In this paper, we present mathematical models, methods of obtaining their solutions, and the model problem studies for wave propagation in compressible thermoelastic (TE) and thermoviscoelastic (TVE) solid media, i.e., this research addresses shock physics in compressible TVES without memory. The mathematical model consists of conservation and balance laws (CBL) of classical continuum mechanics (CCM) derived using the contravariant second Piola–Kirchhoff stress tensor and the convected time derivative of the covariant Green’s strain tensor up to order n. Constitutive theories are derived using conjugate pairs in the entropy inequality augmented with strain rates up to order n and the representation theorem. The dissipation mechanism in this theory is due to ordered rates of Green’s strain tensor up to order n. This mathematical model permits finite deformation, finite strain, as well as finite strain rate deformation physics and is thermodynamically and mathematically consistent. The solutions of the initial value problem (IVP) described by this mathematical model are obtained using a space-time coupled variationally consistent space-time finite element method based on the space-time residual functional for a space-time strip with time marching. The p-version hierarchical space-time local approximations of higher degree as well as higher-order global differentiability are considered in higher-order scalar product spaces. This permits accurate computations of a posteriori errors in the solution measured in the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11012_2024_1893_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$L_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> norm of the space-time residual functional and provides means of improving the accuracy of computed solutions. In the research presented in this paper, no assumptions or approximations are made regarding shock wave, shock structure, or its analytic (or non-analytic) nature. This work relies on the mathematical models and the computational infrastructure presented in this paper to reveal and simulate shock physics details: deviatoric stress wave, density wave, their propagations, reflections, and interactions in compressible TVES medium without rheology. Detailed model problem studies are presented to illustrate all aspects of the shock physics. To our knowledge, this work is not available in the published literature. This paper is the first presentation of this research.

Shock physics in compressible thermoelastic and thermoviscoelastic solids

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发表日期：

2025年2月26日

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K. S. Surana,
E. Abboud

摘要

In this paper, we present mathematical models, methods of obtaining their solutions, and the model problem studies for wave propagation in compressible thermoelastic (TE) and thermoviscoelastic (TVE) solid media, i.e., this research addresses shock physics in compressible TVES without memory. The mathematical model consists of conservation and balance laws (CBL) of classical continuum mechanics (CCM) derived using the contravariant second Piola–Kirchhoff stress tensor and the convected time derivative of the covariant Green’s strain tensor up to order n. Constitutive theories are derived using conjugate pairs in the entropy inequality augmented with strain rates up to order n and the representation theorem. The dissipation mechanism in this theory is due to ordered rates of Green’s strain tensor up to order n. This mathematical model permits finite deformation, finite strain, as well as finite strain rate deformation physics and is thermodynamically and mathematically consistent. The solutions of the initial value problem (IVP) described by this mathematical model are obtained using a space-time coupled variationally consistent space-time finite element method based on the space-time residual functional for a space-time strip with time marching. The p-version hierarchical space-time local approximations of higher degree as well as higher-order global differentiability are considered in higher-order scalar product spaces. This permits accurate computations of a posteriori errors in the solution measured in the $L_2$ $L_{2}$ norm of the space-time residual functional and provides means of improving the accuracy of computed solutions. In the research presented in this paper, no assumptions or approximations are made regarding shock wave, shock structure, or its analytic (or non-analytic) nature. This work relies on the mathematical models and the computational infrastructure presented in this paper to reveal and simulate shock physics details: deviatoric stress wave, density wave, their propagations, reflections, and interactions in compressible TVES medium without rheology. Detailed model problem studies are presented to illustrate all aspects of the shock physics. To our knowledge, this work is not available in the published literature. This paper is the first presentation of this research.