Stress Waves in a Microperiodic Layered Elastic Solid Revisited

A one-dimensional pure stress initial boundary value problem of linear elastodynamics for amicroperiodic layered semi-space in which amicrostructural length is taken into account is revisited. Also, the plane stress harmonic waves propagating in a microperiodic layered infinite elastic space are discussed. It is shown that (i) for a particular system of the length and time units the transient stress waves in the microperiodic layered semi-space are independent of the microstructural length, and (ii) there are two dispersive plane stress harmonic waves propagating in a microperiodic layered infinite elastic space. The graphs illustrating the transient stress waves in the semi-space and the dispersion of harmonic waves in the infinite space are included. 2000 Mathematics Subject Classification. 74H05, 74H25, 74H45, 74J05, 74Q10. 1. Basic field equations for a microperiodic layered elastic semi-space. Consider a layered semi-infinite elastic solid composed of an infinite number of identical subunits that are mechanically bonded to form a spatially periodic pattern as shown in Figure 1.1. Each subunit consists of two layers that, in general, have different dimensions and are made of different homogeneous isotropic elastic materials. Let li, ρi, λi, and μi (i= 1,2), respectively, denote the physical dimension, density, Lamé modulus, and shearmodulus of the ith layer in a subunit. If the interface conditions between any two adjacent layers are assumed to be of an ideal mechanical contact type, that is, the displacement and stress vectors are continuous across an interface, and a mechanical load is uniformly distributed over the boundary x = 0 for every time t ≥ 0, an elastic process in the layered semi-space can be described by a solution to a one-dimensional initial boundary value problem of classical elastodynamics. In such a problem the field equations of homogeneous isotropic elastodynamics are to be satisfied for each layer and suitable initial, interface, and boundary conditions at x = 0 and x =∞ are to be met. Since an exact solution to the problem is not feasible, the classical formulation for the layered semi-space is replaced by the approximate one of a refined average theory (RAT), (see [1, 2, 4]). The field equations of the approximate theory read: the h-approximation of the displacement field u(x,t)=U(x,t)+h(x)V(x,t). (1.1) The equations of motion Sx−〈ρ〉Utt = 0, H+ 〈 ρh 〉 Vtt = 0. (1.2) 654 J. IGNACZAK AND E. MRÓWKA-MATEJEWSKA x3 0 x2 ρ2,λ2,μ2 ρ1,λ1,μ1 l l1 l2