Reprillled [romDEVELOPMENTS IN THEORETICAL AND APPLIED MECHANICS

Volume 8Edited by

DANIEL FREDERICKPERGAMON PRESS - OXFORD & NEW YORK - 1970

A GENERALIZATION OF THE FINITE ELEMENT CONCEPTAND ITS APPLICATION TO A CLASS OF PROBLEMS IN

NONLINEAR VISCOELASTICITY

J. T. ODEN

University of Alabama at Huntsville

ABSTRACT

A generalization of the finite element concept is presented which leads to the construction of discrete modelsfor studying the behavior of nonlinear continua. The notion of topological simplexes is used to obtain discreterepresentations of continuous vector fields. Invariant forms of the equations of motion of finite elementsof nonlinear continua are derived from energy balances associated with isothermal deformations. Specificforms of these equations for phenomenologically simple materials. which include a large class of nonlinearviscoelastic materials. are examined. It is shown that the equations of the discrete model are exact for certainhomogeneous deformations. Simple shear of a nonlinearly viscoelastic cuboid is investigated.

NOTATION

Upper case indices indicate nodal points in space and ranges from 1 to k + 1. k beingthe dimension of the space. Lower case indices indicate components of vectors and tensorsor elements in an array. The summation convention applies to all repeated indices unlessnoted otherwise. Pertinent symbols used are listed as follows:'

bRij

CmN

EeFi.F;gi' giKi(t - (, ... )kN

MHM• mN.\/

mn

PMi

p(t)

Multi-dimensional arrayCoefficients in simplex modelNumber of finite elementsComponents of body forceBase vectors associated with initial coordinatesHeredity kernelsCoefficients in simplex modelsConsistent mass matricesNumber of nodes in global systemGlobal components of generalized force at node MHydrostatic pressureLocal components of generalized force at node N of element eComponents of surface tractionsInternal energyGlobal components of generalized displacement velocity, and accelerationat node MLocal components of generalized displacement, velocity. and accelerationat node N of element eGlobal components of a vector field at node MLocal components of a vector field at node N of element e

581

582

Xi

r;k}'ijLl(t)

(NMRS' (ilk,y.

p·POqij

f

t/lNQ

QNMe~ij. (£i)

J. T. aDEN

Convected coordinatesChristoffel symbol of the second kindStrain tensorPrescribed displacementPermutation symbolsTime variableKinetic energyMass densitiesStress tensorVolume of finite elementNodal function in simplicial representationsPower of external forcesMapping functionTensor functionals

INTRODUCTION

Finite element procedures represent continua by discrete models conslstmg of anassembly of a finite number of elements of finite dimensions. Local equations of motioncan be obtained which describe the behavior of each finite element, indcpendent of itslocation or mode of connection in the model. These local equations also involve only afinite number of dcpendent variablcs, and their relation to the global representation isobtaincd through simple transformations. This makes it possible to examinc the behaviorof typical finite elemcnts independently, and to then connect them so as to representcontinuous bodies of arbitrary shape with arbitrary boundary conditions.

During the last dccade, the finite element mcthod has been applied successfully tonumerous problems in linear elasticity as well as to certain nonlinear structural problcms.Examples of these applications, with references to earlier work. can be found in the book byZicnkiewicz and Cheung [1]. Recently. extensions of the mcthod to the analysis of finitedeformations of three-dimensional elastic solids [2]. elastic membrancs [3. 4, 5. 6] andviscoelastic solids [7, 8. 9] have also been presented,

In the present paper, a generalization of the finite element concept is presented whichis applicable to the analysis of finite deformations of a large class of nonlincar continuaThe concept of a topological simplex is shown to lead systematically to finite element modelsof continuous vector and scalar fields, which arc valid for any choice of initial coordinates.Physically, the concept is related to the observation that the deformation of a small finiteregion of continuous body differs from a state of homogeneous deformation only by anamount dependent on the dimensions of the region itself Indeed, it is shown that in the caseof homogeneous deformations, the finite element representation yields exact solutions.General equations of motion for isothermal deformations of finitc clcments of continuousbodics arc dcrivcd from energy considcrations. and applications to constitutively simplematcrials. including nonlincar viscoelastic materials with memory. are bricOy cxamincd.

SIMPLICIAL REPRESENTATIONS OF VECTOR FIELDS

In thc following, we review briefly the notions of simplicial representations of vcctorfields p, 10. III and extend these so as to apply fields with components referrcd to a general

A GEl'lERALIZA nON OF THE FINITE ELEMENT CONCEPT 583

systcm of curvilinear coordinates. Specifically. we are to construct finite element modelswhich depict a given field within the element as a topological simplex.t Thus, for one-dimensional problems the finite element is a line element connecting two nodes: for two-dimensional problems "curvilinear" triangular elements with three nodes are used:for three-dimensional problems, tetrahedral elements with four nodes, one at each vertex.are uscd: etc.

Consider a region ~ in (say) three-dimensional space which we represent by a discretemodel consisting of a total of Itln nodes and Ee finite elements. Let xi(i = l. 2. 3) denote asystem of curvilincar coordinates embedded in 9t with covariant and contravariant basevectors gi and gi. A typical finite element e is then a curvilinear tetrahedral subregion of ,?,f

with four nodes. one corresponding to each vertex of the tetrahedron (see Fig. 1).

FtG. I. Curvilinear finite element for three-dimensional space.

Let V be a continuous vcctor field ovcr?J. with componcnts l'i and Vi with respect to thcbase vectors gi and gi: i.e.

(1)

Thc portion of V defined throughout clement e is denoted Ve and is referred to as the localfield corresponding to element e. In the discrete model of gJ. thc finitc point sets

t A simplex in k-dimensional space is a convex set S determined by a collection of k + I vertices (nodes)"l' "2" ... 111+ I which do not lie in the (k - [) - dimensional hyperplane. and S consists of all points y such that

HI

.r = L Ci"ii~ 1

for which

C, ~ 0 andHI

L Ci= Ii= 1

The coefficients c, are generally linear in coordinates x,{i = I. .... k~ If the Xi are cartesian and k = 1, the simplexis a straight line connecting II, and "2' For k = 2. a simplex consists of the points within a triangle. For k = 3.a simplex consists of points within a tetrahedron: etc. See, for example. Graves [10].

Sygne [12] refers to simplical representations of scalar fields as "polyhedral functions". and shows that theseapproximate a function II with the properties: (1) 1/ is cominuous. (2) 1/. i are continuous. (3) u. iJ are bounded withinthe simplex. "as closely as we like by taking a sufficiently fine triangulation".

584 J. T. aDEN

i = 1. 2, 3 (2)

where J.-:\fi and V:\I are. respectively. the covariant and contravariant components of Vat node 1\;1. define the global representation of V. Thc finite point sets

\'e -+ {VNi(e)} or {v~(e)} N = 1. 2. 3,4: i = 1. 2. 3:

e = 1. 2..... Ee (3)

where VNie and v~e are. respectively. the covariant and contravariant components of thelocal fielas at node N of element e. define the local representations of V. The local sets arcrepresented by simplicial models for each element and arc temporarily regarded as beingindependent of the global sets.

Considering now a typical finitc element e isolated from gf., the local field is approximatedby

(4)

in which(Sa. b)

Herc the repeated node index N is to be summed throughout its range (N = 1. 2, 3. 4 forthree-dimcnsional e1cmcnts). The functions t/IN are defined by

(6)

whcrc

(7a)

(7b)

(7c)i=l:j

{

"i"R

bRij = 1 i=j

Here (NRST and [ijk (N. R. S. T = I. 2, 3. 4: i. j. k = 1. 2, 3) are the four-dimensional andthree-dimensional permutation symbols. x~. are the curvilinear coordinates of node Nof the element.. and Vo is determinant of the four order matrixt

xl x2 x3. 1 · 1

x~ x~ x3

Vo = 11 • 2xj x~ x3

· 3

xi x~ xl

(7d)

Note that the functions t/IN have the properties4

L t/lN = 1N=l

(8a, b)

throughout the finite element.

-t If the x' are cartesian coordinates. and if the nodes are numbered so as to form a right-handed sequence.then 1'0 can be taken as simply the volume of the finite element.

A GENERALIZATION OF THE FINITE ELEMENT CONCEPT 585

At this point. the local and global sets {UNi(e)} and {VMi} (or {V:~'Ie)} and {V~t}) are inde-pendent. The conncctivity of the discrete modcl is established by mapping {VNi} onto{VNi(e'} by the transformations [2. 7. 11. 13]

whcrein

[:Nie = QN.llcV\/i

if node N of clemcnt e coincides with node M of thc global systcm

if otherwise

(9)

(to)

Herc N = 1.2.3.4: i = 1.2, 3: e = 1.2 ..... Ee: and M = 1.2 ..... nln. The transformationEq. (9) is refcrrcd to as an incidence relation. The function QN.\te establishes depcndcnciesbetwecn {VNilrl} and {V\ti} which. in elTect. connect all of the elements togcther in thc appro-priate manner so as to form a single discrete model of the field V over gp.

Similarly, for the contravariant components, we have

(tI)

(t2)

Equations (5). (9) and (II) define the finite element representation of the vector field V.These relations also hold for one and two-dimensional fields if the ranges of the indicesare appropriately rcdefined. Considering. for example. Eq. (9): for two-dimensional fieldsN = 1. 2. 3 and i = 1, 2: for one-dimensional fields, N = 1. 2: and i = 1.

MOTION OF A FINITE ELEMENT

We now considcr the motion of a typical finite e1emcnt independent of the bchavior ofother elements in thc model. The element identification index is temporarily dropped forsimplicity.

For a given finite elemcnt, we postulate a law of conservation of energy for isothermaldeformations which. in a global sense. is of the form [14]

DDt(Y. + U) = Q

Herc D/Dt indicates the material derivativc. Yo is the kinetic energy of thc elemcnt U isthe internal encrgy. and Q is the power of the external forces. That is.

U = Splilpo df,Q = SpFil~i df + SSil!ids

t •

(l3a)

(t3b)

(t3c)

in which p and Po are mass densities in the deformed and undeformed element, Vi and Vi

are vclocity components, f is the element volume. 'I is the internal energy per unit of unde-formed vol ume, Fi are components of body force per unit mass. and Si are components of

586 J. T. aDEN

(14)

surfacc traction per unit surface area s. It can be shown [14. 15] that Eq. (12) implies alocal energy balance of the form

Dtl Po ij DYij-=-(1 -Dt p Dr

in which (1ij is the stress tensor per unit of dcformcd area referred to the convected coordi-nate lincs Xi. and Yi) is the strain tensor. Thus

1f DViVi J" Di'ij J . J'- p-df + (1'J-df = flF'L"df+ S·,,'ds2 Dr Dr ' •

T j l' S

(IS)

In what follows, we follow the general plan of [14] and refcr thc displacemcnt. vclocity.and acceleration fields to base vectors gi and gi of the coordinates .-t at a time to corre-sponding to the initial. undcformed state of the element The coordinatcs Xi move with thebody as it deforms and are. therefore. convected coordinates, Moreover. for this choice ofreferencc. the quantities Ii) in Eq. (14) are components of Grcen's strain tensor (or. theLagrangian strain tensor) and are given in terms of the displacement vector u by

(16)

where the commas indicate partial differentiation with respect to the Xi. From the symmetryof (1ij it thcn follows that

.. DYij _ (Jij(o. + u) .U'i(1'J- - >-OJDr

(17)

in which the superposed dot indicates ordinary partial differentiation with rcspect to time.We now return to the simplicial representations recorded in thc previous section. In

view of Eqs. (5). the displacement. velocity. and acceleration fields associated with thefinite elcmcnt under consideration arc given by

u = t/J,,"INigi = t/JNII~gi

V = U = t/JNUNigi = t/JNli~gi

ii = t/JNiiNigi = t/JNU: ...gi

(IRa)

(18b)

(l8c)

where t/J... is deli ned in Eq. (6) and UNi' UNi, ii."li (u~, li~, u~) are the covariant (contravariant)componcnts of displacement. velocity, and acceleration at node N.

Noting that. for example.

(19)

where CiN is delined in Eq. (7b) and r~iare the Christoffel symbols of the sccond kind for theframc Xi, wc introduce Eqs. (17) and (18) into (15) and lind that for a typical linitc e1emcnt

.. • m [j' ij(. ./. rr )d f ij( "kmN,HuMmu,\' + (1 ciNYjm + 'l'N migrj f + U.\/k (J cj.\/ciN"mr r

./. rk ./. rk, ./ •• /. rk rr )d ] 'm _ 'm- 'I'M mJ.ciN + 'I' N n:i'j.\f - 'I' N'I' M jr mi f liN - PNml'N (20a)

A GENERALIZA nON OF THE FINITE ELEMENT CONCEPT

in which gjm = gj . gm is the covariant metric tensor.

mNJI == Jpt/l"t/lJI dtr

andP""m == JpF mt/l ....df + JSml/tN ds

r s

587

(20b)

The array mNM is the consistentt mass matrix and PNm are the covariant components ofthe generalized force at node N. Since Eq. (20a) holds for arbitrary node velocities. we have

(20c)

(21)

Equation (21) represents the general equations of motion for isothermal deformations ofa finite element. If the coordinates Xi are initially cartesian, the strait) tensor is uniformthroughout the element. If, in addition. the element is homogeneous.t then Eq. (21) reducesto [5. 7]

(22)

in which f is the volume of the undeformed element.The above relations apply to a single finite element. To obtain the equations of motion

of the entire discrete model (the global system). we refer to the transformations indicatedin Eqs. (9) and (11). Let VMi andU Mi denote global values of the acceleration and velocityand let P.\/i denote global values of the generalized force (i.e. PMi are the covariant com-ponents of generalized force at node M of the assembly of finite elements). Then

iiNi(~) = QNM~t) Mi

Further. because of the invariance of Q.

with N = 1. 2. 3. 4: M = 1. 2, ... , Inn. we observe thatE,

P.\/i = L Q,.,..\/ePNi(e)~= 1

(23a, b)

(24)

(25)

Introducing Eqs. (21) and (23) into (25), we arrive at the global equations of motion for theentire assembly of finite elements:

wherein

MK1.t) l.i + F Ki + GiLP Lk = PKi

E".M K/. == L QNKelnNM(e)Q.\fL~

e

(26)

(27a)

t Consistent. in that the mass is distributed to each node in a manner which is consistent with the simplicialapproximation of the lields iJ and ii.

t By taking the dimensions of the element sufficiently small. it is permissible to treat each element as homo-geneous. even though the global system may be non·homogeneous. Non-homogeneity is then accounted for byassigning different material properties to each finite element.

588 J. T. aDEN

Ee

Fli.i == 'LQNIi."JqmjCmN(gji + t/lNr~m{/rj)df (27b)e '.

E.

G~Li == 'L QNIi..[J qmj(Cj.\/cmNc57 - r~it/l.W:mN + rt"CjMt/lN - r',rimt/l Nt/I.\/) df J Q.\/Le (27c)e tL'

Here Te is the volume of element e: K. L = 1,2, ... ,11ln: N, M = ].2, 3. 4: e = 1,2, ... , Ee:and ;.}, k, m, r = l. 2. 3.

Equation (26) completes the formulation of the equations of motion of the discrete modelof the continuum. Notc that no restrictions have been placed on the order of magnitude ofthe strains or displaccments. To apply thesc rcsults to specific materials, constitutiveequations must bc introduced so that the stress tcnsor can be eliminated from the equationsof motion'of the discretc Inodel. '"

NONLINEAR VISCOELASTIC MATERIALS

Although Eqs. (21) and (26) are valid for a broader class of materials. for illustrationpurposes. we confine our attention to simple materials for which the constitutive equationsare of the form [16-20].

(28)-00

in which tyij arc tcnsor functionals and' is a time parameter. Since, for the discrete modclof the continuum,

we have for a finite element

where

qiJ = (£ij[U~i mJ-'"

(29)

(30a)

(30b)

Thus, the equations of motion for a finite element of this type of material are of the form

mN.\fUMn + J(Ci ...iJjm + t/lNr~Srj) (£/j[U~i (mdf+ U.\fkJ(CjMCi,v<5~r -00 t

- r~njCiNt/I.\/ + r~iCjMt/I.\f - t/lNt/lMrV":..;) (£:ij[U~i mJ dr = PNm(l) (31)-00

Equations (28) and (31) apply to a large class of matcrials and numerous special cases of(31) can be considered. If l'Yij (or (£ij) satisfies certain continuity conditions [16-18]. then,for materials of the hcrditary type, the constitutivc functionals can be approximatcd byfinite sums of multiple integrals by the Frechet expansion

r I

Fij = f fn ••

N

dij~ L F~

n=t

I ,J Kilr,., ... r"." (c - ~I' t - '2'···. t - ,.) I'" ••('d 1r,.,('2)

(32a)

-00-00 -CCl

A GENERALIZATION OF THE FINITE ELEMENT CONCEPT 589

wherein the kernels Kijr,s, ... r"s" arc invariant functions of the strain history. Assumingthat the material is isotropic in its initial state, the above kernels are expressible as poly-nomials in integral invariants of the strains. Then Eq. (32) can be written in the form[21-23]

I

g;ij = HDijK1(f - 'dYmm(O + K2(1 - OgimgjnYm.(O]d'0, I.

+ J J {DijK3(c - '1' C - (2) Ymm(' th'nn('2) + ~ijK4(C - C, t - '2)Ymn(' 1)1::'('2)00'

+ Ks(/ - '1.1 - '2) Ymm('t) yij('2) + K6(t - 'I- t - (2) yU"('t) tm('2)} d'l d'2 + ...(33)

It is now possible to obtain approximate constitutive equations for a number of specialtypes of materials or deformations (e.g. small, short. or slow motions, etc.) by using the firstfew terms of Eq. (33) and by introducing additional restrictions on the kernels Ki and thestrain rates. Various approximate constitutive equations derived from expansions of thetype in Eq. (33) are discussed elsewhere [21. 23,24]. Whatever the form of g;ij, the equationsof motion for a finite element of the matcrial it characterizes are obtained by simply cal-culating (£ij with the aid of Eq. (30b) and incorporating the result into Eq. (31).

For future reference, we record here only one such approximation, it being understoodthat the same procedure ap~lies to any other approximation we care to examine. Specifically.for incompressible materials,

(34)

where p(c) is an arbitrary hydrostatic pressure, Gij is the contravariant metric tensor in thedeformed body (i.e. Gij is the inverse of g ij + 2Yij), and sij is a tensor functional of the typein Eq. (32b) or (33). Further, if only the first two terms of Eq. (33) are used (second-ordertheory) and if the material is undisturbed prior to t = O. it can be shown [25] that for shorttime ranges,

I

sij::::: KI(O)yij(r) - f :,K1(C - ')yij(Od' + K2(0, 0) yim(t) Im(t)o

I- f :,K2(t - ,. 0) [1t~(O r!..(/) tn,(O] d( (35)o

Thus. for the finite element.

mNMi~,\/m + J(ci,v{/jm + t/lNr .....;{Jr) (sij + p(t) Gij) dT + UMKJ(CjMCi~~r t

In addition to Eq. (36a). the nodal displacements of each element must satisfy the incom-pressibilty condition -,

(36b)

590 1. T. aDEN

EXACT SOLUTIONS

If the displacement field D is a linear function of the initial coordinates Xi. we refer tothe associated motion as homogeneous. If, for example. the initial coordinates are rec-tangular cartesian.t the relation

(37)

where the aij are constants or functions oftimc only, describes a homogeneous deformation.Returning to Eqs. (5). (6) and (18a). we see that. apart from a rigid-body translation re-presented by the tcrm kNIINi, the displacement field of a finite element is of the form

(38)

wherecjN is defined in Eq. (7b). Thus. Eqs. (37) and (38) arc identical if aij = CjNUNi'

It follows that the simplicial model of rhe deformation of a cominl/ous media depicts thedeformation of each finite element as a homogeneol/s deformation. Moreover. if the fieldsD, Ii, ii. etc. and gradients D,j' Ii'i' ii'j etc .. are finite and continuous throughout the domainfJe of the continuum. we havc noted earlier that the response of the discrete model convergesto that of the continuum as the finite element network is refmed. Thus. if these continuityconditions are satisfied, one can find a neighborhood (j about every point in a continuumin which the deformation can be represented as II + fW, where II is a homogeneous defor-mation and (, -+ 0 as c5 -+ O. Keeping (j finite is, in fact. the underlying philosophy ofsimplicial finite element representations.

Homogeneous deformations encompass several interesting classes of problems. Theseinclude pure homogeneous strain, simple cxtcnsion, and simple shear. From the aboveobservations, it is clear that if a finite elemem is subjected to a prescribed homogeneol/s

(a)

(b)

FIG. 2. Simple shear of a finite element model of a cuboid.

t A motion which is homogeneous with respect to one reference configuration. of course. may not be homo-geneous with respect to another with reference configuration.

A GENERALIZATION OF THE FINITE ELEMENT CONCEPT 591

deformation. thell Eq. (21) [or (22)] is exact: hence, we can use these equations of motionto write down immediately the exact solutions to problems of homogeneous deformationif the constitutive equations of the material are known.

As an example. consider the problem of simple shear of a cuboid of nonlinearly visco-elastic material. A model of the cuboid of thc type in Fig. 2a might be constructed. but it isquickly verified that this is not necessary. for. apart from a rigid translation. the motionof every element is thc same. Thus only two (actually, just one) finite elements are required.Using the model indicated in Fig. 2b, we have for element 1.

i = 1. 2

UIII = UII = ab(t)

where b(t) is a prescribed displacement and a is a dimension of the element at t = O. FromEq. (7b). the coefficients in the simplicial approximation are

1[0Ilc.d =-. a 1

Hence, the nodal displacements are

-I Il-1 oj

U 2 11 = 113 1 I = U 3 t2 = U 22 I = !I32 I = 0

l/t21 = Ut22 = U222 = U321 = !l322 = 0

U II I = !II 12 = l/ 2 I 2 = ab( t)

where

c5(t) = bo(1 - e-~)

and e is a dimensionless time parameter. Thus. for clement (I),

II"N• .I1 = aboll - e-'I ~]Introducing these values into Eq. (16). we find. as cxpected.

"12 = "21 = b(t)/2 f22 = cF(t)j2

all other components being zero.To obtain the stresses corresponding to these strains. we simply introduce the above

functions along with an appropriatc constitutive equation into Eq. (31). As an example ofthe result of such calculations, the time variation of the shear functional Sl2 of Eq. (35)for ql1asistatic, short-time deformations of a nonlinearly viscoelastic solid is given in Fig. 3.The kernel functions given by Haung and Lee [25], based on experiments of Ward andOnat [26], were used in these calculations In this case. the incompressibility conditionEq. (39b) is automatically satisfied It is interesting to note that. while a second-ordernonlinear theory is used in this example. equivalent results for higher-order nonlineartheories can be obtained at the expense of only lengthier integrations.

592

060

0,55

050

OA5

0·40

/- 035N 0

~ ;Z 0,30

0·25

020

O'i~

0·10

J. T. aDEN

--

- -- LInear theory- S<cond-order theory

80'0"

03 OA 0·5 0,6 0·7 0'8 0,9 1·0,

•

FrG. 3. Variation in shear functional with non-dimensionalized time.

REFERENCES

I. ZIENKIEWICZ. O. C. and CHEUNG. Y. K., The Finite Elemelll Methot! iI/ S/rIIclllral and Con/illlllllll Mechanics,McGraw-Hill Publishing Co .. London. 1967.

2. ODEN. J. T .. "Numerical Formulation of Nonlincar Elasticity Problcms," Journal of the Struclllral Division.ASCE, Vol. 93. No. sn. pp. 235-255. June 1967.

3. ODEN. J. T. and SATO. T., "Finite Strains and Displacements of Elastic Membranes by the Finite ElementMethod," International Journal of Solids and Structures. July 1967.

4. ODEN. J. T. and KUBITZA. W. K., "Numerical Analysis of Nonlinear Pneumatic Structures:' Proceedings.Internatiol/al Colloquium 01/ Pnellfllll/ic Structures, Stullgart, Germany. May 1967.

5. ODEN. J. T. and SATO. T .. "Structural analysis of Aerodynamic Deceleration Systems by the Finite ElementMethod," Advances in Astrol/amical Sciences. Vol. 24. June 1967.

6. ODEN, J. T .. "Analysis of Large Deformations of Elastic Membranes by the Finite Elcment Method,"Proceedings, lASS COl/gress on Large-spall Sflells. Leningrad. 1966.

7. ODEN. J. T .. "Numerical Formulation of a Class of Problems in Nonlinear Viscoelasticity," Ad~'ances inAs/rol/autical Sciences. Vol. 24. June 1967.

8. TAYLOR. R. L. and CHANG. T. Y., "An Approximate Method for Thermoviscoelastic Stress Analysis,"Nue/ear Engineering and Design. Vol. 4. pp. 21-28,1966.

9. CHANG, T. Y .. "Approximate Solutions in Linear Viscoelasticity:' Structural Engineering Laboratory ReportNo. 66--8. University of California, Berkeley. California, 1966.

10. GRAVES, L. M .. The Theory of Functions of Real Variables, 2nd Ed .. pp. 146-150. McGraw-Hili Book Co ..New York. 1956.

tl. WISSMANN. J. W .. "Nonlinear Structural Analysis, Tensor Formulation," Proceedings of COliference onMatrix Methods in Structural Mechanics. pp. 679-fJ96. Wright-Pallerson Air Force Base. Dayton. Ohio.December 1965.

12. SYNGE,J. L.. The Hypercircle in Mathematical Physics. pp. 209-213. Cambridge University Press. Cambridge.G.B .. 1956.

13. WISSMANN. J. W .. "Numerische Berechnung nichtlinearer elastischer Korper." Dissertation. TechnischeHochschule, Hannover. 1963.