On surface instability of incompressible elastic layer exposed to the finite initial deformation

Abstract

The problem of normal waves propagation in a pre-deformed incompressible elastic layer is considered. To study the propagation of Lamb waves in a elastic layer, we will use prestressed body model and the three-dimensional linearized equations of finite deformations for the elastic body. We will use a problem formulation and a method that are based on the general solutions of the linearized equations of motion of a prestressed body. Using the Fourier method, we arrive at the eigenvalue problem for the equation of motion of an elastic body. Solving it, we determine the corresponding eigenfunctions. After substituting the obtained general solutions in the boundary conditions, we obtain a homogeneous system of linear algebraic equations with respect to arbitrary constants. Based on the condition for the existence of a nontrivial solution to this system, we obtain the dispersion equation. A dispersion equation, which describes propagation of harmonic waves in elastic layer in a wide range of frequencies is obtained. On the basis of three-dimensional linearized equations of the elasticity theory of finite deformations for a incompressible elastic layer the dispersion curves of Lamb normal waves are constructed in a wide range of frequencies. The influence of finite initial deformations in an incompressible elastic layer on phase velocities, dispersion of the Lamb modes and surface instability is analyzed. It follows from the graphical material presented that in the case of compression with when shortening equal 0.54, i.e., with a 46 percent’s reduction in the length of the highly elastic incompressible body, the phase velocities of the surface waves vanish. This indicates that surface instability develops at when shortening equal 0.54 for a highly elastic incompressible non-Hookean body initially in a plane stress-strain state. We should point out that these figures agree with results obtained earlier in the theory of stability and correspond to the critical value of the contraction parameter. In the case of highly elastic incompressible bodies, linearized wave theory makes it possible to study not only general and several specific wave processes, but also the conditions under which surface instability begins in elastic bodies. The numerical results are presented in the form of graphs and their analysis is given.