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 as well as the constitutive theories are derived for finite deformation and finite strain using second Piola-Kirchoff stress tensor and Green’s strain tensor and their material derivatives [1]. Fourier heat conduction law with constant conductivity is used as the constitutive theory for heat vector. Numerical studies are performed using space-time variationally consistent finite element formulations derived using space-time residual functionals and the non-linear equations resulting from the first variation of the residual functional are solved using Newton’s Linear 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. 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.
Highlights
Linear and Nonlinear Waves, Second Piola-Kirchoff Stress, Green's Strain, Constitutive Theories, Dissipation, Memory, Rheology, Finite Strain
In the published works cited and discussed here we address four basic questions: 1) what is the source of nonlinearity, 2) type of material considered, 3) constitutive theories, 4) methodology or approach used to obtain numerical solution of the resulting mathematical model
The mathematical models are first presented in R3
Summary
The work presented here considers nonlinear wave propagation, reflection and interaction in thermoelastic solid continua and thermoviscoelastic solid continua with and without memory. Rate constitutive theory of order zero is used in which the contravariant second Piola-Kirchoff stress is a linear function of the Green’s strain tensor. The constitutive theory for deviatoric second Piola-Kirchoff stress tensor for thermoviscoelastic solids without memory is considered as a first order rate theory [1] in which the deviatoric second Piola-Kirchoff stress tensor is a linear function of the Green’s strain tensor and its material derivative. Explicit forms of the mathematical models are presented in R1 These models are used to study one dimensional nonlinear wave propagation, reflection, and interaction in the three types of solid continua considered here. Linear forms of these mathematical models based on small-strain small-deformation assumptions are considered in the numerical studies.
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