J' ,(;'/;-'.: ,- ' ,',.~>~;"~':'.:, '~L ",.

L·

~ l_'.

10

1. Introduction

J.T. ODEN (*) and J.E. KEY (t)

ANALYSIS OF FINITE DEFORMATIONS OF ELASTICSOLIDS BY THE FINITE ELEMENT METHOD

Since its beginnings over a century ago. the theory of finite elastic defor-mation has been regarded as largely an academic subject. Although it representsa generalization of the classical infinitesimal theory, it has stood apart from thattheory chiefly because its inherent mathematical difficulties have discouragedits application to may significant practical problems. The few exact solutionsto problems of finite elasticity that are available, however, have often beenmotivated by a need for quantitative information in certain areas of technologyThe series of papers by Rivlin (1), largely responsible for the renewed interestin the subject since 1948, were the results of research on the mechanical behaviorof natural and synthetic rubbers; the theory of finite deformation of elasticbodies reinforced by inextensible cords, for example, was developed by Adkinsand Rivlin (2) in connection with problems encountered with the design ofreinforced rubber tires. Solutions to problems of finite inflation of membranesand hollow cylinders are of interest in the design of inflatable structures andsolid-propellant rocket motors, and many problems of stress concentrationand stability, inaccessible by the classical theory, await treatment by moregeneral theories.

While closed-form solutions to basic problems such as extension of rods,torsion of circular cylinders, inflation of hollow spheres, etc. have beendiscussed by Rivlin (1), Green and AIkins (3), Truesdell and Noll (4), Eringen(5), and others, exact solutions to more general problems of practical interestare not available; nor are they likely to be solved by classical methods of mathe-matical analysis. In the case of finite elastic strains, Hooke's law is not appli-cable, loading surfaces change in area orientation during deformation, incom-pressibility of the material must often be taken into account, stresses are deve-loped on deformed material surfaces whose geometry is not known a prioriand techniques of superposition are generally not valid. It ii natural, therefore,to seek approximate solutions to problems of this type. For irregular geome-

1

"

• I

(",

/ -,

.-' \1

I' ..

. ,\

, .

.. ,'-\ "

.i

',' .

'.

, .'. '

'. ,',

'-'

.-

..,

j 1-',

.!

....1,.

'-, ,~.. r-' l ; • ,

-,'~:':;-.;::,,- ", I.. " " '- ,'" "', , I"::~>i;:",:'!::~~:" 'I'

'...•', ..~, , ']~ J- _" j

I.- ,-' ,'. . j :

, I

.-IIiI

:~

f, ,",': :-i _.•~-1_·,."l

( ':, .)1, :.::,\),,,- ,'.J

"

-r

" ,

4' • -. ~

.~,

~. I

'J\. ......(

,-', ,. \ ~

- '

~(

,'I,

'1

-'...... ~

\

:--.::; -

,,

),~

r-·

.'

",

(.) Professor of Engineering Mechanics, Research Institute, University of Alabama inHuntsville, U.S.A.

t Aerospace Engineer, Marshall Space Flight Center, N.A.S.A., Huntsville, Alabama,U.S.A.

Kinematics.

2. Finite elasticity

The displacement vector u can be expressed in terms of covariant and contra-variant components WI and w' with respect to the biorthogonal basis gl, gj as

Here we record for future reference certain basic ideas of finite elasticity.Further details can be found in the books by Green and Adkins (3) and Greenand Zerna (20).

Consider the motion of a continuous body ~. The body is a collectionof material particles which, at some time t = t may be put in one-to-onecorrespondence with points x in three-dimensional Euclidean space. At t = 0,the collection of points is called the reference configuration and is denoted Co'We establish in Co a fixed cartesian reference frame Xi(i = 1,2,3) defined by aset of orthonormal base vectors ii; we also associate with each material par-ticle P three intrinsic particle labels (Xl, X2, X3) which, at t = 0, are equatedto spatial coordinates Xl of the point corresponding to the particle P. Weshall refer to the lables Xl as the material coordinates of the particle P. A timet = t, t > 0, P occupies the current configuration C. The motion of P relativeto the frame xI in Co is given by the one-parameter family of configurationsdefined by Xl = Xi(X, t). It is convenient to refer to the body as undeformedwhen it occupies the reference configuration and as deformed in other confi-gurations.

The geometry of the body corresponding to Co may make it convenientto use an alternate collection of particle labels al = ai (X), X' = Xi (9) whichdefine a system of curvilinear material coordinates. Then, if r (X) is the positionvector of a particle in Co (i.e. r = Xl(X. 0) i1 = Xl (9) ii) we may construct abiorthogonal basis gj = ar/aa', gl = Voa' = (aa'/aXj) ij such that the vectorsgl are tangent to the material lines ai, g' are normal to the coordinate surfacesG-i, at = constant (j, k -:f:. i), and gl . gJ = 81. At time t = t, t > 0, the locationof the particle P is given by R = Xi(X (9), t) ij and the corresponding biortho-gonal basis is given by GI = aR/aei, Gj = (ael/axJ)ij.

The deformation of the body is described by comparing the metric tensorG = (Gj . GJ) GI ® Gj = (G' . GJ) GI ® GJ or its components GIj = Gj • GJ,GIJ = G' . GJ with the tensor g = gij gi ® gJ = gij gl ® gJ or its componentsgjj = gi . gJ' gij = gl. gj. We choose to use the components Gij, referred to asthe Green-St. Venant deformation tensor, or the components of the Green-St.Venant strain tensor 2 Yij = Gij - glj' Since GI = aR/aal = gl + Uj, whereD (9, t) = R (9, t) - r (9) is the displacement vector and U,i = ou/aal, we canwrite

(2.1)2 YIj = U. I . gj + D, j . gl + U. I • U, J

tries and boundary conditions, the finite-element method appears to be themost attractive method-of-attack.

In recent years, the advent of electronic computation and the develop-ment of general methods of discretation, such as finite elements, have madepossible the solution of certain problems of finite elasticity numerically. Thefinite-element method has been applied successfully to problems of finitedeformation of thin elastic membranes (6, 7, 8, 9), finite stretching of thinelastic sheets (10, 11), finite plane strain of incompressible solids (12), andfinite axisymmetric deformations of incompressible elastic solids of revolu-tion (13). Formulations of the general three-dimensional problem using simplexelements have also been presented (14, 15), and important applications ofthe method have been recently made to the analysis of test specimens inexperiments designed to determine mechanical properties of highly elasticmaterials (16). Other references to related work can be found in a recent surveyarticle (17).

The present paper is concerned with an extension and partial review ofprevious finite element applications in finite elasticity wherein difficultiesboth in the formulation of certain problems and in their numerical solutionare investigated. Following this introduction, we give a brief account of basicnotions of finite elasticity and cite specific forms of the strain energyfunction for several highly-elastic materials. The third section contains asummary of certain features of general finite-element approximations and anextension of the theory of conjugate approximation functions (18) to higher-order finite-element approximations. Next we present general formulations ofthe equations governing finite deformations of elastic elements. Here we recastthe formulation so that it applies to both higher-order approximations andcurvilinear elements. Following a quite different line of reasoning based onthe theory of conjugate approximations, we are able to extend the conclusionsof Hughes and Allik (19) to finite deformations of incompressible solids.In this connection, two types of incompressibility conditions arc obtained;one which is applied locally. element by element. and another which is appliedat each node in the connected model and which insures an average satisfac-tion of incompressibility in the neighborhood of each node. Section 5 containsa brief discussion of nonconservative generalized forces and consistent stressesin finitely deformed elements, and certain difficulties, peculiar to finite defor-mations, encountered in plane stress and plane strain problems are reviewedin section 6. Section 7 and 8 of the paper deal with computational problems andnumerical results. We restate certain incremental loading algorithms andpresent a computational scheme for handling unbanded matrices encounteredin the analysis of incompressible solids. We also describe a computerprogram designed for finite elasticity problems and present several numericalexamples.

wherein the semi colon denotes covariant differentiation :

Thus, (2.1) can be rewritten in the form

2 Ylj = WI;j + Wj;I + W~I Wk;j

a2x aek Ik m._

rij = aei aej aXm T~O

U = Wi gi = ~ gi

(2.8)

(2.9)

(2.10)

13 = Gigl'12 = gij G J 13IjII = g Gij

Globally, energy is conserved if

1 d f -I, d f W·d f F~i' d fl' dA'2<it Po W WI Uo + Uo = Po Wi Uo + to WI 0

00 Uo Uo Ao

wherein g = det (gij)' Then, in accordance with (2.7)1'

tlj = 2 [aw gij + aw (I gij _ glr .,..is G ) + I aw Gij]all 012 I 5 rs 3 01

3

In the case of isotropic bodies, W is written as a function of the principalinvariants 11,12,13 of the ten sot Gij

(2.4)

(2.3)

(2.2)U· glWi

awl_ + rl kaej jk WI

W;j

Wi = U· gl

krij Wk

aWi

aejWI;j

follows

Here r;k are the Christoffel symbols of the second kind associated with theundeformed body.

In the case of incompressible materials. 13 = 1, W = W (II' 12), and the strainenergy determines tlj only to within a scaler-valued function h called thehydrostatic pressure. Then

The Strain Energy Function.

Mechanics of Elastic Solids.

Let to denote the stress vector in the deformed body measured per unit areaAo of undeformed material surface area. Then, if fi = Ui gl = ill gi is a unitvector normal to Ao in the undeformed body,

tij 2 aw ij 2 aw (ij rs Gall g + aI

2g g rs glr gjO Gro) + hGij (2.11)

where tlj are the contravariant components of the stress tensor measured perunit area Ao referred to the basis Gj in the deformed body, and, assumingangular momentum is balanced, tlj = tji. The principles of conservation ofmass and balance of linear momentum require that at each particle (2.12)

to = tij ~l Gj (2.5) In selecting appropriate forms of tbe strain-energy function for variousmaterials, it is common practice to assume that W is an analytic function ofthe strains or of various strain invariants. In general

W - 1 Eljkm + 1 Eijkmop +- '2 Yij Ykl T Yij YkmYop '"

~GI t- [ir (s:J j )] F-j ..jPo = P V ~ v g t Vr + w; r ; j + Po = Po W (2.6) where Eijkm, E1jkmnp, ... are elasticities of order 1, 2 ..... For isotropic mate-rials.

wherein Po and P are the mass densities in the undeformed and deformed bodyGig = det (Gij)/det (gij), pj are components of body force per unit mass ofundeformed body, and superposed dots indicate time rates-of-change.

We shall confine our attention to adiabatic or isothermal processes onhomogeneous hyperelastic solids. Then the stress depends only on the currentdeformation and is derivable from a potential function W (Yij), associated withthe internal or the free energy, and referred to as the strain energy per unitundeformed volume uO' Energy is conserved locally if

etc, where A. and µ are the Lame constants and EI, E2, E3' E4 are second-orderelasticities. Since the material is assumed hyperelastic, it can be shown thatE2 + E3 = 2 (A. - µ) (4). By truncating (2.12) to only quadratic terms, weobtain the strain energy function for a Hookean material. Note that if theCauchy stress tensor Tij = trs x" IX.,j is used in (2.10) instead of tij, theHookean law can hold only if A. = - µ, which is impossible (1).

W (Yij) = tij Ylj tij = aw (Yij)aYiJ

(2.7)

EijkrnnpEijkrn = 1..gij gkrn + µ (gik gjrn + girn gjk)

EI glj gkrn gnp + E2 (gij gkrn gPn _ glj tp gmn) ++ E3 gik gjrn gnp + E4 gik gPrn gnJ (2.13)

Kavanaugh (16) proposed a polynomial approximation for compressiblematerials of the form

= a, Ki + a2 Ki + a3 K; + a4 K1 K2 + as K3 + a6 K1 ++ a7 Ki K2 + as K1 K3 + ag Ki (2.14)

Not all proposed forms of W for rubber-like materials have regarded thefunction F (12 - 3) in (2.18) as a polynomial in (12 - 3). Using a non-Gaussianmolecular theory as a guide, Gent and Thomas (27) assumed that of (12 - 3)/012= C/l2 where C is a material constant. Hart-Smith (28) elaborated on thistheory and proposed the exponential-hyperbolic law

where a" ... , ag are material constants and K" K2' K3 are invariants definedby

w = C {f ekdl,-3)2 dIl + k2 In ( i )} (2.22)

Most materials regarded as elastic at finite strain are treated as incom-~pressible. In the case of isotropic incompressible materials, W = W (I,. 12)can sometimes be represented as a power series in 11 ,and 12 :

~ ." '"W = L L Cr. (I, - 3)' (12 - 3)"

r=O .=0

K, = 'Yrr K2 = 'Y.. 'Yr. K3 = 'Y1j 'Yir 'Yjr

Coo = 0

(2.15)

(2.16)

Similarly, Alexander (29) proposed the forms

[12 - 3 + kJW = CI (I, - 3) + C2 (12 - 3) + C3 In k (2.23)

and

W = c, f ek(1,-3)2 dl, + C2 (12 - 3) + C3 In [12 - :1+ klJ (2.24)

Among materials of this type, the most widely used is the Mooney material(21), which follows from (2,16) by retaining only linear terms in I, and 12 :

W = C1 (I, - 3) + C2 (I2 - 3) (2.17)

wherein Ct, C2, C3, k, and k, are material constants. Equation (2.33) conformsto the Rivlin-Saunders form (2.18) whereas (2.24) combines the characteris-tics of (2.18) and (2.22) and apparently gives good agreement with experimentsperformed on neoprene film.

wherein the form of F (12 - 3) may vary from one type of material to another.Various polynomial approximation of F(l2 - 3) have been proposed. Forexample, adding a quadratic term to (2.17) yields (24).

Here C, = CIO, C2 = Co, are material constants. When C2 = 0, (2.17)reduces to the strain energy function of a neo-Hookean material suggested byTreloar (22) on the basis of a Gaussian kinetic theory for rubber-like ma-terials.

Based on more extensive experiments with rubbers, Rivlin and Saunders(23) suggest as a more general form of strain energy function

3. Finite element approximations

General

We now construct a discrete model of the body by representing it as acollection of a finite number E of material elements connected togetherat prescribed nodes X'\ !J. = 1, 2, ... ,G The collection of connected elementsis denoted ~ Let F (X) denote a vector-valued function which, with its firstr derivatives, is continuous on ~. We construct an approximation F (X) ofF (X) on ~ which is defined as a linear combination of G linearly independentfonctions <1> 4 (X) :

(3.1)F (X) = F4<1>4 (X)

(2.18)~W = C, (11 - 3) + F (12 - 3)

VI = C, (11 - 3) + C2 (12 - 3) + C3 (12 - 1)2 (2.19) The repeated index !J. is summed from 1 to G If the basis functions <1> 4 (X)are normalized in the sense that

then the components F4 = F (X4) are the values of F (X) at each node X4.

Since (3.1) involves only the values of F (X) and not the values of derivatives,we refer to it as a first order approximation; higher-order approximationsare discussed later. For additional fectures of such approximations, consult(30)

Biderman (25) suggests for a sulfur-filled rubber

VI = C1 (II - 3) + B, (I, - 3)2 + B2 (11 - 3)3 + C2 (12 - 3) (2.20)

while, more recently, Klosner and Segal (26) proposed that F (12 - 3) of (2.18)be a cubic in (12 - 3) :

Vi = C, (I, - 3) + C2 (12 - 3) + C3 (12 - 3)2 + C4 (12 - 3)3 (2.21)

<1>4 (XT) = o~ (3.2)

and its inverse is denoted erA We can now generate a system of conjugateapproximation functions (18)

where Uo is the volume of the body ~ (or ~). Then, the fundamental matrixerA is defined by

~~ 11 ... Ir (Xr) = 0

a~~i1 ... ir (X r)axJ = 0

. r<I)~ (X) = 0 n.

. ra«l>~ (X ) i r- = oJ 04 .n

wherein the interpolation functions are normalized in the sense that

Similar relations hold for higher-order approximations Let FA, ~I' .•. ,

~It h...lr; i, ii, ... , ir = 1,2,3; if;;;:::iz ;;;:::... ;;;:::ir denote the value of F (X)and its r partial derivatives at each node XA Then. instead of (3.1) we use

F (X) = FA ~ (X) + FA ~I (X) + .n + FA. «1>1. h ... Ir (X) (3.12)A • I A • I. '1 ... Ir A

Higher-Order Approximations

<1).1 (Xr) = o~OcI>A (Xr)

axl = 0

(3.3)

(3.4)

(3.5)

erA = (<<I>r, <1).1)

11>.1(X) = eAr I1>r (X)

Following the work of Brauchli and Oden (18), we assume that an inner-product (F, G) is defined on the subspace spanned by the functions cf> A (X)In the present paper, we define

(F, G) = L. F (X) . G (X) duo

which provide a basis for the space conjugate to that to which F (X) belongsand which satisfy the biorthogonality condition

Then

where

(~4, 11>~) = o~

- A AF(X) = F ~A (X) = F4 11> (X)

(3.6)

(3.7)

ar 11> (Xr) 0cI>1 (Xr) or 11>1,i1 ... ir (Xr) . .A A 4 II·n 'r r

axJ• axh n. axJr = 0 axJ, axh ... axJr = 0 ... ~yJ. ~yh ~yJr = OJ, .n Ojr 0.1

(3.13)

~ = (F, 11>.1) and FA = (F, «1>.1) (3.8) Introducing a new set of functions and coefficients

Various other properties of <1>4 (X) can be found in (18) and (31).

In finite-element approximations. the functions <I) A (X) are generated bysumming local approximations over individual elements. If te

) (X) is therestriction of F (X) to element e, then locally

FA f ~A (X)a'

F4 I1>l (X) I.l'.J

F4' = 1 ~2 \jIA' (X) = ~i (X) a'

1,2 , ... , G

G + 1, n., 2 G

2 G + 1, n., 3 G

M, N = 1, 2, ... , Ne, Ne being the number of nodes of element e. Here\jIl-:) are the local interpolation functions corresponding to element e. Then,except for a set of zero measure,

j(e) (X) = ~) "'l-:) (X)

EF (X) = L j(e) (X)

"'l-:) (XM) = O~

E (e)

$.1 (X) = 2: n~\jIl-:) (X)

(3.9)

(3.10)F~rrr. .. r

(3.14)

cI>~rr ... r (X) a' = (N - 1) G + 1. ,n, NG

(e)

where n~are elements of the Boolian transformations from local to global(e) (e)

coordinates [i.e. n~ = 1 if node N of element e is incident on node a and n~= 0 if otherwise; see (30)] It follows that

E (e)

<1)4 (X) = eAr 2: n~ \jIl-:) (X)e

(3.11)

(3.12) can then be rewritten in the form

- -A' •F (X) = F \jIA' (X)

The conjugate functions are now given by

\jIA' (X) = CAT' \jIr- (X) C4T' NA" "'H-I

(3,15)

(3.16)

4. General equations for elastic: elements Henceforth, we confine our attention to static and quasi-static behaviorof isotropic, homogeneous, hyperelastic elements. Then (4.4) reduces to

in which u~) is the displacement vector at node N of element e and X =(X 1, X2, X3) are the local. material cartesian coordinates of particles in theundeformed element. Then, omitting the element identification label (e) forsimplicity, the strain tensor is given by

General

Considering only first-order approximations for the moment, we isolatea typical finite element e of the body, and construct a local approximationof the displacement field of the form

O(e) = VN (X) or.) (4.1)

f { oW Ij oW [l:lj ( (1 p) R)J ".0 2 all 0 + 2 aI2

u 3 + 2 VR. r on + 2" Vp, r Us U; -

- oir ojl (On + 21 ..)] + 13 ~~ GIJ } o/N,i (Okj + VM.j U~) duo = PNk (4.6)

wherein 1.. is given by (4.2), M,N,P,R = 1,2, ... , Ne, and for the finite ele-ment

Gn) = [(Oiro + VN,; u~ (Ojro + VM,j ur)]-l (4.7)

Incompressible Materials

Here uo(e)is the undeformed element volume and mNMand PNK= PNK(t) are theconsistent mass matrices and the generalized nodal forces, respectively:

mNM = f Po o/No/Mduo PNk = f Po Fk o/Nduo + f Sk o/NdAo (4,5)00 (.) Uo (.) Ao

we introduce (4.1) in the global form of the conservation of energy (2.8) for theelement and, requiring that the result hold for arbitrary nodal velocities, obtainfor the equations of motion of an elastic finite element (32, 33),

"M f aW(e) MmNM Uk + a- o/N,l (Okj + o/M,j ud duo = PNk (4.4)

"0 (0) 11j

Here the dependence of o/Non X and u~ on t is understood; o/N,; = 8o/N/aX1;

M, N = 1. 2, ... , Ne and i, j, k = 1,2.3.Observing that

. oW. Ij Orlj'N iJ M 'NW = -;-- 11J = t o~ Uk = t o/N,; (Okj + o/M,j Uk) Uk (4.3)

u1ij Uk

(4.8)heel = o/N (X) h~)

Most materials capable of undergoing finite elastic deformations areassumed to be incompressible. In formulating finite-element models of suchmaterials, it becomes necessary to construct approximations of the localhydrostatic pressure h (e) in addition to the local displacement field :

The hydrostatic pressures ~) represent additional unknowns in the pro-blem. However, we must suppplement the equations of equilibrium (4.9)with incompressibility conditions which insure that the volume of each ele-ment is conserved during deformation. Two different methods can be used toformulate the incompressibility conditions (19).

Here h~ is the hydrostatic pressure at node N of the element. The equilibriumequations for a typical isotropic element then become

f. [2 oW oiJ + 2 oW (Oij oyl + oir OJI) (on + 2 "frs) +"0 (0) oIt 012

+ o/R hR G~~)Jo/N,; (Ojk + o/M,j u~) duo = PNk (4.9)

(4.2)2 1ij = VN,I uf + o/N,j u~ + o/N, io/M,j u~ u~

The quantities PNkare the cartesian components of generalized force at node Nof the element referred to the orthonormal basis ik in the undeformed bodyHowever, the surface tranctions SdAo, are developed on material surfaces inthe deformed body so that their components Sk with respect to ik are, ingeneral, functions of the nodal displacements u~; ie, the forces PNk aregenerally nonconservative We discuss specific forms of these forces in the nextsection

(i) Uniform Pressure

We adopt as a general rule that the order of the approximation (4.8) shallbe less than or equal to that of the local displacement field. In addition, ifu(e) = o/NUNand the o/N (X) are polynomials of degree n, then heel = o/~)~)and o/~) are to be polynomials of degree m ~ n. The simples t approximationof heel (X) is the zero-order approximation in which heel is assumed to be uni-form over each element (I 2, 13, 14). Them o/RhR in (4.9) is replaced by h~e)

= constant and only one incompressibility condition is needed for each ele-ment. Locally, the incompressibility condition is

Equations (4.9) (with "'RhR = ho) and (4.11) represent 3Nc + 1 equations inthe 3 No + 1 unknowns u~, ho

13 = det (Gij) = [det (olj + "'N, I Uf)]2 = 1Consequently, for a finite element we use

f det (Olj + "'N, I uf) duo = 1\)0 (G)

(4.10)

(4.11)

ciably alter the values of u~. For finite deformations, however, procedure (ii)may lead to significant errors unless a very fine network of elements is used.

Higher-Order Approximations

Except for lengthier algebra, generation of higher-order finite-elementapproximations follows essentially the same procedure as that used to obtain(4.4). For example, consider the third-order approximation

U1 = WN u~ + q>~ u~J + X~k U~jk (4.15)

(ii) Variable Pressures

(4.17)

(4.18)

. N )Uk,mn

OYij---rraU..,mn

jk "NXN Ui,jk

P(I) . NN p(lI). N p(lll) 0NI U1 + Nij Ul,j + Nijk =

.1, . N J. N jk . NUl = 'I'N Ul + q>N Ui,j + XN Ui,jk

.I,"N j "NU1 = 'I'N Ul + q>N ul, j +

aw (OYlj . N aYlj. N;-- -;-N Uk + -a N Uk,m +uYij UUk Uk, m

w

where j ~ k and

WN (XM) = o~a"'N (XM

)= 0

a2

WN (XM)= 0axi axi axJ

q>~ (XM) = 0aq>~ (XM)

= ot O~a

2WN (XM)

= 0axiaxi axr

X~ (XM) = 0

OX~ (XM)

= 002

X~ (XM)= ot o~ o~ (4.16)oxi

axi axr

Note that the functions o/N (X) are, in general, different than those in (4.1)since the first-order approximations need not possess continuous or nonzerofirst and second partial derivatives. Moreover, we may sometimes find itconvenient to eliminate certain of the functions WN, q>~, and X~k in approxi-mations of ui.

Observing that

we find for the global form of the principal of conservation of energy of thefinite element an equation of the form

. hi h p(l) p(lI) d p(lII) f . f N N d h' .m w c NI' Nlj' an Nijk are unctiOns 0 Ul; ui, j' etc. an t elr tImederivatives. Since (4.18) must hold for arbitrary values of ii~, i1~j' and i1~jk'

(4.12)

(4.13)

(4.14)

E (0)

Qd = L n~q~)

q~) = f (13 - 1) WN duoVo (ft)

Let Q A denote the value of q at node A of the connected model; i.e.

Then the incompressibility condition 13 = 1 is satisfied in an average sensein the neighborhood of node A if

E (c) fL n~ (I~c) - 1) "'~) duo = 0e Va (<<')

If the element hydrostatic pressures are allowed to vary over each element.it becomes necessary to formulate nodal incompressibility conditions at eachnode of the connected finite element model. To obtain these equations, wemake use of the fact that h(c) belongs to the space conjugate to u (0) and u~~).Since the projection of the function (13 - I) on the subspace spanned by"'N (X) is (13 - 1, "'N) ",N (X) and WN (X) (the conjugate basis) are linearlyindependent, we introduce

These equations, together with the global form of (4.9), provide, after applyingboundary conditions, a complete system for the determination of the nodal

(c)

displacements and the global hydrostatic pressures hN = n~Hd.

Ordinarily, the use of uniform hydrostatic pressures as suggested in (i)leads to a more accurate (and sometimes exact) satisfaction of the incompres-sibility condition over the finite element model than that obtained usingprocedure (ii). However, the corresponding approximation of pressures and,consequently, stresses in (i) is less accurate than that of (ii). In the case ofinfinitesimal deformations procedure (ii) is probably preferrable becauseminor errors in satisfying the incompressibility condition should not appre-

we conclude that ~: = p~m= P(N~11<= O. This leads to the three systems ofequations

- -M I' -M TTl -M f 'OW .1. d -mNM Uk + NM Uk., + ..INMUk.rs + ::\...,'l'N,J Xi.k 00 = PNku. (0) v r 'J

Alternately, we can useu2 = WN(0) UN

wherein

UN = WN1 gi (eN) = wf gl (ON)

(4.25)

(4.26)

I' "M II'· -M 111"1 -M f 'OW, d d' (4 19)NMUk + NMUk,s + NM Uk.sl + a <PN,jXi,k 00 = Nk .Uo (0) 1jl

where XI,k are the deformation gradients

'OW Ij (~ ,I. M + ' M ,s M) (4 20)~ XI.k = t Vik + 'I'M.i Uk <PM,I Uk" + XM,I Uk." .v flJ

Jrs"M IIIrsl ··M Krsmn ..M f 'OW rs dNMUk + NMUk.1 + NM Uk,mn + a XN.j Xi,k 00

Un (0) YiJµ~k

Thus u1 and u2 coincide at each node. However, at an interior point in theelement u1 is referred to the actual basis gl' gl at that point whereas u2 isreferred to an approximate basis representing an average of the nodal valuesof gj and gl at the nodes. While u1 generally represents a much better appro-ximation than 0

2 ,the displacement gradients corresponding to u2 are consi-derably simpler than those of u1 since the use of an average constant biortho-gonal basis over the element results in zero values of the Christoffel symbols.Hence

u~i = [WN,j (0) wN, + WN(0) wNj rj;] g, U~i = WN.I (0) UN (4.27)

and

IDNM= f PoWNWMdoo I~ = f PoWN<P~doo J~M = f PoWNX~ doo00(.) \>0(.) Uo(.)

IIrs f 's d mrsl f ' sl d Krsmn f .. mndNM= Po<PN<PM00 NM= Po<PNXM 00 NM = PoXNXM 00

\>0 (.) Uo (.) 00 (.)

(4.21)

PNk = i Po Fk WNdoo + J Sk WNdoou" (0) A. (0)

d~k = j Po Fk <PNdoo + f Sk <P~dAouo (0) Ao (0)

µ~k = f Po Fk X~ doo + f Sk X~ dAo (4.22)Uo c.) Ao (.)

The equations of motion of an element corresponding to approximation u2

are essentially of the same form as (4.4) except that WN (X) and a"'N (X)jaX,are replaced by "'N (0) and a"'N (O)jael and u~, PNk are replaced by ~ and q~,where q~ are now contravariant components of PN' In the case of approxima-tions of the type in (4.24), we have, for example, instead of (4.4), (33)

. f. 'Ow(c) [~I ,I. .1. rj (.1. .1. ~i .1. .1. rinlNMwM' + ~ Vj 'I'N,q - 'I'N jq + 'I'M,j 'I'N,q Vm+ 'I'M'I'N,q mj-

Uo(.) Vrqj

- "'M "'N r:nj r;q) WMm] doo = q~ (4.28)

In spite of its complexity, we prefer to use (4.28) in most cases owing to thegreater accuracy with which it allows stresses to be approximated

5. Generalized forces and consistent stresses

(4.24)

Nonconservative Forces

However, the quantities sj are themselves functions of the displacementgradients. For example, in the case of an applied pressure normal to A,

We commented earlier that the generalized forces PNkof (4.5) are dependentupon the components §k of the surface traction per unit Ao, referred to thebasis Ik in Co, but developed on material surfaces in the deformed body.If SJ = tijUI are the contravariant components of S with respect to Gj, it canbe shown (32) that

(5.1)

(5.2)sj = - q .jG <JIk nk

"J MSk = S (Ojl< + "'M,j Uk)

Curvilinear Elements

Two different types of approximations suggest themselves in the case ofcurvilinear elements. First, we consider approximations of the componentsWi and Wi of (2.2) :

N i NIWI = Wi WN(0) W = W WN(0) (4.23)

N = 1, 2, ... , Nc, WN (e~ = o~. Then1 N I .. .NiU = WI WN(0) g = W WN(0) gj

Essentially the same procedure can be used for fourth-and higher-order appro-

ximations

Thus, in general,

f f· N MPNk = Po F k *N duo + SJ (p, U ) (Ojk + *M. j Uk) \jfN ciA., (5.3)Uo (.) Ao (l1li')

Let t:~) = aW<C)/aYiJ denote the local stress tensor in element e derivedfrom the local constitutive equations and TIJ the corresponding discontinuousstress field over the connected model :

where p is some load parameter.E

-'j " tijTI = £., (c)c

(5.8)

and XN, UN are the position vectors and nodal displacements of node N of asubelement (N = 1, 2, 3)

In the case of a uniform applied load normal to a, (5.4) yields

Owing to the complexity of (5,3), we shall adopt the approximate procedurefor calculating nonconservative forces discussed in (35) and represent the defor-med material surfaces area of each boundary element as a network of tlat,generally triangular ,elements over which the applied tractions are assumed to beuniform. Let n denote a unit vector normal to a subelement of the networkof area a and (n, e1, e2) an orthonormal triad on a. Let qo' 81, and S2 denotethe uniform normal and tangential components of applied force over a. Then thecomponents of generalized force at node N of a triangular subelement are

B = X3 + u3- (Xl + ul) (5.5)

6. Plane stress, bodies of revolution, plane strain

(5.9)

(5.10)

(5.11)

E (c)

<Dd (X) = L n~*~)(X)c=l

Recalling that

we use, instead of (5.8), the consistent stress distribution

Tij = T~ <Dd (X)

where <Dd (X) is defined by (3.5) and

T~ = f. Tij <D4 (X) duoUQ

The quantities T~ are the consistent components of stress at node aof the connected finite-element model. In practice, it is sufficient to determineonly the quantities T~ in order to adequately assess the stress distribution.

(5.4)

A = X2 + u2 _ '(Xl + ut)

where

~ 1[ 1 ~ - - ]PNk = '6 qo A x B + fAT (2 a SI + A· B S2) A - S2 I A I B

qo (X2 2 Xl I) (X3 3 Xli)PN" = - e..~ . + u· - . - u, ,+ u· - . - uj.. 6 IJ.. I I I I J J J

The net force at a node joining E' elements is

(5.6) We record here certain special formulations pertaining to problems ofplane stress, plane strain and axisymmetric deformations without proof.Details can be found in (7, 12, 13).

Once the nodal displacement components u~ have been determined, thelocal strains, strain energy and stresses can be computed using (2.7) and (4.2)Stresses computed in this manner, however, will generally suffer finite disconti-nuities at interelement boundaries and will represent, at best, only rough ave-rages of the actual stresses in an element. To overcome these difficulties, wecompute continuous consistent stresses using the procedure outlined in (18)and (31)

Plane Stress

+ [1.2

sail + f'1l (1 - 2 1.4 - 2 1.4 rlJJ ~~ } *N.a (Opi +

+ *M.1l Ur) dA: = PNi (6.1)

Consider a thin elastic sheet of isotropic, incompressible elastic material,initially in the Xl, X2-plane, for which t33 = 0 and A is the extension ratio ofa material line originally normal to the undeformed sheet. The equilibriumequations of a typical finite element are

1.4 f'1l) awm+I

2 do Lo { (S~1l

(5.7)1, 2, 3N'E' (c)

PNk = L n~'PN' kc=l

Consistent Stresses

(6.6)

7. Computational schemes

In this section, we describe an existing computer program designed forthe finite-element analysis of finite deformations of isotropic elastic bodies

Ao = 21 I f SMNP s'1l

3~ X~ IA = ~ I f SMNP sall3 ~ + u~t) (X~ + u~) I

P=1 psI

(6.7)

(6.8)

Plane Strain

The equations corresponding to the case of finite plane strain followdirectly from (6.6) by setting').. equal to a constant (generally unity). It isinteresting to note that in the case of finite plane strain of compressible mate-rials, the behavior of an element is determined by the nodal displacementswhile for incompressible materials, incompressibility conditions must beimposed to yield a sufficient number of equations to determine the hydro-static pressure. On the other hand, for plane stress of compressible materialsthe transverse extension ratio ')..2becomes an independent unknown and mustbe determined from auxiliary conditions such as (6.5). Conversely, for casesof plane stress of incompressible elements. the hydrostatic pressure is deter-mined immediately from the boundary condition on t33 (e.g. t33 = 0 or aprescribed function ) and ')..2is determined by the nodal displacements.

where h = WNhNis the element hydrostatic pressure, duo = 2mdrdz,

<p = 1 + 2 r: + 2 sa~ s~I' ra)"r~I'

Types of Elements

In the case of finite deformations of inelastic solids, numerical integrationof stiffnesses involves complications of unusual proportion. For certainmaterials, for example, it is necessary to evalute volume integral which involveexponentials and logarithms of complicated functions of the unknown nodal

and').. = 1 + utJr. We must also impose the discrete incompressibility condi-tions of the form (4.11) or (4.14)

In various applications, it is customary to use simplified versions of (6.6)arising from the use of simplex elements together with an acceptable appro-ximation of ')..2. These are discussed in (13), wherein').. is approximated as1 + uJi; U1 is the average radial displacement of the nodes and r is the averageradial coordinate of the nodal poinys Also h = ho and the incompressibilitycondition is obtained by simply comparing deformed and undeformed volumes:

2 1t r Ao - 2 1t (r1 + 0,) A = 0

Axisymmetric Deformations of Solids of RevolutionIn the case of a body of revolution, we use intrinsic cylindrical coordinates

e1 = r, e2 = z. e3 = e. Then g" = g22 = 1, g33 = r2, and gij = 0 i #- j:

and the Riemann-Christoffel symbols are r:3 = r;1 = l/r, r~3 = - r, all. a

other rjk = O. Since U3 = a~3 = 0, we have u~~ = u~ II = ua, II' Hence, upon

introducing these quantities into (4.28) and simplifying, we arrive at the equi-librium equations for an isotropic, incompressible, finite element of revolution:

f { oW oy~ oW alp [OW II OW] 0')..22 ill a N + aI ~+ aI + 2(1 + y~)ill ~+

110 (.) 1 u. 2 Ua 1 2 Ua

oW oyt 2 ( 2 alp 0')..2

) } d _+ 2 aI2

au~ ').. + h ').. o~ + lp au~ \)0 - PN.

and

')..2 = [1 + 2 "'N.a (Qak+ "'M,a u~) u~ + sa)"s~I' ("'N,a u~ + "'N.~ u~ ++ "'N,a "'M,~ u~ u~) ("'R. ~ u~ + "'R. I' u~ + "'R.~ "'p, I' u~ Uf)] -1 (6.3)

Here i, j. k = 1,2,3; ex, ~, ')...µ = L 2,; N, M, P, R. = 1,2, ... , Ne

In the case of compressible materials, a quite different formulation isnecessary. The function ')..2is then independent of u~ and must be representedapproximately over the element by ')..te) = "'NµN where µN = ')..2 (xN). Then,instead of (6.1), we have

f [oW 2 R oW.~ oW2 do ill + \jIR (')..) ill 15 + ill +

Ao (.) 1 2 2

R oW f"~] M •+ \jIR µ 013

\jIN, a (Q~l + \jiM.~ U1) dAo = PNI (6.4)

The Ne

values µR = (')..2)Rare determined from the condition that t33 mustvanish in an average sense over the element; i.e., at each node N,

f (oW oW, f"~ OW) • -Ao' 011 + 01

2(lall + 013 A \jIN dAo - 0 (6.5)

The 3Nc equations (6.4) together with the Nc equations (6.5) constitute 4Ne

equations in the 4Ne unknowns u~ and µN. If ')..2is assumed to be uniform overeach element, say ')..2= µo' we simply replace \jIRµRin (6.4) by µo and add to the3Ne equilibrium equations the single condition SAO t33 d A: = 0, which followsfrom (6.5) by replacing \jINby unity.

where d" is the undeformed thickness, A.: the element area in the Xl, X2_plane,

f"~ = Qa~ + sa~ s~I' ("'N. I' u~ + "'N. ~ u~ + "'N. I' "'M.~ u~ u~) (6.2)

3. Composite Quadrilateral Elements.

Ai

(b)

~

~

(d)

5,

~

X,

KZ

(e)Fig. I. - Types of elements.

Xz

(0 )

X3

(7.1)

I, 2, 3, Cl, ~, A., µ = 1,2. See Fig. la.

. NuJ = (aN + bNi Xl) Uj

2. Plane Stress.

where i, j, k, M, N, P, R

A plane stress element for plane deformations of elastic sheets is obtainedfrom (7.1) and (7.2) by simply changing the range of the indices i, j k, from3 to 2 (fig. 1.b).

By arranging four triangular elements as indicated in Fig. l.c, a compositequadrilateral element for membranes and plane deformations of sheets isobtained.

+ £,a), £,lIp (bN: u~ + bNII u~ + bNtt bMIl u~ U~) tbn u~ ++ bRp u~ + bRA bpp U~ uj)r 1 (7.2)

where aN' bNi are the usual simplex coefficients which depend only on thecoordinates of the nodes, and N, i = 1,2,3. Here WN.1 = bNj, t33 is assumedto be zero, and the transverse extension ratio A. (for incompressible materials)is determined by the equation

A.2 = [1 + 2 bNa (Oak + bMa u~) U~ +

The displacement components over a thin triangular, membrane elementare assumed to be of the form

1. Triangular Membrane Element.

displacements. To minimize such difficulties, it is natural to first examinethe simplest finite-element approximations; i.e. the simplex elements. Ourexperience with such elements has been encouraging; the nonlinear stiffnessrelations are more manageable, numerical results agree well with the experi-mental data that is available, and many of the traditional criticisms of suchelements in so far as computing stresses are concerned are largely overcomeby the use of the consistent stress approximations described in Section 5 ofthis paper. We consider here the following types of elements

4. Triangular Ring Element, Composite Quadrilateral.

For the analysis of axisymmetric deformations of incompressible elasticsolids, we employ a triangular element of the type indicated in Fig. Ld, or a

composite quadrilateral consisting of four such triangles. Displacement fieldsobey (7.1) with Xl = r, x2 = z, x3 = 0 and incompressibility conditions areobtained by comparing deformed and undeformed element volumes, thedeformed volume being uniquely defined by the displacements of each node.

To simplify integrations, we use the following approximation of the circum-ferential extension ratio :

i, j = 1,2, ... , n. Since both (x. p) and (x + Ox, p + op) are assumed to satisfy(7.5), we have

where J is the rom jacobian matrix and G is a load-correction matrix :

A. = 1 + (f u~/ f rN)N=l N=l

(7.3)

Jij (x, p) = ofj (x, p)/oxj Gjj (x, p) = afj (x, p)/apJ (7.7)

where rN is the radial coordinate of node N. Ox = r1 Gop (7.8)

f (x + Ox, p + op) ,.., f (x, p) + J (x, p) ox + G (x, p) op (7.6)

By setting A. = 1, r = 00, the ring element described in 4 reduces to atriangular (or composite quadrilateral) element that can be used in the analysisof finite plane strain of incompressible elastic solids.

6. Isoparametric Element.

We also consider an isoparametric quadrilateral element of the type inFig. i.e. Here, following the usual procedure [e.g., (36)], we set

where ex = 1,2; N = 1,2,3,4. Approximate incompressibility conditions areobtained by computing the deformed volume of a quadrilateral obtained byjoining displaced nodes with straight lines. This element is not yet incorpora-ted into program to bt: described subsequently.

(7.9)

(7.10)pr = L opi1=0

Here

which leads to the algorithm for the r + I iterate :

x'+ 1 = x' - J-1 (x', PJ G (x', p') op

It is interesting to note that, unlike conventional cases, the incrementalloading process is not initiated at a zero starting value for many applicationsin finite elasticity. In the case of incompressible bodies, the equilibriumequations yield specific constant values for element hydrostatic pressures atzero applied loads. These must be introduced as starting values to guaranteea convergent process. For finite deformations of initially flat, thin membranes,the instantaneous, stiffness matrix may be singular; it is then necessary tospecify a realistic set of nodal displacements to initiale the process.

Since some error in the nodal displacements tends to accumulate duringthe incremental loading process, it is worthwhile to perform several cycles ofNewton-Raphson iteration using the current, accumulated solution x' as astarting value. Our experience indicates that such Newton-Raphson correctionsare best introduced near the end of the complete incremental loading processrather than after each load increment. Ordinarily, only a few cycles of Newton-Raphson iteration need be performed to equilibrate forces on all elements.

Computer Program Description.

We now describe briefly certain basic features of a computer programwhich employs the incremental-loading technique described above. The pro-gram consists of the seven basic subroutines indicated in the flow chart in Fig. 2,and is capable of analyzing finite deformations of the following types ofproblems involving compressible and incompressible, isotropic, elastic bodies:thin elastic membranes, plane deformations of elastic sheets, axisymmetricdeformations of bodies of revolution. and finite plane strain. The program isdesigned to accomodate afI arbitrary strain energy function W; i.e., the formof W is designated in appropriate FORTRAN statements. The nonlinearequations are solved using the algorithm (7.9), and output can be generated as

(7.4)

(7.5)f (x, p) = 0

XN (I;) = aN + bN2 1;" + cN ~1 ~2

u" = o/N (I;) u~

x" = o/N (I;) ?"

5. Plane Strain Elements.

Incremental Loading - Newton Raphson Methods.

All of the numerical results presented in the next section were obtainedby solving the large systems of nonlinear equations generated in the analysisby the method of incremental loading. To review briefly the ideas behind thebasic incremental-loading algorithm (13), note that each of the formulationsdescribed previously leads to a system of nonlinear equations of the form

Here x is an n-vector of unknowns, the first members of which are nodal dis-placements and the last are hydrostatic pressures, p is a vector of applied loads,and f is an n-vector of nonlinear equilibrium equations and incompressibilityconditions. Let (x, p) and (x + Ox, p + op) denote two neighboring solutions to(7.5) ,where Ox, op denote increments which are small in the sense that II Ox II< £, £ being a prescribed real number. Then

(7.11)

(7.12)

fj (x.. + DELXj, p) - fj (xu, p)DELXj

J ..IJ

The first columa of Jij is generated on cycle 2, the second column on cycle 3,.. '. the last column on cycle (ND + 1). The last cycle of the loop (NU + 2)generates the incremental load array.

G .. 0 . = [fl (x, Pj + DELXj) - fj (x, pj)J DELP1J PJ DELXj

Subroutine JACOB. - This subroutine generates the approximate jacobianof the incremental stiffness matrix, and is primarily a loop which indexes from1 to (NU + 2) each load increment, where NU represents the number ofunknown displacements and hydrostatic pressures. Subroutines STIFF andAPLOAD are called at the beginning of each cycle. On the first cycle the non-linear stiffness equations are evaluated to generate f (xo' p) which is stored inthe FI array, Xo is taken as the accumulated value of the displacement andhydrostatic pressure vector, and p is taken as the accumulated applied load.Cycles 2 to (NU + 1) evaluate the nonlinear stiffness equations to computethe jacobian :

(V~

Fig. 2. - Flow chart.

~

stresses and plots of the deflected profile at the end of each load increment.In applying (7.9), the matrices J and G of (7.7) are computed each cycle usingfinite difference approximations.

The main program initializes the plot tape and calls the subroutines whichinitiate the solution process. It allows for stacking of several independentproblems and contains comment cards explaining the symbols associated withthe input data, dimensioning of arrays and the order that the subroutines arecalled. The subroutines which form the working elements of the program arelabeled INITIAL, INCLD, JACOB, STIFF, APLOAD, RITE and SIMULT,and are described briefly below :

Subroutine INITIAL - This subroutine reads the input data cards and thenwrites the data on the program output. The unknown displacements, unknownhydrostatic pressures, and various arrays are initialized to the required startingvalues.

Subroutine INCLD. - This subroutine applies the load in the specifiednumber of increments. Subroutines JACOB and SIMULT are called, respec-tively. The incremental solution generated by SIMULT is added to the cummu-lative solution, and these values are printed and stored on tape which is inputto the plot and stress program.

Subroutine STIFF. - This subroutine evaluates the stiffness and incom-pressibility equations. In evaluating these equations, the boundary conditionsare applied to the elemental stiffness equations, which are then assembled togenerate the instantaneous stiffness matrices.

Subroutine APLOAD. - This subroutine generates the applied loadvectors for two independent loading conditions. One is a load vector whichis incremented in magnitude; however, the direction is not allowed to change.The other loading condition computes the total force on the deformed surfacefor each increment of pressure and distributes this force into node components.

Subroutine RITE. - This subroutine writes the incremental stiffness arrayfor specified load increments. The arrays are printed in the conventionalmatrix arrangement.

Subroutine SIMULT. - This subroutine is a linear simultaneous equationsolver based on the Gaussian Elimination technique. The storage scheme andthe section of the pivotal elements were modified to solve efficiently sparce,unsymmetric arrays. (For plane stress problems in which banded incrementalmatrices are encoutered, an alternate subroutine based on a modified Choleskyscheme is used).

The storage scheme used in this subroutine stores all non-zero numbersand a limited number of zeros. Two arrays are input into the subroutine.

une array contatns tbe non-zero elements, and the other array contains thecolumn index of the corresponding elements. For example, the array

3.0 0.0 - 2.0 0.0

001is stored as

[1 11

[30 - 2.00.0 2.0 0.0 0.0 l.0 2.0 1.00.0 2.0 4.0 0.0 0.0 2.0 4.0l.0 0.0 0.0 2.0 0.0 l.0 2.00.0 0.0 2.0 0.0 3.0 2.0 3.0

For large arrays with a small number of entries in each row, this technique isvery efficient.

The pivotal element is selected by sequentially searching for the row withthe minimum number of entries; then, from this row, the column with the leastnumber of entries is selected. In the event that more than one row or columnwith the same number of entries is encountered, the row or column with thesmallest row or column index is chosen. (This pivot selection technique waschosen because it minimizes the number of columns required in the solutionprocess which, in turn, minimizes storage. In addition, the number of elementsthat must be eliminated is a near minimum so as to minimize the number ofarithmetic operations. This technique appears to be very efficient for thesparce unsymmetrical arrays resulting from the incompressibility conditions.However, the technique may prove to be inefficient for arrays which differ inthe distribution of the non-zero elements).

8. Numerical results

In this section, we cite numerical results obtained from application ofthe program described previously to representative problems in finite elasticity.

We consider here a finite~lement analysis of the same problem it: which thesimplex model shown in Fig. 3, is used. After generating the connected model,generalized forces at the boundary nodes of the interior circle were equatedto zero and those acting at boundary nodes on the exterior circle were prescribedso as to represent a uniformly applied radial load.

Figure 3. - Finite-element model of a sheet containing a circular hole.

Figure 4 indicates the computed vanatIon in the displacements of theinterior and exterior nodes with the applied edge force P acting on one-eighthof the exterior boundary. We observe that the material softens with increasingload; under small loads, the material appears to be rather stiff, the slope Sof the interior curve being around 77 1bs./in., while at larger strains the responseis almost linear and S .....,22 lbs./in. We note that the use of a linearized theoryin this analysis would lead to a displacement of the exterior mode which is only28 percent of the displacement predicted by the nonlinear theory.

Fig. 4. - Variation in nodal displacement with applied edge force.

Figure 5 contains a comparison of the computed displacement profileswith those determined experimentally by Rivlin and Thomas for various values

Sheet with a Circular Hole.

Rivlin and Thomas (37) obtained experimental data on the behavior of athin circular sheet of rubber containing a small, centrally located circularhole, and subjected to finite, axisymmetric stretching in the plane of the sheet.The experiments of Rivlin and Saunders (23) on the same material indicatethat it can be adequately characterized by a Mooney form of the strain energyfunction with material constants C1 = C2/O.08 = 18.35 psi. The experimentsof Rivlin and Thomas (37) concerned a specimen of this material, which, in itsundeformed state, was 5.0in. in diameter, 0.0625 in. thick, and which containeda 1.0 in. diameter hole. Radial loads were applied at the outside edge of thesheet, and the displacement profiles for various radial extension ratios weremeasured.

24

_18

~

o

t-·

~

: ·,~..-pI -

: ~,I--

: ;,/lo-;---

2 3 4EDGE DEFLECTIONS (Inch..,)

6

01 the radial extension ratio A. at r = co. The dimensionless ordinates of thecurves are the ratios of the deformed to undeformed radii while the abscissasare dimensionless distances ria from the center of the hole, r being the radialcoordinate of a point in the undeformed body and a being the undeformed radiusof the central hole. We observe that the values computed using the finite-ele-ment model are in very good agreement with those obtained experimentally.

of the sheet at the hole is less than one-fourth the undeformed thickness.Figure 6 shows the stress concentration factor k versufothe radial extensionratio for the radially-loaded sheet of Mooney material considered here.For small strains, we see that the value of k = 2 corresponding to the lineartheory is obtained, However, for increases in extension ratios k also increases,reaching a value of approximately 6 when Aao = 2.0. This trend is in agreementwith the conclusions of Yang (38) on stress concentrations in Mooney materials.

6.0...... Experimental Results (Rivlin a lllomosl

RAOIAL EXTENSION RATIO A,

Fig.6. - Stress concentration factor as a function of deformation

Uniaxial Stretching of a Sheet with a Circular Hole.

We consider the problem of uniform stretching of a thin, initially square,homogeneous sheet containing a centrally located circular hole. The originof an Xl, x2 material coordinate system is established at the center of the hole,with the Xl and x2 axes parallel to the sides of the sheet, and the sheet is stret-ched in successive stages in the Xl - direction while its lenght in the x2 -

direction is held constant.This problem is motivated by the experiments of Segal and Klosner (39)

on elastomeric sheets for which the strain energy function deviated substan-tially from the Mooney form. After a series of tests on samples of a certainincompressible elastomer (26), they proposed the form of the strain energyfunction, cubic in 12, given by (2.21). Thus. the strain energy function involvesfour material constants. Approximate values of these constants of CI = 20.28psi. C2 = 5.808 psi. C3 = - 0.7200 pis, and C4 = 0.04596 psi were determinedfrom experiments on samples of a natural rubber. Tests were then run to deter-mine the displacement field and the deformed shapes of 6.5 in. square specimenof this material, 0.079 in. thick, containing a 0.5 in. diameter circular hole. Thespecimen was subjected to the program of uniaxial extension described above.

A finite-element model of the sheet is shown in Fig. 7.a. Its analysis wascarried out using twenty load increments, the jacobian JIj of (7.11) beingevaluated at the beginning of each increment using .:\Xj = 0.0001. The analysisinvolved 199simultaneous nonlinear equations, each containing terms of 12th-degree in the nodal displacements. Their solution required approximately 27minutes of on the UNIVAC 1108 computer.

The deformed finite-element netwotk calculated at Al = 2 is shown inFig. 7.b. We observe that the circular hole is stretched in an elliptical shape andthat lines, originally radial from the hole in the first octant of the sheet,flatten, but acquire a very little curvature. An enlarged view of the deformedshape of a quarter of the hole is shown in Fig. 8, wherein the calculated profilesare compared with those determined experimentally by Segal and Klosner (39)for extension ratios Ax. = Al = 1.0, 1.36, and 1.46. Agreement between compu-ted and measured deformations is excellent. Figure 9 contains a comparison ofthe calculated variation in displacements along the x I - and x2 - axes with the

109.080

30

1,0

,-~

- Finite Element

~,..1-1-,

zo

-----~.:-,~

1.0

~ 70

~ 60.,r1.z 50Q

!:i~ 40

~30

LO 1.4 IS 2.2 2.6 3.0 34

ria

Fig. 5. - Comparison of computed displacements with those determined experimentallyby Rivlin and Thomas (37).

For the geometry and loading in this example, the linear theory of elasticityyields a stress concentration factor of 2 [i.e. crll (CO)O'22 (a) = 2]. Tnthe generalcase considered here, however. the stress concentration factor is stronglydependent on the deformation and the character of the material. One factorcontributing to the change in stress concentration is, of course, the significantdecrease in thicknses of the sheet when subjected to large extensional strains;for example, when radial strains reach approximately 150percent, the thickness

2.4

1l29~d

.~. 1.2T"_~

11.!l6 eon.>uled

X· 1.~4 Measured

)(2/a

2.01.6

U,Ia

-<>--~~ Finite Element

Experimental Results(Segal a Klosner)

x.,-1.00

0.8 1.2I

X/a

A-ute

- __ - Fifti,.. a....t

- Ekpwimentol

"*""" ..IS~oll!l KJot.n.,)

0.4

2481111012

0.0

1.6

1.2

0.4

2X 14

0,2

0.8

••,.••'0..

":fa u

Fig. 8. - Comparison of computed shapes of deformed hole with shapes determined experi-mentally by Segal and Klosncr (39).

Fig. 9. - Comparison of computed displacements with those determined experimentally bySegal and Klosncr (39).

Bending and Inflation of a Circular Plate.

An interesting finite-element formulation for application of the axisym-metric-solids concerns the quasi-static behavior of a simply-supported, flat,circular plate subjected to a piecewise linear varying external pressure of the

r.:

I. Ab .1I.. b .1

undeformed boundary

Fig. 7. - Undeformed and deformed finite-element model of finite uniaxial stretching of asquare sheet containing a circular hole.

variations measured by Segal and Klosner for various values of the longitudinalextension ratio 1.1' Again, good agreement is obtained. Some differences occurbetween the calculated and measured vertical displacements of nodes on thex2-axis at low values of 1.1, Our computed displacements appear to be. 01 in.larger than those measured experimentally; some differences, of course, canbe expected since all computed and measured extension ratios did not coincideexactly and at low extension ratios, less accuracy in experimentally determi-ned displacements can be expected.

Fig. 10.

(a) Load historyon an initiallyflat circularplate and(b) a finite elementmodelof the plate.

IB 0 ,

A C II.)

_-_..0LDIVEII<lEN:E

llU£ TO MOOELDETERIORATiON

~ ro ~ ~ ~DISPLACEMENT OF CENTER (inches)

'? soIt....~w

~tl~~

10

Fig.11.- Variationinthetransversedeflectionof thecenteroftheplatewithappliedpressure.

e.~-;:::;;-,It)

10 20 30 .0 !lO eo 70 80time t

(al

!lO

10

.0

j~30

'""'20f

type indicated in Fig. 10. a. The undeformed plate is 5 in. in diameter, 0.5 in.thick, and is constructed of an isotropic, incompressible material of the Mooneytype with material constants of C1 = 80 psi, C2 = 20 psi. The finite-elementmodel of the plate is shown in Fig. 10.b; note that the thickness is exaggeratedin this figure for clarity. The analysis was performed using the incrementalloading procedure described previously; the external pressure was appliedin increments (corresponding to one unit of time in Fig. lO.a) from 0 psi to apeak value of 43.7 psi. We remark that the applied loads in this problem arenonconservative; element loading surfaces change markedly in magnitude andorientation with increases in applied pressure. Unloading of the plate followedthe same loading history in reverse.

A number of preliminary analyses of the problem revealed an extremesensitivity of the predicted response to the choice of load increment size.This is illustrated in Fig. 11 wherein the variation of the transverse deflectionof the center of the plate with internal pressure is shown. We observe that theresponse quickly departs from that predicted by the linear theory after thepressure reaches only 1psi; the curve then swings upward, being approximatelylinear from 3 to 28 psi, and then acquires a gradually decreasing slope forincreases in applied pressure. Since it is difficult, if not impossible, to anticipatethis type of behavior prior to selecting a load increment size, several trial solu-tions using coarse finite-element models were obtained using various constantload increment sizes. The results of these analyses are indicated by the dashed

lines in Fig. 11; typically, the results of too large a load increment are charac-terized by a large initial displacement. followed by essentially no increasein the displacements of certain nodes with an indefinitely large increase inpressure. Also, local equilibrium of element forces is violated and the deformedfinite-element network assumes unrealistic distortions. In this particular pro-blem, it was necessary to use at least 30 load increments to depict the responsecorresponding to an external pressure of only 3 psi; an additional externalpresssure of 40 psi was then applied in 25 increments.

It was impossible to predict the response of the plate beyond 42 psi withoutmodifying the finite-element model. This was due to the incompressibility of theelements : at strains on the order of 500 percent. the cross-sectional areas ofcertain finite elements shrink to nearly zero in order to maintain a constantvolume of the ring element. This results in poor conditioning of the jacobianmatrices and, ultimately, divergence of the iterative solution process. Toovercome this difficulty, it is necessary to stop the incremental loading processprior to such instabilities and to construct a new, refined finite-element modelof the deformed body. Starting values of the nodal displacements of the newmodel can be obtained from the deformed shape of the initial model using linearor quadratic interpolation. If such a procedure is employed, it is advisableto perfom several cycles of Newton- Raphson iteration to further correct nodaldisplacements and hydrostatic pressures prior to returning to the incrementalloading process.

The assumed incompressibility of the material also leads to other compu-tational difficulties in problems of this type. For example, suppose edge ABin Fig. 10. b is clamped rather than simply supported and that the plate issubjected to essentially the same program of external pressure indicated in Fig.10.a. Under small increases in pressure, the nodes along, say, CD will displaceapproximately vertically, so that all elements in the rectangle ABCD will beapproximately in a state of homogeneous, simple shear. This being the case,any value of the vertical displacement of nodes along CD will lead to isochoric

P=42.0

p= 35.0

P=31.0

P= 25.0

p= 14.0

p= 7.70

P=3.0P=1.5

Fig.12.- Deformed shapes of the finite-element model of the circular plate for various valuesof external pressure.

(volume preserving) deformations. Consequently, the system of nonlinearequations (equilibrium and incompressibility conditions) will then not possess aunique solution. While it is possible to reformulate the equations in such casesso as to obtain a determinate system, it is generally more practical to use analternate finite-element network, typically of an irregular pattern, at theboundaries to avoid such singularities.

The computed deformed shapes of the finite element model of the platefor various values of applied pressure are shown in Fig. 12. In this particularproblem, linear theory appears to be adequate for only around eight 0,1 psiload increments. At pressures of around 0.8-1.0, extensional strains are stillsmall but rotations begin to be appreciable. Strains appear to remain small(less that 5 percent) until the external pressure reaches a value of approximately4.0 psi. At this level of pressure, rotations of certain line elements reach valuesof 50 degrees. Shear deformations and transverse strains at this level of loadingare not appreciable; straight lines originally normal to the underformed middlesurface are still approximately straight and normal to the deformed middlesurface; i.e., the Kirchhoff-Love hypotheses appear to hold for relativelylarge deflections. Results indicate that nonlinear plate theories of the VonKarman type (i.e., nonlinear theories based on the Kirchhoff hypothesis whichassume moderate rotations but infinitesimal strains) should be adequate forpressures up to around 4.0 psi. After psi. rotations become large, and at.6.0-8.0 psi extensional strains of the middle surface attain values of approxima-tely 20 percent. At this load level, membrane action begins to take a dominantrole in the behavior of the plate; i.e., the plate begins to inflate rather than bend.At 25 psi, transverse extensional strains reach -20 percent and the rotationsof certain line elements exceed 90 degrees. At a peak pressure of 42 psi, strainsof the order of 500 percent are obtained, line elements near the support havesuffered rotations of 128 degrees, the thickness of the plate has shrunk from0.5 inches to approximately 0.15 inches, and the plate assumes the considerablydistorted shape indicated in the figure.

A Plane Strain Problem.

As a final example, we describe briefly numerical results obtained byapplying the theory and computer program described previously to the pro-blem of finite plane strain of an incompressible, square specimen of Mooney-type rubber containing a centrally located circular hole. The specimen is clampedalong two opposite edges, the remaining edges being free, and is subjectedto a concentrated force at the center of the bottom free edge as indicated inFig. 13. Dimensions of the undeformed plate are 10 in. x 10 in., the centralhole being initially 4.0 in. in diameter, and the material constants of the platewere assumed to be C1 = 80 psi, Cz = 20 psi. The finite-element model

p

PFig. 13. - Finite-element model of a plane, incompressible body subjected to a concentrated

edge force.

indicated in Fig. 13was used in the analysis, and an external load of 2000 poundswas applied in twenty 50-pound increments. The deformed shape of the finite-element model at various values of P is shown in Fig. 14, and the final deformed

p-=P'I!lOOP'IOOOp·eoop·o

l''l.''c'''c'i':o'l:.\\ + "1\~,. J\:~~:::::;.f/i

'./

.t'''':::~~::>;7'~'~Fig. 14. - Computed deformed profiles for various values of applied force.

shape at P = 2000 pounds is shown to scale in Fig. 15. As expected, strainsingularities at the point of application of the load are evident. Strains on theorder of 50 percent are developed and the principal «diameter» of the dis-

Fig. 15. - Deformed shape of the finite-element model at a load P of 2000 pounds.

torted, heart-shaped, hole reached 5 inches. The analysis required 31 minutesof UNIVAC 1108 computing time.

Acknowledgements.

Support of this work by the United States Air Force Office of ScientificResearch through Contract F44620-69-C-0124 is gratefully acknowledged.We also wish to express gratitude to Miss Marion Smith and Mr. J.N. Reddy forassistance in the preparation of certain subroutines in our computer program,

9. References

(1) R.S. RNLIN : «Large Elastic Deformations of Isotropic Materials, I. FundamentalConcepts - N. Further Developments of the General Theory ». Philosophical Transac-tions of the Royal Society. Vol. A-240, pp. 459-490, 491,508, 509-525, Vol. A-241 ,pp. 379-397, 1958.

(2) J.E. ADKINS and R.S. RIVLIN : « Large Ela~tic Deformations ofIsotropic Materials-X.Reinforcement by Inextensible Cords ». Philosophical Transactions of the Royal Society,Vol. A-248, pp. 201-223, 1955.

(3) A.E. GREEN and J.E. ADKINS: Large Elastic Deformations, Oxford UniversityPress, Oxford, 1960.

(4) C.A. TRUESDELL and W. NOLL : The Non-Linear Field Theories of Mechanics.Encyclopedia of Physics, (ed. S. F1iigge Springer-Verlag, Berlin-Heidelbery-New York,1965.

(S) A.C. ERINGEN : Mechanics of Continua, John Wiley and Sons, New York, 1967.

(6) J.T. ODEN : «Analysis of Large Defonnations of Elastic Membranes by the FiniteElement Method ». Proceedings, lASS Congress on Large-Span Shells. Leningrad, 1966.

(7) J.T. ODEN and T. SATO : « Finite Defonnations of Elastic Membranes by the FiniteElement Method ». International Journal of Solids and Structures, Vol. 1, pp. 471-488,1967.

(8) J.T. ODEN and T. SATO : « Structural Analysis of Acrodynamic Deceleration Systems ».Advances in the Astronautical Sciences, Vol. 24, pp. PS6-3-PS6-19, 1967.

(9) J.T. ODEN and W.K. KUBITZA : « Numerical Analysis of Nonlinear PneumaticStructures ». Proceedings, Colloquium on Pneumatic Structures, Stuttgart, pp. 82-107,1967.

(10) E.B. BECKER: « A Numerical Solution of a Class of Problems of Finite Elastic Defor-mation ». Ph. D. Thesis, University of California, Berkeley, 1966.

(11) F.E. PETERSON, D.M. CAMPBELL and L.R. HERMANN : « Nonlinear PlaneStress Applicable to Solid Propellant Rocket Grains ». Bul/etin, 5th Meeting of theICPRG Working Group on Mechanical Behavior, CPIA Publication No. 119, Vol. 1,The Johns Hopkins University Applied Physics Laboratory, pp. 421-455, 1966.

(12) J.T. ODEN : « Finite Plane Strain of Incompressible Elastic Solids by the Finite ElementMethod ». Aeronautical Quarterly. Vol. XIX, pp. 254-264, August, 1968.

(13) J.T. ODEN and J.E. KEY: « Numerical Analysis of Finite Asixymmetric Deformationsof Incompressible Elastic Solids of Revolution ». International Journal of Solids andStructures, VoL 6, pp. 497-518, 1970.

(14) J.T. ODEN : « Numerical Formulation of Nonlinear Elasticity Problems ». Journal ofthe Structural Division, ASCE, Vol. 93, No. ST3, pp. 235-255, Junc 1967.

(15) S. NEMAT-NASSER and H.D. SHATOFF : « A Consistent Numerical Method for theSolution of Nonlinear Elasticity Problems at Finite Strains ». Technical Report-Dept.of Aerospace and Mechanical Engineering Sciences, No.2, University of California,San Diego, 1970.

(16) K.T. KAVANAUGH: « The Finite Element Analysis of Physically and KinematicallyNonlinear Elastic Solids ». Ph.D. Thesis, University of California, Berkeley, 1969.

(17) J.T. ODEN : « Finite Element Applications in Nonlinear Structural Analysis ». Procee-dings of the Symposium on Application of Finite Element Methods in Civil Engineering,Vanderbilt University, pp. 419-456, 1969.

(18) H.J. BRAUCHLI and J.T. ODEN : « Conjugatc Approximation Functions in FiniteElement Analysis ». Quarterly of Applied Mathematics (in press).

(19) T.J.R. HUGHES and H. ALLlK. : « Finite Elements for Compressible and Incompress-ible Media ». Proceedings of the Symposium on Applications of Finite Elements in CivilEngineering, Vanderbilt University. pp. 27-62, 1969.

(20) A.E. GREEN and W. ZERNA : Theoretical Elasticity, Second Edition, University ofOxford Press, Oxford, 1968.

(21) M. MOONEY: « A Theory of Large Elastic Deformation ». Journal of Applied Physics,VoL 11, pp. 582-592. 1940.

(22) L.R.G. TRELOAR: The Physics of Rubber Elasticity, Second Ed., Oxford UniversityPress. Oxford, 1958.

(23) R.S. RlVLIN and D.W. SAUNDERS: « Large Elastic Deformations of IsotropicMaterials, VII. Experiments on the Deformation of Rubber ». Philosophical Transac-tions of the Royal Society, VoL A243, pp. 251·288, 1951.

(2<1) A. ISIHARA, N. HASHlTSUME and M. TA TIBANA : « Statistical Theory of Rubber-Like Elasticity - IV (Two-dimensional Stretching) ». Journal of Chemical Physics, Vol.19, pp. 1508-1512, 1951.

(25) V.L. BIDERMAN : « Calculation of Rubber Parts (in Russian) ». Rascheti naProchnost,Moscow, 1958.

(26) J.M. KLOSNER and A. SEGAL: « Mechanical Characterization of a Natural Rubber ».PIBAL Report, No. 68-42, Polytechnic Institute of Brooklyn, New York, 1969.

(27) A.N. GENT and A.G. THOMAS : « Forms of Stored (Strain) Energy Function forVulcanized Rubber ». Journal of Polymer Science, Vol. 28, pp. 625-628.

(28) L.J. HART-SMITH : « Elasticity Parameters for Finite Deformations of Rubber-LikeMaterials ». Zeitschrift fur Angewandte Mathematik und Physik, Vol. 17, pp. 608-625,1966.

(29) H. ALEXANDER : «A Constitutive Relation for Rubber-Like Materials ». Interna-tional Journal of Engineering Science, Vol. 6, No.9, pp. 549-563, 1968.

(30) J.T. ODEN : «A General Theory of Finite Elements. I Topological Considerations ».International Journal of Numerical Metlwds in Engineering, Vol. 1, No.2, pp. 205-221,1969.

(31) J.T. ODEN and H.J. BRAUCHLI : « On the Calculation of Consistent Stress Distri-butions in Finite Element Applications ». International Journal of Numerical Methodsin Engineering, (to appear).

(32) J.T. ODEN and G. AQUIRRE-RAMIREZ: « Formulation of General Discrete Modelsof the Thermomechanical Bchavior of Materials with Mcmory ». International Journalof Solids and Structures, Vol. 5, pp. 1077-1093. 1969.

(33) J.T. ODEN : «Finite-Element Formulation of Problems of Finite Deformation andIrreversible Thermodynamics of Nonlinear Continua ». Proceedings. U.S.-Japan Seminaron Matrix Methods in Structural Analysis alld Desigll (ed. R.H. Gallagher, Y. Yamada.and J.T. Oden) University of Alabama Press. Tuscaloosa. (in press).

(34) J.T. ODEN : «A Generalization of the Finite Element Concept and its Application to aOass of Problems in Nonlinear Viscoelasticity ». Developments in Theoretical and AppliedMechanics (proceedings Fourth SEcrAM, New Orleans, 1968), Vol. 4, pp. 581-593,Pergamon Press, Oxford, 1970.

(35) J.T. ODEN : « An Approximate Method for Computing Nonconservative GeneralizedForces on Finitely Deformed Finite Elements ». (in review).

(36) I. ERGATOUDIS, B.M. IRONS ans O.C. ZIENKIEWIEZ : « Curved, Isoparametric,,Quadrilateral' Elements for Finite Element Analysis ». International Journal of Solidsand Structures, Vol. 4, pp. 31-42, 1968.

(37) R.S. RIVLIN and A.G. THOMAS: « Large Elastic Deformations of Isotropic MaterialsVIII. Strain Distribution Around a Hole in a Sheet ». Philosophical Transactions of theRoyal Society, Vol A243. pp. 389-398, 1%1.

(38) W.H. YANG : «Stress Concentration in a Rubber Sheet Under Axially SymmetricStretching ». Journal of Applied Mechanics, Vol. 34, pp. 942-946. 1967.

(39) A. SEGAL and J.M. KLOSNER : « Stress Concentration in an Elastomeric SheetSubject to Large Deformations». PIBAL Report, No. 70-11, Polytechnic Institute ofBrooklyn, New York, 1970.