Analysis of a dynamic viscoelastic-viscoplastic piezoelectric contact problem

Dynamic contact problems for viscoelastic materials have been studied in numerous publications. For instance, we could refer the papers [12, 18, 20, 19, 31, 32, 35, 36, 37, 40, 44] where these problems were considered assuming different friction laws and types of contact (deformable and rigid obstacles or bilateral contact, Coulomb’s friction law, slip dependent friction, etc). Moreover, the numerical approximation of these problems were also done, including effects as the adhesion, the piezoelectricity or the damage (see, e.g., [1, 2, 4, 7, 13, 14, 21, 22, 26]).

These viscoelastic materials have been utilized in many engineering applications since they can be customized to meet a desired performance while maintain low cost. An important issue concerning such materials is that they may exhibit time-dependent and inelastic deformations. The viscoelastic strain component consists of a recoverable-reversible part (elastic strain) and a recoverable-dissipative deformation part (inelastic strain). When an inelastic strain is assumed to depend only on the magnitude of the stress or strain, the term plastic strain is used. When the plastic deformation also changes with time, like in the viscous component of the viscoelastic part, the term viscoplastic strain is used. Therefore, a combined viscoelastic-viscoplastic constitutive relationships should be considered. Recently, new models coupling both viscoplastic and viscoelastic effects have been proposed (see, for instance, [3, 10, 15, 16, 27, 29, 33, 39, 41]).

Piezoelectricity is the ability of certain cristals, like the quartz (also ceramics (BaTiO3, KNbO3, LiNbO3, PZT-5A, etc.) and even the human mandible or the human bone), to produce a voltage when they are subjected to mechanical stress. This effect is characterized by the coupling between the mechanical and the electrical properties of the material. We note that this kind of materials appears usually in the industry as switches in radiotronics, electroacoustics or measuring equipments. Since the first studies by Toupin ([47, 48]) and Mindlin ([42, 43]) a large number of papers have been published dealing with related models (see, for instance, [30, 11, 45] and the references cited therein). During the last ten years, numerous contact problems involving this piezoelectric effect have been studied from the variational and numerical points of view (see, i.e., [6, 7, 8, 44, 37, 38]).

In this paper, the contact is assumed to be with a deformable obstacle and so, the classical normal compliance contact condition is used ([34, 40]). Moreover, a viscoelastic-viscoplastic material is considered including, for the sake of generality in the modelling, piezoelectric effects. Both variational and numerical analyses are then performed, providing the existence of a unique weak solution to the continuous problem and an a priori error analysis for the fully discrete approximations. Finally, some numerical simulations are presented in two-dimensional examples to demonstrate the accuracy of the approximation and the behaviour of the solution.

The outline of this paper is as follows. In Section 2, we describe the mathematical problem and derive its variational formulation. An existence and uniqueness result is proved in Section 3. Then, fully discrete approximations are introduced in Section 4 by using the finite element method for the spatial approximation and an Euler scheme for the discretization of the time derivatives. An error estimate result is proved from which the linear convergence is deduced under suitable regularity assumptions. Finally, in Section 5 some two-dimensional numerical examples are shown to demonstrate the accuracy of the algorithm and the behaviour of the solution.

Denote by $\mathbb{S}^{d}$, $d=1,2,3$, the space of second order symmetric tensors on $\mathbb{R}^{d}$ and by “$\cdot$” and $\|\cdot\|$ the inner product and the Euclidean norms on $\mathbb{R}^{d}$ and $\mathbb{S}^{d}$.

Let $\Omega\subset\mathbb{R}^{d}$ denote a domain occupied by a viscoelastic-viscoplastic piezoelectric body with a Lipschitz boundary $\Gamma=\partial\Omega$ decomposed into three measurable parts $\Gamma_{D}$, $\Gamma_{F}$, $\Gamma_{C}$, on one hand, and on two measurable parts $\Gamma_{A}$ and $\Gamma_{B}$, on the other hand, such that $\hbox{meas }(\Gamma_{D})>0$, $\hbox{meas }(\Gamma_{A})>0$, and $\Gamma_{C}\subseteq\Gamma_{B}$. Let $[0,T]$, $T>0$, be the time interval of interest. The body is being acted upon by a volume force with density $\mbox{\boldmath{$f$}}_{0}$, it is clamped on $\Gamma_{D}$ and surface tractions with density $\mbox{\boldmath{$f$}}_{F}$ act on $\Gamma_{F}$. Moreover, an electrical potential is prescribed on $\Gamma_{A}$ and electric charges are applied on $\Gamma_{B}$. Finally, we assume that the body may come in contact with a deformable obstacle on the boundary part $\Gamma_{C}$, which is located, in its reference configuration, at a distance $s$, measured along the outward unit normal vector $\mbox{\boldmath{$\nu$}}=(\nu_{i})_{i=1}^{d}$.

Let $\mbox{\boldmath{$x$}}\in\Omega$ and $t\in[0,T]$ be the spatial and time variables, respectively. In order to simplify the writing, some times we do not indicate the dependence of various functions and unknowns on $x$ and $t$. Moreover, a dot above a variable represents its first derivative with respect to the time variable and two dots indicate derivative of the second order.

Let $\mbox{\boldmath{$u$}}=(u_{i})_{i=1}^{d}\in\mathbb{R}^{d}$, $\varphi\in\mathbb{R}$, $\mbox{\boldmath{$\sigma$}}=(\sigma_{ij})_{i,j=1}^{d}\in\mathbb{S}^{d}$ and $\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}})=(\varepsilon_{ij}(\mbox{\boldmath{$u$}}))_{i,j=1}^{d}\in\mathbb{S}^{d}$ denote the displacements, the electric potential, the stress tensor and the linearized strain tensor, respectively. We recall that

$$\varepsilon_{ij}(\mbox{\boldmath{$u$}})=\frac{1}{2}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right),\quad i,j=1,\ldots,d.$$

The body is assumed to be made of a viscoelastic-viscoplastic piezoelectric material and it satisfies the following constitutive law (see, for instance, [23, 30]),

$$\begin{array}[]{l}\displaystyle\mbox{\boldmath{$\sigma$}}(\mbox{\boldmath{$x$}},t)=\mathcal{A}\mbox{\boldmath{$\varepsilon$}}(\dot{\mbox{\boldmath{$u$}}}(\mbox{\boldmath{$x$}},t))+\mathcal{B}\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}(\mbox{\boldmath{$x$}},t))+\int_{0}^{t}\mathcal{G}(\mbox{\boldmath{$\sigma$}}(\mbox{\boldmath{$x$}},s),\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}(\mbox{\boldmath{$x$}},s)))\,ds\\ \qquad\qquad-\mathcal{E}^{*}{\bf E}(\varphi(\mbox{\boldmath{$x$}},t)),\end{array}$$

(2.1)

where $\mathcal{A}=(a_{ijkl})$ and $\mathcal{B}=(b_{ijkl})$ are the fourth-order viscous and elastic tensors, respectively, $\mathcal{G}$ is a viscoplastic function whose properties will be detailed later, ${\bf E}(\varphi)=(E_{i}(\varphi))_{i=1}^{d}$ represents the electric field defined by

$$E_{i}(\varphi)=-\frac{\partial\varphi}{\partial x_{i}},\quad i=1,\ldots,d,$$

and $\mathcal{E}^{*}=(e_{ijk}^{*})_{i,j,k=1}^{d}$ denotes the transpose of the third-order piezoelectric tensor $\mathcal{E}=(e_{ijk})_{i,j,k=1}^{d}$. We recall that

$$e_{ijk}^{*}=e_{kij},\quad\hbox{for all}\quad i,j,k=1,\ldots,d.$$

Following [11] the following constitutive law is satisfied for the electric potential,

$$\mathcal{D}=\mathcal{E}\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}})+\mathbf{\beta}{\bf E}(\varphi),$$

where $\mathcal{D}=(D_{i})_{i=1}^{d}$ is the electric displacement field and $\mathbf{\beta}=(\beta_{ij})_{i,j=1}^{d}$ is the electric permittivity tensor.

We turn now to describe the boundary conditions.

On the boundary part $\Gamma_{D}$ we assume that the body is clamped and thus the displacement field vanishes there (and so $\mbox{\boldmath{$u$}}=\mbox{\boldmath{$0$}}$ on $\Gamma_{D}\times(0,T)$). Moreover, since the density of traction forces $\mbox{\boldmath{$f$}}_{F}$ is applied on the boundary part $\Gamma_{F}$, it follows that $\mbox{\boldmath{$\sigma$}}\mbox{\boldmath{$\nu$}}=\mbox{\boldmath{$f$}}_{F}$ on $\Gamma_{F}\times(0,T).$

The contact is assumed with a deformable obstacle and so, the well-known normal compliance contact condition is employed for its modeling (see [34, 40]); that is, the normal stress $\sigma_{\nu}=\mbox{\boldmath{$\sigma$}}\mbox{\boldmath{$\nu$}}\cdot\mbox{\boldmath{$\nu$}}$ on $\Gamma_{C}$ is given by

$$-\sigma_{\nu}=p(u_{\nu}-s),$$

where $u_{\nu}=\mbox{\boldmath{$u$}}\cdot\mbox{\boldmath{$\nu$}}$ denotes the normal displacement, in such a way that, when $u_{\nu}>s$, the difference $u_{\nu}-s$ represents the interpenetration of the body’s asperities into those of the obstacle. The normal compliance function $p$ is prescribed and it satisfies $p(r)=0$ for $r\leq 0$, since then there is no contact. As an example, one may consider

$$p(r)=c_{p}\,r_{+},$$

where $c_{p}>0$ represents a deformability constant (that is, it denotes the stiffness of the obstacle), and $r_{+}={\rm max}\,\{0,r\}.$ Formally, the classical Signorini nonpenetration conditions are obtained in the limit $c_{p}\to\infty$. We also assume that the contact is frictionless, i.e. the tangential component of the stress field, denoted by $\mbox{\boldmath{$\sigma$}}_{\tau}=\mbox{\boldmath{$\sigma$}}\mbox{\boldmath{$\nu$}}-\sigma_{\nu}\mbox{\boldmath{$\nu$}}$, vanishes on the contact surface.

Let $\Omega$ be subject to a prescribed electric potential on $\Gamma_{A}$ and to a density of surface electric charges $q_{F}$ on $\Gamma_{B}$, that is,

$$\begin{array}[]{l}\varphi=\varphi_{A}\quad\hbox{on}\quad\Gamma_{A}\times(0,T),\\ \mathcal{D}\cdot\mbox{\boldmath{$\nu$}}=q_{F}\quad\hbox{on}\quad\Gamma_{B}\times(0,T).\end{array}$$

For the sake of simplicity, we assume that no electric potential is imposed on the boundary $\Gamma_{A}$ (i.e. $\varphi_{A}=0$), and that $q_{F}=0$ on $\Gamma_{C}$; that is, the foundation is supposed to be insulator. We note that it is straightforward to extend the results presented below to more general situations by decomposing $\Gamma$ in a different way and by introducing some modifications in the analyses shown in the following sections.

The mechanical problem of the dynamic deformation of a viscoelastic-viscoplastic piezoelectric body in contact with a deformable obstacle is then written as follows.

Problem P. Find a displacement field $\mbox{\boldmath{$u$}}:\overline{\Omega}\times[0,T]\rightarrow\mathbb{R}^{d}$, a stress field $\mbox{\boldmath{$\sigma$}}:\overline{\Omega}\times[0,T]\rightarrow\mathbb{S}^{d}$, an electric potential field $\varphi:\overline{\Omega}\times(0,T)\rightarrow\mathbb{R}$ and an electric displacement field $\mathcal{D}:\overline{\Omega}\times(0,T)\rightarrow\mathbb{R}^{d}$ such that,

	$\displaystyle\mbox{\boldmath{$\sigma$}}(\mbox{\boldmath{$x$}},t)=\mathcal{A}\mbox{\boldmath{$\varepsilon$}}(\dot{}\mbox{\boldmath{$u$}}(\mbox{\boldmath{$x$}},t))+\mathcal{B}\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}(\mbox{\boldmath{$x$}},t))+\int_{0}^{t}\mathcal{G}(\mbox{\boldmath{$\sigma$}}(\mbox{\boldmath{$x$}},s),\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}(\mbox{\boldmath{$x$}},s)))\,ds$
	$\displaystyle\qquad\qquad-\mathcal{E}^{*}{\bf E}(\varphi(\mbox{\boldmath{$x$}},t))\quad\hbox{for a.e.}\quad\mbox{\boldmath{$x$}}\in\Omega,\,t\in(0,T),$		(2.2)
	$\displaystyle\mathcal{D}=\mathcal{E}\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}})+\mathbf{\beta}{\bf E}(\varphi)\quad\hbox{in}\quad\Omega\times(0,T),$		(2.3)
	$\displaystyle\rho\ddot{\mbox{\boldmath{$u$}}}-\mbox{\rm Div\,}\mbox{\boldmath{$\sigma$}}=\mbox{\boldmath{$f$}}_{0}\quad\hbox{in}\quad\Omega\times(0,T),$		(2.4)
	$\displaystyle\mbox{\rm div\,}\mathcal{D}=q_{0}\quad\hbox{in}\quad\Omega\times(0,T),$		(2.5)
	$\displaystyle\mbox{\boldmath{$u$}}=\mbox{\boldmath{$0$}}\quad\hbox{on}\quad\Gamma_{D}\times(0,T),$		(2.6)
	$\displaystyle\mbox{\boldmath{$\sigma$}}\mbox{\boldmath{$\nu$}}=\mbox{\boldmath{$f$}}_{F}\quad\hbox{on}\quad\Gamma_{F}\times(0,T),$		(2.7)
	$\displaystyle\mbox{\boldmath{$\sigma$}}_{\tau}=\mbox{\boldmath{$0$}},\quad-\sigma_{\nu}=p(u_{\nu}-s)\quad\hbox{on}\quad\Gamma_{C}\times(0,T),$		(2.8)
	$\displaystyle\varphi=0\quad\hbox{on}\quad\Gamma_{A}\times(0,T),$		(2.9)
	$\displaystyle\mathcal{D}\cdot\mbox{\boldmath{$\nu$}}=q_{F}\quad\hbox{on}\quad\Gamma_{B}\times(0,T),$		(2.10)
	$\displaystyle\mbox{\boldmath{$u$}}(0)=\mbox{\boldmath{$u$}}_{0},\quad\dot{\mbox{\boldmath{$u$}}}(0)=\mbox{\boldmath{$v$}}_{0}\quad\hbox{in}\quad\Omega.$		(2.11)

Here, $\rho>0$ is the density of the material (which is assumed constant for simplicity), and $\mbox{\boldmath{$u$}}_{0}$ and $\mbox{\boldmath{$v$}}_{0}$ represent initial conditions for the displacement and velocity fields, respectively. $\mbox{\boldmath{$f$}}_{0}$ is the density of the body forces acting in $\Omega$ and $q_{0}$ is the volume density of free electric charges. Moreover, Div  and div  represent the divergence operators for tensor and vector-valued functions, respectively.

In order to obtain the variational formulation of Problem P, let us denote by $H=[L^{2}(\Omega)]^{d}$ and define the variational spaces $V$, $W$ and $Q$ as follows,

$$\begin{array}[]{l}V=\{\mbox{\boldmath{$w$}}\in[H^{1}(\Omega)]^{d}\,;\,\mbox{\boldmath{$w$}}=\mbox{\boldmath{$0$}}\quad\hbox{on}\quad\Gamma_{D}\},\\ W=\{\psi\in H^{1}(\Omega)\,;\,\psi=0\quad\hbox{on}\quad\Gamma_{A}\},\\ Q=\{\mbox{\boldmath{$\tau$}}=(\tau_{ij})_{i,j=1}^{d}\in[L^{2}(\Omega)]^{d\times d}\,\,\,;\,\,\,\tau_{ij}=\tau_{ji},\,\;i,j=1,\ldots,d\}.\end{array}$$

We will make the following assumptions on the problem data.

The viscosity tensor $\mathcal{A}(\mbox{\boldmath{$x$}})=(a_{ijkl}(\mbox{\boldmath{$x$}}))_{i,j,k,l=1}^{d}:\mbox{\boldmath{$\tau$}}\in\mathbb{S}^{d}\rightarrow\mathcal{A}(\mbox{\boldmath{$x$}})(\mbox{\boldmath{$\tau$}})\in\mathbb{S}^{d}$ satisfies:

$$\begin{array}[]{l}\mathrm{(a)\ }a_{ijkl}=a_{klij}=a_{jikl}\quad\hbox{for}\quad i,j,k,l=1,\ldots,d.\\ \mathrm{(b)\ }a_{ijkl}\in L^{\infty}(\Omega)\quad\hbox{for}\quad i,j,k,l=1,\ldots,d.\\ \mathrm{(c)\ \mathrm{T}h\mathrm{ere}\ exists\ }m_{\mathcal{A}}>0\mathrm{\ such\ that\ }\mathcal{A}(\mbox{\boldmath{$x$}})\mbox{\boldmath{$\tau$}}\cdot\mbox{\boldmath{$\tau$}}\geq m_{\mathcal{A}}\,\|\mbox{\boldmath{$\tau$}}\|^{2}\\ {}\qquad\forall\,\mbox{\boldmath{$\tau$}}\in\mathbb{S}^{d},\mathrm{\ a.e.\ }\mbox{\boldmath{$x$}}\in\Omega.\end{array}$$

(2.13)

The elastic tensor $\mathcal{B}(\mbox{\boldmath{$x$}})=(b_{ijkl}(\mbox{\boldmath{$x$}}))_{i,j,k,l=1}^{d}:\mbox{\boldmath{$\tau$}}\in\mathbb{S}^{d}\rightarrow\mathcal{B}(\mbox{\boldmath{$x$}})(\mbox{\boldmath{$\tau$}})\in\mathbb{S}^{d}$ satisfies:

$$\begin{array}[]{l}\mathrm{(a)\ }b_{ijkl}=b_{klij}=b_{jikl}\quad\hbox{for}\quad i,j,k,l=1,\ldots,d.\\ \mathrm{(b)\ }b_{ijkl}\in L^{\infty}(\Omega)\quad\hbox{for}\quad i,j,k,l=1,\ldots,d.\\ \mathrm{(c)\ \mathrm{T}h\mathrm{ere}\ exists\ }m_{\mathcal{B}}>0\mathrm{\ such\ that\ }\mathcal{B}(\mbox{\boldmath{$x$}})\mbox{\boldmath{$\tau$}}\cdot\mbox{\boldmath{$\tau$}}\geq m_{\mathcal{B}}\,\|\mbox{\boldmath{$\tau$}}\|^{2}\\ {}\qquad\forall\,\mbox{\boldmath{$\tau$}}\in\mathbb{S}^{d},\mathrm{\ a.e.\ }\mbox{\boldmath{$x$}}\in\Omega.\end{array}$$

(2.14)

The piezoelectric tensor $\mathcal{E}(\mbox{\boldmath{$x$}})=(e_{ijk}(\mbox{\boldmath{$x$}}))_{i,j,k=1}^{d}:\mbox{\boldmath{$\tau$}}\in\mathbb{S}^{d}\rightarrow\mathcal{E}(\mbox{\boldmath{$x$}})(\mbox{\boldmath{$\tau$}})\in\mathbb{R}^{d}$ satisfies:

$$\begin{array}[]{l}\mathrm{(a)\ }e_{ijk}=e_{ikj}\quad\hbox{for}\quad i,j,k=1,\ldots,d.\\ \mathrm{(b)\ }e_{ijk}\in L^{\infty}(\Omega)\quad\hbox{for}\quad i,j,k=1,\ldots,d.\end{array}$$

(2.15)

The permittivity tensor $\mathbf{\beta}(\mbox{\boldmath{$x$}})=(\beta_{ij}(\mbox{\boldmath{$x$}}))_{i,j=1}^{d}:\mbox{\boldmath{$w$}}\in\mathbb{R}^{d}\rightarrow\mathbf{\beta}(\mbox{\boldmath{$x$}})(\mbox{\boldmath{$w$}})\in\mathbb{R}^{d}$ verifies:

$$\begin{array}[]{l}\mathrm{(a)\ }\beta_{ij}=\beta_{ji}\quad\hbox{for}\quad i,j=1,\ldots,d.\\ \mathrm{(b)\ }\beta_{ij}\in L^{\infty}(\Omega)\quad\hbox{for}\quad i,j=1,\ldots,d.\\ \mathrm{(c)\ \mathrm{T}h\mathrm{ere}\ exists\ }m_{\mathbf{\beta}}>0\mathrm{\ such\ that\ }\mathbf{\beta}(\mbox{\boldmath{$x$}})\mbox{\boldmath{$w$}}\cdot\mbox{\boldmath{$w$}}\geq m_{\mathbf{\beta}}\,\|\mbox{\boldmath{$w$}}\|^{2}\\ {}\qquad\forall\,\mbox{\boldmath{$w$}}\in\mathbb{R}^{d},\mathrm{\ a.e.\ }\mbox{\boldmath{$x$}}\in\Omega.\end{array}$$

(2.16)

The normal compliance function $p:\Gamma_{C}\times\mathbb{R}\longrightarrow\mathbb{R}^{+}$ satisfies:

$$\left.\begin{array}[]{ll}{\rm(a)\ There\ exists\ }L_{p}>0\ {\rm such\ that}\\ {}\qquad|p(\mbox{\boldmath{$x$}},r_{1})-p(\mbox{\boldmath{$x$}},r_{2})|\leq L_{p}\,|r_{1}-r_{2}|\quad\forall\,r_{1},r_{2}\in\mathbb{R},{\rm\ a.e.\ }\mbox{\boldmath{$x$}}\in\Gamma_{C}.\\ {\rm(b)\ The\ mapping\ }\mbox{\boldmath{$x$}}\mapsto p(\mbox{\boldmath{$x$}},r)\ {\rm is\ Lebesgue\ measurable\ on\ }\Gamma_{C},\\ \qquad\forall r\in\mathbb{R}.\\ {\rm(c)\ }(p(\mbox{\boldmath{$x$}},r_{1})-p(\mbox{\boldmath{$x$}},r_{2}))\cdot(r_{1}-r_{2})\geq 0\quad\forall\,r_{1},r_{2}\in\mathbb{R},{\rm\ a.e.\ }\mbox{\boldmath{$x$}}\in\Gamma_{C}.\\ {\rm(d)\ The\ mapping\ }\mbox{\boldmath{$x$}}\mapsto p(\mbox{\boldmath{$x$}},r)=0\quad{\rm for\ all\ }r\leq 0.\end{array}\right.$$

(2.17)

The viscoplastic function $\mathcal{G}:\Omega\times\mathbb{S}^{d}\times\mathbb{S}^{d}\rightarrow\mathcal{G}(\mbox{\boldmath{$x$}})(\mbox{\boldmath{$\tau$}},\mbox{\boldmath{$\varepsilon$}})\in\mathbb{S}^{d}$ satisfies:

$$\begin{array}[]{rl}&\hskip-19.91684pt\mathrm{(a)\ \mathrm{T}h\mathrm{ere}\ exists\ }L_{\mathcal{G}}>0\mathrm{\ such\ that\ }\\ &\hskip 0.0pt\left|\mathcal{G}\left(\mbox{\boldmath{$x$}},\mbox{\boldmath{$\sigma$}}_{1},\mbox{\boldmath{$\varepsilon$}}_{1}\right)-\mathcal{G}\left(\mbox{\boldmath{$x$}},\mbox{\boldmath{$\sigma$}}_{2},\mbox{\boldmath{$\varepsilon$}}_{2}\right)\right|\leq L_{\mathcal{G}}\left(\left|\mbox{\boldmath{$\varepsilon$}}_{1}-\mbox{\boldmath{$\varepsilon$}}_{2}\right|+\left|\mbox{\boldmath{$\sigma$}}_{1}-\mbox{\boldmath{$\sigma$}}_{2}\right|\right)\\ &\hskip 0.0pt\mbox{ for all }\mbox{\boldmath{$\varepsilon$}}_{1},\mbox{\boldmath{$\varepsilon$}}_{2},\mbox{\boldmath{$\sigma$}}_{1},\mbox{\boldmath{$\sigma$}}_{2}\in\mathbb{S}^{d},\quad\mbox{ a.e. }\ \mbox{\boldmath{$x$}}\in\Omega.\\ &\hskip-19.91684pt\mathrm{(b)\ }\mbox{The function }\;\mbox{\boldmath{$x$}}\rightarrow\mathcal{G}\left(\mbox{\boldmath{$x$}},\mbox{\boldmath{$\sigma$}},\mbox{\boldmath{$\varepsilon$}}\right)\mbox{ is measurable.}\\ &\hskip-19.91684pt\mathrm{(c)\ }\hbox{The mapping }\mbox{\boldmath{$x$}}\rightarrow\mathcal{G}\left(\mbox{\boldmath{$x$}},\mathbf{0},\mathbf{0}\right)\hbox{ belongs to }Q.\end{array}$$

(2.18)

The following regularity is assumed on the density of volume forces and tractions:

$$\begin{array}[]{l}\mbox{\boldmath{$f$}}_{0}\in C([0,T];H),\quad\mbox{\boldmath{$f$}}_{F}\in C([0,T];[L^{2}(\Gamma_{F})]^{d}),\\ q_{0}\in C([0,T];L^{2}(\Omega)),\quad q_{F}\in C([0,T];L^{2}(\Gamma_{B})).\end{array}$$

(2.19)

Finally, we assume that the initial displacement and velocity satisfy

$$\mbox{\boldmath{$u$}}_{0},\mbox{\boldmath{$v$}}_{0}\in V.$$

(2.20)

Moreover, we denote by $V^{\prime}$ the dual space of $V$. We identify $H$ with its dual and consider the Gelfand triple

$$V\subset H\subset V^{\prime}.$$

We use the notation $<\cdot,\cdot>_{V^{\prime}\times V}$ to denote the duality product and, in particular, we have

$$<\mbox{\boldmath{$v$}},\mbox{\boldmath{$u$}}>_{V^{\prime}\times V}=(\mbox{\boldmath{$v$}},\mbox{\boldmath{$u$}})_{H}\quad\forall\mbox{\boldmath{$u$}}\in V,\ \mbox{\boldmath{$v$}}\in H.$$

Using Riesz’ theorem, from (2.19) we can define the elements $\mbox{\boldmath{$f$}}(t)\in V^{\prime}$ and $q(t)\in W$ given by

$$\begin{array}[]{l}\displaystyle\langle\mbox{\boldmath{$f$}}(t),\mbox{\boldmath{$w$}}\rangle_{V^{\prime}\times V}=\int_{\Omega}\mbox{\boldmath{$f$}}_{0}(t)\cdot\mbox{\boldmath{$w$}}\,d\mbox{\boldmath{$x$}}+\int_{\Gamma_{F}}\mbox{\boldmath{$f$}}_{F}(t)\cdot\mbox{\boldmath{$w$}}\,d\Gamma\quad\forall\mbox{\boldmath{$w$}}\in V,\\ \displaystyle(q(t),\psi)_{W}=\int_{\Omega}q_{0}(t)\psi\,d\mbox{\boldmath{$x$}}+\int_{\Gamma_{B}}q_{F}(t)\psi\,d\Gamma\quad\forall\psi\in W,\end{array}$$

and then $\mbox{\boldmath{$f$}}\in C([0,T];V^{\prime})$ and $q\in C([0,T];W)$. Now, let us define the contact functional $j:V\times V\rightarrow\mathbb{R}$ by

$$j(\mbox{\boldmath{$u$}},\mbox{\boldmath{$v$}})=\displaystyle\int_{\Gamma_{C}}p(u_{\nu}-s)\,v_{\nu}\,d\Gamma\quad\forall\mbox{\boldmath{$u$}},\mbox{\boldmath{$v$}}\in V,$$

where we let $v_{\nu}=\mbox{\boldmath{$v$}}\cdot\mbox{\boldmath{$\nu$}}$ for all $\mbox{\boldmath{$v$}}\in V$. Moreover, from properties (2.17) let us conclude that

$$j(\mbox{\boldmath{$u$}},\mbox{\boldmath{$w$}})-j(\mbox{\boldmath{$v$}},\mbox{\boldmath{$w$}})\leq C\|\mbox{\boldmath{$u$}}-\mbox{\boldmath{$v$}}\|_{V}\|\mbox{\boldmath{$w$}}\|_{V}\ \forall\,\mbox{\boldmath{$u$}},\mbox{\boldmath{$v$}},\mbox{\boldmath{$w$}}\in V.$$

(2.21)

Plugging (2.2) into (2.4), (2.3) into (2.5) and using the previous boundary conditions, applying a Green’s formula we derive the following variational formulation of Problem P, written in terms of the velocity field $\mbox{\boldmath{$v$}}(t)=\dot{\mbox{\boldmath{$u$}}}(t)$ and the electric potential $\varphi(t)$.

The proof of Theorem 3.1 will be carried in several steps. First, let $\mbox{\boldmath{$M$}}\in L^{2}(0,T;Q)$ and consider the auxiliary problem.

To show the proof of this theorem we have to proceed in several steps as well. Let $\mbox{\boldmath{$\eta$}}\in L^{2}(0,T;V^{\prime})$ be given and consider the following additional auxiliary problem.

Now, we can apply a result proved in [46, p. 107] which we can reformulate here as follows.

We now consider the auxiliary problem for the electric part.

We have the following result.

Proof.   We define a bilinear form $b(\cdot,\cdot):W\times W\to{{\rm I}\mkern-3.0mu{\rm R}}$ such that

$$b(\varphi,\psi)=(\beta\nabla\varphi,\nabla\psi)_{H}\quad\forall\varphi,\psi\in W.$$

We use (2.16) to show that the bilinear form is continuous, symmetric and coercive on $W$. Moreover, using (3.11) and (2.15), the Riesz’ representation theorem allows us to define an element $q_{\eta}:[0,T]\to W$ such that

$$(q_{\eta}(t),\psi)_{W}=(\mathcal{E}\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}_{M\eta}(t)),\nabla\psi)_{H}+(q(t),\psi)_{W}\quad\forall\psi\in W.$$

We apply the Lax-Milgram theorem to deduce that there exists a unique element $\varphi_{M\eta}(t)$ such that

$$b(\varphi_{M\eta}(t),\psi)=(q_{\eta}(t),\psi)_{W}\quad\forall\psi\in W.$$

We conclude that $\varphi_{M\eta}(t)$ is the solution to variational equation (3.11). Moreover, by using (2.19) and the regularity of $\mbox{\boldmath{$u$}}_{M\eta}$ and $q$, we conclude straightforwardly that $\varphi_{M\eta}\in C([0,T];W)$. ∎

Now, let $\Lambda\mbox{\boldmath{$\eta$}}(t)$ denote the element of $V^{\prime}$ defined by

	$\displaystyle<\rho\Lambda\mbox{\boldmath{$\eta$}}(t),\mbox{\boldmath{$w$}}>_{V^{\prime}\times V}$
	$\displaystyle\qquad=(\mathcal{B}\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}_{M\eta}(t)),\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$w$}}))_{Q}+(\mathcal{E}^{*}\nabla\varphi_{M\eta}(t),\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$w$}}))_{Q}+j(\mbox{\boldmath{$u$}}_{M\eta}(t),\mbox{\boldmath{$w$}}),$		(3.12)

for all $\mbox{\boldmath{$w$}}\in V$ and $t\in[0,T]$. We have the following result.

Proof.   The continuity of $\Lambda\mbox{\boldmath{$\eta$}}$ is a straightforward consequence of the continuity of $\varphi_{M\eta}$ and $\mbox{\boldmath{$u$}}_{M\eta}$. Let now $\mbox{\boldmath{$\eta$}}_{1},\mbox{\boldmath{$\eta$}}_{2}\in L^{2}(0,T;V^{\prime})$ and $t\in[0,T]$. We use the shorter notation $\mbox{\boldmath{$u$}}_{i}=\mbox{\boldmath{$u$}}_{M\eta_{i}}$, $\mbox{\boldmath{$v$}}_{i}=\mbox{\boldmath{$v$}}_{M\eta_{i}}$, $\varphi_{i}=\varphi_{M\eta_{i}}$, for $i=1,2$. Then, taking $\mbox{\boldmath{$\eta$}}=\mbox{\boldmath{$\eta$}}_{i}$ for $i=1,2$ successively in (3.12) and subtracting the resulting equations, we have, for all $\mbox{\boldmath{$w$}}\in V$ and $t\in(0,T),$

	$\displaystyle\hskip-19.91684pt<\rho\Lambda\mbox{\boldmath{$\eta$}}_{1}(t)-\rho\Lambda\mbox{\boldmath{$\eta$}}_{2}(t),\mbox{\boldmath{$w$}}>_{V^{\prime}\times V}=\left(\mathcal{B}(\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}_{1}(t))-\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}_{2}(t))),\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$w$}})\right)_{Q}$
	$\displaystyle\qquad\qquad+\left(\mathcal{E}^{*}\nabla(\varphi_{1}(t)-\varphi_{2}(t)),\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$w$}})\right)_{Q}+j(\mbox{\boldmath{$u$}}_{1}(t),\mbox{\boldmath{$w$}})-j(\mbox{\boldmath{$u$}}_{2}(t),\mbox{\boldmath{$w$}}).$

By using (2.14), (2.15) and (2.21), we find that

$$\|\Lambda\mbox{\boldmath{$\eta$}}_{1}(t)-\Lambda\mbox{\boldmath{$\eta$}}_{2}(t)\|_{V^{\prime}}\leq C(\|\mbox{\boldmath{$u$}}_{1}(t)-\mbox{\boldmath{$u$}}_{2}(t)\|_{V}+\|\varphi_{1}(t)-\varphi_{2}(t)\|_{W}).$$

(3.13)

Here and below, $C$ stands for a positive constant depending on data whose specific value may change from place to place. On the other hand, from (3.10) we know that

$$\|\mbox{\boldmath{$u$}}_{1}(t)-\mbox{\boldmath{$u$}}_{2}(t)\|_{V}\leq\int_{0}^{t}\|\mbox{\boldmath{$v$}}_{1}(s)-\mbox{\boldmath{$v$}}_{2}(s)\|_{V}\,ds.$$

(3.14)

Also, from (3.11) and using (2.15) and (2.16) we deduce

$$\|\varphi_{1}(t)-\varphi_{2}(t)\|_{W}\leq C\|\mbox{\boldmath{$u$}}_{1}(t)-\mbox{\boldmath{$u$}}_{2}(t)\|_{V},$$

which, combined with (3.14), gives

$$\|\varphi_{1}(t)-\varphi_{2}(t)\|_{W}\leq C\int_{0}^{t}\|\mbox{\boldmath{$v$}}_{1}(s)-\mbox{\boldmath{$u$}}_{2}(s)\|_{V}\,ds.$$

(3.15)

Thus, by using consecutively (3.15) and (3.14) in (3.13), we obtain

$$\|\Lambda\mbox{\boldmath{$\eta$}}_{1}(t)-\Lambda\mbox{\boldmath{$\eta$}}_{2}(t)\|_{V^{\prime}}\leq C\int_{0}^{t}\|\mbox{\boldmath{$v$}}_{1}(s)-\mbox{\boldmath{$v$}}_{2}(s)\|_{V}\,ds,$$

which implies

$$\|\Lambda\mbox{\boldmath{$\eta$}}_{1}(t)-\Lambda\mbox{\boldmath{$\eta$}}_{2}(t)\|_{V^{\prime}}^{2}\leq C\int_{0}^{t}\|\mbox{\boldmath{$v$}}_{1}(s)-\mbox{\boldmath{$v$}}_{2}(s)\|_{V}^{2}\,ds.$$

(3.16)

Taking $\mbox{\boldmath{$\eta$}}=\mbox{\boldmath{$\eta$}}_{i}$ for $i=1,2$ successively in (3.9) with $\mbox{\boldmath{$w$}}=\mbox{\boldmath{$v$}}_{1}(t)-\mbox{\boldmath{$v$}}_{2}(t)$ and subtracting the resulting expressions, we find

	$\displaystyle\hskip-28.45274pt<\rho\dot{\mbox{\boldmath{$v$}}}_{1}(t)-\rho\dot{\mbox{\boldmath{$v$}}}_{2}(t),\mbox{\boldmath{$v$}}_{1}(t)-\mbox{\boldmath{$v$}}_{2}(t)>_{V^{\prime}\times V}$
	$\displaystyle\qquad\qquad\quad+(\mathcal{A}(\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}}_{1}(t))-\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}}_{2}(t))),\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}}_{1}(t))-\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}}_{2}(t)))_{Q}$
	$\displaystyle\qquad\quad=<\mbox{\boldmath{$\eta$}}_{2}(t)-\mbox{\boldmath{$\eta$}}_{1}(t),\mbox{\boldmath{$v$}}_{1}(t)-\mbox{\boldmath{$v$}}_{2}(t)>_{V^{\prime}\times V}.$

By integrating in time, using the ellipticity of $\mathcal{A}$, the fact that $\mbox{\boldmath{$v$}}_{1}(0)=\mbox{\boldmath{$v$}}_{2}(0)=\mbox{\boldmath{$v$}}_{0}$ and Korn’s inequality, we find

	$\displaystyle m_{\mathcal{A}}\int_{0}^{t}\|\mbox{\boldmath{$v$}}_{1}(s)-\mbox{\boldmath{$v$}}_{2}(s)\|_{V}^{2}ds\leq\int_{0}^{t}<\mbox{\boldmath{$\eta$}}_{2}(s)-\mbox{\boldmath{$\eta$}}_{1}(s),\mbox{\boldmath{$v$}}_{1}(s)-\mbox{\boldmath{$v$}}_{2}(s)>_{V^{\prime}\times V}ds$
	$\displaystyle\qquad\leq\frac{1}{m_{\mathcal{A}}}\int_{0}^{t}\|\mbox{\boldmath{$\eta$}}_{2}(s)-\mbox{\boldmath{$\eta$}}_{1}(s)\|_{V^{\prime}}^{2}ds+\frac{m_{\mathcal{A}}}{4}\int_{0}^{t}\|\mbox{\boldmath{$v$}}_{1}(s)-\mbox{\boldmath{$v$}}_{2}(s)\|_{V}^{2}ds,$

where we used several times Young’s inequality

$$ab\leq\epsilon a^{2}+\frac{1}{4\epsilon}b^{2},\,a,b,\epsilon\in\mathbb{R},\,\epsilon>0.$$

(3.17)