1 Introduction

A piezoelectric body is one that produces an electric charge when a mechanical stress is applied (the body is squeezed or stretched). Conversely, a mechanical deformation (the body shrinks or expands) is produced when an electric field is applied. This kind of materials appears usually in the industry as switches in radiotronics, electroacoustics, or measuring equipments. Piezoelectric materials for which the mechanical properties are elastic are also called electro-elastic materials, and those for which the mechanical properties are viscoelastic are also called electro-viscoelastic materials. Different models have been developed to describe the interaction between the electrical and mechanical fields( see, e.g., [2, 14, 16–19, 29–31] and the references therein). General models for elastic materials with piezoelectric effect, called electro-elastic materials, can be found in [2, 4, 14]. A static frictional contact problem for electric-elastic materials was considered in [1, 15], under the assumption that the foundation is insulated. Contact problems involving elasto-piezoelectric materials [1, 15, 28], viscoelastic piezoelectric materials [5, 25] have been studied.

Adhesion may take place between parts of the contacting surfaces. It may be intentional, when surfaces are bonded with glue, or unintentional, as a seizure between very clean surfaces. The adhesive contact is modeled by a bonding field on the contact surface, denoted in this paper by \(\beta \); it describes the pointwise fractional density of active bonds on the contact surface, and sometimes referred to as the intensity of adhesion. Following [10, 11], the bonding field satisfies the restrictions \(0\le \beta \le 1\); when \(\beta =1\) at a point of the contact surface, the adhesion is complete and all the bonds are active; when \(\beta =0\) all the bonds are inactive, severed, and there is no adhesion; when \(0<\beta <1\) the adhesion is partial and only a fraction \(\beta \) of the bonds is active. Basic modeling can be found in [10–12]. Analysis of models for adhesive contact can be found in [7, 8] and in the monographs [24, 27]. An application of the theory of adhesive contact in the medical field of prosthetic limbs was considered in [22, 23]; there, the importance of the bonding between the bone-implant and the tissue was outlined, since debonding may lead to decrease in the persons ability to use the artificial limb or joint.

In this work we continue in this line of research, where we extend the result established in [3, 20] for contact problem described with the Signorini conditions into contact problem described with the Signorini conditions with adhesion where the obstacle is a perfect insulator and the resistance to tangential motion is generated by the glue, in comparison to which the frictional traction can be neglected. Therefore, the tangential contact traction depends only on the bonding field and the tangential displacement.

The paper is structured as follows. In Sect. 2 we present the electro-viscoelastic contact model with adhesion and provide comments on the contact boundary conditions. In Sect. 3 we list the assumptions on the data and derive the variational formulation. In Sect. 4, we present our main existence and uniqueness result, Theorem 4.1, which states the unique weak solvability of the Signorini adhesive contact problem. The proof of the theorem is provided in Sect. 5, where it is carried out in several steps and is based on a general result on evolution equations with maximal monotone operators and fixed-point theorem.

2 The Model

We consider a body made of a piezoelectric material which occupies the domain \(\Omega \subset \mathbb {R} ^{d}\) \((d=2,3)\) with a smooth boundary \(\partial \Omega =\Gamma \) and a unit outward normal \(\nu \). The body is acted upon by body forces of density \( f_{0}\) and has volume free electric charges of density \(q_{0\,}\). It is also constrained mechanically and electrically on the boundary. To describe these constraints we assume a partition of \(\Gamma \) into three open disjoint parts \(\Gamma _{1}\), \(\Gamma _{2}\), and \(\Gamma _{3}\), on the one hand, and a partition of \(\Gamma _{1}\cup \) \(\Gamma _{2}\) into two open parts \(\Gamma _{a}\) and \(\Gamma _{b}\), on the other hand. We assume that meas \(\Gamma _{1}>0\) and meas \(\Gamma _{a}>0\); these conditions allow the use of coercivity arguments in the proof of the unique solvability of the model. The body is clamped on \(\Gamma _{1}\), and therefore, the displacement field vanishes there. Surface tractions of density \(f_{2}\) act on \( \Gamma _{2}\). We also assume that the electrical potential vanishes on \( \Gamma _{a}\) and a surface electrical charge of density \(q_{2\,}\)is prescribed on \(\Gamma _{b}.\) On \(\Gamma _{3}\) the body is in adhesive contact with an insulator obstacle, the so-called foundation. The contact is frictionless and, since the foundation is assumed to be rigid, we model it with the Signorini condition.

We are interested in the deformation of the body on the time interval \(\left[ 0\,T\right] \). The process is assumed to be quasistatic, i.e., the inertial effects in the equation of motion are neglected. We denote by \(x\in \Omega \cup \Gamma \) and \(t\in \left[ 0\,T\right] \) the spatial and the time variable, respectively, and, to simplify the notation, we do not indicate in what follows the dependence of various functions on x and t. Here and everywhere in this paper, i, j, k, \(l=1,\ldots ,d,\) summation over two repeated indices is implied, and the index that follows a comma represents the partial derivative with respect to the corresponding component of x. The dot above variable represents the time derivatives.

We denote by \(\mathbb {S}^{d}\) the space of second-order symmetric tensors on \( \mathbb {R} ^{d}\) \((d=2,3)\) and by \(``.'',\) \(\Vert .\Vert \) the inner product and the norm on \(\mathbb {S}^{d}\) and \( \mathbb {R} ^{d}\), respectively, that is \(u.\upsilon =u_{i\,}\upsilon _{i}\), \( \Vert \upsilon \Vert =(\upsilon .\upsilon )^{1/2}\) for \(u=(u_{i})\), \( \upsilon =(\upsilon _{i})\in \mathbb {R} ^{d}\), and \(\sigma .\tau =\sigma _{ij\,}\tau _{ij}\), \(\Vert \sigma \Vert =(\sigma .\sigma )^{1/2}\) for \(\sigma =(\sigma _{ij})\), \(\tau =(\tau _{ij})\in \mathbb {S}^{d}.\) We also use the usual notation for the normal components and the tangential parts of vectors and tensors, respectively, given by \(\upsilon _{\nu }=\upsilon \cdot \nu \), \(\upsilon _{\tau }=\upsilon -\upsilon _{\nu }\nu \), \(\sigma _{\nu }=\sigma _{ij\,}\nu _{i\, }\nu _{j}\), and \(\sigma _{\tau }=\sigma \nu -\sigma _{\nu }\nu \).

With these assumptions, the classical model for the process is the following.

Problem 1

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

$$\begin{aligned}&\sigma =\mathcal {A}\varepsilon \mathcal {(}\dot{u})+\mathcal {F}\varepsilon (u)-\mathcal {E}^{*}\mathbf {E}(\varphi )&&&\qquad \quad \mathrm{in}&&\Omega \times ( 0,T), \end{aligned}$$
(2.1)
$$\begin{aligned}&D =\mathcal {B}\mathbf {E}(\varphi )+\mathcal {E}\varepsilon \mathcal {(}u)&&&\qquad \quad \mathrm{in}&&\Omega \times ( 0,T), \end{aligned}$$
(2.2)
$$\begin{aligned}&Div\,\sigma +f_{0} =0&&&\qquad \quad \mathrm{in}&&\Omega \times ( 0,T), \end{aligned}$$
(2.3)
$$\begin{aligned}&div\,D =q_{0}&&&\qquad \quad \mathrm{in}&&\Omega \times ( 0,T), \end{aligned}$$
(2.4)
$$\begin{aligned}&u =0&&&\qquad \quad \mathrm{on}&&\Gamma _{1}\times ( 0,T), \end{aligned}$$
(2.5)
$$\begin{aligned}&\sigma \nu =f_{2}&&&\qquad \quad \mathrm{on}&&\Gamma _{2}\times ( 0,T), \end{aligned}$$
(2.6)
$$\begin{aligned}&\left\{ \begin{array}{l} u_{\nu }\le 0,\, \\ \sigma _{\nu }-\gamma _{\nu }\beta ^{2}R_{\nu }(u_{\nu })\le 0, \\ \,(\sigma _{\nu }-\gamma _{\nu }\beta ^{2}R_{\nu }(u_{\nu }))u_{\nu }=0 \end{array} \right.&&&\qquad \quad \mathrm{on}&&\Gamma _{3}\times ( 0,T), \end{aligned}$$
(2.7)
$$\begin{aligned}&-\sigma _{\tau } =p_{\tau }(\beta )R_{\tau }(u_{\tau })\,&&&\qquad \quad \mathrm{on}&&\Gamma _{3}\times ( 0,T), \end{aligned}$$
(2.8)
$$\begin{aligned}&\dot{\beta }(t) =-(\gamma _{\nu }\beta (t)R_{\nu }(u_{\nu }(t)){{}^{2}} -\varepsilon _{a})_{+}\,&&&\qquad \quad \mathrm{on}&&\Gamma _{3}\times ( 0,T), \end{aligned}$$
(2.9)
$$\begin{aligned}&\varphi =0&&&\qquad \quad \mathrm{on}&&\Gamma _{a}\times ( 0,T), \end{aligned}$$
(2.10)
$$\begin{aligned}&D.\nu =q_{2}&&&\qquad \quad \mathrm{on}&&\Gamma _{b}\times ( 0,T), \end{aligned}$$
(2.11)
$$\begin{aligned}&D.\nu =0&&&\qquad \quad \mathrm{on}&&\Gamma _{3}\times ( 0,T), \end{aligned}$$
(2.12)
$$\begin{aligned}&u(0) =u_{0}\,&&&\qquad \quad \mathrm{in}&&\Omega , \end{aligned}$$
(2.13)
$$\begin{aligned}&\beta (0) =\beta _{0}&&&\qquad \quad \mathrm{on}&&\Gamma _{3}. \end{aligned}$$
(2.14)

We now provide some comments on equations and conditions (2.1)–(2.14).

First, equations (2.1) and (2.2) represent the electro-viscoelastic constitutive law in which \(\sigma =(\sigma _{ij})\) is the stress tensor, \(\varepsilon (u)=(\varepsilon _{ij}(u))\) denotes the linearized strain tensor, \(\mathbf {E}(\varphi )=-\nabla \varphi \) is the electric field, \(\mathcal {A}\) and \(\mathcal {F}\) are the viscosity and elasticity operators, respectively, \(\mathcal {E}\) \(=(e_{ijk})\) represents the third-order piezoelectric tensor, \(\mathcal {E}^{*}=(e_{ijk}^{*})\), where \(e_{ijk}^{*}=e_{kij}\), is its transpose, \(\mathcal {B=}(\mathcal { \mathcal {B}}_{ij})\) denotes the electric permittivity tensor, and \( D=(D_{1},\ldots ,D_{d})\) is the electric displacement vector. Details on the constitutive equations of the form (2.1) and (2.2) can be found, for instance, in [1, 2, 13, 21] and the references therein.

Next, Eqs. (2.3) and (2.4) are the equilibrium equations for the stress and electric displacement fields, respectively, in which Div” and “div ” denote the divergence operators for tensor and vector valued functions, respectively.

Conditions (2.5) and (2.6) are the displacement and traction boundary conditions, whereas (2.10) and (2.11) represent the electric boundary conditions. Note that we need to impose assumption (2.12) for physical reasons. Indeed, this condition models the case when the obstacle is a perfect insulator and was used in [1, 9, 15, 25, 26]. The evolution of the bonding field is governed by the differential Eq. (2.9) with given positive parameters \(\gamma _{\nu }\) and \( \varepsilon _{a}\) where \(r_{+}=\max \{0,r\}\).

Condition (2.7) represents the Signorini contact condition with adhesion, where \(u_{\nu }\) is the normal displacement, \(\sigma _{\nu }\) represents the normal stress, \(\gamma _{\nu }\) denotes a given adhesion coefficient, and \(R_{\nu }\) is the truncation operator defined by

$$\begin{aligned} R_{\nu }(s)=\left\{ \begin{array}{ll} -L &{} \quad \text {if}\ s<-L, \\ -s &{} \quad \text {if}\ -L\le s\le 0, \\ 0 &{} \quad \text {if}\ s>0. \end{array} \right. \end{aligned}$$

Here \(L>0\) is the characteristic length of the bond, beyond which it does not offer any additional traction (see [27]).

We assume that the resistance to tangential motion is generated only by the glue, and is assumed to depend on the adhesion field and on the tangential displacement, but, again, only up to the bond length L (see (2.8)), where the truncation operator \(R_{\tau }\) is defined by

$$\begin{aligned} R_{\tau }(\upsilon )=\left\{ \begin{array}{l@{\quad }l} \upsilon &{}\text {if}\ \Vert \upsilon \Vert \le L, \\ L\dfrac{\upsilon }{\Vert \upsilon \Vert }&{}\text {if}\ \Vert \upsilon \Vert >L. \end{array} \right. \end{aligned}$$

Then, \(p_{\tau }(\beta )\) acts as the stiffness or spring constant, increasing with \((\beta )\), and the traction is in the direction opposite to the displacement. The maximal modulus of the tangential traction is \(p_{\tau }(1)L\).

Finally, (2.13) and (2.14) represent the initial conditions in which \(u_{0}\) and \(\beta _{0}\) are the prescribed initial displacement and bonding fields, respectively.

3 Variational Formulation and Preliminaries

In this section, we list the assumptions on the data and derive a variational formulation for the contact problem. To this end we need to introduce some notation and preliminaries.

Everywhere below, we use the classical notation for \(L^{p}\) and Sobolev spaces associated to \(\Omega \) and \(\Gamma \). Moreover, we use the notation \(L^{2}(\Omega )^{d}\), \(H^{1}(\Omega )^{d}\), \(\mathcal {H}\), and \( \mathcal {H}_{1}\) for the following spaces

$$\begin{aligned}&\left. L^{2}(\Omega )^{d}=\{\ \upsilon =(\upsilon _{i})\ |\ \upsilon _{i}\in L^{2}(\Omega )\ \},\quad H^{1}(\Omega )^{d}=\{\ \upsilon =(\upsilon _{i})\ |\ \upsilon _{i}\in H^{1}(\Omega )\ \},\right. \\&\left. \mathcal {H}=\{\ \tau =(\tau _{ij})\ |\ \tau _{ij}=\tau _{ji}\in L^{2}(\Omega )\ \},\quad \mathcal {H}_{1}=\{\ \tau \in \mathcal {H}\ |\ \tau _{ij,j}\in L^{2}(\Omega )\ \}.\right. \end{aligned}$$

The spaces \(L^{2}(\Omega )^{d}\), \(H^{1}(\Omega )^{d}\), \(\mathcal {H}\), and \( \mathcal {H}_{1}\) are real Hilbert spaces endowed with the canonical inner products given by

$$\begin{aligned}&\left. (u,\upsilon )_{L^{2}(\Omega )^{d}}=\int _{\Omega }u\cdot \upsilon \,dx,\quad (u,\upsilon )_{H^{1}(\Omega )^{d}}=\int _{\Omega }u\cdot \upsilon \,dx+\int _{\Omega }\nabla u\cdot \nabla \upsilon \,dx,\right. \\&\left. (\sigma ,\tau )_{\mathcal {H}}=\int _{\Omega }\sigma \cdot \tau \,dx,\quad (\sigma ,\tau )_{\mathcal {H}_{1}}=\int _{\Omega }\sigma \cdot \tau \,dx+\int _{\Omega }\text {Div}\,\sigma \cdot \text {Div}\,\tau \,dx,\right. \end{aligned}$$

and the associated norms \(\Vert \cdot \Vert _{L^{2}(\Omega )^{d}}\), \(\Vert \cdot \Vert _{H^{1}(\Omega )^{d}}\), \(\Vert \cdot \Vert _{\mathcal {H}}\), and \( \Vert \cdot \Vert _{\mathcal {H}_{1}}\), respectively. Here and below we use the notation

$$\begin{aligned} \left. \begin{array}{l} \nabla \upsilon =(\upsilon _{i,j}),\quad \varepsilon (\upsilon )=(\varepsilon _{ij}(\upsilon )),\quad \varepsilon _{ij}(\upsilon )=\frac{1}{ 2}(\upsilon _{i,j}+\upsilon _{j,i})\quad \forall \,\upsilon \in H^{1}(\Omega )^{d}, \\ Div\,\tau =(\tau _{ij,j})\quad \forall \,\tau \in \mathcal {H} _{1}. \end{array} \right. \end{aligned}$$

For every element \(\upsilon \in H^{1}(\Omega )^{d}\) we also write \(\upsilon \) for the trace of \(\upsilon \) on \(\Gamma \) and we denote by \(\upsilon _{\nu }\) and \(\upsilon _{\tau }\) the normal and tangential components of \(\upsilon \) on \(\Gamma \).

We now list the assumptions on the problem’s data. The viscosity operator \( \mathcal {A}\) and the elasticity operator \(\mathcal {F}\) are assumed to satisfy the conditions

$$\begin{aligned}&\left\{ \begin{array}{ll} \text {(a)}&{}\mathcal {A}:\,\Omega \times \mathbb {S} ^{d}\longrightarrow \mathbb {S}^{d}. \\ \text {(b)}&{}\mathcal {A(}x,\tau )=(a_{ijkl}(x)\tau _{kl}\,) \,\,\forall \tau \in S^{d}\,\text {a.e. }x\in \Omega . \\ \text {(c)}&{}a_{ijkl}=a_{klij}=a_{jikl}\in L^{\infty }(\Omega ). \\ \text {(d)}&{}\hbox {there exists } m_{\mathcal {A}}>0\hbox { such that} \\ &{} a_{ijkl\,}\tau _{ij}\tau _{kl}\ge m_{\mathcal {A}}\Vert \tau \Vert ^{2},\,\forall \tau =(\tau _{ij})\in \mathbb {S}^{d}, \text { a.e. } x\in \,\Omega . \end{array}\right. \end{aligned}$$
(3.1)
$$\begin{aligned}&\left\{ \begin{array}{ll} \text {(a)}&{} \mathcal {F}:\Omega \times \mathbb {S}^{d}\rightarrow \mathbb {S} ^{d}. \\ \text {(b)}&{} \text {There exists }L_{\mathcal {F}}>0\text { such that} \\ &{}\Vert \mathcal {F}(x,\tau _{1})-\mathcal {F}(x,\tau _{2})\Vert \le L_{\mathcal {F}}\Vert \tau _{1}-\tau _{2}\Vert \\ &{}\forall \,\tau _{1},\tau _{2}\in \mathbb {S}^{d},\, \text { a.e.}\ \mathbf {x}\in \Omega . \\ \text {(c)}&{} \text {The mapping }x\mapsto \mathcal {F}(x,\tau )\, \text {is measurable on }\Omega , \\ &{}\text {for each }\,\tau \in \mathbb {S}^{d}. \\ \text {(d)}&{} \text {The mapping }x\mapsto \mathcal {F}(x,\varvec{0})\ \text {belongs to } \mathcal {H}. \end{array} \right. \end{aligned}$$
(3.2)

The piezoelectric tensor \(\mathcal {E}\) and the electric permittivity tensor \(\mathcal {B}\) satisfy

$$\begin{aligned}&\left\{ \begin{array}{ll} \text {(a) }&{}\mathcal {E}:\Omega \times \mathbb {S}^{d}\rightarrow \mathbb {R} ^{d}. \\ \text {(b) }&{}\mathcal {E}(x,\tau )=(e_{ijk}(x)\tau _{jk})\quad \forall \tau =(\tau _{ij})\in \mathbb {S}^{d},\ \text { a.e.}\, x\in \Omega . \\ \text {(c)}&{}e_{ijk}=e_{ikj}\in L^{\infty }(\Omega ). \end{array} \right. \end{aligned}$$
(3.3)
$$\begin{aligned}&\left\{ \begin{array}{ll} \text {(a)}&{}\mathcal {B}:\Omega \times \mathbb {R}^{d}\rightarrow \mathbb {R} ^{d}. \\ \text {(b)}&{}\mathcal {B}(x,\mathbf {E})=(\mathcal {B}_{ij}(x)E_{j})\quad \forall \mathbf {E}=(E_{i})\in \mathbb {R}^{d},\, \text { a.e.}\, \mathbf {x} \in \Omega . \\ \text {(c) }&{}\mathcal {B}_{ij}=\mathcal {B}_{ji}\in L^{\infty }(\Omega ). \\ \text {(d)}&{}\text {There exists}\ m_{\mathcal {B}}>0\ \,\text {such that}\ \,\mathcal {B}_{ij}(x)E_{i}E_{j}\ge m_{\mathcal {B}}\Vert \mathbf {E}\Vert ^{2} \\ &{}\forall \mathbf {E}=(E_{i})\,\in {\mathbb {R}}^{d},\ \text {a.e. }\ x\in \Omega . \end{array} \right. \end{aligned}$$
(3.4)

As in [8] we assume that the tangential contact function satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} \text {(a)}&{}p_{\tau }:\Gamma _{3}\times \mathbb {R} \longrightarrow \mathbb {R} _{+}. \\ \text {(b)} &{}\text {There exists}\quad L_{\tau }>0\,\text {such that} \\ &{}\mid p_{\tau }(x,\beta _{1})-p_{\tau }(x,\beta _{2})\mid \le L_{\tau }\mid \beta _{1}-\beta _{2}\mid \\ &{}\forall \beta _{1},\,\beta _{2}\in \mathbb {R},\ \text {a.e.}\, x\in \Gamma _{3}. \\ \text {(c)}&{} \text {There exists}\quad M_{\tau }>0\,{ \text {such that}} \\ &{}\mid p_{\tau }(x,\beta )\mid \le M_{\tau }\quad \forall \beta \in \mathbb {R},\ \text { a.e.}\ x\in \Gamma _{3}. \\ \text {(d)}&{}\text {For any }\beta \in \mathbb {R}, \,x\mapsto p_{\tau }(x,\beta )\,\text {is measurable on} \,\Gamma _{3}. \\ \text {(e)}&{}\text {The mapping } x\mapsto p_{\tau }(x,0)\text { belongs to } L^{2}(\Gamma _{3}). \end{array} \right. \end{aligned}$$
(3.5)

The forces, tractions, volume, and surface-free charge densities satisfy

$$\begin{aligned}&\left. f_{0}\in W^{1,1}(0,T;L^{2}(\Omega )^{d}),\quad f_{2}\in W^{1,1}(0,T;L^{2}(\Gamma _{2})^{d}),\right. \end{aligned}$$
(3.6)
$$\begin{aligned}&\left. q_{0}\in W^{1,1}(0,T;L^{2}(\Omega )),\quad \ q_{2}\in W^{1,1}(0,T;L^{2}(\Gamma _{b})).\right. \end{aligned}$$
(3.7)

The adhesion coefficient \(\gamma _{\nu }\) and the limit bound \(\varepsilon _{a}\) satisfy the conditions

$$\begin{aligned} \gamma _{\nu }\in L^{\infty }(\Gamma _{3}),\quad \epsilon _{a}\in L^{2}(\Gamma _{3}),\quad \gamma _{\nu },\,\gamma _{\tau },\,\epsilon _{a}\ge 0\quad \text {a.e. on}\ \Gamma _{3}. \end{aligned}$$
(3.8)

Also, we assume that the initial bonding field satisfies

$$\begin{aligned} \beta _{0}\in L^{2}(\Gamma _{3}),\quad 0\le \beta _{0}\le 1\ \ \text {a.e. on}\ \Gamma _{3}. \end{aligned}$$
(3.9)

Moreover, the tensor \(\mathcal {E}\) and its transpose \(\mathcal {E}^{*}\) satisfy the equality

$$\begin{aligned} \mathcal {E}\sigma .\upsilon =\sigma .\mathcal {E}^{*}\upsilon \ \quad \forall \sigma \in \mathbb {S}^{d},\,\upsilon \in \mathbb {R}^{d} \end{aligned}$$
(3.10)

Let now consider the closed subspace of \(H^{1}(\Omega )^{d}\) defined by

$$\begin{aligned} V=\{\ \upsilon \in H^{1}(\Omega )^{d}\ |\ \upsilon =0\ \text {on}\, \Gamma _{1}\ \}. \end{aligned}$$

Since \(meas\,(\Gamma _{1})>0\) and the viscosity tensor satisfies assumption (3.1), it follows that V is a real Hilbert space endowed with the inner product

$$\begin{aligned} (u,\upsilon )_{V}=(\mathcal {A}\varepsilon (u),\varepsilon (\upsilon ))_{ \mathcal {H}}, \end{aligned}$$
(3.11)

and let \(\Vert \cdot \Vert _{V}\) be the associated norm.

We also introduce the following spaces

$$\begin{aligned} W=\{\ \psi \in H^{1}(\Omega )\ |\ \psi =0\ \text {on}\ \Gamma _{a} \},\,\mathcal {W}=\{\ D=(D_{i})\ |\ D_{i}\in L^{2}(\Omega ),\ div\,D\in L^{2}(\Omega )\ \}. \end{aligned}$$

Since \(meas\,(\Gamma _{a})>0\) it is well known that W is a real Hilbert space endowed with the inner product

$$\begin{aligned} (\varphi ,\psi )_{W}=(\nabla \varphi ,\nabla \psi )_{L^{2}(\Omega )^{d}}\,, \end{aligned}$$

and the associated norm \(\Vert \cdot \Vert _{W}\). Also we have the following Friedrichs-Poincaré inequality

$$\begin{aligned} \Vert \nabla \psi \Vert _{L^{2}(\Omega )^{d}}\ge c_{F}\,\Vert \psi \Vert _{H^{1}(\Omega )}\quad \forall \,\psi \in W, \end{aligned}$$
(3.12)

where \(c_{F}>0\) is a constant which depends only on \(\Omega \) and \(\Gamma _{a}\). The space \(\mathcal {W}\) is a real Hilbert space endowed with the inner product

$$\begin{aligned} (D,\mathbf {E})_{\mathcal {W}}=\int _{\Omega }D\cdot \mathbf {E} \,dx+\int _{\Omega }div\,D\cdot div\,\mathbf {E}\,dx, \end{aligned}$$

and the associated norm \(\Vert \cdot \Vert _{\mathcal {W}}\). Moreover, by the Sobolev trace theorem, there exist two positive constants \(c_{0}\) and \(\tilde{c}_{0}\) such that

$$\begin{aligned} \Vert \upsilon \Vert _{L^{2}(\Gamma _{3})^{d}}\le c_{0}\Vert \upsilon \Vert _{V}\quad \forall \upsilon \in V, \,\Vert \psi \Vert _{L^{2}(\Gamma _{3})}\le \tilde{c}_{0}\Vert \psi \Vert _{W}\quad \forall \,\psi \in W. \end{aligned}$$
(3.13)

Next, we define the two mappings f : [0 \(T]\longrightarrow V\) and q : [0 \( T]\longrightarrow W\), respectively, by

$$\begin{aligned}&\left. (f(t),\upsilon )_{V}=\int _{\Omega }f_{0}(t)\cdot \upsilon \,dx+\int _{\Gamma _{2}}f_{2}(t)\cdot \upsilon \,da,\right. \end{aligned}$$
(3.14)
$$\begin{aligned}&\left. (q(t),\psi )_{W}=\int _{\Omega }q_{0}(t)\psi \,dx-\int _{\Gamma _{b}}q_{2}(t)\psi \,da,\right. \end{aligned}$$
(3.15)

for all \(\upsilon \in V,\psi \in W\), and \(t\in [0,T]\). We note that the definitions of f and q are based on the Riesz representation theorem. Moreover, it follows from assumptions (3.6) and (3.7) that

$$\begin{aligned}&\left. f\in W^{1,1}(0,T;V),\right. \end{aligned}$$
(3.16)
$$\begin{aligned}&\left. q\in W^{1,1}(0,T;W).\right. \end{aligned}$$
(3.17)

For Signorini problem, we use the convex subset of admissible displacements fields given by

$$\begin{aligned} U_{ad}=\left\{ \upsilon \in V/\,\upsilon _{\nu }\le 0\text { on } \Gamma _{3}\right\} , \end{aligned}$$

and we make the regularity assumption

$$\begin{aligned} u_{0}\in U_{ad}, \end{aligned}$$
(3.18)

on the initial data. Also, we introduce the set

$$\begin{aligned} \mathcal {Q}=\{\,\beta \in L^{\infty }(0,T;L^{2}(\Gamma _{3}))\ |\ 0\le \beta (t)\le 1\ \ \forall \,t\in [0,T],\ a.e.\ on\ \Gamma _{3}\,\}. \end{aligned}$$

Next, we define the functional \(j:L^{2}(\Gamma _{3})\times V\times V\longrightarrow \mathbb {R} \) by

$$\begin{aligned} j(\beta ,u,\upsilon )=\int \limits _{\Gamma _{3}}-\gamma _{\nu }\beta ^{2}R_{\nu }(u_{\nu })\upsilon _{\nu }da+\int \limits _{\Gamma _{3}}p_{\tau }(\beta )R_{\tau }(u_{\tau })\cdot \upsilon _{\tau }da. \end{aligned}$$
(3.19)

It follows from assumptions (3.5)–(3.8) that the integrals in (3.14), (3.15), and (3.19) are well defined.

Using a standard procedure based on Green’s formulas and equalities (3.14), (3.15), (3.19), it is easy to see that if \((u,\sigma ,\varphi ,\beta ,D)\) are sufficiently regular functions which satisfy (2.3)–(2.12) then

$$\begin{aligned} u(t)\in U_{ad},\,(\sigma (t),\varepsilon (\upsilon )-\varepsilon (u(t)))_{\mathcal {H}}+j(\beta (t),u(t),\upsilon -u(t)\ge (f(t),\upsilon -u(t))_{V}, \end{aligned}$$
(3.20)
$$\begin{aligned} (D(t),\nabla \psi )_{L^{2}(\Omega )^{d}}+(q(t),\psi )_{W}=0, \end{aligned}$$
(3.21)

for all \(\upsilon \in U_{ad},\) \(\psi \in W\)   and \(t\in [0,T]\). We substitute (2.1) in (3.20), (2.2) in (3.21), keeping in mind that \(\mathbf {E}(\varphi )=-\nabla \varphi \) and use the initial condition (2.13) to derive the following variational formulation of Problem (\(\mathcal {P}\)).

Problem 2

(\(\mathcal {P}^{V}\)). Find a displacement field \( u:[0,T]\longrightarrow V\), an electric potential field \(\varphi :[0,T]\rightarrow W\), and a bonding field \(\beta :[0,T]\rightarrow L^{2}(\Gamma _{3})\) such that

$$\begin{aligned}&\left. \begin{array}{l} u(t)\in U_{ad},\,(\mathcal {A}\varepsilon (\dot{u}(t)),\varepsilon (\upsilon )-\varepsilon (u(t)))_{\mathcal {H}}+(\mathcal {F}\varepsilon (u(t)),\varepsilon (\upsilon )-\varepsilon (u(t)))_{\mathcal {H}}\\ \qquad +\, (\mathcal {E}^{*}\nabla \varphi (t),\varepsilon (\upsilon )-\varepsilon (u(t)))_{\mathcal {H} }+j(\beta (t),u(t),\upsilon -u(t)) \\ \quad \ge (f(t),\upsilon -u(t))_{V}\qquad \forall \upsilon \in U_{ad},\,\,\text {a.e. }t\in (0\,T), \end{array} \right. \end{aligned}$$
(3.22)
$$\begin{aligned}&\left. \begin{array}{c} (\mathcal {B}\nabla \varphi (t),\nabla \psi )_{L^{2}(\Omega )^{d}}-(\mathcal {E }\varepsilon (u(t)),\nabla \psi )_{L^{2}(\Omega )^{d}}=(q(t),\psi )_{W}\, \\ \times \forall \psi \in W,\,\forall \,t\in \left[ 0\,T\right] , \end{array} \right. \end{aligned}$$
(3.23)
$$\begin{aligned}&\left. \dot{\beta }(t)=-(\gamma _{\nu }\beta (t)R_{\nu }(u_{\nu }(t)){{}^{2}} -\varepsilon _{a})_{+}\,\text {.a.e. }\,t\in (0\,T),\right. \end{aligned}$$
(3.24)
$$\begin{aligned}&\left. u(0)=u_{0},\right. \end{aligned}$$
(3.25)
$$\begin{aligned}&\left. \beta (0)=\beta _{0}.\right. \end{aligned}$$
(3.26)

4 Existence and Uniqueness Result

Our main existence and uniqueness result is the following.

Theorem 4.1

Assume that (3.1)–(3.9) and (3.18) hold. Then, there exists a unique solution \((u,\varphi ,\beta )\) to Problem (\( \mathcal {P}^{V}\)). Moreover, the solution satisfies

$$\begin{aligned}&\varvec{u}\in W^{1,\infty }(0,T;V), \end{aligned}$$
(4.1)
$$\begin{aligned}&\varphi \in W^{1,\infty }(0,T;W), \end{aligned}$$
(4.2)
$$\begin{aligned}&\beta \in W^{1,\infty }(0,T;L^{2}(\Gamma _{3}))\cap \mathcal {Q}. \end{aligned}$$
(4.3)

A “quintuple; of functions \((u,\,\sigma ,\,\varphi ,\,D,\,\beta )\) which satisfies (2.1), (2.2) and (3.22)–(3.26) is called a weak solution of the contact Problem (\( \mathcal {P}\)). We conclude by Theorem 4.1 that, under the assumptions (3.1)–(3.9) and (3.18), there exists a unique weak solution of Problem (\(\mathcal {P}\)).

To precise the regularity of the weak solution we note that the constitutive relations (2.1) and (2.2), the assumptions (3.1)–(3.4) and the regularities (4.1), (4.2) imply that \(\mathbf { \sigma }\in L^{\infty }(0,T;\mathcal {H})\), \(\varvec{D}\in W^{1,\infty }(0,T;L^{2}(\Omega )^{d})\). By taking \(\upsilon =u(t)\pm \xi ,\) where \(\xi \in C_{0}^{\infty }(\Omega )^{d},\) in (3.20) and \(\psi \in C_{0}^{\infty }(\Omega )\) in (3.21) and using the notation (3.14), (3.15), (3.19) we find

$$\begin{aligned} Div \,\sigma \,(t)+f_{0}(t)=0,\,div \, D(t)=q_{0}(t), \end{aligned}$$

for all \(t\in [0,T]\). It follows now from the regularities (3.6 ), (3.7) that Div \(\sigma \in L^{\infty }(0,T;L^{2}(\Omega )^{d})\) and div \(D\in W^{1,\infty }(0,T;L^{2}(\Omega ))\), which shows that

$$\begin{aligned} \sigma\in & {} L^{\infty }(0,T;\mathcal {H}_{1}), \end{aligned}$$
(4.4)
$$\begin{aligned} D\in & {} W^{1,\infty }(0,T;\mathcal {W}). \end{aligned}$$
(4.5)

We conclude that the weak solution \((u,\sigma ,\varphi ,D,\beta )\) of the piezoelectric contact problem (\(\mathcal {P}\)) has the regularity (4.1 )–(4.5).

The proof of Theorem 4.1 will be carried out in several steps and is based on the following abstract result.

Let X be a real Hilbert space with the inner product \((\cdot ,\cdot )_{X}\) and the associated norm \(\Vert \cdot \Vert _{X}\), and let \( A:D(A)\subset X\longrightarrow 2^{X}\) be a multivalued operator, where D(A) is the domain of A given by

$$\begin{aligned} D(A)=\left\{ x\in X:Ax\ne \varnothing \right\} , \end{aligned}$$

and \(2^{X}\) represents the set of the subsets of X. The graph of A denoted by Gr(A) is given by

$$\begin{aligned} Gr(A)=\left\{ (x,y)\in X\times X:y\in Ax\right\} . \end{aligned}$$

The operator \(A:X\longrightarrow 2^{X}\) is called

  1. (i)

    monotone if       \(\forall (x_{1},y_{1})\in Gr(A),\,\forall (x_{2},y_{2})\in Gr(A):(y_{1}-y_{2},x_{1}-x_{2})_{X}\ge 0.\)

  2. (ii)

    \(\text {maximal monotone if }\,A\,\text { is monotone and there is no monotone}\) \(\text {operator }B:X\longrightarrow 2^{X}\text {such that }Gr(A)\text { is a proper subset of }Gr(B),\) \(\text {which is equivalent to the following implication}\)       \(\left[ (y_{1}-y_{2},x_{1}-x_{2})_{X}\ge 0,\,\forall (x_{1},y_{1})\in Gr(A)\right] \Rightarrow (x_{2},y_{2})\in Gr(A).\)

For a function \(\left. \left. \phi :X\longrightarrow \right] -\infty ,+\infty \right] \) we use the notation \(D(\phi )\) and \(\partial \phi \) for the effective domain and the subdifferential of \(\phi \), i.e.,

$$\begin{aligned}&\left. D(\phi )=\left\{ u\in X:\phi (u)<\infty \right\} ,\right. \end{aligned}$$
(4.6)
$$\begin{aligned}&\left. \partial \phi (u)=\left\{ f\in X:\phi (\upsilon )-\phi (u)\ge (f,\upsilon -u)_{X}\quad \forall \upsilon \in X\right\} ,\, \forall u\in X.\right. \end{aligned}$$
(4.7)

Finally, let \(\left. \left. \phi _{K}:X\rightarrow \right] -\infty ,+\infty \right] \) denote the indicator function of the set K, i.e.,

$$\begin{aligned} \phi _{K}(\upsilon )=\left\{ \begin{array}{l@{\quad }l} 0&{} \text {if }\upsilon \in K, \\ \infty &{} \text {if }\upsilon \notin K. \end{array} \right. \end{aligned}$$

It can be shown that the subdifferential of the indicator function \(\partial \phi _{K}:X\longrightarrow \) \(2^{X}\) of a closed convex K of the space X is a maximal monotone operator. We can also show that the sum of a maximal monotone operator and a single-valued monotone Lipschitz continuous operator is a maximal monotone operator.

Finally, we use the usual notation for the Lebesgue spaces \( L^{p}(0,T;X)\) and Sobolev spaces \(W^{k,p}(0,T;X)\) where \(1\le p\le \infty \) and \(k\in \mathbb {N}\). We will need the following result for existence and uniqueness proofs.

Theorem 4.2

Let X be a real Hilbert space and let \(A:D(A)\subset X\longrightarrow 2^{X}\) be a multivalued operator such that the operator \( A+\omega I_{X}\) is maximal monotone for some real \(\omega \). Then, for every \(f\in W^{1,1}(0,T;X)\) and \(u_{0}\in D(A),\) there exists a unique function \( \varvec{u}\in W^{1,\infty }(0,T;X)\) which satisfies

$$\begin{aligned}&\left. \dot{u}(t)+Au(t)\ni f(t)\text { a.e. }t\in (0\,T),\right. \end{aligned}$$
(4.8)
$$\begin{aligned}&\left. u(0)=u_{0}.\right. \end{aligned}$$
(4.9)

A proof of Theorem 4.2 may be found in ([6], p. 32). Here and below \(I_{X}\) is the identity map on X.

5 Proof of Theorem 4.1

We assume in the following that the conditions of Theorem 4.1 hold and below we denote by c a generic positive constant which is independent of time and whose value may change from place to place.

By the Riesz representation theorem we can define the following operators \(\mathcal {G}:W\longrightarrow W\) and \(\mathcal {R}:V\longrightarrow W\), respectively, by

$$\begin{aligned}&\left. (\mathcal {G}\varphi ,\psi )_{W}=(\mathcal {B}\nabla \varphi ,\nabla \psi )_{L^{2}(\Omega )^{d}}\qquad \forall \varphi ,\psi \in W,\right. \end{aligned}$$
(5.1)
$$\begin{aligned}&\left. (\mathcal {R}\upsilon ,\varphi )_{W}=(\mathcal {E}\varepsilon (\upsilon ),\nabla \varphi )_{L^{2}(\Omega )^{d}}\qquad \forall \varphi \in W,\,\upsilon \in V.\right. \end{aligned}$$
(5.2)

We can show that \(\mathcal {G}\) is a linearly continuous symmetric positive definite operator. Therefore, \(\mathcal {G}\) is an invertible operator on W. We can also prove that \(\mathcal {R}\) is a linear continuous operator on V. Let \(\mathcal {R}^{*}\) the adjoint of \(\mathcal {R}\). Thus, from (3.10) we can write

$$\begin{aligned} (\mathcal {R}^{*}\varphi ,\upsilon )_{V}=(\mathcal {E}^{*}\nabla \varphi ,\varepsilon (\upsilon ))_{\mathcal {H}}\qquad \forall \varphi \in W,\,\upsilon \in V. \end{aligned}$$
(5.3)

Let \(t\in \left[ 0\,T\right] \). By introducing (5.1), (5.2) in (3.23) we get

$$\begin{aligned} (\mathcal {G}\varphi (t),\psi )_{W}=(\mathcal {R}u(t),\psi )_{W}+(q(t),\psi )_{W}\,\forall \psi \in W, \end{aligned}$$
(5.4)

where we obtain

$$\begin{aligned} \mathcal {G}\varphi (t)=\mathcal {R}u(t)+q(t), \end{aligned}$$

for all \(t\in [0,T]\). On the other hand, \(\mathcal {G}\) is invertible where the previous equality gives us

$$\begin{aligned} \varphi (t)=\mathcal {G}^{-1}\mathcal {R}u(t)+\mathcal {G}^{-1}q(t). \end{aligned}$$
(5.5)

Now, using (5.3), (5.5), and (3.22) we obtain

$$\begin{aligned}&u(t)\in U_{ad},\,(\mathcal {A}\varepsilon (\dot{u}(t)),\varepsilon (\upsilon )-\varepsilon (u(t)))_{\mathcal {H}}+(\mathcal {F}\varepsilon (u(t)),\varepsilon (\upsilon )-\varepsilon (u(t)))_{\mathcal {H}} \nonumber \\&\qquad +\,(\mathcal {R}^{*}\mathcal {G}^{-1}\mathcal {R}u(t),\upsilon -u(t))_{V}+j(\beta (t),u(t),\upsilon -u(t))_{V} \nonumber \\&\quad \ge (f(t)-\mathcal {R}^{*}\mathcal {G}^{-1}q(t),\upsilon -u(t))_{V}\qquad \forall \upsilon \in U_{ad}\,\text {a.e. }t\in (0\,T). \end{aligned}$$
(5.6)

Let \(\eta \in W^{1,\infty }(0,T;V)\) be given. In the first step we prove the following existence and uniqueness result for the displacement field.

Lemma 5.1

There exists a unique function \(u_{\eta }\in W^{1,\infty }(0,T;V)\) such that

$$\begin{aligned}&u_{\eta }(t)\in U_{ad},\,(\mathcal {A}\varepsilon (\dot{u}_{\eta }(t)),\varepsilon (\upsilon )-\varepsilon (u_{\eta }(t)))_{\mathcal {H}}+( \mathcal {F}\varepsilon (u_{\eta }(t)),\varepsilon (\upsilon )-\varepsilon (u_{\eta }(t)))_{\mathcal {H}}\nonumber \\&\qquad + \,(\mathcal {R}^{*}\mathcal {G}^{-1}\mathcal {R}u_{\eta }(t),\upsilon -u_{\eta }(t))_{V}+(\eta (t),\upsilon -u_{\eta }(t))_{V} \nonumber \\&\quad \ge (f(t)-\mathcal {R}^{*}\mathcal {G}^{-1}q(t),\upsilon -u_{\eta }(t))_{V}\qquad \forall \upsilon \in U_{ad}\,\text { a.e. }t\in (0\,T), \end{aligned}$$
(5.7)
$$\begin{aligned}&u_{\eta }(0)=u_{0}. \end{aligned}$$
(5.8)

Proof

Let now the operator \(L:V\rightarrow V\) defined by

$$\begin{aligned} L(\upsilon )=\mathcal {R}^{*}\mathcal {G}^{-1}\mathcal {R}(\upsilon ),\,\forall \upsilon \in V. \end{aligned}$$
(5.9)

Using the properties of the operators \(\mathcal {G}\), \(\mathcal {R}\), and \( \mathcal {R}^{*}\) we deduce that L is a continuous linear operator on V . Thus we have

$$\begin{aligned} \Vert Lu_{1}-Lu_{2}\Vert _{V}\le \Vert L\Vert \Vert u_{1}-u_{2}\Vert _{V} \quad \forall u_{1},\,u_{2}\in V. \end{aligned}$$
(5.10)

By the Riesz representation theorem we can define an operator \( \mathbf {G}:V\longrightarrow V\) by

$$\begin{aligned} (\mathbf {G}u,\upsilon )_{V}=(\mathcal {F}\varepsilon (u),\varepsilon (\upsilon ))_{\mathcal {H}}+(Lu,\upsilon )_{V}\quad \forall u,\, \upsilon \in V. \end{aligned}$$
(5.11)

Now, taking into account (3.1), (3.2), (3.11) and (5.11) it follows

$$\begin{aligned} \Vert \mathbf {G}u_{1}-\mathbf {G}u_{2}\Vert _{V}\le \left( \frac{L_{\mathcal {F}}}{ m_{\mathcal {A}}}+\Vert L\Vert \right) \Vert u_{1}-u_{2}\Vert _{V}\qquad \forall u_{1},\,u_{2}\in V, \end{aligned}$$
(5.12)

that is, \(\mathbf {G}\) is a Lipschitz continuous operator. Moreover, the operator

$$\begin{aligned} \mathbf {G+}\left( \frac{L_{\mathcal {F}}}{m_{\mathcal {A}}}+\Vert L\Vert \right) I_{V}:V\rightarrow V, \end{aligned}$$

is a monotone Lipschitz continuous operator on V.

Let the function \(\mathbf {f}\) \(:\left[ 0\,T\right] \longrightarrow V\) given by

$$\begin{aligned} \mathbf {f}(t)=f(t)-\mathcal {R}^{*}\mathcal {G}^{-1}q(t)-\eta (t),\,\forall t\in \left[ 0\,T\right] . \end{aligned}$$
(5.13)

Keeping in mind that \(\eta \in W^{1,\infty }(0,T;V)\), using (3.16), (3.17) and the fact that \(\mathcal {R}^{*}\mathcal {G}^{-1}\) is linearly continuous, it follows from (5.13) that

$$\begin{aligned} \mathbf {f\in }W^{1,1}(0,T;V). \end{aligned}$$
(5.14)

Let \(\left. \left. \phi _{U_{ad}}:V\rightarrow \right] -\infty ,+\infty \right] \) denote the indicator function of the set \(U_{ad}\) and let \(\partial \phi _{U_{ad}}\) be the subdifferential of \(\phi _{U_{ad}}\). Since \(U_{ad}\) is a nonempty, convex, closed part of V, it follows that \(\partial \phi _{U_{ad}}\) is a maximal monotone operator on V and \(D(\partial \phi _{U_{ad}})=U_{ad}\). Moreover, the sum

$$\begin{aligned} \partial \phi _{U_{ad}}+\mathbf {G+}\left( \frac{L_{\mathcal {F}}}{m_{\mathcal {A}}} +\Vert L\Vert \right) I_{V}:U_{ad}\subset V\rightarrow 2^{V}, \end{aligned}$$

is a maximal monotone operator. Thus, conditions (3.18) and (5.14 ) allow us to apply Theorem 4.2 with \(X=V,\) \(A=\partial \phi _{U_{ad}}+\mathbf {G:}D(A)=U_{ad}\subset V\rightarrow 2^{V}\), and \(\omega = \frac{L_{\mathcal {F}}}{m_{\mathcal {A}}}+\Vert L\Vert .\) We deduce that there exists a unique element \(u_{\eta }\in W^{1,\infty }(0,T;V)\) such that

$$\begin{aligned}&\left. \dot{u}_{\eta }(t)+\partial \phi _{U_{ad}}(u_{\eta }(t))+\mathbf {G} u_{\eta }(t)\ni \mathbf {f(}t\mathbf {)}\quad \text {a.e.}\,t\in (0\, T),\right. \end{aligned}$$
(5.15)
$$\begin{aligned}&\left. u_{\eta }(0)=u_{0}.\right. \end{aligned}$$
(5.16)

Since for any elements u,  \(g\in V\), the following equivalence holds

$$\begin{aligned} g\in \partial \phi _{U_{ad}}(u)\Leftrightarrow u\in U_{ad},\,\, (g,\upsilon -u)_{V}\le 0\,\,\forall \,\,\upsilon \in U_{ad}, \end{aligned}$$

the differential inclusion (5.15) is equivalent to the following variational inequality

$$\begin{aligned} \left. \begin{array}{l} u_{\eta }(t)\in U_{ad},\,(\dot{u}_{\eta }(t),\upsilon -u_{\eta }(t))_{V}+(\mathbf {G}u_{\eta }(t),\upsilon -u_{\eta }(t))_{V} \\ \qquad \ge (\mathbf {f}(t),\upsilon -u_{\eta }(t))_{V} \,\,\forall \upsilon \in U_{ad}\,\,\text {a.e.}\,t\in (0\,T). \end{array} \right. \end{aligned}$$
(5.17)

We use now (5.17), (5.11), (3.11) to see that \(u_{\eta }\) satisfies the following inequality

$$\begin{aligned}&u_{\eta }(t)\in U_{ad},\,(\mathcal {A}\varepsilon (\dot{u}_{\eta }(t)),\varepsilon (\upsilon )-\varepsilon (u_{\eta }(t)))_{\mathcal {H}}+( \mathcal {F}\varepsilon (u_{\eta }(t)),\varepsilon (\upsilon )-\varepsilon (u_{\eta }(t)))_{\mathcal {H}}\nonumber \\&\quad +\, (Lu_{\eta }(t),\upsilon -u_{\eta }(t))_{V}\ge (\mathbf {f}(t),\upsilon -u_{\eta }(t))_{V}\,\forall \upsilon \in U_{ad}\,\text {a.e.}\,t\in (0\,T). \end{aligned}$$
(5.18)

It follows now from (5.18), (5.13), (5.9), and (5.16) that \(u_{\eta }\) satisfies (5.7) and (5.8), which concludes the proof of Lemma 5.1. \(\square \)

In the second step we use the displacement field \(u_{\eta }\) obtained in Lemma 5.1 to obtain the following existence and uniqueness result for the electric potential field.

Lemma 5.2

There exists a unique function \(\varphi _{\eta }\in W^{1,\infty }(0,T;W)\) such that

$$\begin{aligned} \left. \begin{array}{l} (\mathcal {B}\nabla \varphi _{\eta }(t),\nabla \psi )_{L^{2}(\Omega )^{d}}-( \mathcal {E}\varepsilon (u_{\eta }(t)),\nabla \psi )_{L^{2}(\Omega )^{d}}=(q(t),\psi )_{W}\, \\ \quad \forall \psi \in W,\,\,\forall \,t\in \left[ 0\,T\right] , \end{array} \right. \end{aligned}$$
(5.19)

Proof

Let \(u_{\eta }\in W^{1,\infty }(0,T;V)\) be the function defined in Lemma 5.1. Clearly, equality (5.19) holds from (5.4), (5.2 ), and (5.1). Moreover, since \(u_{\eta }\in W^{1,\infty }(0,T;V)\) it follows from (5.5), (3.17) that \(\varphi _{\eta }\in W^{1,\infty }(0,T;W)\). Now, using (5.5) we deduce that the uniqueness of \(\varphi _{\eta }\) follows from the uniqueness of the function \(u_{\eta }\).

In the third step, we use again the displacement field \(u_{\eta }\) obtained in Lemma 5.1 and we consider the following initial value problem.

Problem 3

\(\mathcal {P}^{\beta _{\eta }}\). Find a bonding field \(\beta _{\eta }:\) \(\left[ 0,T\right] \rightarrow L^{2}(\Gamma _{3})\) such that

$$\begin{aligned}&\left. \dot{\beta }_{\eta }(t)=-(\gamma _{\nu }\beta _{\eta }(t)R_{\nu }(u_{\eta \nu }(t)){{}^{2}}-\varepsilon _{a})_{+}\text { a.e. }t\in (0\, T),\right. \end{aligned}$$
(5.20)
$$\begin{aligned}&\left. \beta _{\eta }(0)=\beta _{0}.\right. \end{aligned}$$
(5.21)

We obtain the following result.

Lemma 5.3

There exists a unique solution \(\beta _{\eta }\) to Problem \(\mathcal {P}^{\beta _{\eta }}\) and it satisfies \(\beta _{\eta }\in W^{1,\infty }(0,T,L^{2}(\Gamma _{3}))\cap \mathcal {Q}\).

Proof

Consider the mapping \(F:\,[0,T]\times L^{2}(\Gamma _{3})\rightarrow L^{2}(\Gamma _{3})\) defined by

$$\begin{aligned} F(t,\beta _{\eta })=-\left( \gamma _{\nu }\beta _{\eta }(t)R_{\nu }(u_{\eta \nu }(t)){{}^{2}}-\varepsilon _{a}\right) _{+}, \end{aligned}$$
(5.22)

for all \(t\in [0,T]\) and \(\beta _{\eta }\in L^{2}(\Gamma _{3})\). It follows from the properties of the truncation operator \(R_{\nu }\) that F is Lipschitz continuous with respect to the second argument, uniformly in time. Moreover, for any \(\beta _{\eta }\in L^{2}(\Gamma _{3})\), the mapping \(t\mapsto F(t,\beta _{\eta })\) belongs to \(L^{\infty }(0,T;L^{2}(\Gamma _{3}))\). Using now a version of Cauchy–Lipschitz theorem (see, e.g., [27], p. 48), we obtain the existence of a unique function \(\beta _{\eta }\in W^{1,\infty }(0,T,L^{2}(\Gamma _{3}))\) which solves (5.20), (5.21). We note that the restriction \(0\le \beta _{\eta }\le 1\) is implicitly included in the Cauchy problem \(\mathcal {P}^{\beta _{\eta }}\). Indeed, (5.20) and (5.21) guarantee that \(\beta _{\eta }(t)\le \beta _{0}\), and therefore, assumption (3.9) shows that \(\beta _{\eta }(t)\le 1\) for \(t\ge 0\), a.e. on \(\Gamma _{3}\). On the other hand, if \(\beta _{\eta }(t_{0})=0\) at \( t=t_{0}\), then it follows from (5.20) and (5.21) that \(\dot{\beta }_{\eta }(t)=0\) for all \(t\ge t_{0}\) and therefore, \(\beta _{\eta }(t)=0\) for all \(t\ge t_{0}\), a.e. on \(\Gamma _{3}\). We conclude that \(0\le \beta _{\eta }(t)\le 1\) for all \(t\in [0,T]\), a.e. on \(\Gamma _{3}\). Therefore, from the definition of the set \(\mathcal {Q}\), we find that \(\beta _{\eta }\in \mathcal {Q}\), which concludes the proof of Lemma 5.3. \(\square \)

Now, for \(\eta \in W^{1,\infty }(0,T;V)\) we denote by \(u_{\eta }\) and \( \beta _{\eta }\) the functions obtained in Lemmata 5.1 and 5.3, respectively. We use Riesz’s representation theorem to define the function \(\Lambda \eta :[0,T]\longrightarrow V\) by

$$\begin{aligned} (\Lambda \eta (t),\upsilon )_{V}=j(\beta _{\eta }(t),u_{\eta }(t),\upsilon ), \end{aligned}$$
(5.23)

for all \(\upsilon \in V\) and \(t\in \left[ 0,T\right] \). We have the following result.

Lemma 5.4

For all \(\eta \in W^{1,\infty }(0,T;V)\) the function \(\Lambda \eta \) belongs to \(W^{1,\infty }(0,T;V)\). Moreover, there exists a unique element \(\eta ^{*}\in W^{1,\infty }(0,T;V)\) such that

$$\begin{aligned} \Lambda \eta ^{*}=\eta ^{*}. \end{aligned}$$
(5.24)

Proof

Let \(\eta \in W^{1,\infty }(0,T;V)\) and let \(t_{1},t_{2}\in \left[ 0,T\right] .\) Using (5.23) and (3.19), we obtain

$$\begin{aligned}&\parallel \Lambda \eta (t_{1})-\Lambda \eta (t_{2})\parallel _{V}\le c\parallel \beta _{\eta }^{2}(t_{1})R_{\nu }(u_{\eta \nu }(t_{1}))\\&\quad -\beta _{\eta }^{2}(t_{2})R_{\nu }(u_{\eta \nu }(t_{2}))\parallel _{L^{2}(\Gamma _{3})}+c\parallel p_{\tau }(\beta _{\eta }(t_{1}))R_{\tau }(u_{\eta \tau }(t_{1}))\\&\quad -p_{\tau }(\beta _{\eta }(t_{2}))R_{\tau }(u_{\eta \tau }(t_{2}))\parallel _{L^{2}(\Gamma _{3})}. \end{aligned}$$

Now, keeping in mind (3.5), (3.13), the inequality \(0\le \beta _{\eta }(t)\le 1\) and the properties of the truncation operators \(R_{\nu }\) and \(R_{\tau },\) we find that

$$\begin{aligned}&\parallel \Lambda \eta (t_{1})-\Lambda \eta (t_{2})\parallel _{V}\le c\parallel u_{\eta }(t_{1})-u_{\eta }(t_{2})\parallel _{V}\nonumber \\&\quad +\, c\parallel \beta _{\eta }(t_{1})-\beta _{\eta }(t_{2})\parallel _{L^{2}(\Gamma _{3})}. \end{aligned}$$
(5.25)

Since \(u_{\eta }\in W^{1,\infty }(0,T;V)\) and \(\beta _{\eta }\in W^{1,\infty }(0,T,L^{2}(\Gamma _{3}))\cap \mathcal {Q}\), we deduce from inequality (5.25) that \(\Lambda \eta \in W^{1,\infty }(0,T;V).\)

Let now \(\eta _{1},\eta _{2}\in W^{1,\infty }(0,T;V)\) and let \(u_{i}=\) \( u_{\eta _{i}},\) \(\dot{u}_{i}=\) \(\dot{u}_{\eta _{i}},\) \(\beta _{i}=\beta _{\eta _{i}}\) for \(i=1,2\). For \(t\in [0,T]\) we integrate (5.20) with the initial conditions (5.21) to obtain

$$\begin{aligned} \beta _{i}(t)=\beta _{0}-\int \limits _{0}^{t}(\gamma _{\nu }\beta _{i}(s)R_{\nu }(u_{i\nu }(s)){{}^{2}}-\varepsilon _{a})_{+}ds. \end{aligned}$$

Using the definition of \(R_{\nu },\) the inequality \(\mid R_{\nu }(u_{\nu })\mid \le L,\) and writing \(\beta _{1}=\beta _{1}-\beta _{2}+\beta _{2}\), we get

$$\begin{aligned}&\parallel \beta _{1}(t)-\beta _{2}(t)\parallel _{L^{2}(\Gamma _{3})}\le c\int \limits _{0}^{t}\parallel \beta _{1}(s)-\beta _{2}(s)\parallel _{L^{2}(\Gamma _{3})}ds \\&\quad + \,c\int \limits _{0}^{t}\parallel u_{1\nu }(s)-u_{2\nu }(s)\parallel _{L^{2}(\Gamma _{3})}ds. \end{aligned}$$

By Gronwall’s inequality, it follows that

$$\begin{aligned} \left. \parallel \beta _{1}(t)-\beta _{2}(t)\parallel _{L^{2}(\Gamma _{3})}\le c\int \limits _{0}^{t}\parallel u_{1\nu }(s)-u_{2\nu }(s)\parallel _{L^{2}(\Gamma _{3})}ds,\right. \end{aligned}$$

and, using (3.13) we obtain

$$\begin{aligned} \left. \parallel \beta _{1}(t)-\beta _{2}(t)\parallel _{L^{2}(\Gamma _{3})}\le c\int \limits _{0}^{t}\parallel u_{1}(s)-u_{2}(s)\parallel _{V}ds.\right. \end{aligned}$$
(5.26)

On the other hand, using arguments similar to those in the proof of (5.25), we find that

$$\begin{aligned} \left. \parallel \Lambda \eta _{1}(t)-\Lambda \eta _{2}(t)\parallel _{V}\le c\parallel u_{1}(t)-u_{2}(t)\parallel _{V} +\, c\parallel \beta _{1}(t)-\beta _{2}(t)\parallel _{L^{2}(\Gamma _{3})}\right. \end{aligned}$$

Then, by (5.26) we have

$$\begin{aligned} \parallel \Lambda \eta _{1}(t)-\Lambda \eta _{2}(t)\parallel _{V}\le c\parallel u_{1}(t)-u_{2}(t)\parallel _{V} +\,c\int \limits _{0}^{t}\parallel u_{1}(s)-u_{2}(s)\parallel _{V}ds. \quad \end{aligned}$$
(5.27)

Next, we use (5.17) and (5.13) to find that

$$\begin{aligned}&(\dot{u}_{1}(t)-\dot{u}_{2}(t),u_{1}(t)-u_{2}(t))_{V}\le (\eta _{2}(t)-\eta _{1}(t),u_{1}(t)-u_{2}(t))_{V} \\&\quad +\,(\mathbf {G}u_{2}(t)-\mathbf {G} u_{1}(t),u_{1}(t)-u_{2}(t))_{V}, \end{aligned}$$

using Cauchy–Schwarz inequality and (5.12) we obtain

$$\begin{aligned}&(\dot{u}_{1}(t)-\dot{u}_{2}(t),u_{1}(t)-u_{2}(t))_{V}\le \Vert \eta _{1}(t)-\eta _{2}(t)\Vert _{V}\Vert u_{1}(t)-u_{2}(t)\Vert _{V} \\&\quad +\left( \frac{L_{\mathcal {F}}}{m_{\mathcal {A}}}+\Vert L\Vert \right) \Vert u_{1}(t)-u_{2}(t)\Vert _{V}^{2}. \end{aligned}$$

We integrate this inequality with respect to time and use the initial conditions \(u_{1}\left( 0\right) =u_{2}\left( 0\right) =u_{0}\) to find that

$$\begin{aligned}&\frac{1}{2}\parallel u_{1}(t)-u_{2}(t)\parallel _{V}^{2}\le \int \limits _{0}^{t}\Vert \eta _{1}(s)-\eta _{2}(s)\Vert _{V}\Vert u_{1}(s)-u_{2}(s)\Vert _{V}ds \\&\quad +\left( \frac{L_{ \mathcal {F}}}{m_{\mathcal {A}}}+\Vert L\Vert \right) \int \limits _{0}^{t}\Vert u_{1}(s)-u_{2}(s)\Vert _{V}^{2}ds. \end{aligned}$$

Applying the inequality

$$\begin{aligned} ab\le \frac{1}{2}a^{2}+\frac{1}{2}b^{2}\,\,a,b\in \mathbb {R}, \end{aligned}$$

we find that

$$\begin{aligned} \frac{1}{2}\parallel u_{1}(t)-u_{2}(t)\parallel _{V}^{2}\le & {} \frac{1}{2} \int \limits _{0}^{t}\Vert \eta _{1}(s)-\eta _{2}(s)\Vert _{V}^{2}ds+\frac{1}{2 }\int \limits _{0}^{t}\Vert u_{1}(s)-u_{2}(s)\Vert _{V}^{2}ds \\&+\left( \frac{L_{\mathcal {F}}}{m_{ \mathcal {A}}}+\Vert L\Vert \right) \int \limits _{0}^{t}\Vert u_{1}(s)-u_{2}(s)\Vert _{V}^{2}ds, \end{aligned}$$

where we obtain

$$\begin{aligned} \parallel u_{1}(t)-u_{2}(t)\parallel _{V}^{2}\le c\int \limits _{0}^{t}\Vert \eta _{1}(s)-\eta _{2}(s)\Vert _{V}^{2}ds+c\int \limits _{0}^{t}\Vert u_{1}(s)-u_{2}(s)\Vert _{V}^{2}ds, \end{aligned}$$

and, after a Gronwall argument, we obtain

$$\begin{aligned} \left. \parallel u_{1}(t)-u_{2}(t)\parallel _{V}^{2}\le c\int \limits _{0}^{t}\Vert \eta _{1}(s)-\eta _{2}(s)\Vert _{V}^{2}ds.\right. \end{aligned}$$
(5.28)

Using (5.27) we find that

$$\begin{aligned}&\parallel \Lambda \eta _{1}(t)-\Lambda \eta _{2}(t)\parallel _{V}^{2}\\&\quad \le c^{2}\parallel u_{1}(t)-u_{2}(t)\parallel _{V}^{2}+\, c^{2}\left( \int \limits _{0}^{t}\parallel u_{1}(s)-u_{2}(s)\parallel _{V}ds\right) ^{2}\\&\qquad +\, 2c\parallel u_{1}(t)-u_{2}(t)\parallel _{V}\cdot c\int \limits _{0}^{t}\parallel u_{1}(s)-u_{2}(s)\parallel _{V}ds, \\&\parallel \Lambda \eta _{1}(t)-\Lambda \eta _{2}(t)\parallel _{V}^{2}\\&\quad \le c^{2}\parallel u_{1}(t)-u_{2}(t)\parallel _{V}^{2}+\,c^{2}\left( \int \limits _{0}^{t}\parallel u_{1}(s)-u_{2}(s)\parallel _{V}ds\right) ^{2}\\&\qquad +\,c^{2}\parallel u_{1}(t)-u_{2}(t)\parallel _{V}^{2}+\,c^{2}\left( \int \limits _{0}^{t}\parallel u_{1}(s)-u_{2}(s)\parallel _{V}ds\right) ^{2}, \\&\parallel \Lambda \eta _{1}(t)-\Lambda \eta _{2}(t)\parallel _{V}^{2}\\&\quad \le \,2c^{2}\parallel u_{1}(t)-u_{2}(t)\parallel _{V}^{2}+\,2\,c^{2}\left( \int \limits _{0}^{t}\parallel u_{1}(s)-u_{2}(s)\parallel _{V}ds\right) ^{2}. \end{aligned}$$

Using Cauchy–Schwarz inequality we find

$$\begin{aligned} \parallel \Lambda \eta _{1}(t)-\Lambda \eta _{2}(t)\parallel _{V}^{2}\le c\parallel u_{1}(t)-u_{2}(t)\parallel _{V}^{2}+\,c\int \limits _{0}^{t}\parallel u_{1}(s)-u_{2}(s)\parallel _{V}^{2}ds. \quad \end{aligned}$$
(5.29)

We combine now (5.28) and (5.29) to see that

$$\begin{aligned} \left. \parallel \Lambda \eta _{1}(t)-\Lambda \eta _{2}(t)\parallel _{V}^{2}\le c\int \limits _{0}^{t}\Vert \eta _{1}(s)-\eta _{2}(s)\Vert _{V}^{2}ds\qquad \forall \,t\in \left[ 0\,T\right] .\right. \end{aligned}$$
(5.30)

Reiterating this inequality n times yields

$$\begin{aligned} \parallel \Lambda ^{n}\eta _{1}(t)-\Lambda ^{n}\eta _{2}(t)\parallel _{L^{\infty }(0,T;V)}^{2}\le \frac{c^{n}T^{n}}{n!}\Vert \eta _{1}(t)-\eta _{2}(t)\Vert _{L^{\infty }(0,T;V)}^{2}. \end{aligned}$$
(5.31)

which implies that, for n sufficiently large, a power \(\Lambda ^{n}\) of \( \Lambda \) is a contraction in the Banach space \(L^{\infty }(0,T;V). \) Then, there exists a unique element \(\eta ^{*}\in L^{\infty }(0,T;V)\) such that \(\Lambda ^{n}\eta ^{*}=\eta ^{*}\) and \(\eta ^{*}\) is also the unique fixed point of \(\Lambda \), i.e., \(\Lambda \eta ^{*}=\eta ^{*}\). The regularity \(\eta ^{*}\in W^{1,\infty }(0,T;V)\) follows from the regularity \(\Lambda \eta ^{*}\) \(\in \) \( W^{1,\infty }(0,T;V)\), which concludes the proof. \(\square \)

Now, we have all the ingredients necessary to prove Theorem 4.1.

Proof of Theorem 4.1

Existence. Let \(\eta ^{*}\in W^{1,\infty }(0,T;V)\) be the fixed point of the operator \(\Lambda \) and let \( (u,\varphi ,\beta )\) be the functions defined in Lemmata 5.1, 5.2, and 5.3, respectively, for \(\eta =\eta ^{*},\) i.e., \( u=u_{\eta ^{*}},\) \(\varphi =\varphi _{\eta ^{*}},\) \(\beta =\beta _{\eta ^{*}}.\) Clearly, equalities (3.23), (3.24), and (3.26) hold from Lemmata 5.2 and 5.3. Moreover, since \( \Lambda \eta ^{*}=\eta ^{*}\) it follows from (5.7), (5.5 ), (5.3), (5.8), and (5.23) that (3.22) and (3.25) hold, too. The regularity of the solution expressed in (4.1), (4.2), and (4.3) follows from Lemmata 5.1, 5.2, and 5.3, respectively.

Uniqueness. The uniqueness of the solution follows from the uniqueness of the fixed point of \(\Lambda \) and the uniqueness part in Lemmata 5.1, 5.2, and 5.3. \(\square \)