UniversalRelations for 'ftansverselyIsotropic Elastic Materials

R. C. BATRA Department of Engineering Science and Mechanics, M/C 0219, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA

(Received 8 March 2002; Final version 13 March 2002)

Dedicated to Millard E Beatty, respected colleague and friend.

Abstract: We adoptHayes and Knops'sapproach and deriveuniversal relations for finite deformationsof a transverselyisotropic elastic material. Explicit universalrelations are obtained for homogeneousdeformations correspondingto triaxial stretches,simple shear,and simultaneousshear and extension.Universal relations are also derivedfor five families of nonhomogeneousdeformations.

1. INTRODUcnON

Truesdell and Noll [1, section 54] and Beatty [2] have lucidly stated the importance of universal relations for finite deformations of isotropic elastic materials; the same remarks apply to transverselyisotropic elastic materials. Beatty [2, 3] has derived a class of universal relations for constrainedand unconstrainedisotropic elastic materials. Pucci and Saccomandi [4], have given a general approach for finding universal relations in continuum mechanics. Saccomandi and Vianello [5] have given a variational characterization of the universal relations for hemitropic materials. They have proved that the classof transverselyhemitropic hyperelasticbodies is characterizedby the condition that the inner product betweenSC - CS and W N vanishes for all deformations of the body; S is the second Piola-Kirchhoff stress tensor, C the right Cauchy-Green tensor and W N the skew-symmetric tensor associated with the unit vector N which points along the axis of transverseisotropy. Here we adopt the approachemployed by Hayes and Knops [6] for deriving universal relations for isotropic elastic materials and use it to deduce universal relations for transversely isotropic elastic materials. The general relations so obtained are applied to three classes, namely triaxial extension,simple shearand simultaneousshear and extension,of homogeneousdeformations to get explicit universal relations for these deformations. We note that homogeneous deformations can be produced by the action of surfacetractions alone in every homogeneous elastic body; Ericksen [7] proved that theseare the only deformations that can be produced in every isotropic compressibleelastic body. We also derive universal relations for five families of nonhomogeneousdeformations which, according to Ericksen [8], can be producedin every isotropic incompressiblehyperelastic body by the action of surface tractions alone. Universal relations given here are properties of the constitutive relation for a transversely isotropic elastic material. They are valid for both static and dynamic deformations, and

Mathematics and Mechanics alSo/ids 7: 421-437,2002 DOl: 10.1177/108128028483 (t;)2002 Sage Publications 422 R. C. BATRA whether or not body forces arerequired to producethe envisageddeformations. Someof these are easily verifiable experimentally. Universal relations for nonhomogeneousdeformations hold pointwise and will require local measurements.

2. UNIVERSAL RELATIONS

The constitutive relation in tenDs of the Cauchy stresstensor T and the left Cauchy-Green strain tensor B for a transverselyisotropic elastic material with the axis of transverseisotropy along the unit vector n in the present configuration is (e.g. see [9})

T = )'11 + )'2B + )'3B2 + )'4n@ n + )'5(n@Bn+Bn@n)

Here the responsefunctions Y 1, Y2 , y 5 are functions of the five principal invariants h,I2,... ,Is definedas

II = tr(B) /2 = tr(B2), 13 = tr(B3), 14= n . Bn, 15 n.B2n. Thetensor product, a@ b, betweenvectors a andb is definedas (a@ b)c = (b. c)a for every vector c, and a . b equalsthe inner productbetween vectors a and b. For an

incompressibletransversely isotropic elastic material, y 1 in (2.1) is replaced by an arbitrary pressure-p, and B must satisfy detB = 1. Equation (2.1) implies that ifn is an eigenvector(or proper or principal vector) of B, then it is also an eigenvectorof T. Also, the other two eigenvectorsof B which are perpendicular to n are also eigenvectorsofT. If any two eigenvaluesofB are equal, then the corresponding eigenvaluesof T are also equal only if the eigenvector of B corresponding to the distinct eigenvalueis parallel to n. However, if n is not an eigenvector of B, then an eigenvector of B is not an eigenvector ofT. For

}'2 +11}'3 > 0, )'3 ~ 0, y 4 = Y5 = 0,

it follows from Batra's [10] theorem that eigenvectors of T and B coincide and if two eigenvaluesof T are equal, then the corresponding two eigenvalues of B are also equal. Note that the responsefunctions y 1, Y2, . . . , y 5 while satisfying (2.2) may still dependupon n through their dependenceon the principal invariants 14 and 15. We first consider homogeneous deformations of a homogeneous body so that the deformation gradient F, the left Cauchy-Green tensor B = FFT and the Cauchy stress tensor T are constants within the body. Hence equations of equilibrium with zero body force are identically satisfied. Without any loss of generality, we use rectangular Cartesian coordinateswith orthonormal basis vectors el, e2 and e3. Following Hayes and Knops [6], we note that if, for a given deformation, there is a relationship of the form

aii Tii = 0 UNIVERSAL RELATIONS FOR TRANSVERSELY ISOTROPIC MATERIALS where a is a symmetric second-order tensor independent of the response functions y 1, Y2, . . . , y 5, and a repeated index implies summation over the range of the index, then equation (2.3) will be a universal relation. Hayes and Knops used it to derive universal relations for an unconstrainedisotropic elastic body: Here we use it to find universal relations for an unconstrainedand also for an incompressibletransversely isotropic elastic body.

Let A. ~,A. ~ and A.~ be eigenvaluesof B with the correspondingorthonormal eigenvectors p, q and r. Thus B and B2 have the representations

2 2 2 Bij ).lPiPj +).2qi~ +).3rirj, (2.41)

(B2)ij ). tPiPj +). iqi~ +). ~ri rj . (2.4V

We assumethat the defonnationgradient F and the left Cauchy-Greentensor B are given. Substitutionfrom (2.4) into (2.1) andthe resultinto (2.3) gives

Ylaii +Y2aij().~PiPj +).~qi~ +).~rirj) +Y3aij ().iPiPj +).~qiqj +).~rirj -I- Y4aijninj +Y5[aij ().~niPjPknk +).~niqjqknk + ).~nirjrknk +).~njPiPknk +).~njqiqknk +).~njrirknk)] = o. ~

In order that equation (2.5) hold for all transverselyisotropic elastic materials, it must be satisfied for all choices of)' 1,)' 2, . . . ,)' 5. Thus

a..I! - 0, aij (A~PiPj +A~qi~ +A~rirj) 0, aij (AfpiPj +A~qiqj +A~rirj) 0, aij ninj 0, aij [A~(niPj+ njPi)Pknk+A~(ni~ + njqi)qknk +A~(nirj + njri)rknk] O. (2.6)

In tenDsof the eigenvectorsof B, let a have the representation

aij = apiPj + Pqi~ yrirj +t5(Piq; +Pjqi)+E(qirj +q;ri)+p(riPj +rjpi), (2.7)

where constantsa, p, }' , c)',to and.u are to be detemlined. Equations (2.6) and (2.7) give

a + P + y = 0, a).~ +P).~ + y).~ = 0, a).f +P).~ + y).~ = 0, a(pjnj)2 + p(qjnj)2 + y (rjnj)2 + 2(j (pjnjqj nj) + 2{ (qjnjrj nj) + 2,u(riniPjnj)=O, a).~(pjnj)2 +p).~(qjnj)2 + y).~(rjnj)2 + (j().~ + ).~)(pjnjqjnj) + {().~ +).~)(rjnj)~nj +Jl().~+).~)(pjnj)(rjnj) = o. 424 R. C. BATRA

We now consider the following three cases:

( "' ) ,2 ,2' ,2 (i) 12 -t. 12 -t. 12. 2 2 2 11.1 ;- 11.2 ;- 11.3' (ii) ).1 =).2 # ).3; III A1=tf.2=:"-3

For case(i), the only solution of equations(2.8)1, (2.8)2 and (2.8)3 is

a=p " = o. (2.10)

Thus equations(2.8)4 and (2.8)5 simplify to

(2 2 2 r5(Pini)~nj2 +E(qini)rjnj~ +,u(rini)pjnj2,G '=[,9, . ').,1+).,2)r5(Pini)Qjnj + ().,2+).,3)f(rini)~nj + ().,3+).,1),u(Pini)rjnj = 0.(2.11)

The problem of fmding universal relations reducesto that of fmding solutions of equation (2.3) for all choices of <5,E and,u that satisfy equations (2.11). Alternatively, introducing Lagrange multipliers Al and A2, the following equation (2.12) must hold for all choices of <5,E and,u:

2t5Tijpj~ +2ETijqirj +2p,Tijrjpj Al [t5(pjnj)qj nj + E(qinj)rjnj +p,(rjnj)pjnj] A2[(A.~+A.~)t5(pjnj)~nj + (A.~+A.~)E(rjnj)qjnj + (A.~+A.~)p,(pjnj)rjnj] = o. (2.12)

Thus

2TijPi~ - A1Pini~nj - A2()'~ +).~)Piniqjnj = 0, 2Tijqirf - A1qinirjnj - A2()'~ +).;)riniqjnj ~,: 0, 2Tijripj -A1riniPjnj -A2().~+).~)Pinirjnj = o. (2.13)

When one of the eigenvectorsof B, say p, is parallel to n, then , qini = rini =0, and equations(2.13) yield

TijPiq,. = Tijqirj = Tijripj =0.

Hence p, q and rare eigenvectorsof T or eigenvectorsof B are also eigenvectorsof T; a property already observedabove. In this case,equations (2.15) are the three universal relations. UNIVERSAL RELATIONS FOR TRANSVERSELY ISOTROPIC MATERIALS 425

If noneof the eigenvectorsorB is parallelto ll, thenpjnj # 0, qjnj # 0, rjnj # O. We can eliminate Aland A2 from equations(2.13) to arrive at the following universal relations:

(2.16)

Substitution from constitutive relation (2.1) into (2.16) reveals that each term equals y 5- Thus relations (2.16) provide a way to determine Y5 from the experimental data. For case(ii) of equations(2.9), the solution of equations(2.81)-{2.83) is

a+fJ o. 11 0, and equations(2.84) and (2.85) reduce to

a[(pini)2 - (qini)2] + 2b(Pini)~nj

~ 2f(qknk)rini + 2.u(rini)(pjnj) = 0, aA.~((Pini)2 - (qini)2) + 2bA.~(qini)pjnj + f(A.~ +A.~)(rini)(qjnj) +.u(A.~ +A.~)(niPi)rjnj = O.

The analogueof equation (2.12) is

aTij (pjPj - qjqj) + 2t5Tijpj~ +2ETij qjrj + 2/.lTijrjpj Al [a((pjni)2 - (qjni)2) + 2t5(Pinj)(~nj) + 2E(qini)(rjnj) + 2/.l(rini)(pjnj)] A2[a).~((Pini)2- (qini)2) + 2t5).~(qini)(pjnj) E().~+).~)(rini)(~nj) +/.l().~ +).~)(niPj)(rjnj)] = O.

The necessaryand sufficient conditions for equation (2.19) to hold for all choices of a, <5"uand E are

Tij (pjpj - qj~) - Al[(pjnj)2 - (qjnj)2] - A2A.~[(pjnj)2 - (qjnj)2] = 0, Tijpj~ - A1(Pknk)(~nj) - A2A.~(qjnj)(pjnj) = 0,

2T ij qjrj - 2Al (qknk )(rjnj) - A2(A.~ + A.~)(qjnj )(rj nj) = 0,

2Tii riPi - 2Al (rknk)( Pinf) - A2(A.~+ A. ~)(nfPf )(rjni) = 0.(2.20) 426 R. C. BATRA

If eigenvector r corresponding to the distinct eigenvalue). ~ of B is parallel to ll, then Pint = 0, qtnt = 0, andequations (2.20) give

Tijpjqj = 0, Tij qjrj - 0, Tij riPj = O~ Tij (pjPj - qjqj) o.

That is, p, q and rare eigenvectorsof T and eigenvaluescorresponding to eigenvectors p and q are equal. However, if eigenvector p correspondingto the repeatedeigenvalue J.. 1 is parallelto n thenqjnj = 0,rjnj = 0,and equations (2.20) reduce to

Tij (PiPj - qiqj -A1-A2)'~ =0, TijPiqj =0, Tijq;rj 0, TIi riPj = O.

That is, p, q and r are eigenvectorsof T but eigenvaluesof T correspondingto eigenvectorsp andq arenot equalfor all transverselyisotropic elastic materials. When none of the eigenvectorsofB is parallel to n and p.n # q'n, then the elimination of Al andA2 from equations(2.20) gives

p .r . ~"IJ I J == ~ , pknk qp;np; q 'p ' T if (PiPj - qiq;) ~"IJ I J - qknk p(n( (Pknk)2 - (q£n£)2

For p . n = q . n, the universal relation is

Pi Tij Pj qjTijqjo

If desired,y 5 can be evaluatedfrom the following relation:

'~" q ' PI IJ J (2.24) Pknkqtnt

For case(iii) of equation (2.9), equations(2.81}-(2.83) have the solution

a+p+y =0. (2.25)

Thus equations(2.84) and (2.85) yield

a[(p;n;)2 - (rjnj)2] +p[(q;n;)2 - (rjnj)2] = 0, J (p;n; )qj nj + f (q;n; )rj n; + # (r;n; )p; nj = o. (2.26) UNIVERSAL RELATIONS FOR TRANSVERSELY ISOTROPIC MATERIALS

The equation which is analogousto equation (2.12) is

aTij (pjpj - rjrj) +pTij (qj~ - rjrj) + b'Tij (pj~ + qjpj)

+ ETij (qjrj +~rj) +/lTij (riPj +rjpj) Al [a((pjnj)2 - (rjnj )2) +p((qjnj)2 - (rjnj )2)] A2[b'(pjnj)qjnj + E(qjnj)rjnj +/l(rinj)pjnj] =0.

The necessaryand sufficient conditions for this equation to hold for all choices of lX,p, t>',E andp, are

Tif (pjPj - rirj) - A1((Pini)2 - (rini)2) - 'Jr! 0, Tij (qiqj - rirj) - A1((qini)2 - (rini)2) =! 0, 2TijPi~ - A2(Pini)qjnj = 0, q.r. - A2(q.n.)r. n. 2~..Y I J I I J :I 0, 2~..Y r.Ip J. - A2(r.I nI, )rJ,n. n:I. o.

Since B is a spherical tensor, every vector is an eigenvector of B. We choose three orthonomlal vectors p, q and r as eigenvectorsofB. When p is parallel to n, thenpjnj = 1, qinj = 0, rinj = 0, and equations(2.28) reduce to

Tij (pjPj - rjrj) - Al 0, Tij (qj~ - rjrj) = 0, Tijpj~ = 0, Tij qjrj = 0, Tij riPj = O.

That is, p, q and r are eigenvectorsofT, and eigenvaluescorresponding to eigenvectors q and r are equal, but these differ from the eigenvalue correspondingto the eigenvector p. The difference between the eigenvalues of T corresponding to eigenvectors p and q is material dependent and hence not universal. For the case of p not parallel to n and p . n # q. n # r . n, the elimination of Al and A2 from equations (2.28) yields the following universal relations:

TIi (PiPj - qi~ ) (Pk nk)2 - (qtnt)2 TliPi~ = (2.301) (Pknk) (q£n£) 428 R. C. BATRA

Ifp . n = q. n, thenthe universalrelation is

pjTijpj = qjTij ~ (2.30?) and similar relations hold when q . n = r . n or p . n = r . n. Thus for the choice of basis vectors p, q, r with p . n = q . n = r . n, a spherical left Cauchy-Green tensor B will result in a spherical Cauchy stresstensor T for a transversely isotropic elastic material. For q' n # r . n, Y5 is given by

1 }'5 (2.31) A.~ (qknk)2 - (r£n£)2

Wenote that the universal relations derived abovefor unconstrainedtransversely isotropic elastic materials also apply to incompressibletransversely isotropic elastic materials provided that deformations consideredare isochoric.

3. UNIVERSAL RELAllONS FOR SIMPLE DEFORMAllONS

3.1. Controllable homogeneousdeformations

We first study three classes of homogeneousdeformations. Thus the stress tensor in a homogeneous transversely isotropic elastic body will be a constant and the balance of linear momentum without body forces will be identically satisfied. These deformations are controllable in the sensethat they can be produced by the action of surfacetractions alone.

3.1.1. Triaxial stretches

Consider a rectangular block made of a transversely isotropic elastic material subjectedto triaxial homogeneousdeformations

Xl = bXI, X2= cX2, Xg = dXg,

where b, c and d are constants.For this deformation

).1 = b, ).2 = c, ).3 = d, p=el, q= e2, r = ea-

Let the unit vector p be parallel to D. For b # c # d, equations(2.15) give

T 12 = T23 = T31 = 0

as the three universal relations. The constitutive relation (2.1) implies that the eigenvalues ofT are distinct. For b # c = d, equations(2.21) imply that (3.2) and T22= T33are the universal relations. However, when q rather than p is parallel to ll, then equations(3.2) still UNIVERSAL RELATIONS FOR TRANSVERSELY ISOTROPIC MATERIALS hold but T22 need not equal T33. For b = c = d,we againget (3.2) andfor p parallelto n, T22 = T33 also as the universal relations. Let the axis n of transverseisotropy be not aligned along anyone of the coordinate axes. For b # c # d, equations(2.16) imply that

1 1 1 ~- T31 - d2 - C2 = b2 - d2 c2-b2 n2n3 n3nl are the three universal relations. When b = c i d and nl i n2,then it followsfrom equations(2.23) that

are the universal relations. For b = c = d and nl # n2 # n3, equations(2.30) imply that the universal relations are

T23

n2n3

For incompressible transversely isotropic materials, the deformation (3.1) must satisfy the relation bcd= 1.

Thus under the conditions stated for equations(3.2), (3.3) and (3.4) and with b, c and d satisfying (3.6), universal relations (3.2), (3.3), and (3.4) hold for incompressibletransversely isotropic elastic materials.

3.1.2. Simpleshear

For simple sheardefonnations in the XI-X2 plane,

Xl =Xl + "X2, x; = X2, X3 = X3, where x is the position of the materialparticle that occupiedplace X in the reference configuration.For the deformation(3.7), 430 R. C. BATRA

1+K2 1C B= K 1 0 0 ~ ]

1 "2 1 + 2"1 2 + " 1 2 ,-2 l+-K /1.2 , ;'3 = 1 A.~ 4

).~

p,q - (2 + !IC 2 =f IC (1+ tIC 2)!)! ' r = ea, (3.8)

Thus A.~ # A.~ # A.~. When either p or q or r is parallel to ll, then p, q and rare eigenvectorsofT, andequations (2.15) give

T 13 = T 23 ~ 0, T11 - T22 = 1CT12 (3.9) as the three well known universal relations which are the same as those for an isotropic material. For r = n, relations (3.9) are intuitive since the material in the XI-X2 plane is isotropic. However, when the axis n of transverseisotropy is parallel to either the xl-axis or the x2-axis, then from equations(2.13) we conclude that

T13 = 0, T23 = 0, (3.10)

and only (3.91) and (3.9V are the universal relations. For the generalcase ofn not parallel to p, q, or r, we obtain from (2.16) the following universal relations for a transverselyisotropic elastic material:

(3.11)

where

-2 2 1C = 1 + 1C / 4. (3.12)

It is implicit in equations(3.11) that terms in the denominator do not vanish. The simple shear deformation (3.7) is isochoric. Thus universal relations (3.9), (3.10) and (3.11) also hold for incompressible transversely isotropic elastic materials under the conditions stated fOT them UNIVERSAL RELATIONS FOR TRANSVERSELY ISOTROPIC MATERIALS

3.1.3. Simultaneous shear and extension

We consider a unifonn shearand extension of a block defmed by

Xl = DXI + EX2, X2 = FX2, X3 = GX3, (3.13) where D, E, F and G are constants. This deformation is isochoric when DFG = For the deformation (3.13),

D2 + E2 EF B EF F2 0 0 J,l

2), ~,2 A~ = G2~

p q = r = H2 = HI = H2 =

For p, q or r parallel to n, it follows from equations(2.15) that p, q and r are eigenvectorsofT, and

T13 = 0, T23= 0, T11 - T22 = MT12; M= (D2 +E2 -F2)/EF (3.15)

The universal relation (3.15) is independentof G, and is the sameas that derived by Beatty [2] for an isotropic elastic material. Universal relations (3.15) will hold for an incompressible transversely isotropic elastic material if DFG = 1. When the axis n of transverseisotropy is parallel to either the X1- or the x2-axis, then (3.151) and (3.152) are the only universal relations. For the caseof n not parallel to p, q or r, the following universal relation follows from equation (2.16): 432 R. C. BATRA

TIIE2F2 + T22(F2+ H2 - D2 - E2)(F2- H2 D2 £2)/4 (EFnl + n2(F2+ H2 - D2 - E2)/2)(EFnl + n2(F2- H2 - D2 - E2)/2) T12EF(F2-D2 -:.-E2)/2 + (EFnl+ n2(F2+H2 - D2 - E2)/2)(EFnl+ n2(F2- H2- D2 - E2)/2) ,(3.16)

For the sakeof brevity, we havenot written out the expressioncorresponding to the secondequality in (2.16).

3.2. Nonhomogeneousdeformations

We now consider five families of nonhomogeneousdeformations that mayor may not be controllable in every homogeneoustransversely isotropic elastic material. Irrespective of whether or not special body forces are required to produce these deformations, universal relations will hold at a point. We use cylindrical coordinates(r, () , z) or spherical coordinates (r, () ,4» as neededfor specifying the deformations; lower-caseletters indicate coordinates of a point in the present configuration and upper-caseletters coordinates of the same point in the reference configuration. The parametersA, B, C, D, E and F are constants; those appearing in the denominator must not vanish. These families of deformations have been described in detail by Truesdell and Noll [1, section 57]. Families 1, 2 and 4 are isochoric, thus these deformations are possible in both compressible and incompressible transversely isotropic elastic materials. Families 3 and 5 are isochoric only when constantsappearing in them satisfy the constraint det F = 1. Below, we use physical componentsof Band T. Family 1: Bending, stretching and shearing of a rectangular block is describedby

r= J2AX;, 8 = BX2, z= X3 - BCX2 AB (3.17) Thus

r ~r2 0 0 B - 0 B2? -B2Cr 0 -B2Cr B2C2+ ~

'=' )..~ A2j?,

2)..~,3 P

q

s

U2

2 H2.3 (3.18) UNIVERSAL RELATIONS FOR TRANSVERSELY ISOTROPIC MATERIALS 433

We have denotedeigenvectors of B by p, q and s rather than by p, q and r in order not to confuse the last one with the radial coordinate r. When the axis n of transverseisotropy is parallel to er, equations(2.15) imply that p, q and s are eigenvectorsof T, and the universal relations are

8 Z ce X2 = X3 = - + (3.20) AB' B AB

From (3.20) we obtain

A2B4R2 0 0 0 1 C B R2A2B2 R2A2B2 C 1 0 R2A2B2 B2

1~ - A2B4R2 1 1 21~,3 - R2A2B2 :B2::1::H; P = el,

n -

s =

H2 -

Hi, 3 -

For the axis n of transverse isotropy aligned along the Xl -coordinate axis, universal relationsdeduced from equations(2.15) are

T12 = 0, T13 = 0, (3.21)

Equations (2.16) give the universal relations when n is not parallel to p, q or So 434 R. C. BATRA

Family 3: Inflation, bending,torsion, extension and shearingof an annularwedge is defmedby

r= VAR2 +B, () =C8+DZ, z=E8 +FZ.

The deformation is isochoric if A (CF - DE) = 1. Fromequation (3.22) we obtain

[ LR2 0 0 B - 0 r(~ + D2) r(~ +DF)

E~ F 2 0 r(~ +DF) ]2+ - ~ ).~ R2'

2), ~,3 (~+F2) + ~ (* +D2):f:H,

P er,

q -

2 J:;r2

2 H2,3 -

For the axis n of transverseisotropy aligned along a radial direction,equations (2.15) give

r2(C2 + D2R2):::::: (£2 + F2R2) Tre = 0, Tn = 0, TIJIJ- Tzz = TlJz (3.24) r( CE + DFR2)

asthe universal relations. For n not aligned along anyone of the eigenvectorsofB, equations (2.16) give the universal relations. Family 4: Inflation or eversionof a sectorof a spherical shell is given by the deformation field

\, 1 r = (:!::R3+A)3 () ::1::8, cf>=

For the defornlation (3.25), the physical componentsorB are

~ UNIVERSAL RELATIONS FOR TRANSVERSELY ISOTROPIC MATERIALS 435

[ -;4'"R4 0 r2 B= ~ R2 0 iR2] If the axis n of transverseisotropy is parallel to er, then we conclude from equations (2.21) that

Tre = Trl/>= Tel/>= 0, T 88 = T t/>t/>

However, if n is parallel to either e/J or etj>,then only the fIrst three relations in (3.27) hold. When n is not parallel to er, e/Jor etj>,then we conclude from equations(2.23) that

Trci> T{Jr -, nci> n{J

(Tthth~ TI/I/)

, ,., ., TlJcI> - ~ 2 2 ~-'.L.~) nOnt{> nt{> - nO are the universal relations. Implicit in (3.28) is the assumptionthat n(J=I nt{>. Family 5: Inflation, bending, extension and azimuthal shearing of an annular wedge is defined by

R r = AR, () = B log D + C8, z = FZ. (3.29)

For this defonnation to be isochoric, A2CF~ 1. The left Cauchy-Green tensor for the defonnation (3.29) is

A2 ~ R 0 82 C2r2 B jl2+R2 0 (3.30) [i 0 F2 Thus p =

q = s =

). ~,2 -

1;'2 ).; 2 A2B2 H2 :=' +4 - R2

H;,2 - (3.31) R.C. BATRA

For the caseofn alignedalong either p or q or ez, we concludefrom equation(2.15)

1 - B2 - C2 Tzr = 0: Tz8 = 0, Trr Tee= Tre (3.32) B When n is not parallel to p, q or s, then universal relations can be derived from equations (2.16).

4. REMARKS

The constitutive relation (2.1) is written in terms of the Cauchy stress tensor T and the left Cauchy-Green tensor B. One could write a similar relation between the secondPiola- Kirchhoff stresstensor S, the right Cauchy-Green tensor C = FTF and the unit vector N along the direction of transverseisotropy in the referenceconfiguration. Universal relations similar to the ones derived above can be deduced in terms of the components of S, and eigenvaluesand eigenvectorsof C. . Even though we have derived explicit forms of universal relations for simple deformations, universal relations (2.15), (2.16), (2.23) and (2.29) etc are applicable for any deformation. These relations, being characteristic of the constitutive relation (2.1), are also applicable to transversely isotropic fluids for which equation (2.1) applies with B replaced by the strain-ratetensor D. Then in equations(2.15), (2.16), (2.23) and (2.29) etc, A~, A~ and A ~ equal eigenvaluesofD, i.e., they are stretchingsrather than squaresof stretches.Whereas stretchesare positive, stretchingsneed not be positive. However, the positivenessof stretches was not used in the derivation of the universal relations.

REFERENCFS

[I] Truesdell, C.A. and Noll, W: The Nonlinear Field Theories of Mechanics, Handbuch der Physik 111/3,Springer \erlag, New York, 1965. [2] Beatty, M.F.: A class of universal relations in isotropic elasticity theory. J: Elasticity, 17, 113-121 (1987). [3] Beatty, M.F.: A class of universal relations for constrained isotropic elastic materials. Acta Mech., 80, 299-312 (1989). [4] Pucci, E. and Saccomandi, G.: On universal relations in continuum mechanics. Continuum Mech. Thermodyn.,9, 61-72 (1997). [5] Saccomandi, G. and Vianello, M.: A universal relation characterizing transversely hemitropic hyperelastic materials. Math. Mech. Solids, 2, 181-188 (1997). [6] Hayes, M. and Knops, R.J.: On universal relations in elasticity theory. Z. Ange\\! Math. Phys., 17,636--639 (1966). [7] Ericksen, J.L.: Deformations possible in every compressible perfectly elastic material. J: Math. Phys., 34,126-128 (1955). [8] Ericksen, J.L.: Deformations possible in every isotropic, incompressible, perfectly elastic body. Z. Ange\\! Math. Phys., 5, 466-489 (1954). [9] Ericksen, J.L. and Rivlin, R.S.: Large elastic deformations of homogeneous anisotropic materials. J: Rat. Mech. Anal., 3, 281-301 (1954). [10] Batra, R.C.: On the coincidence of the principal axes of stress and strain in isotropic elastic bodies. Left. Appl. Eng. Sci., 3, 435-439 (1975). fIll Rivlin. R.S.: Universal relations for elastic materials. Rendiconti di Matematica. 20. 35-55 (200m UNIVERSAL RELATIONS FOR TRANSVERSELY ISOTROPIC MATERIALS 437

NOTE ADDED IN PROOF

When I presentedthis work in the Symposium honoring Professor Millard E Beatty at the 14th U.S. National Congressof Theoretical and Applied Mechanics, Blacksburg, June 23- 28,2002, Dr. P:Saccomandi kindly told me that R.S. Rivlin [11] had recently given universal relations for transversely isotropic elastic materials. Dt: Saccomandiwas kind enough to send me a copy of Rivlin's papet: In the terminology of this paper, Rivlin assumedthat ni = t5i3,solved five out of six equations(2.1) for}'l, }'2,..., }'5, and substitutedfor them in the remaining equation to obtain a universal relation. He gave explicit universal relations for three classesof simple shearing deformations superposedon a homogeneousdeformation. The approach adopted here is different from but equivalent to Rivlin's. We have studied general deformations, and given explicit universal relations for several deformation fields.