Nonlinear Waves in Solid Continua with Finite Deformation

K. S. Surana¹, J. Knight¹, J. N. Reddy²

¹Department of Mechanical Engineering, University of Kansas, Lawrence, KS, USA.
²Department of Mechanical Engineering, Texas A & M University, College Station, TX, USA.

DOI: 10.4236/ajcm.2015.53032                

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.

Keywords

Linear and Nonlinear Waves, Second Piola-Kirchoff Stress, Green's Strain, Constitutive Theories, Dissipation, Memory, Rheology, Finite Strain

Conflicts of Interest

The authors declare no conflicts of interest.

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