Wave Propagation in Initially-stressed Compressible Hyperelastic Materials

Abstract

Study of elastic waves in solids has been an interesting area of research due to its enormous applications in fields as diverse as bio-mathematics, materials, earth sciences, to name a few. In linear elasticity, materials are not assumed to be stressed in their undeformed state but this is contrary to the real instances. In this thesis, non-linear theory of elasticity is used to study the propagation of surface waves in compressible materials, which are considered to be stressed in their initial state. In the presence of initial stress, general constitutive equations for compressible hyperelastic materials are developed. These equations are derived by using the theory of invariants, which depends upon initial stress and finite deformations. In general, for incompressible materials, this derivation involves nine invariants whereas for compressible materials it uses ten invariants. Making use of these invariants, a general form of elasticity tensor for compressible materials is derived. The equations governing infinitesimal motion superimposed on a finite deformation are used to observe the effect of initial stress and finite deformation on the surface wave propagation. In this research, a prototype strain energy function is used illustrate the significance of the results thus obtained. Moreover, in the absence of initial stress, findings are compared with classical results of linear theory. In the literature review given in the introduction of this thesis, it was found that various scientists have taken up the problems related to wave propagation in incompressible materials.However, wave propagation in initially-stressed compressible materials has not been studied todate due to the complex nature of compressible elastic materials. In this thesis, some novel problems related to wave propagation in initially-stressed compressible solids are presented. These include studies related to Love, Rayleigh and interfacial waves. This work will hopefully promote an interesting and challenging direction for theoretical and numerical analysis of initially-stressed compressible materials. It may also be applicable in a number of diverse fields of engineering, ranging from improvement of structural design to prediction of faults and many more. The problem of Love wave propagation in an initially-stressed compressible hyperelastic materials is considered and respective results are presented graphically for different strain energy functions to analyze the effect of initial stress on the wave speed. It is observed that when initial stress parameter is positive, wave speed increases and when it is negative, wave speed decreases. Similar studies are also carried out for Rayleigh and interfacial waves to observe the effect of initial stress. A detailed analysis of the secular equation in both cases is presented. The dimensionless form of the secular equation presented when wave speed is a function of deformation and initial stress. In the absence of initial stress, this equation is reduced to classical results. It is also found that certain conditions are imposed on the set of values for the governing parameters which guarantee the existence of a unique surface wave. For Rayleigh wave, a stability region is obtained which gives the admissible values of stretch ratios and initial stress components, for which solution is obtained in a particular interval. A special case of biaxial initial stress is considered to observe the effect of initial stress on interfacial wave propagation. It is further concluded that the effect of initial stress and governing parameters is considerable on the dimensionless wave speed, which is shown in several graphs using a specific but arbitrary strain energy function.