Barnett-Lothe tensors and their associated tensors for monoclinic materials with the symmetry plane at x 3=0

Abstract

The three Barnett-Lothe tensors S, H, L and the three associated tensors S(θ), H(θ), L(θ) appear frequently in the real form solutions to two-dimensional anisotropic elasticity problems. Explicit expressions of the components of these tensors are derived and presented for monoclinic materials whose plane of material symmetry is at x 3=0. We use the algebraic formalism for these tensors but the results are derived not by the straight-forward substitution of the complex matrices A and B into the formulae. Instead, we find the product −AB -1, whose real and imaginary parts are SL -1 and L -1, respectively. The tensors S, H, L are then determined from SL -1 and L -1. For S(θ), H(θ), L(θ) we again avoid the direct substitution by employing an alternate approach. The new approaches require minimal algebra and, at the same time, provide simple and concise expressions for the components of these tensors. Although the new approaches can be extended, in principle, to monoclinic materials whose plane of symmetry is not at x 3=0 and to materials of general anisotropy, the explicit expressions for these materials are too complicated. More studies are needed for these materials.