The inhomogeneous strain distributions within finite cylinders under the axial point load tests and the effects on the valence band of Si_1-xGe_x alloy

An exact solution for inhomogeneous strain and stress distributions within a finite cubic isotropic cylinder of Si_1-xGe_x alloy under the axial Point Load Strength Test (PLST) is analytically derived. Lekhnitskii's stress function is used to uncouple the equations of equilibrium, and a new expression for the stress function is proposed so that all of the governing equations and boundary conditions are satisfied exactly. The solution for isotropic cylinders under the axial PLST is covered as a special case. Numerical results show that the strain and stress distributions in the central region within half height and radius are relatively homogeneous, but strain and stress concentrations are usually developed near the point loads. The largest tensile strain and stress are always induced along the line joining the point loads, which gives theoretical explanation why most of the cylindrical specimens are split apart along the line joining the point loads under the axial PLST. In addition, by using envelope-function method, the effect of strain on the valence-band structure of Si_1-xGe_x alloy is analyzed. It is found that strain changes the band quantum gap and the shape of constant energy surfaces of the heavy-hole and the light-hole bands of Si_1-xGe_x alloy.