Explicit expressions of Barnett-Lothe tensors and their associated tensors for orthotropic materials

Abstract

The three Barnett-Lothe tensors S, H, L , appear very often in the solutions to two-dimensional anisotropic elasticity problems. So do their associated tensors S ^ ( θ ) , H ^ ( θ ) , L ^ ( θ ) \hat S\left ( \theta \right ), \hat H\left ( \theta \right ), \hat L\left ( \theta \right ) for line forces and dislocations and S ( υ ) , H ( υ ) , L ( υ ) S\left ( \upsilon \right ), H\left ( \upsilon \right ), L\left ( \upsilon \right ) in the problem of surface waves. Explicit expressions of the components of these tensors are derived and presented for orthotropic materials in which the planes of material symmetry coincide with the coordinate planes. With minor modifications, the results for S, H, L and S ^ ( θ ) , H ^ ( θ ) , L ^ ( θ ) \hat S\left ( \theta \right ), \hat H\left ( \theta \right ), \hat L\left ( \theta \right ) can be applied to orthotropic materials in which only the x 3 = 0 {x_3} = 0 plane coincides with one of the planes of material symmetry.