Nonlinear Waves in Solid Continua with Finite Deformation

Abstract

This work considers initiation of nonlinear waves, their propagation, reflection, and their interactions in thermoelastic solids and thermoviscoelastic solids with and without memory. The conservation and balance laws constituting the mathematical models are derived for finite deformation and finite strain using second Piola-Kirchoff stress tensor and Green's strain tensor. The constitutive theories for thermoelastic solids express the second Piola-Kirchoff stress tensor as a linear function of the Green's strain tensor. In the case of thermoviscoelastic solids without memory, the constitutive theory for deviatoric second Piola-Kirchoff stress tensor consists of a first order rate theory in which the deviatoric second Piola-Kirchoff stress tensor is a linear function of the Green's strain tensor and its material derivative. For thermoviscoelastic solids with memory, the constitutive theory for deviatoric second Piola-Kirchoff stress tensor consists of a first order rate theory in which the material derivative of the deviatoric second Piola-Kirchoff stress is expressed as a linear function of the deviatoric second Piola-Kirchoff stress, Green's strain tensor, and its material derivative. For thermoviscoelastic solids with memory, the constitutive theory for deviatoric second Piola-Kirchoff stress tensor consists of a first order rate theory in which the material derivative of the deviatoric second Piola-Kirchoff stress is expressed as a linear function of the deviatoric second Piola-Kirchoff stress, Green's strain tensor, and its material derivative. Fourier heat conduction law with constant conductivity is used as the constitutive theory for heat vector. The mathematical models are derived using conservation and balance laws. Alternate forms of the mathematical models are presented and their usefulness is illustrated in the numerical studies of the model problems with different boundary conditions. Nondimensionalized mathematical models are used in the computations of the numerical solutions of the model problems. All numerical studies are performed using space-time variationally consistent finite element formulations derived using space-time residual functionals in which the second variation of the residuals is neglected in the second variation of the residual functional and the non-linear equations resulting from the first variation of the residual functional are solved using Newton's Linear Method (Newton-Raphson method) with line search. Space-time local approximations are considered in higher order scalar product spaces that permit desired order of global differentiability in space and time. Extensive numerical studies are presented for different boundary conditions. Computed results for non-linear wave propagation, reflection, and interaction are compared with linear wave propagation to demonstrate significant differences between the two, the importance of the nonlinear wave propagation over linear wave propagation as well as to illustrate the meritorious features of the mathematical models and the space-time variationally consistent space-time finite element process with time marching in obtaining the numerical solutions of the evolutions.