Wave Propagation in Solids with Finite Deformation and Finite Strain

Abstract

This work investigates one dimensional wave propagation in thermoelastic and ther- moviscoelastic solids with and without memory. The work considers the solid matter to be compressible with finite deformation and finite strain. The mathematical model utilizes Contravariant second Piola-Kirchhoff stress and Green’s strain as work con- jugate pair in the conservation and balance laws. For thermoviscoelastic solids the second Piola-Kirchhoff stress is decomposed into equilibrium and deviatoric stress. The constitutive theory for deviatoric stress is expressed in terms of Greens’s strain tensor. The thermodynamic pressure in the constitutive theory for equilibrium second Piola-Kirchhoff stress is defined as a function of density using the published experi- mental data for rubber. In case of thermoelastic solids the constitutive theories consists of total second Piola-Kirchhoff stress as a function of Green’s strain tensor. The math- ematical model consisting of conservation, balance laws and the constitutive theories are first presented in R3, then explicitly given in R1 followed by the dimensionless form in R1 . The nonlinear partial differential equation describing 1D wave propaga- tion for finite deformation and finite strain are numerically solved using space-time finite element method based on space-time residual functional in which the local ap- proximation function for a space-time element are p-version hierarchical with higher order global differentiability in space and time. For an increment of time the solu- tion is computed for a space-time strip and then time marched to obtain the evolution for desired value of time. The solutions computed in the present work are compared with recently published work in which the thermodynamic pressure is approximated by mean normal stress.