Stress characterization of elastodynamics for an inhomogeneous transversely isotropic infinite cylinder under plane strain conditions

Abstract

Stress characterization of isothermal elastodynamics for an inhomogeneous transversely isotropic infinite cylinder under plane strain conditions is presented. The cylinder, referred to the Cartesian coordinates xi (i=1,2,3) and denoted here by Bi(3), is identified with an infinite solid cylinder of which the rotational symmetry axis coincides with the x3-axis, and the geometrical axis coincides with the xi-axis (i=1,2,3); and the plane strain conditions exist in the plane xi=0, (i=1,2,3). A cross-section of the cylinder Bi(3) with the plane xi=0 is denoted by Ci(3). It is shown that a 3D stress wave Sij (i,j=1,2,3) propagating in the cylinder Bi(3) is generated by a solution to a 2D pure stress initial-boundary value problem for Ci(3), and a uniqueness theorem for the 2D problem is established. In particular, a pure stress initial boundary value problem for C3(3) involving only two (out of five) elastic moduli: c11 and c12 is formulated, and it is shown that the problem accommodates two types of the surface stress wave problems for a transversely isotropic semi-space with a traction free boundary and with an inhomogeneity depending on its depth. The first type is obtained when C3(3) is the semi-space: x1<∞, 0<x2<∞; and the density ρ, and moduli c11, and c12 depend on x2 only, while the second type is obtained when C3(3) is the semi-space: 0<x1<∞, x2<∞; and the density ρ, and moduli c11, and c12 depend on x1 only. Also, it is shown that the two pure stress initial boundary value problems for the cross-sections C2(3) and C1(3) involve only four (out of five) elastic moduli c11, c33, c13 and c44; and since the (x1,x2) is a plane of isotropy, a formulation for C1(3) may be obtained from that of C2(3) by suitable transformation of indices for the strain and stress tensor fields. Also, it is shown that each of the two problems accommodates two types of the surface stress wave problems for a transversely isotropic semi-space with a traction-free boundary and with an inhomogeneity depending on its depth.The results obtained should prove useful for both the analytical and numerical studies of the surface stress waves in an inhomogeneous transversely isotropic elastic semi-space under plane strain conditions.