A Semi-Analytical Finite Element Framework for Lamb Waves in Soft Compressible Plates Considering Strain Stiffening Effect

Abstract

This paper investigates the elastic wave propagation through soft materials that are being subjected to finite deformations. The nonlinear elastic and linearized incremental theories have been exploited to formulate governing wave equations and elastic moduli in Lagrangian space. Semi-analytical finite element (SAFE) method, a numerical approach has been formulated for computing dispersive relations of guided waves in compressible hyper-elastic plates. This framework requires finite element discretization of the cross section of the waveguide and harmonic exponential function assumes the motion along the wave propagation direction. Here, explicit phase velocity results have been shown for soft materials with a prominent stiffening effect by employing the Gent model, and these results are analyzed for elastic wave propagation through compressible materials. It has been noticed that Lamb waves have a strong dependence on the frequency-thickness product, prestretch, and direction of wave propagation. Moreover, with the strain stiffening effect, the dependence becomes stronger, especially for fundamental symmetric and anti-symmetric modes. The numerical results display that at certain prestretch the Gent material encounter snap-through instability resulting from geometrical and material nonlinearities. The influence of material properties like Gent constant and direction of wave propagation on snap-through instability has been discussed. The proposed SAFE framework reveals that finite deformations can affect elastic wave propagation through stiffness and compressibility.