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Thermodynamics of viscoelastic solids, its Eulerian formulation, and existence of weak solutions

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Abstract

The thermodynamic model of viscoelastic deformable solids at finite strains is formulated in a fully Eulerian way in rates. Also, effects of thermal expansion or buoyancy due to evolving mass density in a gravity field are covered. The Kelvin–Voigt rheology with a higher-order viscosity (exploiting the concept of multipolar materials) is used, allowing for physically relevant frame-indifferent stored energies and for local invertibility of deformation. The model complies with energy conservation and Clausius–Duhem entropy inequality. Existence and a certain regularity of weak solutions are proved by a Faedo–Galerkin semi-discretization and a suitable regularization. Subtle physical limitations of the model are illustrated on thermally expanding neo-Hookean materials or materials with phase transitions.

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Abbreviations

\({\varvec{v}}\) :

Velocity (in m/s),

\(\varrho \) :

Mass density (in kg/m\(^3\)),

\(\varrho _\textsc {r}^{}\) :

Referential mass density,

\({\varvec{F}}\) :

Deformation gradient,

\(\theta \) :

Temperature (in K)

\({\varvec{T}}\) :

Cauchy stress (symmetric, in Pa),

\(\mathcal {H}\) :

Hyper-stress (in Pa m),

\({\varvec{j}}\) :

Heat flux (in W/m\(^2\)),

\({\varvec{f}}\) :

Traction load,

\(\det (\cdot )\) :

Determinant of a matrix,

\(\textrm{Cof}(\cdot )\) :

Cofactor matrix,

\(\nu _\flat >0\) :

A boundary viscosity

\({\mathbb {R}}_\textrm{sym}^{d\times d}\) :

\(=\{A\in {\mathbb {R}}^{d\times d};\ A^\top =A\}\).

\(\psi =\psi ({\varvec{F}},\theta )\) :

Referential free energy (in J/m\(^3\)=Pa)

\(\varphi =\varphi ({\varvec{F}})\) :

Referential stored energy (in J/m\(^3\)=Pa),

\(\gamma =\gamma ({\varvec{F}},\theta )\) :

Referential heat part of free energy,

\({\varvec{e}}({\varvec{v}})=\frac{1}{2}{\nabla }{\varvec{v}}^\top \!{+}\frac{1}{2}{\nabla }{\varvec{v}}\) :

Symmetric velocity gradient (in s\(^{-1}\)),

\({\varvec{D}}={\varvec{D}}({\varvec{F}},\theta ;{\varvec{e}}({\varvec{v}}))\) :

Dissipative part of Cauchy stress,

\(w\) :

Heat part of internal energy (enthalpy, in J/m\(^3\)),

\((^{_{_{\bullet }}})\!\mathchoice{{\buildrel {\text {.}}\over {^{}}}}{{\buildrel {\text {.}}\over {^{}}}}{{\buildrel {\text {.}}\over {^{}}}}{{\buildrel {\text {.}}\over {^{}}}}=\frac{\partial {}}{\partial t}{^{_{_{\bullet }}}}+({\varvec{v}}{\cdot }{\nabla })^{_{_{\bullet }}}\) :

Convective time derivative,

\(\cdot \) or  : :

Scalar products of vectors or matrices,

\(\vdots \ \ \) :

Scalar products of third-order tensors,

\(\kappa =\kappa ({\varvec{F}},\theta )\) :

Thermal conductivity (in W/m\(^{-2}\)K\(^{-1}\)),

\(c=c({\varvec{F}},\theta )\) :

Heat capacity (in Pa/K),

\(\nu >0\) :

A bulk hyper-viscosity coefficient,

\({\varvec{g}}\) :

External bulk load (gravity acceleration in m/s\(^{2}\)).

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  1. Mathematical Institute, Charles University, Sokolovská 83 186 75 Praha 8, Czech Republic

    Tomáš Roubíček

  2. Institute of Thermomechanics of the Czech Academy of Sciences, Dolejškova 5 182 00 Praha 8, Czech Republic

    Tomáš Roubíček

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Roubíček, T. Thermodynamics of viscoelastic solids, its Eulerian formulation, and existence of weak solutions. Z. Angew. Math. Phys. 75, 51 (2024). https://doi.org/10.1007/s00033-023-02175-7

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