' ~.\

::1II

-40

·X xDISTANCE ALONG ROD - Inches

Figure 10. Change in Stress During Shock Formation.

Time <INI sec.

Figure 11. Computed Characteristic Field