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 for malism of two-dimensional deformations of anisotropic elastic material s [1-3]. They also appear in certain three-dimensional problems [4, 5] The algebraic representation of H, L, S requires computation of the e igenvalues p(upsilon)(upsilon 1, 2, 3) and the normalized eigenvectors (a,b). The integral representation of H, L, S circumvents the need fo r computing p(upsilon)(upsilon 1, 2, 3) and (a, b), but it is not simp le to integrate the integrals except for special materials. Ting and L ee [6] have recently obtained an explicit expression of H for general anisotropic materials. We present here the remaining tensors L, S usin g the algebraic representation. The key to our success is the obtainin g of the normalization factor for (a, b) in a simple form. The derivat ion 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 nu merical computation. Even though the tensor H given in [6] is in terms of the elastic stiffnesses C-mu upsilon while the tensors L, S presen ted 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 derivat ion of H, we also present alternate expressions of L, S that remain va lid for the degenerate cases p(1) = p(2) and p(1) = p(2) = p(3). One m ay 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 conve rted to an expression in s(mu upsilon)', and vice versa. As an applica tion of the conversion, we present explicit expressions of the sextic equation for p in C-mu upsilon and s(mu upsilon)'.