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On the thermomechanics of solids surrounded by liquid media: balance equations, free energy and nonlinear diffusion


External media surround almost all structures and bodies. Caused by thermodynamical reasons, the ambient medium, a fluid or a gas in this essay, is diffusing into or out of the solid. In consequence, the solid experiences changes in mass, volume and material properties. In order to formulate the thermodynamical balance equations for open systems of this type, it is taken into account that the diffusing liquid carries mass, linear and angular momentum, internal energy and entropy into or out of the solid. In this study, chemical reactions are omitted. All equations to be developed in this work are formulated in dependence on the spatial coordinates of the material points of the solid in the current configuration. In order to derive thermodynamically motivated models for the stress tensor, the specific entropy, the chemical potential and the fluid flux vector, an exemplary constitutive model for the Helmholtz free energy per unit mass is formulated and the second law of thermodynamics for open systems is evaluated by standard methods. Numerical simulations and analytical computations of a simple model problem display some fundamental properties of the theory.

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Orthonormal unit vectors:

\( {\mathop {\mathbf{e }}\limits ^\rightharpoonup {}}_1, {\mathop {\mathbf{e }}\limits ^\rightharpoonup {}}_2, {\mathop {\mathbf{e }}\limits ^\rightharpoonup {}}_3\)


\( {\mathop {\mathbf{a }}\limits ^\rightharpoonup {}}=\sum \limits _{{k}=1}^3 {a_{{k}} {\mathop {\mathbf{e }}\limits ^\rightharpoonup {}}_{{k}}} \)

Tensor of second order:

\(\mathbf{A}=\sum \limits _{{i,k}=1}^3 {A_{{ik}} {\mathop {\mathbf{e }}\limits ^\rightharpoonup {}}_{{i}} \otimes {\mathop {\mathbf{e }}\limits ^\rightharpoonup {}}_{{k}} } \)

Transpose of a tensor:

\(\mathbf{A}^{\mathrm {T}}=\sum \limits _{{i,k}=1}^3 {A_{{ki}} {\mathop {\mathbf{e }}\limits ^\rightharpoonup {}}_{{i}} \otimes {\mathop {\mathbf{e }}\limits ^\rightharpoonup {}}_{{k}} } \)

Unit tensor:

\(\mathbf{1}=\sum \limits _{{k}=1}^3 { {\mathop {\mathbf{e }}\limits ^\rightharpoonup {}}_{{k}} \otimes {\mathop {\mathbf{e }}\limits ^\rightharpoonup {}}_{{k}} } \)

Inverse of a tensor:

\(\mathbf{A}^{-1}\), \(\mathbf{A}^{-1}{} \mathbf{A}=\mathbf{AA}^{-1}=\mathbf{1}\)

Scalar product between vectors:

\( {\mathop {\mathbf{a }}\limits ^\rightharpoonup {}}\cdot {\mathop {\mathbf{b }}\limits ^\rightharpoonup {}}=\sum \limits _{{k}=1}^3 {a_{{k}} b_{{k}} } \)

Vector product:

\( {\mathop {\mathbf{a }}\limits ^\rightharpoonup {}}\times {\mathop {\mathbf{b }}\limits ^\rightharpoonup {}}=\left( {a_{\mathrm{2}} b_{\mathrm{3}} -a_3 b_2 } \right) {\mathop {\mathbf{e }}\limits ^\rightharpoonup {}}_1 +\left( {a_{\mathrm{3}} b_{\mathrm{1}} -a_1 b_3 } \right) {\mathop {\mathbf{e }}\limits ^\rightharpoonup {}}_2 +\left( {a_{\mathrm{1}} b_{\mathrm{2}} -a_2 b_1 } \right) {\mathop {\mathbf{e }}\limits ^\rightharpoonup {}}_3 \)

Dyadic product:

\(\left( { {\mathop {\mathbf{a }}\limits ^\rightharpoonup {}}\otimes {\mathop {\mathbf{b }}\limits ^\rightharpoonup {}}} \right) {\mathop {\mathbf{c }}\limits ^\rightharpoonup {}}= {\mathop {\mathbf{a }}\limits ^\rightharpoonup {}}\left( { {\mathop {\mathbf{b }}\limits ^\rightharpoonup {}}\cdot {\mathop {\mathbf{c }}\limits ^\rightharpoonup {}}} \right) \)

Tensor algebraic rule:

\(\left( {\mathbf{1}\times {\mathop {\mathbf{a }}\limits ^\rightharpoonup {}}} \right) {\mathop {\mathbf{b }}\limits ^\rightharpoonup {}}= {\mathop {\mathbf{a }}\limits ^\rightharpoonup {}}\times {\mathop {\mathbf{b }}\limits ^\rightharpoonup {}}\)

Scalar product between tensors:

\(\mathbf{A}:\mathbf{B}=\sum \limits _{{k}=1}^3 {A_{{ik}} B_{{ik}} } \)

Divergence of a vector:

\(\hbox {div}\left( {\mathop {\mathbf{u }}\limits ^\rightharpoonup {}}\left( {\mathop {\mathbf{x }}\limits ^\rightharpoonup {}}_{\mathrm{S}} \right) \right) =\sum \limits _{{k}=1}^3 {\frac{\partial u_{{k}} }{\partial x_{\mathrm {Sk}}}} \)

Divergence of a tensor:

\(\hbox {div}\left( \mathbf{A}\left( {\mathop {\mathbf{x }}\limits ^\rightharpoonup {}}_{\mathrm{S}} \right) \right) =\sum \limits _{{k}=1}^3 {\frac{\partial A_{{ik}} }{\partial x_{\mathrm {Sk}}} {\mathop {\mathbf{e }}\limits ^\rightharpoonup {}}_{{i}} } \)

Differentiation rules:

\(\hbox {div}\left( {\mathop {\mathbf{x }}\limits ^\rightharpoonup {}}_{\mathrm{S}} \times \mathbf{A}\left( {\mathop {\mathbf{x }}\limits ^\rightharpoonup {}}_{\mathrm{S}} \right) \right) =\sum \limits _{{r,k=1}}^3 {A_{{rk}} {\mathop {\mathbf{e }}\limits ^\rightharpoonup {}}_{{k}} \times {\mathop {\mathbf{e }}\limits ^\rightharpoonup {}}_{{r}} } + {\mathop {\mathbf{x }}\limits ^\rightharpoonup {}}_{\mathrm{S}} \times \hbox {div}\left( \mathbf{A} \right) \) \(\hbox {div}\left( { {\mathop {\mathbf{a }}\limits ^\rightharpoonup {}}\otimes {\mathop {\mathbf{b }}\limits ^\rightharpoonup {}}} \right) = {\mathop {\mathbf{a }}\limits ^\rightharpoonup {}}\hbox {div}\left( { {\mathop {\mathbf{b }}\limits ^\rightharpoonup {}}} \right) +\hbox {grad}\left( { {\mathop {\mathbf{a }}\limits ^\rightharpoonup {}}} \right) {\mathop {\mathbf{b }}\limits ^\rightharpoonup {}}\) \(\hbox {div}\left( {\mathbf{A}^{\mathrm {T}} {\mathop {\mathbf{a }}\limits ^\rightharpoonup {}}} \right) =\hbox {div}\left( \mathbf{A} \right) \cdot {\mathop {\mathbf{a }}\limits ^\rightharpoonup {}}+\mathbf{A}:\hbox {grad}\left( { {\mathop {\mathbf{a }}\limits ^\rightharpoonup {}}} \right) \) \(\hbox {div}\left( {\varphi {\mathop {\mathbf{a }}\limits ^\rightharpoonup {}}} \right) =\varphi \hbox {div}\left( { {\mathop {\mathbf{a }}\limits ^\rightharpoonup {}}} \right) + {\mathop {\mathbf{a }}\limits ^\rightharpoonup {}}\cdot \hbox {grad}\left( \varphi \right) \) \( {\mathop {\mathbf{b }}\limits ^\rightharpoonup {}}\cdot \hbox {grad}\left( {{ {\mathop {\mathbf{a }}\limits ^\rightharpoonup {}}^{2}}/2} \right) =\left( \hbox {div}\left( { {\mathop {\mathbf{a }}\limits ^\rightharpoonup {}}\otimes {\mathop {\mathbf{b }}\limits ^\rightharpoonup {}}} \right) - {\mathop {\mathbf{a }}\limits ^\rightharpoonup {}}\hbox {div}\left( {\mathop {\mathbf{b }}\limits ^\rightharpoonup {}}\right) \right) \cdot {\mathop {\mathbf{a }}\limits ^\rightharpoonup {}}\)

Gradient of a vector:

\(\hbox {grad}\left( {\mathop {\mathbf{a }}\limits ^\rightharpoonup {}}\left( {\mathop {\mathbf{x }}\limits ^\rightharpoonup {}}_{\mathrm{S}} \right) \right) =\frac{\partial a_{{i}} }{\partial x_{\mathrm {Sk}}} {\mathop {\mathbf{e }}\limits ^\rightharpoonup {}}_{{i}} \otimes {\mathop {\mathbf{e }}\limits ^\rightharpoonup {}}_{{k}} \) \(\hbox {GRAD}\left( {\mathop {\mathbf{p }}\limits ^\rightharpoonup {}}\left( {\mathop {\mathbf{X }}\limits ^\rightharpoonup {}}_{\mathrm{S}} \right) \right) =\frac{\partial p_{{i}} }{\partial X_{\mathrm {Sk}}} {\mathop {\mathbf{e }}\limits ^\rightharpoonup {}}_{{i}} \otimes {\mathop {\mathbf{e }}\limits ^\rightharpoonup {}}_{{k}} \)


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Communicated by Michael Johlitz, Lucien Laiarinandrasana and Yann Marco.

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Lion, A., Johlitz, M. On the thermomechanics of solids surrounded by liquid media: balance equations, free energy and nonlinear diffusion. Continuum Mech. Thermodyn. 32, 281–305 (2020).

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  • Liquid diffusion in solids
  • Balance equations
  • Open system
  • Entropy of mixing
  • Swelling