Additional Universal Relations for Transversely Isotropic Elastic Materials

For the case of distinct eigenvalues of the left Cauchy-Green tensor universal relations for a transversely isotropic elastic material are expressed in a compact form. Also, two new universal relations are derived for an incompressible transversely isotropic elastic material. Key Wolds: universal manifold, incompressible materials, simple shear, annular wedge 1. INTRODUcnON A universal relation is an equation that holds for every material in a specified class. Universal relations are basic tools in the analysis of continuum theories of the mechanical behavior of materials [1,2]. Recently, Batra [3] adopted the Hayes and Knops [4] approach to derive universal relations for finite homogeneous and inhomogeneous de~ormations of a transversely isotropic elastic material. The first part of this note discusses relationships between results in [3] and those in [5] and [6]. Moreover, we give additional universal relations for incompressible transversely isotropic elastic materials which cannot be found by using the Hayes and Knops method. They are found by employing the universal manifold method of Pucci and Saccomandi [7,8] which generalizes the Hayes and Knops technique. A brief historical development of universal relations is given in [7]. For unconstrained transversely isotropic elastic materials, Rivlin [9] has derived universal relations for three classes of simple shearing ~ deformations superimposed on a homogeneous deformation. 2. BASIC EQUAll0NS AND THE MAIN UNIVERSAL RELAll0N The constitutive equation for a transversely isotropic elastic material with the axis of transverse isotropy along the unit vector n in the current configuration is Mathematics and Mechanics of Solids. 9: 167-174,2004 001: 10.1177/1081286504035614 ~ 2004 Sage Publications 168 G. SACCOMANDI and R. C. BATRA T = ')Ill + ')I2B+')I3B2+')I4n@n+')I5 (n@ Bn + Bn@ n), (1) where T is the Cauchy stress tensor, I the identity tensor, and B the left Cauchy-Green strain tensor. The response coefficients ')I i (i = 1, . . . , 5) are functions of the five principal invariantsll,12,... ,15 defined as h = trB, 12 = trB2, 13 = trB3, 14 = n. Bn, 15 = n. B2n. (2) The tensor product @ between vectors a and b is defined as (a @ b) c = (b. c) a for all vectors c. For an incompressible transversely isotropic elastic material B must satisfy detB = 1 and ')11 in (1) is replaced by an arbitrary pressure -po When y 4 = ')I 5 = 0 in (1) then eigenvectors of B are also eigenvectors of T. For Y2 + 13')13 > 0 and ')13 < 0, Batra's [10] theorem implies that eigenvectors of T are also eigenvectors of B. When the eigenvalues A. i, A. ~ and A. ~ of B corresponding to eigenvectors p, q, and r are such that A. i # A. ~ # A. ~ all universal relations may be found from the relation TB BT = 0, (3) as has been discussed in detail by Beatty [11]. Isotropic materials are a special case of transversely isotropic materials. For an isotropic material, Y 4 = Y 5 = 0 and ')I 1, ')I 2 and ')I 3 are functions of II, 12 and 13. Results in [3] for A. i # A. ~ # A. ~ may be obtained directly by following the approach of [5] and [6]. From (1) we get TB BT = ')14 [(n @ n) B B (n @ n)] (4) cf+ ')I5[(n@Bn+Bn@n)B-B(n@Bn+Bn@n)]. Because B = BT we have (n @ n) B = n @ Bn. Thus we write (4) as TB BT = Y4 [n@Bn Bn@nJ + ')15 [n@B2n B2n@n] . (5) Here we omit the trivial case of n being an eigenvector of B because in this case universal relations are given by (3). Recall that an anti symmetric tensor W corresponds to an axial vector W x == u, such that for any vector v, we have Wv = u xv. Moreover, for vectors a and b, we have the identity (a @ b b @ a)x = -a x b. Using these, Equation (5) simplifies to (TB BT)x ~ -nx (Y4Bn + ')I5B2n). (6) Therefore the universal relation we are searching for is (TB BT)x . n =0, (7) ADDITIONAL UNIVERSAL RELATIONS 169 which is Equation (2.16) of [3]. The compact form (7) allows for a more direct comparison with the results of [5] and [6]. Moreover, whereas for an isotropic material the meaning of the three general universal relations is the coaxiality between the Cauchy stress and the left Cauchy-Green strain tensor here the meaning of the only general universal relation is the orthogonality between the axial vector associated with TB BT and the axis n of transverse isotropy. It also provides a way to find the axis of transverse isotropy for these materials. We now use the universal manifold method [7] to prove that (7) is the only universal relation. We consider a universal solution and the corresponding strain B, and fix a point X in the reference configuration. Introduce an II-dimensional space S with coordinates 'T = {7l,72,73,74,75,76}, f = {)'1,)'2,)'3,)'4,)'5}, where '7"1 = Tll, 72 == n2, 73 = T13, 74 = T22, 75 = T23, 76 = T33. In this space (1) represents a linear homogeneous manifold V of dimension three parametrized by )'. This manifold may be projected in the subspace S* ( 'T) of S on the manifold n (V) whose dimension k equals the rank of the matrix 1 0 O. 1 0 1 Bll B12 B13 B22 B23 B33 Bil Bi2 Bi3 B~2 B~3 B~3 , (8) 2 2 2 nl nln2 nln3 n2 n2n3 n3 $11 $12 $13 $22 $23 $33 where $ = n @ Bn + Bn @ n, and B~ are components of the tensor B2. The universal manifold n (V) is linear and homogeneous and may be represented by a set of linear homogeneous equations (the universal relations). The number of these equations is 6 k. Since the rank of the matrix (8) is 5, therefore we have only one universal relation. .. The rank of the matrix (8) may be lower (i.e. k < 5) because either).; = ). J for i # j or n is an eigenvector of B. In these cases new universal relations are possible and these can be found by the Hayes and Koops method. However, universal relations that are valid , when the structure of the space S changes cannot be found by the Hayes and Koops approach. This holds for special deformations that introduce a relationship among the )' i or for a special class of materials, e.g. constrained materials. On the other hand, universal relations may be a powerful tool to understand quantitatively the significance of a constitutive parameter, For example, let us consider a transversely isotropic material with )'5 = O. For this class of materials a new universal relation, namely (TB BT) x . Bn = 0, (9) is valid because the structure of S has changed. If experimental data for a transversely isotropic elastic material do not satisfy (9), then clearly )'5 # O. 170 G. SACCOMANDI and R. C. BATRA 3. INCOMPRESSmLE TRANSVERSELY ISOTROPIC MATERIALS For an incompressible transversely isotropic material, Yl in Equation (1) is replaced by an arbitrary pressure -po To show that for these materials additional universal relations can be found, we deduce them for a homogeneous deformation and a nonhomogeneous deformation; universal relations may be found for other isochoric deformations. 3.1. Simple Shear We consider the simple shear deformation x=X+KY, y= Y, z=Z, (10) of a block bounded by the pair of surfaces X = ::I::Xo, Y= ::J::Yo andZ = ::J::Zo in the reference configuration, and the axis n of transverse isotropy in theXZ(or xz) plane. Here (x,y,z) are the current rectangular Cartesian coordinates of the material point initially at (x, Y, Z) in a common Cartesian frame cp = {o; i, j, k} . A straightforward calcu~ation gives (TB BT)x = K {T13i (KT13 + T23)j + (KT12 + T22 Tll) k}. (11) The universal relation (7) becomes T13nl ~ (KTI3 + T23) nz+ (KT12 + T22 Tll) n3 = O. (12) To simplify the computations, we set n = COB ai + sin ak and obtain from (12) TI3 cosa + (KT12 + T22 TIt) sin a = O. (13) For the simple shear deformation (10), the referential plane X = Xl is deformed into the current plane x Ky == Xl. The normal vector to this surface is m = COB ri sin rj where r = tan -1 K is the angle of shear. The normal traction, N, and the shear traction, T, on this inclined plane are given by N = Tll +K2T22 2T12K T = K(TII T22) + T12(1K2) 1 + K2 ' 1 + K2 . (14) We now choose the pressure so that the normal traction N vanishes on the inclined plane. That is, p = (1 +K2)-1 [Y2 +Y3 + (Y4 +2Y5)cos2a]. (15) In this case the structure of the space S has changed because the pressure p cannot be considered as an independent constitutive parameter;p is related to Yi through Equation (15). In the new space S we introduce coordinates ADDITIONAL UNIVERSAL RELATIONS 171 T = {TI,T2,T3,T4,T5,T6}, "Y = {Y2,Y3,Y4,Y5}, (16) and the linear homogeneous manifold V may be projected on to the manifold n (V) whose dimension k is given by the rank of the (6 x 4) matrix BII + ~ Bl2 Bl3 B22 + ~ B23 B33 + ~ B~l + ~ B~2 B~3 B~2 + ~ B~3 B~3 + ~ 2 cos2 a 2 cos2 a 2 COS2 a (17) nl + T+K2 nln2 nln3 n2 + T+K2 n2n3 n3 + T+K2 m. + 2 cos2 a m. m. m. + 2 cos2 a m. m. + 2 cos2 a llPII ~ llPl2 llPl3 llP22 ~ llP23 llP33 ~ Therefore, the following new universal relation not found in [3] exists (because 6 4 = 2): (K2 + 2) T22 TII + 2cotaTl3 = O. (18) If a different boundary condition were used to find the hydrostatic pressure p, then we would have obtained a different universal relation. 3.2. Inflation, Extension and Azimuthal Shearing of an Annular Wedge Another interesting example is the nonhomogeneous deformation r = AR, () = C8, z = FZ, (19) which is a particular case of the Singh-Pipkin universal deformations with constant invariants. Here (r, (), z) are coordinates of a point in the present configuration that occupied the place (R, 8, Z) in the reference configuration. We assume that the axis of transverse isotropy is perpendicular to the z-axis. Physical components of the left Cauchy-Green tensor . are given by ( A2 0 0 ) , B = 0 A2C2 0 . (20) 0 0 F2 For an incompressible material constants A, C and F are related by A2CF = 1. (21) The deformation (19) describes the inflation, extension and azimuthal shearing of an annular wedge and has been studied in [12]. The universal relation in terms of the physical components of T and n that can be obtained using the Hayes and Knops method is