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Universal Relations for Transversely Isotropic Elastic Materials

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Universal Relations for Transversely Isotropic Elastic Materials 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.
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