Coupled thermoelastic theory and associated variational principles based on decomposition of internal energy

Abstract

A new framework of coupled thermoelasticity is constructed in this paper based on a decomposition of internal energy into free internal energy and dissipative energy. Fundamental equations (i.e. divergence and gradient equations, constitutive equations including evolving laws) along with proper boundary conditions are all included in this framework, from which two novel dual-complementary variational principles with explicit functionals are proposed. Unlike the conventional thermoelastic theory, the proposed variational principles for thermoelasticity are established without the approximate assumption of small temperature changes. To verify the usefulness of the proposed variational principles in simulating coupled thermoelastic problems, a coupled thermoelastic example is numerically implemented.

Appendices

Appendix A

The constitutive relation in conventional coupled thermoelasticity is expressed as [8]

$$\left\{ \begin{aligned} {\dot{\varvec{\sigma }}} & = {\varvec{C}}:{\dot{\varvec{\varepsilon }}} - {\varvec{\beta}}\dot{T}, \\ \dot{\eta } & = {\varvec{\beta}}:{\dot{\varvec{\varepsilon }}} + {{c\dot{T}} \mathord{\left/ {\vphantom {{c\dot{T}} {T,}}} \right. \kern-0pt} {T,}} \\ \end{aligned} \right.$$

(A.1)

where $\dot{\eta } = {{\left( {\dot{e} - {\varvec{\upsigma}}:{\dot{\varvec{\varepsilon }}}} \right)} \mathord{\left/ {\vphantom {{\left( {\dot{e} - {\varvec{\upsigma}}:{\dot{\varvec{\varepsilon }}}} \right)} T}} \right. \kern-0pt} T}$ is the total entropy density [8]. Hence, the irreversible entropy $\eta^{\text{i}} = \eta - \eta^{\text{r}}$ renders $T\dot{\eta }^{\text{i}} = \dot{\psi } = {\varvec{J}}_{\text{q}} \cdot {\varvec{X}}_{\text{q}}$, according to Eq. (4). ${\varvec{C}} = \frac{{\partial^{2} h}}{{\partial {\varvec{\upvarepsilon}}\partial {\varvec{\upvarepsilon}}}}$ denotes the stiffness, $c = - T\frac{{\partial^{2} h}}{\partial T\partial T}$ is the heat capacity, and coupling coefficient ${\varvec{\upbeta}} = - \frac{{\partial^{2} h}}{{\partial {\varvec{\upvarepsilon}}\partial T}}$ where $h = e - T\eta$ is the Helmholtz free energy density, rendering $\dot{h} = \dot{h}\left( {{\varvec{\upvarepsilon}},T} \right) = {\varvec{\upsigma}}:{\dot{\varvec{\varepsilon }}} - \eta \dot{T}$ [22, 23].

One can prove that (see Appendix B for details)

$$\left\{ \begin{aligned} {\varvec{\beta}} & = - \frac{{C_{\text{q}}^{\text{r}} }}{T}{\varvec{\beta^{\prime}}}, \\ {\varvec{C}} &= {\varvec{C^{\prime}}} - \frac{{C_{\text{q}}^{\text{r}} }}{T}{\varvec{\beta^{\prime}\beta^{\prime}}} = {\varvec{C^{\prime}}} + {\varvec{\beta \beta^{\prime}}}, \\ \end{aligned} \right.$$

(A.2)

so that Eq. (14) can be rewritten as

$$\left\{ \begin{aligned} {\dot{\varvec{\sigma }}} & = {\varvec{C^{\prime}}}:{\dot{\varvec{\varepsilon }}} + \frac{{C_{\text{q}}^{\text{r}} }}{T}{\varvec{\beta^{\prime}}}\left( {\dot{T} - {\varvec{\beta^{\prime}}}:{\dot{\varvec{\varepsilon }}}} \right) = {\varvec{C}}:{\dot{\varvec{\varepsilon }}} - {\varvec{\beta}}\dot{T}, \\ \dot{\eta }^{\text{r}} & = - \frac{{C_{\text{q}}^{\text{r}} }}{T}{\varvec{\beta^{\prime}}}:{\dot{\varvec{\varepsilon }}} + \frac{{C_{\text{q}}^{\text{r}} }}{T}\dot{T} = {\varvec{\beta}}:{\dot{\varvec{\varepsilon }}} + \frac{{C_{\text{q}}^{\text{r}} }}{T}\dot{T}, \hfill \\ \end{aligned} \right.$$

(A.3)

which yields the first equation in (A.1). Moreover, we have from Eqs. (A.3)₂ and (A.1)₂

$$\dot{\eta }^{\text{i}} = \dot{\eta } - \dot{\eta }^{\text{r}} = {{C_{\text{q}}^{\text{i}} \dot{T}} \mathord{\left/ {\vphantom {{C_{\text{q}}^{\text{i}} \dot{T}} T}} \right. \kern-0pt} T} ,$$

(A.4)

where $C_{\text{q}}^{\text{i}} = c - C_{\text{q}}^{\text{r}}$, and it can be proved that (see Appendix C)

$$C_{\text{q}}^{\text{i}} = \frac{\partial \psi }{\partial T},$$

(A.5)

which measures the energy dissipated per unit of temperature increase and can be referred to as the capacity of irreversible dissipating heat or the irreversible heat capacity. Accordingly, $C_{\text{q}}^{\text{r}}$ can be referred to as the reversible heat capacity.

Hence, we conclude that constitutive relations (A.1) and (A.3) are inherently consistent.

Appendix B

According to $T = T\left( {{\varvec{\upvarepsilon}},\eta^{\text{r}} } \right)$, we have ${\text{d}}T = \left( {\frac{\partial T}{{\partial {\varvec{\upvarepsilon}}}}} \right)_{{\eta^{\text{r}} }} {\text{d}}{\varvec{\upvarepsilon}} + \left( {\frac{\partial T}{{\partial \eta^{\text{r}} }}} \right)_{{\varvec{\upvarepsilon}}} {\text{d}}\eta^{\text{r}}$ and

$$\left( {\frac{{\partial \eta^{\text{r}} }}{{\partial {\varvec{\upvarepsilon}}}}} \right)_{T} = - \frac{{\left( {\frac{\partial T}{{\partial {\varvec{\upvarepsilon}}}}} \right)_{{\eta^{\text{r}} }} }}{{\left( {\frac{\partial T}{{\partial \eta^{\text{r}} }}} \right)_{{\varvec{\upvarepsilon}}} }},$$

(B.1)

where the subscript outside the parenthesis indicates which variable is being held constant during differentiation. Hence, in terms of ${\varvec{C^{\prime}}}$, ${\varvec{\beta^{\prime}}}$ and $C_{\text{q}}^{\text{r}}$ [as defined in Eq. (14)], we have

$${\varvec{\beta}} = - \frac{{\partial^{2} h}}{{\partial {\varvec{\upvarepsilon}}\partial T}} = \left( {\frac{\partial }{{\partial {\varvec{\upvarepsilon}}}}\left( { - \frac{\partial h}{\partial T}} \right)_{{\varvec{\upvarepsilon}}} } \right)_{T} = \left( {\frac{\partial \eta }{{\partial {\varvec{\upvarepsilon}}}}} \right)_{T} = \left( {\frac{{\partial \eta^{\text{r}} }}{{\partial {\varvec{\upvarepsilon}}}}} \right)_{T} = - \frac{{\frac{{\partial^{2} e^{\text{r}} }}{{\partial {\varvec{\upvarepsilon}}\partial \eta^{\text{r}} }}}}{{\frac{{\partial^{2} e^{\text{r}} }}{{\partial \eta^{\text{r}} \partial \eta^{\text{r}} }}}} = - \frac{{C_{\text{q}}^{\text{r}} }}{T}{\varvec{\beta^{\prime}}} ;$$

(B.2)

then

$$\begin{aligned} {\varvec{C}} & = \left( {\frac{{\partial^{2} h}}{{\partial {\varvec{\upvarepsilon}}\partial {\varvec{\upvarepsilon}}}}} \right)_{T} = \left[ {\frac{{\partial^{2} \left( {e - T\eta } \right)}}{{\partial {\varvec{\upvarepsilon}}\partial {\varvec{\upvarepsilon}}}}} \right)_{T} = \left( {\frac{{\partial^{2} \left( {e^{\text{r}} + \psi - T\eta } \right)}}{{\partial {\varvec{\upvarepsilon}}\partial {\varvec{\upvarepsilon}}}}} \right]_{T} \\ & = \left( {\frac{{\partial^{2} e^{\text{r}} }}{{\partial {\varvec{\upvarepsilon}}\partial {\varvec{\upvarepsilon}}}}} \right)_{T} + \left( {\frac{{\partial^{2} \psi }}{{\partial {\varvec{\upvarepsilon}}\partial {\varvec{\upvarepsilon}}}}} \right)_{T} - T\left( {\frac{{\partial^{2} \eta }}{{\partial {\varvec{\upvarepsilon}}\partial {\varvec{\upvarepsilon}}}}} \right)_{T} \\ & = \left\{ {\frac{\partial }{{\partial {\varvec{\upvarepsilon}}}}\left[ {\left( {\frac{{\partial e^{\text{r}} }}{{\partial {\varvec{\upvarepsilon}}}}} \right)_{{\eta^{\text{r}} }} + T\left( {\frac{{\partial \eta^{\text{r}} }}{{\partial {\varvec{\upvarepsilon}}}}} \right)_{T} } \right]} \right\}_{T} - T\left[ {\frac{\partial }{{\partial {\varvec{\upvarepsilon}}}}\left( {\frac{\partial \eta }{{\partial {\varvec{\upvarepsilon}}}}} \right)_{T} } \right]_{T} \\ & = \frac{{\partial^{2} e^{\text{r}} }}{{\partial {\varvec{\upvarepsilon}}\partial {\varvec{\upvarepsilon}}}} + \frac{{\partial^{2} e^{\text{r}} }}{{\partial {\varvec{\upvarepsilon}}\partial \eta^{\text{r}} }}\left( {\frac{{\partial \eta^{\text{r}} }}{{\partial {\varvec{\upvarepsilon}}}}} \right)_{T} + T\left( {\frac{{\partial {\varvec{\upbeta}}}}{{\partial {\varvec{\upvarepsilon}}}}} \right)_{T} - T\left( {\frac{{\partial {\varvec{\upbeta}}}}{{\partial {\varvec{\upvarepsilon}}}}} \right)_{T} \\ & = {\varvec{C^{\prime}}} + {\varvec{\beta \beta^{\prime}}}, \\ \end{aligned}$$

(B.3)

where use is made of $\left( {\frac{{\partial \eta^{\text{r}} }}{{\partial {\varvec{\upvarepsilon}}}}} \right)_{T} = {\varvec{\beta}}$ from Eq. (B.2) and

$$\left[{\frac{{\partial e^{\text{r}} \left( {{\varvec{\upvarepsilon}},\eta^{\text{r}} } \right)}}{{\partial {\varvec{\upvarepsilon}}}}} \right]_{T} = \left( {\frac{{\partial e^{\text{r}} }}{{\partial {\varvec{\upvarepsilon}}}}} \right)_{{\eta^{\text{r}} }} + \left( {\frac{{\partial e^{\text{r}} }}{{\partial \eta^{\text{r}} }}} \right)_{{\varvec{\upvarepsilon}}} \left( {\frac{{\partial \eta^{\text{r}} }}{{\partial {\varvec{\upvarepsilon}}}}} \right)_{T} = \left( {\frac{{\partial e^{\text{r}} }}{{\partial {\varvec{\upvarepsilon}}}}} \right)_{{\eta^{\text{r}} }} + T\left( {\frac{{\partial \eta^{\text{r}} }}{{\partial {\varvec{\upvarepsilon}}}}} \right)_{T} ,$$

(B.4)

$$\left[ {\frac{\partial }{{\partial {\varvec{\upvarepsilon}}}}\left( {\frac{{\partial e^{\text{r}} }}{{\partial {\varvec{\upvarepsilon}}}}} \right)_{{\eta^{\text{r}} }} } \right]_{T} = \left[ {\frac{\partial }{{\partial {\varvec{\upvarepsilon}}}}\left( {\frac{{\partial e^{\text{r}} }}{{\partial {\varvec{\upvarepsilon}}}}} \right)_{{\eta^{\text{r}} }} } \right]_{{\eta^{\text{r}} }} + \left[ {\frac{\partial }{{\partial \eta^{\text{r}} }}\left( {\frac{{\partial e^{\text{r}} }}{{\partial {\varvec{\upvarepsilon}}}}} \right)_{{\eta^{\text{r}} }} } \right]_{{\varvec{\upvarepsilon}}} \left( {\frac{{\partial \eta^{\text{r}} }}{{\partial {\varvec{\upvarepsilon}}}}} \right)_{T} = \frac{{\partial^{2} e^{\text{r}} }}{{\partial {\varvec{\upvarepsilon}}\partial {\varvec{\upvarepsilon}}}} + \frac{{\partial^{2} e^{\text{r}} }}{{\partial {\varvec{\upvarepsilon}}\partial \eta^{\text{r}} }}\left( {\frac{{\partial \eta^{\text{r}} }}{{\partial {\varvec{\upvarepsilon}}}}} \right)_{T} ,$$

(B.5)

(Note: ${\varvec{\beta \beta^{\prime}}}$ represents dyadic production of ${\varvec{\beta}}$ and ${\varvec{\beta^{\prime}}}$, which is a fourth-order tensor.)

In conclusion,

$$\left\{ \begin{aligned} {\varvec{\beta}} & = - \frac{{C_{\text{q}}^{\text{r}} }}{T}{\varvec{\beta^{\prime}}}, \\ {\varvec{C}} & = {\varvec{C^{\prime}}} - \frac{{C_{\text{q}}^{\text{r}} }}{T}{\varvec{\beta^{\prime}\beta^{\prime}}} = {\varvec{C^{\prime}}} + {\varvec{\beta \beta^{\prime}}}, \\ \end{aligned} \right.$$

(B.6)

i.e. Eq. (A.2) holds true.

Appendix C

Recall that $c = - T\frac{{\partial^{2} h}}{\partial T\partial T}$ and $C_{\text{q}}^{\text{r}} = \frac{T}{{L_{\text{q}}^{\text{r}} }},L_{\text{q}}^{\text{r}} = \frac{{\partial^{2} e^{\text{r}} }}{{\partial \eta^{\text{r}} \partial \eta^{\text{r}} }}$; then

$$\begin{aligned} C_{\text{q}}^{\text{i}} & = c - C_{\text{q}}^{\text{r}} \\ &= -\, T\frac{{\partial^{2} h}}{\partial T\partial T} - \frac{T}{{L_{\text{q}}^{\text{r}} }} = - T\frac{{\partial^{2} h}}{\partial T\partial T} - \frac{T}{{\frac{{\partial^{2} e^{\text{r}} }}{{\partial \eta^{\text{r}} \partial \eta^{\text{r}} }}}} = - \,T\frac{{\partial^{2} \left( {e - T\eta } \right)}}{\partial T\partial T} - \frac{T}{{\frac{\partial }{{\partial \eta^{\text{r}} }}\frac{{\partial e^{\text{r}} }}{{\partial \eta^{\text{r}} }}}} \\ & = -\, T\left[ {\frac{{\partial^{2} e\left( {{\varvec{\upvarepsilon}},\eta } \right)}}{\partial T\partial T} - \frac{{\partial^{2} \left( {T\eta } \right)}}{\partial T\partial T}} \right] - \frac{T}{{\left( {\frac{\partial T}{{\partial \eta^{\text{r}} }}} \right)_{{\varvec{\upvarepsilon}}} }} \quad \left( {{\text{where}}{\kern 1pt} {\kern 1pt} :\left( {\frac{{\partial e^{\text{r}} }}{{\partial \eta^{\text{r}} }}} \right)_{{\varvec{\upvarepsilon}}} = T} \right) \\ &= - T\frac{\partial }{\partial T}\left[ {\left( {\frac{{\partial e\left( {{\varvec{\upvarepsilon}},\eta } \right)}}{\partial \eta }\frac{\partial \eta }{\partial T}} \right) - \left( {\eta + T\frac{\partial \eta }{\partial T}} \right)} \right]_{{\varvec{\upvarepsilon}}} - T\left( {\frac{{\partial \eta^{\text{r}} }}{\partial T}} \right)_{{\varvec{\upvarepsilon}}} \quad \left( {{\text{where}}{\kern 1pt} {\kern 1pt} :\left( {\frac{\partial T}{{\partial \eta^{\text{r}} }}} \right)_{{\varvec{\upvarepsilon}}} \left( {\frac{{\partial \eta^{\text{r}} }}{\partial T}} \right)_{{\varvec{\upvarepsilon}}} = 1} \right) \\ & = - T\left[ {\left( {\frac{{\partial^{2} e}}{\partial \eta \partial \eta }\frac{\partial \eta }{\partial T} - 1 - 1} \right)\left( {\frac{\partial \eta }{\partial T}} \right)_{{\varvec{\upvarepsilon}}} + \left( {\frac{\partial e}{\partial \eta } - T} \right)\left( {\frac{{\partial^{2} \eta }}{\partial T\partial T}} \right)_{{\varvec{\upvarepsilon}}} } \right] - T\left( {\frac{{\partial \eta^{\text{r}} }}{\partial T}} \right)_{{\varvec{\upvarepsilon}}} \\ & = T\left( {\frac{\partial \eta }{\partial T}} \right)_{{\varvec{\upvarepsilon}}} - T\left( {\frac{{\partial \eta^{\text{r}} }}{\partial T}} \right)_{{\varvec{\upvarepsilon}}} \quad \left( {{\text{where}}{\kern 1pt} {\kern 1pt} :\left( {\frac{\partial e}{\partial \eta }} \right)_{{\varvec{\upvarepsilon}}} = T,\left( {\frac{\partial T}{\partial \eta }} \right)_{{\varvec{\upvarepsilon}}} \left( {\frac{\partial \eta }{\partial T}} \right)_{{\varvec{\upvarepsilon}}} = 1} \right) \\ & = \left( {\frac{\partial e}{\partial \eta }} \right)_{{\varvec{\upvarepsilon}}} \left( {\frac{\partial \eta }{\partial T}} \right)_{{\varvec{\upvarepsilon}}} - \left( {\frac{{\partial e^{\text{r}} }}{{\partial \eta^{\text{r}} }}} \right)_{{\varvec{\upvarepsilon}}} \left( {\frac{{\partial \eta^{\text{r}} }}{\partial T}} \right)_{{\varvec{\upvarepsilon}}} = \left( {\frac{{\partial \left( {e - e^{\text{r}} } \right)}}{\partial T}} \right)_{{\varvec{\upvarepsilon}}} = \frac{\partial \psi }{\partial T}. \\ \end{aligned}$$

(C.1)

Taking advantage of Eqs. (5), (15) and $\psi = \psi \left( T \right)$, we have

$$C_{\text{q}}^{\text{i}} \dot{T} = \dot{\psi } = \dot{\phi } = {\varvec{K}}^{TT} \cdot {\varvec{X}}_{\text{q}} \cdot {\varvec{X}}_{\text{q}} .$$

(C.2)

Appendix D

Theorem

The first variation of $\varPi$ (or $\varPi^{ *}$ ) vanishes if and only if ${\varvec{u}},{\varvec{J}}_{\text{q}}$ (or ${\varvec{\upsigma}},{\kern 1pt} \;T$ ) are the exact solution of the BVP I (or BVP II).

Proof

• i. Necessity:

  
  

  Under $\delta \varPi = 0$ in its admissible states with Precondition I taken as the constraint conditions, we have

  $$\begin{aligned} \delta \varPi \left( {{\varvec{u}},{\varvec{J}}_{q} } \right) & = \delta E + \delta \varPhi - \delta W = 0 \\ & = \int_{\varOmega } {\left[ {\delta e^{\text{r}} + \delta \left( {{\varvec{\upxi}}_{\text{q}} \cdot {\varvec{X}}_{\text{q}} } \right) - \delta \phi^{ *} - \delta \left( {{\varvec{f}} \cdot {\varvec{u}}} \right)} \right]{\text{d}}V} \\ &\quad - \delta \int_{{\partial \varOmega_{\sigma } }} {{\bar{\varvec{T}}} \cdot {\varvec{u}}{\text{d}}S} + \delta \int_{{\partial \varOmega_{T} }} {\int_{t} {\bar{T}{\varvec{J}}_{\text{q}} \cdot {\varvec{n}}{\text{d}}t} {\text{d}}S} \\ & = \int_{\varOmega } {\left[ {{\varvec{\upsigma}}:\delta {\varvec{\upvarepsilon}} + T\delta \eta^{\text{r}} + {\varvec{X}}_{\text{q}} \cdot \delta {\varvec{\upxi}}_{\text{q}} - {\varvec{f}} \cdot \delta {\varvec{u}}} \right]{\text{d}}V} \\&= - \int_{{\partial \varOmega_{\sigma } }} {{\bar{\varvec{T}}} \cdot \delta {\varvec{u}}{\text{d}}S} + \int_{{\partial \varOmega_{T} }} {\int_{t} {\bar{T}\delta {\varvec{J}}_{\text{q}} \cdot {\varvec{n}}{\text{d}}t} {\text{d}}S} \\ & = \int_{\varOmega } {\left[ {\nabla \cdot \left( {{\varvec{\upsigma}} \cdot \delta {\varvec{u}}} \right) - \nabla \cdot {\varvec{\upsigma}} \cdot \delta {\varvec{u}} - \nabla \cdot \left( {T\delta \int_{t} {{\varvec{J}}_{\text{q}} {\text{d}}t} } \right)} \right.} \\ &\quad \left. { + \nabla T \cdot \delta \int_{t} {{\varvec{J}}_{q} {\text{d}}t} + {\varvec{X}}_{q} \cdot \delta {\varvec{\upxi}}_{q} - {\varvec{f}} \cdot \delta {\varvec{u}}} \right]{\text{d}}V \hfill \\ &\quad - \int_{{\partial \varOmega_{\sigma } }} {{\bar{\varvec{T}}} \cdot \delta {\varvec{u}}{\text{d}}S} + \int_{{\partial \varOmega_{T} }} {\int_{t} {\bar{T}\delta {\varvec{J}}_{\text{q}} \cdot {\varvec{n}}{\text{d}}t} {\text{d}}S} \\ & = \int_{\varOmega } {\left[ { - \left( {\nabla \cdot {\varvec{\upsigma}} + {\varvec{f}}} \right) \cdot \delta {\varvec{u}} + \left( {{\varvec{X}}_{q} + \nabla T} \right) \cdot \int_{t} {\delta {\varvec{J}}_{\text{q}} \cdot {\varvec{n}}{\text{d}}t} } \right]{\text{d}}V} \hfill \\ & \quad + \int_{{\partial \varOmega_{\sigma } }} {\left( {{\varvec{\upsigma}} \cdot {\varvec{n}} - {\bar{\varvec{T}}}} \right) \cdot \delta {\varvec{u}}{\text{d}}S} + \int_{{\partial \varOmega_{T} }} {\int_{t} {\delta {\varvec{J}}_{\text{q}} \cdot {\varvec{n}}{\text{d}}t} \left( {\bar{T} - T} \right){\text{d}}S} . \\ \end{aligned}$$

  (D.1)

  Considering the arbitrariness of $\delta {\varvec{u}},\delta {\varvec{J}}_{\text{q}} ,$$\delta \varPi = 0$ requires the following equations

  $$\left\{ \begin{aligned} \nabla \cdot {\varvec{\upsigma}} + {\varvec{f}} = 0 \\ {\varvec{X}}_{\text{q}} + \nabla T = 0 \\ \end{aligned} \right.\;\left( {{\text{in}}\;\;\varOmega } \right) \quad \left\{ \begin{aligned} {\varvec{\upsigma}} \cdot {\varvec{n}} = {\bar{\varvec{T}}} \\ T = \bar{T} \\ \end{aligned} \right.\;\left( {{\text{on}}\;\;\partial \varOmega } \right),$$

  (D.2)

  i.e. Eqs. (16₂–17)₁ and Eqs. (18₂–19)₁. Under prescribed Eqs. (14–16)₁ and Eq. (17)₂, Eq. (D.2) can be rewritten as

  $$\left\{ \begin{aligned} \nabla \cdot \left( {{\varvec{C}}:\nabla {\dot{\varvec{u}}}} \right) + \nabla \cdot \left( {{\varvec{\upbeta}}\dot{T}} \right) + {\dot{\varvec{f}}}{ = }0 \hfill \\ C_{\text{q}}^{\text{r}} \dot{T} = T\nabla \cdot \left( {{\varvec{K}}^{TT} \cdot \nabla T} \right) - T{\varvec{\upbeta}}:{\dot{\varvec{\varepsilon }}} \hfill \\ \end{aligned} \right.\;\left( {{\text{in}}\;\;\varOmega } \right),\;\;\;\;\;\;\left\{ \begin{aligned} &{\varvec{\upsigma}} \cdot {\varvec{n}} = {\bar{\varvec{T}}} \hfill \\& T = \bar{T} \hfill \\ \end{aligned} \right.\;\left( {{\text{on}}\;\;\partial \varOmega } \right),$$

  (D.3)

  which is exactly the boundary value problem described by governing equations (Eqs. (20) and (23)) and boundary conditions (Eqs. (18)₂ and (19)₁), i.e. BVP I.

  Thus, the necessary condition for $\delta \varPi = 0$ is that ${\varvec{u}},{\varvec{J}}_{\text{q}}$ are the exact solution of the boundary value problem (D.3).

• ii. Sufficiency:

  When ${\varvec{u}},{\varvec{J}}_{\text{q}}$ are the exact solution of the boundary value problem (D.3), according to Eq. (D.1), we instantly have $\delta \varPi = 0$, i.e. the fact that ${\varvec{u}},{\varvec{J}}_{\text{q}}$ are the exact solution of BVP I is the sufficient condition for $\delta \varPi = 0$.

  Similarly, in the admissible states with Precondition II being the constraint conditions, then

  $$\begin{aligned} \hfill \delta \varPi^{ *} \left( {{\varvec{\upsigma}},T} \right) = \int_{\varOmega } {\left\{ {\left[ {{\varvec{\upvarepsilon}} - \frac{1}{2}\left( {\nabla {\varvec{u}} + {\varvec{u}}\nabla } \right)} \right]:\delta {\varvec{\upsigma}} + \int_{t} {\left( {\dot{\eta }^{\text{r}} { + }\nabla \cdot {\varvec{J}}_{q} } \right){\text{d}}t} \delta T} \right\}{\text{d}}V} \\ - \int_{{\partial \varOmega_{u} }} {\left( {{\bar{\varvec{u}}} - {\varvec{u}}} \right) \cdot \delta {\varvec{\upsigma}} \cdot {\varvec{n}}{\text{d}}S} - \int_{{\partial \varOmega_{{J_{\text{q}} }} }} {\int_{t} {\left( {{\varvec{J}}_{\text{q}} \cdot {\varvec{n}} - \bar{J}_{q} } \right){\text{d}}t\delta } T{\text{d}}S} . \\ \end{aligned}$$

  (D.4)

  Hence, the fact that ${\varvec{\upsigma}},{\kern 1pt} \;T$ are the exact solution of the BVP II is the necessary and sufficient condition for $\delta \varPi^{ *} = 0$.

  To sum up, the first variation of $\varPi$ (or $\varPi^{ *}$) vanishes if and only if ${\varvec{u}},{\varvec{J}}_{\text{q}}$ (or ${\varvec{\upsigma}},{\kern 1pt} \;T$) are the exact solution of the BVP I, i.e. Eq. (D.3) (or BVP II). Thus, the theorem is proved.