New explicit expression of Barnett-Lothe tensors for anisotropic linear elastic materials

The three Barnett-Lothe tensors H, L, S appear often in the Stroth formalism of two-dimensional deformations of anisotropic elastic materials [1-3]. They also appear in certain three-dimensional problems [4, 5] The algebraic representation of H, L, S requires computation of the eigenvalues p(upsilon)(upsilon 1, 2, 3) and the normalized eigenvectors (a,b). The integral representation of H, L, S circumvents the need for computing p(upsilon)(upsilon 1, 2, 3) and (a, b), but it is not simple to integrate the integrals except for special materials. Ting and Lee [6] have recently obtained an explicit expression of H for general anisotropic materials. We present here the remaining tensors L, S using the algebraic representation. The key to our success is the obtaining of the normalization factor for (a, b) in a simple form. The derivation of L and S then makes use of (a, b) but the final result does not require computation of (a, b), which makes the result attractive to numerical computation. Even though the tensor H given in [6] is in terms of the elastic stiffnesses C-mu upsilon while the tensors L, S presented here are in terms of the reduced elastic compliances s(mu upsilon)' the structure of L, S is similar to that of H. Following the derivation of H, we also present alternate expressions of L, S that remain valid for the degenerate cases p(1) = p(2) and p(1) = p(2) = p(3). One may want to compute H, L, S using either C-mu upsilon or s(mu upsilon)', but not both. We show how an expression in C-mu upsilon can be converted to an expression in s(mu upsilon)', and vice versa. As an application of the conversion, we present explicit expressions of the sextic equation for p in C-mu upsilon and s(mu upsilon)'.