First, we require the elasticity tensors occurring in (13) and (14) to satisfy the following condition: there is a constant \(c_{0}>0\) so that
$$\begin{aligned}& C_{ijmn}u_{ij}u_{mn}+ 2G_{ijmn}u_{ij}v_{mn}+ 2F_{ijmnr}u_{ij}w_{mnr}+ B_{ijmn}v_{ij}v_{mn} \\& \quad {}+2D_{ijmnr}v_{ij} w_{mnr}+ A_{ijkmnr}w_{ijk}w_{mnr} \ge c_{0} ( u _{ij}u_{ij}+v_{ij}v_{ij}+w_{ijk}w_{ijk} ) \end{aligned}$$
(30)
for any three tensors \(u_{ij}, v_{ij}\) and \(w_{ijk}\).
A result regarding the uniqueness of solution to the problem \({\mathcal {P}}\) is given in the next theorem.
Theorem 1
Assume that
-
(i)
\(\varrho (x)>0, \beta (x)>0, T_{0}(x)>0, \forall x\in \Omega \);
-
(ii)
\(I_{ij}(x)\)
is a positive definite tensor;
-
(iii)
condition (30) takes place.
Then the problem
\({\mathcal {P}}\)
has at most one solution.
Proof
Suppose, by contradiction, that the mixed problem \({\mathcal {P}}\) would have two solutions \(( u_{i}^{(1)},\varphi_{ij} ^{(1)},\theta^{(1)} ) \) and \(( u_{i}^{(2)},\varphi_{ij}^{(2)}, \theta^{(2)} ) \). Because of linearity, their difference is also a solution to the problem \({\mathcal {P}}\). Let us use the notation \(u_{i}^{(d)}=u_{i}^{(2)}-u_{i}^{(1)}\), \(\varphi_{ij}^{(d)}=\varphi _{ij}^{(2)}-\varphi_{ij}^{(1)}\), \(\theta^{(d)}=\theta^{(2)}-\theta ^{(1)}\). Of course, \(( u_{i}^{(d)},\varphi_{ij}^{(d)},\theta^{(d)} ) \) satisfies the problem \({\mathcal {P}}\) in the particular case when all the elements of external data are zeros, the initial data are null and also with null boundary conditions. To obtain an estimate of the difference \(( u_{i}^{(d)},\varphi_{ij}^{(d)},\theta^{(d)} ) \), we will use the integral
$$ \int_{\Omega } ( \tau_{ij}\dot{\varepsilon }_{ij}+\sigma_{ij} \dot{\gamma }_{ij}+ \mu_{ijk}\dot{\chi }_{ijk} )\,dV. $$
Here and until the end of the demonstration, because there is no likelihood of confusion, we give up writing the superscript \(^{(d)}\) for all functions.
Using the equations of motion (8)2 and (8)3 for the difference solutions and applying the divergence theorem, because of null boundary conditions, we are led to
$$\begin{aligned} \int_{\Omega } ( \tau_{ij}\dot{\varepsilon }_{ij}+\sigma_{ij} \dot{\gamma }_{ij}+ \mu_{ijk}\dot{\chi }_{ijk} )\,dV=- \int_{ \Omega } ( \varrho \ddot{u}_{i} \dot{u}_{i}+I_{jk}\ddot{\varphi } _{js}\dot{\varphi }_{ks} )\,dV. \end{aligned}$$
(31)
This identity can be rewritten in the form
$$\begin{aligned}& \frac{1}{2} \frac{d}{dt} \biggl[ \int_{\Omega } \bigl( C_{ijmn} \varepsilon_{ij} \varepsilon_{mn}+ 2G_{ijmn}u\varepsilon_{ij}\gamma _{mn}+ 2F_{ijmnr}\varepsilon_{ij}\chi_{mnr} \\& \quad {}+B_{ijmn}\gamma_{ij}\gamma_{mn}+ 2D_{ijmnr} \gamma_{ij} \chi_{mnr}+ A _{ijkmnr}\chi_{ijk} \chi_{mnr} \\& \quad {} +\varrho \dot{u}_{i}\dot{u}_{i}+I_{jk} \dot{\varphi }_{js} \dot{\varphi }_{ks}+\beta \theta^{2} \bigr)\,dV \biggr] = \int_{\Omega} \dot{\eta }\theta \,dV \end{aligned}$$
(32)
upon adding \(\beta \theta \dot{\theta }\) and considering the constitutive equations (14) and the symmetry relations (15).
On the other hand, taking into account inequality (17) in the particular case \(q=0\) and the constitutive relation (14)5, we obtain
$$\begin{aligned} \int_{\Omega }\frac{1}{T_{0}}q_{i,i}\theta \,dV=- \int_{\Omega } \dot{\eta }\theta \ge 0, \end{aligned}$$
(33)
and this inequality with (32) lead to
$$\begin{aligned}& \frac{1}{2} \frac{d}{dt} \biggl\{ \int_{\Omega } \bigl( C_{ijmn} \varepsilon_{ij} \varepsilon_{mn}+ 2G_{ijmn}u\varepsilon_{ij}\gamma _{mn}+ 2F_{ijmnr}\varepsilon_{ij}\chi_{mnr} \\& \quad {}+B_{ijmn}\gamma_{ij}\gamma_{mn}+ 2D_{ijmnr} \gamma_{ij} \chi_{mnr}+ A _{ijkmnr}\chi_{ijk} \chi_{mnr} \\& \quad {}+\varrho \dot{u}_{i}\dot{u}_{i}+I_{jk} \dot{\varphi }_{js} \dot{\varphi }_{ks}+\beta \theta^{2} \bigr)\,dV\frac{}{} \biggr\} \le 0 \end{aligned}$$
(34)
for all \((x,t)\in \Omega \times [0, \infty )\).
Clearly, from (34) it easy to obtain \(\dot{u}_{i}=0\), \(\dot{\varphi } _{ij}=0\) and \(\theta =0\). As such, if we consider the fact that for difference of solutions, the initial conditions are null, we deduce \(u_{i}=0\) and \(\varphi_{ij}=0\), which concludes the proof of the theorem. □
Let us consider two different systems of external data acting on our dipolar body, namely
$$\begin{aligned} {\mathcal {S}}^{(\nu )}&= \bigl\{ F_{i}^{(\nu )} , G_{jk}^{(\nu )} , R^{(\nu )} , \tilde{u}_{i}^{(\nu )} , \tilde{\varphi }_{ij}^{(\nu )} , \tilde{\alpha }^{(\nu )}, \\ &\quad {}\tilde{t}_{i}^{(\nu )} , \tilde{m}_{ij} ^{(\nu )} , \tilde{q}^{(\nu )} , u_{i}^{0(\nu )} , u_{i}^{1(\nu )} , \varphi_{ij}^{0(\nu )} , \varphi_{ij}^{1(\nu )} , \theta^{0( \nu )} , \theta^{1(\nu )} \bigr\} , \end{aligned}$$
(35)
where \(\nu =1,2\). The solutions of mixed problem corresponding to each system of external data will be denoted by \(s^{(\nu )}\), that is,
$$ s^{(\nu )}= \bigl\{ u_{i}^{(\nu )}, \varphi_{ij}^{(\nu )}, \theta^{( \nu )}, \alpha^{(\nu )} \bigr\} , \quad \nu =1,2. $$
The link between charging systems \({\mathcal {S}}^{(\nu )}\) and corresponding solutions \(s^{(\nu )}\) is given in the following theorem.
Theorem 2
Between loading systems
\({\mathcal {S}}^{(\nu )}\)
and corresponding solutions
\(s^{(\nu )}\), the next reciprocal relation of Betti type holds
$$\begin{aligned}& \int_{\Omega } \biggl( \varrho F_{i}^{(1)}*u_{i}^{(2)}+ \varrho G_{ij} ^{(1)}*\varphi_{ij}^{(2)}-t*R^{(1)}* \theta^{(2)}-\frac{1}{T_{0}}t*q _{i}^{(1)}* \beta_{i}^{(2)} \biggr)\,dV \\& \qquad {}+ \int_{\partial \Omega } \biggl( t*t_{i}^{(1)}*u_{i}^{(2)}+t*m_{ij}^{(1)}* \varphi_{ij}^{(2)}+\frac{1}{T_{0}}t*q^{(1)}* \alpha^{(2)} \biggr)\,dA \\& \quad = \int_{\Omega } \biggl( \varrho F_{i}^{(2)}*u_{i}^{(1)}+ \varrho G_{ij} ^{(2)}*\varphi_{ij}^{(1)}-t*R^{(2)}* \theta^{(1)}-\frac{1}{T_{0}}t*q _{i}^{(2)}* \beta_{i}^{(1)} \biggr)\,dV \\& \qquad {}+ \int_{\partial \Omega } \biggl( t_{i}^{(2)}*u_{i}^{(1)}+m_{ij}^{(2)}* \varphi_{ij}^{(1)}+\frac{1}{T_{0}}t*q^{(2)}* \alpha^{(1)} \biggr)\,dA. \end{aligned}$$
(36)
Proof
With the help of constitutive equation (14)1 and symmetry relations (15), we can write
$$\begin{aligned}& \int_{\Omega } \bigl( t*\tau_{ij}^{(1)}* \varepsilon_{ij}^{(2)}-t*\tau _{ij}^{(2)}* \varepsilon_{ij}^{(1)} \bigr)\,dV \\& \quad = \int_{\Omega } \bigl( t*G_{ijmn}\gamma_{mn}^{(1)}* \varepsilon_{ij} ^{(2)}+t*F_{ijmnr}\chi_{mnr}^{(1)}* \varepsilon_{ij}^{(2)}- t*a_{ij} \theta^{(1)}\varepsilon_{ij}^{(2)} \bigr)\,dV \\& \quad \quad {}- \int_{\Omega } \bigl( t*G_{ijmn}\gamma_{mn}^{(2)}* \varepsilon_{ij}^{(1)}+t*F _{ijmnr}\chi_{mnr}^{(2)}* \varepsilon_{ij}^{(1)}- t*a_{ij}\theta^{(2)} \varepsilon_{ij}^{(1)} \bigr)\,dV. \end{aligned}$$
(37)
Also, with (14)2 and symmetry relations (15), we can write
$$\begin{aligned}& \int_{\Omega } \bigl( t*\sigma_{ij}^{(1)}* \gamma_{ij}^{(2)}-t*\sigma _{ij}^{(2)}* \gamma_{ij}^{(1)} \bigr)\,dV \\& \quad = \int_{\Omega } \bigl( t*G_{ijmn}\gamma_{ij}^{(1)}* \varepsilon_{mn} ^{(2)}+t*D_{ijmnr}\chi_{mnr}^{(1)}* \gamma_{ij}^{(2)}- t*b_{ij}\theta ^{(1)} \gamma_{ij}^{(2)} \bigr)\,dV \\& \quad\quad {}-\int_{\Omega } \bigl( t*G_{ijmn}\gamma_{mn}^{(2)}* \varepsilon_{ij}^{(1)}+t*D _{ijmnr}\chi_{mnr}^{(2)}* \gamma_{ij}^{(1)}- t*a_{ij}\theta^{(2)} \varepsilon_{ij}^{(1)} \bigr)\,dV. \end{aligned}$$
(38)
Similarly, with the help of (14)3 and symmetry relations (15), we can write
$$\begin{aligned}& \int_{\Omega } \bigl( t*\mu_{ijk}^{(1)}* \chi_{ijk}^{(2)}-t*\mu_{ijk} ^{(2)}* \chi_{ijk}^{(1)} \bigr)\,dV \\& \quad = \int_{\Omega } \bigl( t*F_{ijkmn}\chi_{ijk}^{(1)}* \varepsilon_{mn} ^{(2)}+t*D_{ijkmn}\chi_{ijk}^{(1)}* \gamma_{mn}^{(2)}- t*c_{ijk} \theta^{(1)} \chi_{ijk}^{(2)} \bigr)\,dV \\& \quad \quad {}- \int_{\Omega } \bigl( t*F_{ijkmn}\chi_{ijk}^{(2)}* \varepsilon_{mn} ^{(1)}+t*D_{ijkmn}\chi_{ijk}^{(2)}* \gamma_{mn}^{(1)}- t*c_{ijk} \theta^{(2)} \chi_{ijk}^{(1)} \bigr)\,dV. \end{aligned}$$
(39)
Now, we will use the constitutive equation for entropy, (14)4, in order to obtain
$$\begin{aligned}& \int_{\Omega } \bigl( t*\eta^{(1)}*\theta^{(2)}-t* \eta^{(2)}*\theta ^{(1)} \bigr)\,dV \\& \quad = \int_{\Omega } \bigl( t*a_{ij} \varepsilon_{ij}^{(1)}* \theta^{(2)}+t*b _{ij}\dot{\gamma }_{ij}^{(1)}* \theta^{(2)}+ t*c_{ijk}\dot{\chi }_{ijk} ^{(1)}*\theta^{(2)} \bigr)\,dV \\& \quad \quad {}- \int_{\Omega } \bigl( t*a_{ij} \varepsilon_{ij}^{(2)}* \theta^{(1)}+t*b _{ij}\dot{\gamma }_{ij}^{(2)}* \theta^{(1)}+ t*c_{ijk}\dot{\chi }_{ijk} ^{(2)}*\theta^{(1)} \bigr)\,dV. \end{aligned}$$
(40)
Finally, if we add member by member equalities (37), (38) and (39) and then subtract equality (40), we obtain
$$\begin{aligned}& \int_{\Omega } \bigl( t*\tau_{ij}^{(1)}* \varepsilon_{ij}^{(2)}-t*\tau _{ij}^{(2)}* \varepsilon_{ij}^{(1)} +t*\sigma_{ij}^{(1)}* \gamma_{ij} ^{(2)}-t*\sigma_{ij}^{(2)}* \gamma_{ij}^{(1)} \\& \quad {}+t*\mu_{ijk}^{(1)}*\chi_{ijk}^{(2)}-t* \mu_{ijk}^{(2)}*\chi_{ijk} ^{(1)} -t* \eta^{(1)}*\theta^{(2)}+t*\eta^{(2)}* \theta^{(1)} \bigr)\,dV=0, \end{aligned}$$
and, obviously, this equality can be rewritten in the form
$$\begin{aligned}& \int_{\Omega } \bigl( t*\tau_{ij}^{(1)}* \varepsilon_{ij}^{(2)} +t* \sigma_{ij}^{(1)}* \gamma_{ij}^{(2)} +t*\mu_{ijk}^{(1)}* \chi_{ijk}^{(2)} -t*\eta^{(1)}*\theta^{(2)} \bigr)\,dV \\& \quad = \int_{\Omega } \bigl( t*\tau_{ij}^{(2)}* \varepsilon_{ij}^{(1)} +t* \sigma_{ij}^{(2)}* \gamma_{ij}^{(1)} +t*\mu_{ijk}^{(2)}* \chi_{ijk}^{(1)} -t*\eta^{(2)}*\theta^{(1)} \bigr)\,dV. \end{aligned}$$
(41)
In (41) we will consider the equations of motion (8)2, (8)3, the energy equation (22) and the kinetic relations (1). Then we apply the divergence theorem and the boundary conditions (24) so that we are led to the identity
$$\begin{aligned}& \int_{\Omega } \bigl( \varrho F_{i}^{(1)}*u_{i}^{(2)}+ \varrho G_{ij} ^{(1)}*\varphi_{ij}^{(2)}-t* R^{(1)}*\theta^{(2)}-t* h_{i}^{(1)}* \theta_{,i}^{(2)} \bigr)\,dV \\& \qquad {}+ \int_{\partial \Omega } \bigl( t*t_{i}^{(1)}*u_{i}^{(2)}+t*m_{ij}^{(1)}* \varphi_{ij}^{(2)}+t*h^{(1)}*\theta^{(2)} \bigr)\,dA \\& \quad = \int_{\Omega } \bigl( \varrho F_{i}^{(2)}*u_{i}^{(1)}+ \varrho G_{ij} ^{(2)}*\varphi_{ij}^{(1)}-t* R^{(2)}*\theta^{(1)}-t* h_{i}^{(2)}* \theta_{,i}^{(1)} \bigr)\,dV \\& \qquad {}+ \int_{\partial \Omega } \bigl( t*t_{i}^{(2)}*u_{i}^{(1)}+t*m_{ij}^{(2)}* \varphi_{ij}^{(1)}+t*h^{(2)}*\theta^{(1)} \bigr)\,dA. \end{aligned}$$
(42)
Considering that \(h_{i}=\frac{1}{T_{0}}\hat{q}_{i}\), \(\alpha = \hat{\theta }\) and \(\dot{\beta }_{i}=\theta_{,i}\), from (42) we immediately obtain the desired identity (36). □
Remark
We want to give an explicit form for the components of the heat flux vector \(q_{i}\). So, if we integrate in (20) and take into account the initial conditions \(q_{i}(0)=0\) and \(\frac{\partial }{ \partial t}q_{i}(0)=0\), we will obtain
$$\begin{aligned} q_{i}=- \bigl( k_{ij}a_{j}+k_{ij}^{*}b_{j} \bigr) , \end{aligned}$$
(43)
in which, we remind that \(k_{ij}\) is the thermal conductivity tensor and \(k_{ij}^{*}\) is the conductivity rate tensor. For \(a_{j}\) and \(b_{j}\), we obtain the expressions
$$\begin{aligned}& a_{j}=\frac{2e^{-t/\tau_{q}}}{\tau_{q}} \biggl[ \sin \frac{t}{\tau_{q}} \int_{0}^{t} \biggl( e^{\frac{t}{\tau_{q}}}\cos \frac{t}{\tau_{q}} \dot{\xi }_{j} \biggr)\,d\tau_{q} - \cos \frac{t}{\tau_{q}} \int_{0}^{t} \biggl( e^{\frac{t}{\tau_{q}}}\sin \frac{t}{\tau_{q}}\dot{\xi }_{j} \biggr)\,d\tau_{q} \biggr], \\& b_{j}=\frac{2e^{-t/\tau_{q}}}{\tau_{q}} \biggl[ \cos \frac{t}{\tau_{q}} \int_{0}^{t} \biggl( e^{\frac{t}{\tau_{q}}}\sin \frac{t}{\tau_{q}} \dot{\varsigma }_{j} \biggr)\,d\tau_{q} - \sin \frac{t}{\tau_{q}} \int _{0}^{t} \biggl( e^{\frac{t}{\tau_{q}}}\cos \frac{t}{\tau_{q}} \dot{\varsigma }_{j} \biggr)\,d\tau_{q} \biggr] , \end{aligned}$$
where ξ and \(\varsigma_{j}\) are defined in (19) and \(\tau_{q}\) is the phase-lag of the heat flux.
Other important result of our paper is a variational principle. We strengthen the known variational principle in order to cover the three-phase-lag dipolar thermoelasticity theory.
Assuming that the tensors \(k_{ij}\) and \(k_{ij}^{*}\) can be reversed, we will use the symmetric tensors \(\lambda_{ij}\) and \(\lambda_{ij}^{*}\), defined by
$$\begin{aligned} \lambda_{ij}= [ k_{ij} ] ^{-1},\qquad \lambda_{ij}^{*}= \bigl[ k _{ij}^{*} \bigr] ^{-1}. \end{aligned}$$
(44)
We will write \(h_{i}\) in the form
$$\begin{aligned} h_{i}=h_{i}^{(I)}+h_{i}^{(II)}+h_{i}^{(III)}+h_{i}^{(IV)} \end{aligned}$$
(45)
so that considering (18) and (26) we obtain
$$ \begin{aligned} &( 1+D_{t} ) \lambda_{ij}h_{j}^{(I)}+ \frac{1}{T_{0}}\beta _{i}=0, \qquad ( 1+D_{t} ) \lambda_{ij}h_{j}^{(II)}+\frac{\tau _{T}}{T_{0}} \theta_{,i}=0, \\ &( 1+D_{t} ) \lambda_{ij}^{*}s_{j}^{(III)}+ \frac{1}{T_{0}} \beta_{i}=0, \qquad ( 1+D_{t} ) \lambda_{ij}^{*}h_{j}^{(IV)}+ \frac{ \tau_{\alpha }}{T_{0}}\beta_{i}=0, \end{aligned} $$
(46)
where \(\beta_{i}=\hat{\theta }_{,i}\) and \(s_{i}^{(III)}=\partial h _{i}^{(III)}/\partial t\).
Motivation of the decomposition (45) will appear later. We will call an admissible process be an ordered array
$$\begin{aligned} p= ( u_{i}, \varphi_{ij}, \alpha , \theta , \varepsilon_{ij}, \gamma_{ij}, \chi_{ijk}, \tau_{ij}, \sigma_{ij}, \beta_{i}, \eta , h _{i}, h_{i,i}, q_{i} ) \end{aligned}$$
(47)
having as components sufficiently regular functions on their domain of definition.
Let us denote by \({\mathcal {A}}\) the set of all admissible processes which is a linear space with addition and scalar multiplication.
On \({\mathcal {A}}\) and for each \(t\in [0,\infty )\), we define the functional \({\mathcal {F}}_{t}(p)\) by
$$\begin{aligned} {\mathcal {F}}_{t}(p)= {}&\frac{1}{2} \int_{\Omega }t* ( C_{ijmn}\varepsilon _{mn}* \varepsilon_{ij}+2G_{ijmn}\varepsilon_{mn}* \gamma_{ij} +2F_{ijmnr} \chi_{mnr}*\varepsilon_{ij} \\ &{} +B_{ijmn}\gamma_{mn}*\gamma_{ij} +2D_{ijmnr}\chi_{mnr}*\gamma _{ij}+A_{ijkmnr} \chi_{mnr}*\chi_{ijk} )\,dV \\ &{}+ \int_{\Omega } \bigl[ \varrho u_{i}*u_{i}+I_{jk} \varphi_{js}*\varphi _{ks}-t*(\eta -R)*\theta \bigr]\,dV \\ &{}+\int_{\Omega } \frac{1}{2\beta } \bigl[ t * ( \eta - a_{ij}\varepsilon_{ij} - b_{ij} \gamma_{ij} - c_{ijk}\chi_{ijk} ) \\ &{}* ( \eta - a_{mn}\varepsilon_{mn} - b_{mn} \gamma_{mn} - c_{mnr}\chi_{mnr} ) \bigr]\,dV \\ &{}-\int_{\Omega } \bigl\{ \bigl[ t* ( \tau_{ij}+ \sigma_{ij} ) _{,j}+\varrho F_{i} \bigr] *u_{i}+t*\tau_{ij}*\varepsilon_{ij} +t* \sigma_{ij}*\gamma_{ij} \bigr\} \,dV \\ & {}- {\int _{\Omega } \bigl[ ( t*\mu _{ijk,i}+t*\sigma _{jk}+\varrho G_{jk} ) *\varphi _{jk}-t*\mu _{ijk}*\chi _{ijk} \bigr]\,dV} \\ & {}+\frac{1}{2}\int_{\Omega } ( 1 + D_{t} ) \biggl( \hat{T}_{0} \lambda_{ij}h_{i}^{(I)} * h_{j}^{(I)} + t * \frac{T_{0}}{\tau_{T}}\lambda_{ij}h_{i}^{(II)} * h_{j} ^{(II)} \\ & {}+ \lambda_{ij}^{*}q_{i}^{(III)} * h_{j}^{(III)} + \hat{\frac{T_{0}}{\tau _{\alpha }}}\lambda_{ij}^{*}h_{i}^{(IV)} * h_{j}^{(IV)} \biggr)\,dV \\ & {}+\int_{\Omega } \biggl( \hat{h}_{i}*\beta_{i}+ \hat{h}_{i}*\alpha_{,i}-t*\frac{1}{T_{0}}q_{i}* \beta_{i}+\hat{h}_{i,i}*\alpha - \frac{1}{T_{0}}t*\hat{q} _{i,i}*\theta \biggr)\,dV \\ & {}+\int_{\partial \Omega_{u}} ( t*t_{i}*\tilde{u}_{i} )\,dA + \int_{\partial \Omega_{u}^{c}} \bigl[ t* ( t_{i}-\tilde{t}_{i} ) *u _{i} \bigr]\,dA+\int_{\partial \Omega_{\varphi }} ( t* m_{ij}* \tilde{\varphi }_{ij} )\,dA \\ & {}+\int_{\partial \Omega_{\varphi }^{c}} \bigl[ t* ( m_{ij}- \tilde{m}_{ij} ) *\varphi_{ij} \bigr]\,dA- \int_{\partial \Omega_{\alpha }} \bigl[ \hat{h}* ( \alpha - \tilde{\alpha } ) \bigr]\,dA-\int_{\partial \Omega_{\alpha } ^{c}} [ \hat{\tilde{h}}*\alpha ]\,dA. \end{aligned}$$
(48)
Now, we can state and prove the convolutional variational principle for the three-phase-lag dipolar thermoelasticity theory.
Theorem 3
If the symmetric tensors
\(k_{ij}\)
and
\(k_{ij}^{*}\)
can be reversed, \(\tau_{\alpha }>0\), \(\tau_{T}>0\)
and the symmetry relations (15) hold on Ω, then
$$\begin{aligned} \delta {\mathcal {F}}_{t}(p)=0,\quad t\ge 0 \end{aligned}$$
(49)
if and only if
p
is a solution of the mixed initial boundary value problem
\(\mathcal {P}\).
Proof
First, we will prove the inverse implication, that is, assuming that p from (47) is a solution of the mixed problem \({\mathcal {P}}\), we must prove that \(\delta {\mathcal {F}}_{t}(p)=0\). Along with the admissible process p, we consider another process
$$ \breve{p}= ( \breve{u}_{i}, \breve{\varphi }_{ij}, \breve{ \alpha }, \breve{\theta }, \breve{\varepsilon }_{ij}, \breve{\gamma }_{ij}, \breve{\chi }_{ijk}, \breve{\tau }_{ij}, \breve{\sigma }_{ij}, \breve{\beta }_{i}, \breve{\eta }, \breve{h} _{i}, \breve{h}_{i,i}, \breve{q}_{i} ) . $$
Of course, we have
$$ p, \breve{p} \in {\mathcal {A}}\quad \Rightarrow\quad p+\kappa \breve{p} \in {\mathcal {A}}, \quad \forall \kappa \in R. $$
Let us compute the variation of the functional \({\mathcal {F}}_{t}(p)\)
$$\begin{aligned} \delta {\mathcal {F}}_{t}(p)={}& \int_{\Omega } \biggl\{ t* \biggl[ C_{ijmn}\varepsilon_{mn}+G_{ijmn}\gamma_{mn}+F_{ijmnr}\chi_{mnr} \\ & {}- \frac{a_{ij}}{\beta } ( \eta -a_{mn}\varepsilon_{mn}-b_{mn}\gamma_{mn}-c_{ijk}\chi_{ijk} ) -\tau_{ji} \biggr] * \breve{\varepsilon }_{ij} \\ &{}+t* \biggl[ G_{ijmn}\varepsilon_{mn}+B_{ijmn}\gamma_{mn}+D_{ijmnr} \chi_{mnr} \\ & {}-\frac{b_{ij}}{\beta } ( \eta -a_{mn}\varepsilon_{mn}-b_{mn}\gamma_{mn}-c_{ijk}\chi_{ijk} ) -\sigma_{ji} \biggr] *\breve{\gamma }_{ij} \\ & {}+ t* \biggl[ F_{ijkmn}\varepsilon_{mn}+D_{mnijk} \gamma_{mn}+A_{ijkmnr} \chi_{mnr} \\ & {} - \frac{c_{ijk}}{\beta } ( \eta -a_{mn}\varepsilon _{mn}-b_{mn}\gamma_{mn}-c_{ijk} \chi_{ijk} ) -\mu_{ijk} \biggr] * \breve{\chi }_{ijk} \biggr\} \,dV \\ & {}+ \int_{\Omega } \biggl\{ t* \biggl[ -\theta +\frac{1}{\beta } ( \eta -a _{mn}\varepsilon_{mn}-b_{mn} \gamma_{mn}-c_{mnr}\chi_{mnr} ) \biggr] * \breve{\eta } \biggr\} \,dV \\ & {}+ \int_{\Omega } \bigl[ t * ( R - \eta - h_{i,i} ) * \breve{ \theta } \bigr]\,dV + \int_{\Omega } \bigl[ \bigl( \varrho u _{i} - t* ( \tau_{ij} + \sigma_{ij} ) _{,j} - \varrho F_{i} \bigr) *\breve{u}_{i} \bigr]\,dV \\ & {}+ \int_{\Omega } \bigl[ ( I_{js}\varphi_{ks}-t* \mu_{ijk,i}-t* \sigma_{jk}-\varrho G_{jk} ) *\breve{ \varphi }_{jk} \bigr]\,dV \\ & {}+ \int_{\Omega } \bigl[ T_{0} ( 1 + D_{t} ) \lambda_{ij}h _{j}^{(I)} + \beta_{i} \bigr] * \breve{h}_{i}^{(I)}\,dV \\ &{} + \int_{\Omega } t * \biggl[ \frac{T_{0}}{\tau_{T}} ( 1 + D_{t} ) \lambda _{ij}h_{j}^{(II)} + \beta_{i} \biggr] * \breve{h}_{i}^{(II)}\,dV \\ & {}+ \int_{\Omega } \bigl[ ( 1 + D_{t} ) \lambda_{ij}^{*}h _{j}^{(III)} + \beta_{i} \bigr]* \breve{h}_{i}^{(III)}\,dV \\ &{}+ \int_{\Omega } t* \biggl[ \frac{T_{0}}{\tau_{\alpha }} ( 1 + D_{t} ) \lambda_{ij}^{*}h_{j}^{(IV)} + \beta_{i} \biggr] * \breve{h}_{i}^{(IV)}\,dV \\ & {}+ \int_{\Omega } \biggl[ \biggl( h_{i} - \frac{q_{i}}{T_{0}} \biggr) * \breve{\alpha }_{,i} + \biggl( h_{i,i} - \frac{q_{i,i}}{T_{0}} \biggr) * \breve{\alpha } \biggr]\,dV \\ &{} + \int_{\Omega } \bigl[ ( \alpha_{,i} - \beta_{i} ) * \breve{h}_{i} + ( \alpha - \hat{\theta } ) * \breve{h}_{i,i} \bigr]\,dV \\ & {}+ \int_{\partial \Omega_{u}} \bigl[ t* ( \tilde{u}_{i}-u_{i} ) * \breve{t}_{i} \bigr]\,dA+ \int_{\partial \Omega_{u}^{c}} \bigl[ t* ( t _{i}-\tilde{t}_{i} ) *\breve{u}_{i} \bigr]\,dA \\ & {}+ \int_{\partial \Omega_{\varphi }} \bigl[ t* ( \tilde{\varphi } _{ij}- \varphi_{ij} ) *\breve{m}_{ij} \bigr]\,dA+\int_{\partial \Omega_{\varphi }^{c}} \bigl[ t* ( m_{ij}-\tilde{m} _{ij} ) *\breve{\varphi }_{ij} \bigr]\,dA \\ & {}+ \int_{\partial \Omega_{\alpha }} \bigl[ t* ( \tilde{\alpha }- \alpha ) *\breve{h} \bigr]\,dA+ \int_{\partial \Omega_{\alpha } ^{c}} \bigl[ t* ( h-\tilde{h} ) *\breve{\alpha } \bigr]\,dA. \end{aligned}$$
(50)
Here, to compute the first variation of \(\hat{h}_{i}*\beta_{i}\), we have used the above decomposition of \(h_{i}\) in four components.
If p is a solution of the mixed problem \(\mathcal {P}\), then the equations of motion (25), the energy equations (20), the initial conditions (23) and the boundary conditions (24) are satisfied. Also, equations (46) are satisfied. If we take into account these equations and conditions in (50), we obtain \(\delta {\mathcal {F}}_{t}(p)=0\).
Now, let us prove the reverse implication, namely, assuming that
$$\begin{aligned} \delta {\mathcal {F}}_{t}(p)=0, \quad t\ge 0, \end{aligned}$$
(51)
we have to prove that p is a solution of the mixed problem \(\mathcal {P}\).
To this aim we use a suggestion given by Gurtin in the paper [28]. We take a displacement \(\breve{u}_{i}\) such that it and its space derivatives vanish on cylinder \(\partial \Omega \times [0,\infty )\) and choose the particular admissible process p̆ of the form
$$\begin{aligned} \breve{p}= ( \breve{u}_{i}, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ). \end{aligned}$$
We substitute p̆ in (50) such that (51) reduces to
$$ \int_{\Omega } \bigl[ \bigl( t* ( \tau_{ij} + \sigma_{ij} ) _{,j} + \varrho F_{i}-\varrho u_{i} \bigr) *\breve{u}_{i} \bigr]\,dV=0 $$
for arbitrary \(\breve{u}_{i}\). According to the fundamental lemma of calculus of variations, we obtain the equation of motion (25)1.
Now we take p̆ of the form (49) but suppose that \(\breve{u} _{i}\) vanishes on \(\partial \Omega_{u}\times [0,\infty )\). With this p̆, (50) and (51) lead to
$$ t* ( t_{i}-\tilde{t}_{i} ) =0 \quad \mbox{on } \partial \Omega_{u}\times [0,\infty ), $$
by using again the fundamental lemma of calculus of variations. This last equality implies the boundary condition (24)2.
We repeat the above procedure by making suitable choices of process p̆. With each choice of p̆, by applying the fundamental lemma of calculus of variations, we get an equation or condition of a mixed problem.
Now we can give a justification for decomposition (45).
So, if we choose \(\breve{p}= ( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \breve{h}_{i}^{(I)}, 0, 0 ) \), from (50) and (51) we deduce the equation \(T_{0} ( 1+D_{t} ) \lambda_{ij}h_{j}^{(I)}+\beta _{i}=0\), which can be rewritten in the form
$$\begin{aligned} ( 1+D_{t} ) h_{j}^{(I)}+ \frac{1}{T_{0}}k_{ij}\beta_{i}=0. \end{aligned}$$
(52)
Let us consider a process p̆ of the form \(\breve{p}= ( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \breve{h}_{i}^{(II)}, 0, 0 ) \). From (49) and (50), with this p̆, we obtain the equation \(( 1+D_{t} ) \lambda_{ij}h_{j}^{(II)}+\tau_{T}/T_{0}\theta _{,i}=0\) which can be rewritten in the form
$$\begin{aligned} ( 1+D_{t} ) h_{j}^{(II)}+ \frac{\tau_{T}}{T_{0}}k_{ij}\theta _{,i}=0. \end{aligned}$$
(53)
Next, we choose a process p̆ of the form \(\breve{p}= ( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \breve{h}_{i}^{(III)}, 0, 0 ) \) and if we use (49) and (50), with this p̆, we are led to the equation \(( 1+D_{t} ) \lambda_{ij}^{*}h_{j}^{(III)}+\beta _{i}=0\), or, in another form,
$$\begin{aligned} ( 1+D_{t} ) h_{j}^{(III)}+ \frac{1}{T_{0}}k_{ij}^{*} \hat{\beta }_{,i}=0. \end{aligned}$$
(54)
Finally, we take the process p̆ in the form \(\breve{p}= ( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \breve{h}_{i}^{(IV)}, 0, 0 ) \). Using (49) and (50), with this p̆, we obtain the equation \(( 1+D_{t} ) \lambda_{ij}^{*}h_{j}^{(IV)}+ \tau_{\alpha }/T_{0}\beta_{i}=0\) which can be rewritten in the form
$$\begin{aligned} ( 1+D_{t} ) h_{j}^{(IV)}+ \frac{\tau_{\alpha }}{T_{0}}k_{ij} ^{*}\beta_{i}=0. \end{aligned}$$
(55)
By adding relations (52)-(55), we obtain (26), and, in this way, decomposition (45) is justified. The proof of Theorem 1 is completed. □