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1 Introduction

Contact phenomena involving deformable bodies abound in industry and everyday life. They lead to nonsmooth and nonlinear mathematical problems. Their analysis, including existence and uniqueness results, was carried out in a large number of works, see for instance [3, 4, 6, 9, 16, 17] and the references therein. The numerical analysis of the problems, including error estimation for discrete schemes and numerical simulations, can be found in [10, 11, 13, 14, 22]. The state of the art in the field, including applications in engineering, could be found in the recent special issue [15].

The study of both the qualitative and numerical analysis of various mathematical models of contact is made by using various mathematical tools, including the theory of variational inequalities. At the heart of this theory is the intrinsic inclusion of free boundaries in an elegant mathematical formulation. Existence and uniqueness results in the study of variational inequalities can be found in [1, 2, 12, 16, 20], for instance. References concerning their numerical analysis of include [5, 11, 13].

The large variety of frictional or frictionless models of quasistatic contact led to different classes of time-dependent or evolutionary variational inequalities which, on occasion, have been studied in an abstract framework. Examples could be found in [7, 8, 19, 20]. Nevertheless, it was recently recognized that some models of contact lead to weak formulations expressed in terms of variational inequalities which are more general than those studied in the above-mentioned papers. Therefore, in order to prove the unique solvability of these models, there is a need to extend these results to a more general classes of inequalities.

The first aim of the present paper is to provide such extension. Thus, we provide here an abstract existence and uniqueness result in the study of a new class of history-dependent variational inequalities. Our second aim is to illustrate how this result is useful in the analysis of a new model of contact with viscoplastic materials.

The rest of the paper is structured as follows. In Sect. 2, we introduce some notation and preliminary material. Then, we state and prove our main abstract result, Theorem 2. In Sect. 3 and we describe the frictional contact problem, list the assumption on the data, derive its variational formulation and state its unique weak solvability, Theorem 3. The proof of Theorem 3, based on the abstract result provided by Theorem 2, is presented in Sect. 4.

2 An Abstract Existence and Uniqueness Result

Everywhere in this paper, we use the notation \(\mathbb {N}\) for the set of positive integers and \(\mathbb R_+\) will represent the set of nonnegative real numbers, i.e. \(\mathbb R_+=[0,+\infty )\). For a normed space \((X,\Vert \cdot \Vert _X)\) we use the notation \(C(\mathbb {R}_+;X)\) for the space of continuously functions defined on \(\mathbb {R}_+\) with values in X. For a subset \(K\subset X\) we still use the symbol \(C(\mathbb {R}_+;K)\) for the set of continuous functions defined on \(\mathbb {R}_+\) with values on K. The following result, obtained in [18], will be used twice in this paper.

Theorem 1

Let \((X,\Vert \cdot \Vert _X)\) be a real Banach space and let \(\varLambda :C(\mathbb R_+;X)\rightarrow C(\mathbb R_+;X)\) be a nonlinear operator. Assume that for all \(n\in \mathbb N\) there exist two constants \(c_n\ge 0\) and \(d_n\in [0,1)\) such that

$$\begin{aligned} \Vert \varLambda u(t)-\varLambda v(t)\Vert _X\le c_n\int \limits _0^t\Vert u(s)-v(s)\Vert _X\,ds+d_n\,\Vert u(t)-v(t)\Vert _X \end{aligned}$$

for all \(u,\,v\in C(\mathbb R_+;X)\) and all \(t\in [0,n]\). Then the operator \(\varLambda \) has a unique fixed point \(\eta ^* \in C(\mathbb R_+;X)\).

The proof of Theorem 1 was carried out in several steps, based on the fact that the space \(C(\mathbb R_+;X)\) can be organized as a Fréchet space with a convenient distance function.

We assume in what follows that X is real Hilbert space and Y is a real normed space. Let K be a subset of X, \(A:K \subset X \rightarrow X\) and \(\mathscr {S}:C(\mathbb R_+;X)\rightarrow C(\mathbb R_+;Y)\). Moreover, let \(j:Y\times X\times K\rightarrow \mathbb R\) and \(f:\mathbb R_+\rightarrow X\). We consider the following assumptions.

$$\begin{aligned}&K\ \mathrm{is\ a\ closed ,\ convex,\ nonempty\ subset\ of }\ X.\end{aligned}$$
(1)
$$\begin{aligned}&\left\{ \begin{array}{ll} \mathrm{(a)\ There\ exists\ } L>0\ \mathrm{such\ that\ }\\ \qquad \Vert Au_1-Au_2\Vert _X\le L\Vert u_1-u_2\Vert _Y\quad \forall \, u_1,\,u_2\in K.\\ \mathrm{(b)\ There\ exists\ } m>0\ \mathrm{such\ that\ }\\ \qquad (Au_1-Au_2,u_1-u_2)_X\ge m\Vert u_1-u_2\Vert ^2_X\quad \forall \, u_1\,,u_2\in K. \end{array}\right. \end{aligned}$$
(2)
$$\begin{aligned}&\left\{ \begin{array}{ll} \mathrm{(a)}\ \mathrm{For\ all}\ y\in Y\ \mathrm{and}\ u \in X, \ j(y, u,\cdot )\ \mathrm{is\ convex\ and\ l.s.c\ on}\ K.\\ \mathrm{(b)}\ \mathrm{There\ exists}\ \alpha >0\ \mathrm{and }\ \beta >0 \mathrm{\ such\ that}\\ {}\qquad j(y_1, u_1, v_2)-j(y_1, u_1, v_1)+j(y_2, u_2, v_1)-j(y_2, u_2, v_2)\\ {}\qquad \quad \le \alpha \Vert y_1-y_2\Vert _Y\,\Vert v_1-v_2\Vert _X+\beta \,\Vert u_1-u_2\Vert _X\,\Vert v_1-v_2\Vert _X\\ \qquad \qquad \forall \, y_1,\, y_2\in Y,\ \forall \, u_1,\, u_2\in X,\ \forall \, v_1,\, v_2\in K. \end{array}\right. \end{aligned}$$
(3)
$$\begin{aligned}&\left\{ \begin{array}{ll} \mathrm{For\ all}\ n\in \mathbb {N}\ \mathrm{there\ exists}\ s_n>0 \mathrm{\ such\ that}\\ \quad \Vert {\mathscr {S}}u_1(t)-{\mathscr {S}}u_2(t)\Vert _Y\le s_n\displaystyle {\int \limits _0^t\Vert u_1(s)-u_2(s)\Vert _X\,ds} \\ \quad \forall \, u_1,\,u_2\in C(\mathbb R_+;X),\ \forall \,t\in [0,n]. \end{array}\right. \end{aligned}$$
(4)
$$\begin{aligned} f\in C(\mathbb R_+;X). \end{aligned}$$
(5)

Concerning these assumptions we have the following comments. First, assumption (2) show that A is a Lipschitz continuous strongly monotone operator on K. Next, in (3) we use the abbreviation l.s.c. for a lower semicontinuous function. Finally, following the terminology introduced in [19] and used in various papers, condition (4) show that the operator \(\mathscr {S}\) is a history-dependent operator. Example of operators which satisfies this condition could be find in [19, 20]. Variational inequalities involving history-dependent operators are also called history-dependent variational inequalities. In their study we have the following existence and uniqueness result.

Theorem 2

Assume that (1)–(5) hold. Moreover, assume that

$$\begin{aligned} m>\beta , \end{aligned}$$
(6)

where m and \(\beta \) are the constants in (2) and (3), respectively. Then, there exists a unique function \(u\in C(\mathbb R_+;K)\) such that, for all \(t\in \mathbb R_+\), the following inequality holds:

$$\begin{aligned}&u(t)\in K,\quad (Au(t),v-u(t))_X+j({\mathscr {S}}u(t),u(t),v)\\&\qquad -j({\mathscr {S}}u(t),u(t),u(t))\ge (f(t),v-u(t))_X\qquad \forall \,v\in K. \nonumber \end{aligned}$$
(7)

Proof

The proof of Theorem 2 is based on argument similar to those presented in [19] and, for this reason, we skip the details. The main step in the proof are the followings.

(i) Let \(\eta \in C(\mathbb R_+;X)\) be fixed and denote by \(y_\eta \in C(\mathbb R_+;Y)\) the function given by

$$\begin{aligned} y_\eta (t)=\mathscr {S} \eta (t) \qquad \forall \,t\in \mathbb R_+. \end{aligned}$$
(8)

In the first step we use standard arguments on time-dependent elliptic variational inequalities to prove that there exists a unique function \(u_\eta \in C(\mathbb R_+;K)\) such that, for all \(t\in \mathbb R_+\), the following inequality holds:

$$\begin{aligned}&u_\eta (t)\in K,\quad (Au_{\eta }(t),v-u_{\eta }(t))_X+j(y_\eta (t),\eta (t),v)\\&\qquad -j(z_\eta (t),\eta (t),u_\eta (t))\ge (f(t),v-u_\eta (t))_X\qquad \forall \,v\in K. \nonumber \end{aligned}$$
(9)

(ii) Next, in the second step, we consider the operator \(\varLambda :C(\mathbb R_+;X)\rightarrow C(\mathbb R_+;K)\subset C(\mathbb R_+;X)\) defined by equality

$$\begin{aligned} \varLambda \eta =u_\eta \qquad \forall \, \eta \in C(\mathbb R_+;X) \end{aligned}$$
(10)

and we prove that it has a unique fixed point \(\eta ^*\in C(\mathbb R_+;K)\). Indeed, let \(\eta _1,\, \eta _2\in C(\mathbb R_+;X)\), and let \(y_i\), be the functions defined by (8) for \(\eta =\eta _i\), i.e. \(y_i=y_{\eta _i}\), for \(i=1,2\). Also, denote by \(u_i\) the solution of the variational inequality (9) for \(\eta =\eta _i\), i.e. \(u_i=u_{\eta _i}\), \(i=1,2\). Let \(n\in \mathbb {N}\) and \(t\in [0,n]\). Then, using (9), (2) and (3) is easy to see that

$$\begin{aligned} m\,\Vert u_1(t)-u_2(t)\Vert _X\le \alpha \, \Vert y_1(t)-y_2(t)\Vert _Y+\beta \,\Vert \eta _1(t)-\eta _2(t)\Vert _X. \end{aligned}$$
(11)

Moreover, by the assumptions (4) on the operator \(\mathscr {S}\) one has

$$\begin{aligned} \Vert y_1(t)-y_2(t)\Vert _Y=\Vert \mathscr {S}\eta _1(t)-\mathscr {S}\eta _2(t)\Vert _Y \le s_n\,\int \limits _0^t\Vert \eta _1(s)-\eta _2(s)\Vert _X\,ds. \end{aligned}$$
(12)

Thus, using (10)–(12) yields

$$\begin{aligned}&\Vert \varLambda \eta _1(t)-\varLambda \eta _2(t)\Vert _X=\Vert u_1(t)-u_2(t)\Vert _X\nonumber \\&\qquad \le \frac{\alpha s_n}{m}\int \limits _0^t\Vert \eta _1(s)-\eta _2(s)\Vert _X\,ds+ \frac{\beta }{m}\,\Vert \eta _1(t)-\eta _2(t)\Vert _X\nonumber \end{aligned}$$

which, together with the smallness assumption (6) and Theorem 1, implies that the operator \(\varLambda \) has a unique fixed point \(\eta ^*\in C(\mathbb R_+;X)\) . Moreover, since \(\varLambda \) has values on \(C(\mathbb R_+;K)\), we deduce that \(\eta ^*\in C(\mathbb R_+;K)\).

(iii) Let \(\eta ^*\in C(\mathbb R_+;K)\) be the fixed point of the operator \(\varLambda \). It follows from (8) and (10) that

$$\begin{aligned} y_{\eta ^*}(t)=\mathscr {S}\eta ^*(t),\qquad u_{\eta ^*}(t)=\eta ^*(t). \end{aligned}$$
(13)

for all \(t\in \mathbb R_+\). Now, letting \(\eta =\eta ^*\) in the inequality (9) and using (13) we conclude that \(\eta ^*\in C(\mathbb R_+;K)\) is a solution to the variational inequality (7). This proves the existence part in Theorem 2.

(iv) The uniqueness part is a consequence of the uniqueness of the fixed point of the operator \(\varLambda \) and can be proved as follows. Denote by \(\eta ^*\in C(\mathbb R_+;K)\) the solution of the variational inequality (7) obtained above, and let \(\eta \in C(\mathbb R_+;K)\) be a different solution of this inequality, which implies that

$$\begin{aligned}&(A\eta (t),v-\eta (t))_X+j({\mathscr {S}}\eta (t),\eta (t),v)\\&\qquad -j({\mathscr {S}}\eta (t),\eta (t),\eta (t))\ge (f(t),v-\eta (t))_X\qquad \forall \,v\in K,\ \ t\in \mathbb {R}_+. \nonumber \end{aligned}$$
(14)

Letting \(y_\eta =\mathscr {S}\eta \in C(\mathbb R_+;Y)\), inequality (14) implies that \(\eta \) is solution to the variational inequality (9). On the other hand, by step (i) this inequality has a unique solution \(u_\eta \) and, therefore,

$$\begin{aligned} \eta =u_\eta . \end{aligned}$$
(15)

This shows that \(\varLambda \eta =\eta \) where \(\varLambda \) is the operator defined by (10). Therefore, by Step (i) it follows that \(\eta =\eta ^*\), which concludes proof.

3 The Contact Model and Main Result

We turn now to an application of Theorem 2 in Contact Mechanics and, to this end, we start by presenting some notations and preliminaries. Let \(\varOmega \) a regular domain of \(\mathbb {R}^d\) (\(d=2,3\)) with surface \(\varGamma \) that is partitioned into three disjoint measurable parts \(\varGamma _1\), \(\varGamma _2\) and \(\varGamma _3\), such that \(\mathrm{meas}\,(\varGamma _1)>0\) and, in addition, \(\varGamma _3\) is plane. We use the notation \(\mathbf {x}=(x_i)\) for a typical point in \(\varOmega \) and \({\varvec{\nu }}=(\nu _i)\) for the outward unit normal at \(\varGamma \). In order to simplify the notation, we do not indicate explicitly the dependence of various functions on the spatial variable \(\mathbf {x}\). Let \(\mathbb {R}^d\) be d-dimensional real linear space and the let \(\mathbb {S}^{d}\) denote the space of second order symmetric tensors on \(\mathbb {R}^d\) or, equivalently, the space of symmetric matrices of order d. The canonical inner products and the corresponding norms on \(\mathbb {R}^d\) and \(\mathbb {S}^{d}\) are given by

$$\begin{aligned} \mathbf {u}\cdot \mathbf {v}&= u_iv_i,\quad \Vert \mathbf {v}\Vert =(\mathbf {v}\cdot \mathbf {v})^{1/2}\qquad \forall \, \mathbf {u}=(u_i),\,\mathbf {v}=(v_i)\in \mathbb {R}^d,\\ \varvec{\sigma }\cdot \varvec{\tau }&=\sigma _{ij}\tau _{ij},\quad \Vert \varvec{\tau }\Vert =(\varvec{\tau }\cdot \varvec{\tau })^{1/2}\qquad \forall \,\varvec{\sigma }=(\sigma _{ij}),\, \varvec{\tau }=(\tau _{ij}) \in \mathbb {S}^{d}, \end{aligned}$$

respectively. Here and below the indices i, j, k, l run between 1 and d and, unless stated otherwise, the summation convention over repeated indices is used.

We use standard notation for the Lebesgue and the Sobolev spaces associated to \(\varOmega \) and \(\varGamma \). Also, we introduce the spaces

$$\begin{aligned}&V=\{\,\mathbf {v}=(v_i)\in H^1(\varOmega )^d: \mathbf {v}=\mathbf {0}\ \text{ a.e. } \text{ on }\ \varGamma _1\,\},\\&Q=\{\,\varvec{\tau }=(\tau _{ij})\in L^2(\varOmega )^{d\times d}: \tau _{ij}=\tau _{ji}\,\},\\&Q_1=\{\,\varvec{\tau }=(\tau _{ij})\in Q: \text {Div}\varvec{\tau }\in L^2(\varOmega )^d\,\}. \end{aligned}$$

Here and below \(\text {Div}\varvec{\tau }= (\tau _{ij,j})\) denotes the divergence of the field \(\varvec{\tau }\), where the index that follows a coma indicates a partial derivative with the corresponding component of the spatial variable \(\mathbf {x}\), i.e. \(\tau _{ij,j}=\partial \tau _{ij}/\partial x_j\). The spaces Q and \(Q_1\) are real Hilbert spaces with the canonical inner products given by

$$\begin{aligned}&(\varvec{\sigma },\varvec{\tau })_Q=\int \limits _\varOmega \varvec{\sigma }\cdot \varvec{\tau }\,dx \qquad \forall \,\varvec{\sigma },\,\varvec{\tau }\in Q,\\&(\varvec{\sigma },\varvec{\tau })_{Q_1}=\int \limits _\varOmega \varvec{\sigma }\cdot \varvec{\tau }\,dx+ \int \limits _\varOmega \text { Div}\varvec{\sigma }\cdot \text { Div}\varvec{\tau }\,dx \qquad \forall \,\varvec{\sigma },\,\varvec{\tau }\in Q_1. \end{aligned}$$

In addition, since meas \((\varGamma _1)>0\), it is well known that V is a real Hilbert space with the inner product

$$\begin{aligned} (\mathbf {u},\mathbf {v})_V=\int \limits _\varOmega \varvec{\varepsilon }(\mathbf {u})\cdot \varvec{\varepsilon }(\mathbf {v})\,dx \qquad \forall \,\mathbf {u},\mathbf {v}\in V \end{aligned}$$

where \(\varvec{\varepsilon }\) is the deformation operator, i.e. \(\varvec{\varepsilon }(\mathbf {u})=\varepsilon _{ij}(\mathbf {u})\), \(\varepsilon _{ij}(\mathbf {u})= \frac{1}{2}\, (u_{i,j} + u_{j,i})\), \(u_{i,j}=\partial u_{i}/\partial x_j.\) The associated norms on the spaces V, Q and \(Q_1\) will be denoted by \(\Vert \cdot \Vert _V\), \(\Vert \cdot \Vert _Q\) and \(\Vert \cdot \Vert _{Q_1}\), respectively.

For all \(\mathbf {v}\in V\) we still write \(\mathbf {v}\) for the trace of \(\mathbf {v}\) to \(\varGamma \). We recall that, by the Sobolev trace theorem, there exists a positive constant \(c_0\) which depends on \(\varOmega \), \(\varGamma _1\) and \(\varGamma _3\) such that

$$\begin{aligned} \Vert \mathbf {v}\Vert _{L^2(\varGamma _3)^d}\le c_0\,\Vert \mathbf {v}\Vert _{V}\qquad \forall \,\mathbf {v}\in V. \end{aligned}$$
(16)

For \(\mathbf {v}\in V\) we denote by \(v_\nu \) and \(\mathbf {v}_\tau \) the normal and tangential components of \(\mathbf {v}\) on \(\varGamma \), in the sense of traces, given by \(v_\nu =\mathbf {v}\cdot \varvec{\nu },\ \ \mathbf {v}_\tau =\mathbf {v}-v_\nu \varvec{\nu }\). Moreover, for \(\varvec{\sigma }\in Q_1\) we denote by \(\sigma _\nu \in H^{-\frac{1}{2}}(\varGamma )\) its normal component, in the sense of traces. Let \(R:H^{-\frac{1}{2}}(\varGamma )\rightarrow L^2(\varGamma )\) be a linear continuous operator. Then, there exists a positive constant \(c_R>0\) which depends on R, \(\varOmega \) and \(\varGamma _3\) such that

$$\begin{aligned} \Vert R\sigma _\nu \Vert _{L^2(\varGamma _3)}\le c_R\,\Vert \varvec{\sigma }\Vert _{Q_1}\qquad \forall \,\varvec{\sigma }\in Q_1. \end{aligned}$$
(17)

Next, we recall that if \(\varvec{\sigma }\) is a regular function, then its normal and tangential components of the stress field \(\varvec{\sigma }\) on the boundary are defined by \( \sigma _\nu =(\varvec{\sigma }\varvec{\nu })\cdot \varvec{\nu },\,\varvec{\sigma }_\tau =\varvec{\sigma }\varvec{\nu }-\sigma _\nu \varvec{\nu } \) and the following Green’s formula holds:

$$\begin{aligned} \int \limits _\varOmega \,\varvec{\sigma }\cdot \varvec{\varepsilon }(\mathbf {v})\,dx+\int \limits _\varOmega \,\text { Div}\varvec{\sigma }\cdot \mathbf {v}\,dx = \int \limits _\varGamma \varvec{\sigma }\varvec{\nu } \cdot \mathbf {v}\,da \qquad \forall \, \mathbf {v}\in V. \end{aligned}$$
(18)

With these notation, we formulate the following problem.

Problem \({\mathscr {P}}\). Find a displacement field \(\mathbf {u}=(u_i):\varOmega \times \mathbb {R}_+\rightarrow \mathbb {R}^d\) and a stress field \(\varvec{\sigma }=(\sigma _{ij}):\varOmega \times \mathbb {R}_+\rightarrow \mathbb {S}^d\) such that

$$\begin{aligned} \dot{\varvec{\sigma }}(t) = {\mathscr {E}} \varvec{\varepsilon }(\dot{\mathbf {u}}(t))+{\mathscr {G}}(\varvec{\sigma }(t),\varvec{\varepsilon }(\mathbf {u}(t))\quad&\text{ in }&\ \varOmega ,\end{aligned}$$
(19)
$$\begin{aligned} \text {Div} \varvec{\sigma }(t) +\mathbf {f}_0={\mathbf {0}} \quad&\text{ in }&\ \varOmega , \end{aligned}$$
(20)
$$\begin{aligned} \mathbf {u}(t) = {\mathbf {0}}\quad&\text{ on }&\ \varGamma _1, \end{aligned}$$
(21)
$$\begin{aligned} \varvec{\sigma }(t)\varvec{\nu } = \mathbf {f}_2(t)\quad&\text{ on }&\ \varGamma _2, \end{aligned}$$
(22)
$$\begin{aligned} -\varvec{\sigma }_\tau (t)=\mu |R\sigma _\nu (t)|\,\mathbf {n}^*\quad&\text{ on }&\ \varGamma _3, \end{aligned}$$
(23)

for all \(t\in \mathbb {R}_+\), there exists \(\xi :\varGamma _3\times \mathbb {R}_+\rightarrow \mathbb {R}\) which satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} u_\nu (t)\le g,\qquad \sigma _\nu (t)+p(u_\nu (t))+\xi (t)\le 0,\\ (u_\nu (t)-g)\big (\sigma _\nu (t)+p(u_\nu (t))+\xi (t)\big )=0,\\ 0\le \xi (t)\le \displaystyle F\Big (\int \limits _0^t u_\nu ^+(s)\,ds\Big ),\\ \xi (t)=0 \ \ \mathrm{if}\ \ u_\nu (t)<0,\\ \xi (t)= \displaystyle F\Big (\int \limits _0^t u_\nu ^+(s)\,ds\Big )\ \ \mathrm{if}\ \ u_\nu (t)>0\end{array}\right. \quad&\text{ on }&\ \varGamma _3, \end{aligned}$$
(24)

for all \(t\in \mathbb {R}_+\) and, moreover,

$$\begin{aligned} \mathbf {u}(0)=\mathbf {u}_0,\qquad \varvec{\sigma }(0)=\varvec{\sigma }_0\quad \mathrm{in}\ \varOmega . \end{aligned}$$
(25)

Problem \({\mathscr {P}}\) represents a mathematical model which describes the quasistatic process of contact between a viscoplastic body and a moving foundation. Here \(\varOmega \) represents the reference configuration of a the body and the dot above denotes the derivative with respect the time variable, i.e. \(\dot{f}=\frac{\partial f}{\partial t}\). Equation (19) represents the viscoplastic constitutive law. Details and various mechanical interpretation concerning such kind of laws can be found in [9, 20], for instance. Equation (20) represents the equation of equilibrium in which \(\mathbf {f}_0\) denotes the density of body forces, assumed to be time-independent. We use this equation since the process is quasistatic and, therefore, the inertial term in the equation of motion is neglected. Conditions (21) and (22) are the displacement and the traction boundary condition, respectively. They describe the fact that the body is fixed on \(\varGamma _1\) and prescribed traction of density \(\mathbf {f}_2\) act on \(\varGamma _2\), during the contact process.

Conditions (23) and (24) represent a sliding version of Coulomb’s law of dry friction and a normal compliance contact condition with unilateral constraint and memory term, respectively. Their are obtained from arguments presented in our recent paper [21] and, for this reason, we do not describe them with details. We just mention that \(\mu \) denotes the coefficient of friction, \(\mathbf {n}^*\) denotes a given unitary vector in the plane on \(\varGamma _3\) and \(v^*<0\) is given. In addition, p and F are given function which describe the deformability and the memory effects of the foundation, \(g>0\) is a given depth and \(r^+\) represent the positive part of r, i.e. \(r^+= {\text {max}}\,\{r,0\}\). Finally, conditions (25) represent the initial conditions for the displacement and the stress field, respectively.

In the study of Problem \({\mathscr {P}}\) we assume that the elasticity operator \(\mathscr {E}\) and the nonlinear constitutive function \(\mathscr {G}\) satisfy the following conditions.

$$\begin{aligned}&\left\{ \begin{array}{ll} \mathrm{(a)\ } {\mathscr {E}}=({\mathscr {E}}_{ijkl}):\varOmega \times \mathbb {S}^d\rightarrow \mathbb {S}^d.\\ \mathrm{(b)\ } {\mathscr {E}}_{ijkl}={\mathscr {E}}_{klij}={\mathscr {E}}_{jikl} \in L^\infty (\varOmega ),\ 1\le i,j,k,l\le d.\\ \mathrm{(c)\ There\ exists}\ m_{\mathscr {E}}>0\ \mathrm{\ such\ that}\\ {} \qquad {\mathscr {E}}\varvec{\tau }\cdot \varvec{\tau }\ge m_{\mathscr {E}} \Vert \varvec{\tau }\Vert ^2\ \ \mathrm{for\ all}\ \varvec{\tau }\in \mathbb {S}^d,\ \mathrm{a.e.\ in\ }\varOmega . \end{array}\right. \end{aligned}$$
(26)
$$\begin{aligned}&\left\{ \begin{array}{ll} \mathrm{(a)\ }{\mathscr {G}}: \varOmega \times \mathbb {S}^d\times \mathbb {S}^d\rightarrow \mathbb {S}^d.\\ \mathrm{(b)\ There\ exists}\ L_{\mathscr {G}}>0 \mathrm{\ such\ that}\\ {} \qquad \Vert {\mathscr {G}}(\mathbf {x},\varvec{\sigma }_1,\varvec{\varepsilon }_1) -{\mathscr {G}}(\mathbf {x},\varvec{\sigma }_2,\varvec{\varepsilon }_2)\Vert \le L_{\mathscr {G}}\,(\Vert \varvec{\sigma }_1-\varvec{\sigma }_2\Vert + \Vert \varvec{\varepsilon }_1-\varvec{\varepsilon }_2\Vert )\\ {}\qquad \quad \ \mathrm{for\ all}\ \varvec{\sigma }_1,\varvec{\sigma }_2,\varvec{\varepsilon }_1,\varvec{\varepsilon }_2\in \mathbb {S}^d,\ \mathrm{a.e.\ }\mathbf {x}\in \varOmega .\\ \mathrm{(c)\ The\ mapping}\ \mathbf {x}\mapsto {\mathscr {G}}(\mathbf {x},\varvec{\sigma },\varvec{\varepsilon }\mathrm{\ is\ measurable\ on\ }\varOmega , \\ {} \qquad \mathrm{for\ all\ }\varvec{\sigma },\varvec{\varepsilon }\in \mathbb {S}^d.\\ \mathrm{(d)\ The\ mapping\ } \mathbf {x}\mapsto {\mathscr {G}}(\mathbf {x},\mathbf {0},\mathbf {0})\ \mathrm{belongs\ to}\ Q. \end{array}\right. \end{aligned}$$
(27)

The densities of body forces and surface traction are such that

$$\begin{aligned} \mathbf {f}_0\in L^2(\varOmega )^d,\quad \mathbf {f}_2\in C(\mathbb {R}_+;L^2(\varGamma _2)^d). \end{aligned}$$
(28)

The normal compliance function p and the surface yield function F satisfy

$$\begin{aligned} \left\{ \begin{array}{ll} \mathrm{(a)\ } p:\varGamma _3\times \mathbb {R}\rightarrow \mathbb {R}_+.\\ \mathrm{(b)\ There\ exists}\, L_p>0 \mathrm{\ such\ that\ }\\ {} \qquad |p(\mathbf {x},r_1)-p(\mathbf {x},r_2)| \le L_p\,|r_1-r_2|\ \ \forall \, r_1,\,r_2\in \mathbb {R},\ \mathrm{a.e.\ } \mathbf {x}\in \varGamma _3.\\ \mathrm{(c)\ } (p(\mathbf {x}, r_{1}) - p(\mathbf {x}, r_{2}))(r_{1} - r_{2}) \ge 0 \ \ \forall \, r_{1}, r_{2} \in \mathbb {R}, \ \mathrm{a.e. }\ \mathbf {x} \in \varGamma _3.\\ \mathrm{(d)\ The\ mapping\ } \mathbf {x}\mapsto p(\mathbf {x},r) \mathrm{\ is\ measurable\ on}\ \varGamma _3,\ \mathrm{for\ any\ }r\in \mathbb {R}.\\ \mathrm{(e)\ }p(\mathbf {x},r)=0\ \mathrm{for\ all\ }r\le 0,\ \mathrm{a.e.}\ \mathbf {x}\in \varGamma _3. \end{array}\right. \end{aligned}$$
(29)
$$\begin{aligned} \left\{ \begin{array}{ll} \mathrm{(a)\ } F:\varGamma _3\times \mathbb {R}\rightarrow \mathbb {R}_+.\\ \mathrm{(b)\ There\ exists}\, L_F>0 \mathrm{\ such\ that\ }\\ {} \qquad |F(\mathbf {x},r_1)-F(\mathbf {x},r_2)| \le L_F\,|r_1-r_2|\ \ \forall \, r_1,\,r_2\in \mathbb {R},\ \mathrm{a.e.\ } \mathbf {x}\in \varGamma _3.\\ \mathrm{(c)\ The\ mapping\ } \mathbf {x}\mapsto F(\mathbf {x},r) \mathrm{\ is\ measurable\ on}\ \varGamma _3,\ \mathrm{for\ any}\ r\in \mathbb {R}.\\ \mathrm{(d)\ }F(\mathbf {x},0)=0\ \mathrm{a.e.}\ \mathbf {x}\in \varGamma _3. \end{array}\right. \end{aligned}$$
(30)

Also, the the coefficient of friction verifies

$$\begin{aligned} \mu \in L^{\infty }(\varGamma _3), \ \, \mu (t, \mathbf {x}) \ge 0 \quad \mathrm{a.e. } \ \mathbf {x} \in \varGamma _3, \end{aligned}$$
(31)

and the initial data are such that

$$\begin{aligned} \mathbf {u}_{0}\in V,\qquad \varvec{\sigma }_{0}\in Q. \end{aligned}$$
(32)

In what follows we consider the set of admissible displacements fields and the set of admissible stress fields defined by

$$\begin{aligned}&U=\{\, \mathbf {v} \in V\ :\ v_{\nu } \le g\ \mathrm{\ on \ } \varGamma _3\, \}, \end{aligned}$$
(33)
$$\begin{aligned}&\varSigma =\{\, \varvec{\tau } \in Q\ :\ \text { Div} \varvec{\tau }+\mathbf {f}_0=\mathbf {0}\ \mathrm{\ in \ } \varOmega \, \} . \end{aligned}$$
(34)

respectively. Note that assumptions \(g>0\) and \(\mathbf {f}_0\in L^2(\varOmega )^d\) imply that U and \(\varSigma \) are closed, convex nonempty subsets of the spaces V and Q, respectively.

Assume in what follows that \((\mathbf {u}, \varvec{\sigma })\) are sufficiently regular functions which satisfy (19)–(24) and let \(\mathbf {v} \in U\) and \(t>0\) be given. First, we use the equilibrium equation (20) and the contact condition (23) to see that

$$\begin{aligned} \mathbf {u}(t)\in U,\qquad \varvec{\sigma }(t)\in \varSigma . \end{aligned}$$
(35)

Then, we use Green’s formula (18), the equilibrium equation (20) and the friction law (23) to obtain that

$$\begin{aligned}&\int \limits _{\varOmega }\varvec{\sigma }(t) \cdot (\varvec{\varepsilon }(\mathbf {v})-\varvec{\varepsilon }(\mathbf {u}(t)))\,dx\\&\quad = \int \limits _{\varOmega } \mathbf {f}_{0}\cdot (\mathbf {v} - \mathbf {u}(t))\,dx + \int \limits _{\varGamma _2}\mathbf {f}_2(t)\cdot (\mathbf {v} - \mathbf {u}(t))\,da \nonumber \\&\qquad + \int \limits _{\varGamma _3}\sigma _\nu (t)(v_\nu - u_\nu (t))\,da-\int \limits _{\varGamma _3}\mu |R\sigma _{\nu }(t)|\,\mathbf {n}^{*}\cdot (\mathbf {v}_\tau - \mathbf {u}_\tau (t))\,da. \nonumber \end{aligned}$$
(36)

We now use the contact conditions (24) and the definition (33) of the set U to see that

$$\begin{aligned} \sigma _\nu (t)(v_\nu -u_\nu (t))\ge -(p(u_\nu (t))+\xi (t))(v_\nu -u_\nu (t))\quad \mathrm{on}\ \varGamma _3. \end{aligned}$$
(37)

Next, we use (24), again, and the hypothesis (30)(a) on function F to deduce that

$$\begin{aligned} F\left( \int \limits _{0}^{t} u_{\nu }^{+}(s)ds\right) \,(v^+_\nu -u^+_\nu (t))\ge \xi (t)(v_\nu -u_\nu (t)) \quad \mathrm{on}\ \varGamma _3. \end{aligned}$$
(38)

We now add the inequalities (37) and (38) and integrate the result on \(\varGamma _3\) to find that

$$\begin{aligned}&\int \limits _{\varGamma _3}\sigma _\nu (t)(v_\nu -u_\nu (t))\,da \ge -\int \limits _{\varGamma _3}p(u_\nu (t))(v_\nu -u_\nu (t))\,da \\&\qquad -\int \limits _{\varGamma _3}F\left( \int \limits _{0}^{t} u_{\nu }^{+}(s)ds\right) \, (v^+_\nu -u^+_\nu (t))\,da.\nonumber \end{aligned}$$
(39)

Finally, we combine (36) and (39) to deduce that

$$\begin{aligned}&\ \ \ \int \limits _{\varOmega }\varvec{\sigma }(t)\cdot (\varvec{\varepsilon }(\mathbf {v})-\varvec{\varepsilon }(\mathbf {u}(t)))\,dx + \int \limits _{\varGamma _3}p(u_\nu (t))(v_\nu -u_\nu (t))\,da \\&\ + \int \limits _{\varGamma _3}F\left( \int \limits _{0}^{t} u_{\nu }^{+}(s)ds\right) \, (v^+_\nu -u^+_\nu (t))\,da + \int \limits _{\varGamma _3}\mu |R\varvec{\sigma }_{\nu }(t)|\mathbf {n}^{*}\cdot (\mathbf {v}_\tau - \mathbf {u}_\tau (t))\,da \nonumber \\&\ \ \ge \int \limits _{\varOmega } \mathbf {f}_{0}\cdot (\mathbf {v} - \mathbf {u}(t))\,dx + \int \limits _{\varGamma _2}\mathbf {f}_2(t).(\mathbf {v} - \mathbf {u}(t))\,da.\nonumber \end{aligned}$$
(40)

We now integrate the the constitutive law (19) with the initial conditions (25), then we gather the resulting equation with the regularity (35) and inequality (40) to obtain the following variational formulation of Problem \({\mathscr {P}}\).

Problem \({\mathscr {P}}_{V}\). Find a displacement field \(\mathbf {u} : \mathbb {R}_+ \rightarrow U\) and a stress field \(\varvec{\sigma } : \mathbb {R}_+ \rightarrow \varSigma \) such that

$$\begin{aligned}&\ \ \ \varvec{\sigma }(t) = {\mathscr {E}} \varvec{\varepsilon }(\mathbf {u}(t))+ \int \limits _0^t {\mathscr {G}}(\varvec{\sigma }(s),\varvec{\varepsilon }(\mathbf {u}(s))\,ds+\varvec{\sigma }_0-{\mathscr {E}}\varvec{\varepsilon }(\mathbf {u}_0),\end{aligned}$$
(41)
$$\begin{aligned}&\ \ \ \int \limits _{\varOmega }\varvec{\sigma }(t)\cdot (\varvec{\varepsilon }(\mathbf {v})-\varvec{\varepsilon }(\mathbf {u}(t)))\,dx + \int \limits _{\varGamma _3}p(u_\nu (t))(v_\nu -u_\nu (t))\,da \\&\ + \int \limits _{\varGamma _3}F\left( \int \limits _{0}^{t}u_{\nu }^{+}(s)ds\right) \,(v^+_\nu -u^+_\nu (t))\,da + \int \limits _{\varGamma _3}\mu |R\varvec{\sigma }_{\nu }(t)|\mathbf {n}^{*}\cdot (\mathbf {v}_\tau - \mathbf {u}_\tau (t))\,da \nonumber \\&\ \ \ \ge \int \limits _{\varOmega } \mathbf {f}_{0}\cdot (\mathbf {v} - \mathbf {u}(t))\,dx + \int \limits _{\varGamma _2}\mathbf {f}_2(t).(\mathbf {v} - \mathbf {u}(t))\,da\nonumber \end{aligned}$$
(42)

for all \( t \in \mathbb {R}_+\).

Our main existence and uniqueness result in the study of the Problem \({\mathscr {P}}\), that we state here and prove in the next section is the following.

Theorem 3

Assume that (26)–(32) hold. Then there exists a positive constant \(\mu _0\) which depends only on \(\varOmega \), \(\varGamma _1\), \(\varGamma _3\), R and \({\mathscr {E}}\) such that Problem \({\mathscr {P}}_{V}\) has a unique solution, if

$$\begin{aligned} \Vert \mu \Vert _{L^{\infty }(\varGamma _3)} < \mu _0. \end{aligned}$$
(43)

Moreover, the solution satisfies \(\mathbf {u} \in C(\mathbb {R}_+; U)\), \(\varvec{\sigma }\in C(\mathbb {R}_+; \varSigma )\).

Note that Theorem 3 provides the unique weak solvability of Problem \({\mathscr {P}}\), under the smallness assumption (43) on the coefficient of friction.

4 Proof of Theorem 3

The proof of the theorem will be carried out in several steps. To present it we assume in what follows that (26)–(32) hold. We start with the following existence and uniqueness result.

Lemma 4

For each function \(\mathbf {u}\in C(\mathbb {R}_+;V)\) there exists a unique function \(\varTheta \mathbf {u}\in C(\mathbb {R}_+;Q)\) such that

$$\begin{aligned} \varTheta \mathbf {u}(t)=\int \limits _0^t{\mathscr {G}}(\varTheta \mathbf {u}(s)+\mathscr {E}\varvec{\varepsilon }(\mathbf {u}(s)),\varvec{\varepsilon }(\mathbf {u}(s)))\,ds +\varvec{\sigma }_0-{\mathscr {E}}\varvec{\varepsilon }({\mathbf {u}}_0)\quad \forall \, t\in \mathbb {R}_+. \end{aligned}$$
(44)

Moreover, the operator \(\varTheta :C(\mathbb {R}_+;V)\rightarrow C(\mathbb {R}_+;Q)\) is history-dependent, i.e. for all \(n\in \mathbb {N} \) there exists \(\theta _n>0\) such that

$$\begin{aligned}&\Vert \varTheta \mathbf {u}_1(t)-\varTheta \mathbf {u}_2(t)\Vert _Q\le \theta _n\,\int \limits _0^t \Vert \mathbf {u}_1(s)-\mathbf {u}_2(s)\Vert _V\,ds\\&\qquad \qquad \forall \, \mathbf {u}_1,\, \mathbf {u}_2\in C(\mathbb {R}_+;V),\,\forall \,t\in [0,n].\nonumber \end{aligned}$$
(45)

Proof

Let \(\mathbf {u}\in C(\mathbb {R}_+;V)\) and consider the operator \(\varLambda :C(\mathbb {R}_+;Q)\rightarrow C(\mathbb {R}_+;Q)\) defined by

$$\begin{aligned}&\varLambda \varvec{\tau }(t)=\int \limits _0^t {\mathscr {G}}(\varvec{\tau }(s)+{\mathscr {E}}\varvec{\varepsilon }(\mathbf {u}(s)), \varvec{\varepsilon }(\mathbf {u}(s)))ds+\varvec{\sigma }_0-{\mathscr {E}}\varvec{\varepsilon }(\mathbf {u}_0)\\&\qquad \qquad \qquad \forall \,\varvec{\tau }\in C(\mathbb {R}_+;Q),\ t\in \mathbb {R}_+.\nonumber \end{aligned}$$
(46)

The operator \(\varLambda \) depends on \(\mathbf {u}\) but, for the sake of simplicity, we do not indicate it explicitly. Let \(\varvec{\tau }_1,\, \varvec{\tau }_2\in C(\mathbb {R}_+;Q)\) and let \(t\in \mathbb {R}_+\). Then, using (46) and (27) we have

$$\begin{aligned}&\Vert \varLambda \varvec{\tau }_1(t)-\varLambda \varvec{\tau }_2(t)\Vert _Q\le L_{\mathscr {G}}\int \limits _0^t\Vert \varvec{\tau }_1(s)-\varvec{\tau }_2(s)\Vert _Q\,ds. \end{aligned}$$

This inequality combined with Theorem 1 shows that the operator \(\varLambda \) has a unique fixed point in \(C(\mathbb {R}_+;Q)\). We denote by \(\varTheta \mathbf {u}\) the fixed point of \(\varLambda \) and we combine (46) with the equality \(\varLambda (\varTheta \mathbf {u})=\varTheta \mathbf {u}\) to see that (44) holds.

To proceed, let \(n\in \mathbb {N}\), \(t\in [0,n]\) and let \(\mathbf {u}_1,\) \(\mathbf {u}_2\in C(\mathbb {R}_+;V)\). Then, using (44) and taking into account (27), (26) we write

$$\begin{aligned}&\Vert \varTheta \mathbf {u}_1(t)-\varTheta \mathbf {u}_2(t)\Vert _Q\\&\qquad = L_0\Big ( \int \limits _0^t\Vert \varTheta \mathbf {u}_1(s)-\varTheta \mathbf {u}_2(s)\Vert _Q\,ds+\int \limits _0^t \Vert \mathbf {u}_1(s)-\mathbf {u}_2(s)\Vert _V\,ds\Big ), \end{aligned}$$

where \(L_0\) is a positive constant which depends on \(\mathscr {G}\) and \(\mathscr {E}\). Using now a Gronwall argument we deduce that

$$\begin{aligned}&\Vert \varTheta \mathbf {u}_1(t)-\varTheta \mathbf {u}_2(t)\Vert _Q\le L_0\,e^{L_0n}\int \limits _0^t \Vert \mathbf {u}_1(s)-\mathbf {u}_2(s)\Vert _V\,ds. \end{aligned}$$

This inequality shows that (45) holds with \(\theta _n=L_0\,e^{L_0n}\).

Next, we consider the operators \(A: V \rightarrow V\) and \({\mathscr {R}}:V\times Q\rightarrow L^2(\varGamma _3)\) defined by

$$\begin{aligned}&(A\mathbf {u}, \mathbf {v})_V= ({\mathscr {E}}\varvec{\varepsilon }(\mathbf {u}), \varvec{\varepsilon }(\mathbf {v}))_Q + \int \limits _{\varGamma _3}p(u_\nu )v_\nu \,da \, \qquad \forall \, \mathbf {u}\,, \mathbf {v} \in V, \end{aligned}$$
(47)
$$\begin{aligned}&{\mathscr {R}}(\mathbf {u}, \mathbf {z})=|R\big (P_\varSigma ({\mathscr {E}}\varvec{\varepsilon }(\mathbf {u}) + \mathbf {z})\big )_\nu | \qquad \forall \, \mathbf {u} \in V,\ \, \mathbf {z} \in Q, \end{aligned}$$
(48)

where \(P_\varSigma :Q\rightarrow \varSigma \) represents the projection operator. Note that, since \(\varSigma \subset Q_1\), the operator \(\mathscr {R}\) is well defined. Denote \(Y=Q \times L^{2}(\varGamma _3) \times Q\) where, here and below, \(X_1\times \ldots \times X_m\) represents the product of the Hilbert spaces \(X_1,\ldots , X_m\) (\(m=2,3\)), endowed with its canonical inner product. Besides the operator \( \varTheta : C(\mathbb {R}_+; V) \rightarrow C(\mathbb {R}_+; Q)\) defined in Lemma 4, let \(\varPhi : C(\mathbb {R}_+; V) \rightarrow C(\mathbb {R}_+; L^{2}(\varGamma _3))\) and \({\mathscr {S}}:C(\mathbb R_+;V)\rightarrow C(\mathbb R_+;Y)\) be the operators given by

$$\begin{aligned}&(\varPhi \mathbf {v})(t)= F\left( \int \limits _{0}^{t}v_{\nu }^{+}(s)ds\right) , \end{aligned}$$
(49)
$$\begin{aligned}&{\mathscr {S}}\mathbf {v}(t)=(\varTheta \mathbf {v}(t), \varPhi \mathbf {v}(t), \varTheta \mathbf {v}(t)) \end{aligned}$$
(50)

for all \(\mathbf {v} \in C(\mathbb R_+;V)\), \( t \in \mathbb {R}_+\). Finally, let \(j:Y\times V \times V \rightarrow \mathbb {R}\) and \(\mathbf {f}: \mathbb {R}_{+} \rightarrow V\) denote the functions defined by

$$\begin{aligned}&j(\mathbf {w}, \mathbf {u}, \mathbf {v})= (\mathbf {x}, \varvec{\varepsilon }(\mathbf {v}))_Q + (y, v_{\nu }^{+})_{L^{2}(\varGamma _3)} + (\mu {\mathscr {R}}(\mathbf {u}, \mathbf {z})\mathbf {n}^{*}, \mathbf {v}_{\tau })_{L^{2}(\varGamma _3)^{d}} \\&\qquad \qquad \forall \,\mathbf {w}=(\mathbf {x}, y, \mathbf {z}) \in Y,\,\ \mathbf {u}, \mathbf {v} \in V, \nonumber \end{aligned}$$
(51)
$$\begin{aligned}&(\mathbf {f}(t), \mathbf {v})_V= \int \limits _{\varOmega } \mathbf {f}_{0}\cdot \mathbf {v}\,dx + \int \limits _{\varGamma _2}\mathbf {f}_2(t) \cdot \mathbf {v}\,da\qquad \forall \,\mathbf {v} \in V,\ t \in \mathbb {R}_+. \end{aligned}$$
(52)

We have the following equivalence result.

Lemma 5

Assume that \(\mathbf {u} \in C(\mathbb {R}_+; U)\) and \(\varvec{\sigma }\in C(\mathbb {R}_+; \varSigma )\). Then, the couple \((\mathbf {u},\varvec{\sigma })\) is a solution of Problem \({\mathscr {P}}_V\) if and only if

$$\begin{aligned}&\varvec{\sigma }(t) = {\mathscr {E}} \varvec{\varepsilon }(\mathbf {u}(t))+\varTheta \mathbf {u}(t),\end{aligned}$$
(53)
$$\begin{aligned}&(A\mathbf {u}(t),\mathbf {v}-\mathbf {u}(t))_V+j({\mathscr {S}}\mathbf {u}(t),\mathbf {u}(t),\mathbf {v})\\&\qquad -j({\mathscr {S}}\mathbf {u}(t),\mathbf {u}(t),\mathbf {u}(t))\ge (\mathbf {f}(t),\mathbf {v}-\mathbf {u}(t))_V\qquad \forall \,\mathbf {v}\in U \nonumber \end{aligned}$$
(54)

for all \( t \in \mathbb {R}_+\).

Proof

Let \((\mathbf {u},\varvec{\sigma }) \in C(\mathbb {R}_+; U\times \varSigma )\), be a solution of Problem \({\mathscr {P}}_V\) and let \(t\in \mathbb {R}_+\). By (41) we have

$$\begin{aligned} {\varvec{\sigma }}(t)-{\mathscr {E}}\varvec{\varepsilon }({\mathbf {u}}(t))=\int \limits _0^t{\mathscr {G}}(\varvec{\sigma }(s)- {\mathscr {E}}\varvec{\varepsilon }({\mathbf {u}}(s))+{\mathscr {E}}\varvec{\varepsilon }({\mathbf {u}}(s)),\varvec{\varepsilon }(\mathbf {u}(s)))\,ds +\varvec{\sigma }_0-{\mathscr {E}}\varvec{\varepsilon }({\mathbf {u}}_0), \end{aligned}$$

and, using the definition (44) of the operator \(\varTheta \), we obtain (53). Moreover, we substitute (41) in (42), then we use (49) and equality \(P_\varSigma \varvec{\sigma }(t)=\varvec{\sigma }(t)\). As a result, we deduce that

$$\begin{aligned}&\int \limits _{\varOmega }{\mathscr {E}}\varvec{\varepsilon }(\mathbf {u}(t))\cdot (\varvec{\varepsilon }(\mathbf {v})- \varvec{\varepsilon }(\mathbf {u}(t)))\,dx + \int \limits _{\varOmega }\varTheta \mathbf {u}(t)\cdot (\varvec{\varepsilon }(\mathbf {v})- \varvec{\varepsilon }(\mathbf {u}(t)))\,dx\\&\quad +\int \limits _{\varGamma _3}p(u_\nu (t))(v_\nu -u_\nu (t))\,da + \int \limits _{\varGamma _3}\varPhi \mathbf {u}(t)\,(v^+_\nu -u^+_\nu (t))\,da \nonumber \\&\qquad + \int \limits _{\varGamma _3}\mu |R\big (P_\varSigma ({\mathscr {E}}\varvec{\varepsilon }(\mathbf {u}(t))+ \varTheta \mathbf {u}(t))\big )_\nu |\,\mathbf {n}^{*}\cdot (\mathbf {v}_\tau - \mathbf {u}_\tau (t))\,da \nonumber \\&\qquad \quad \ge \int \limits _{\varOmega } \mathbf {f}_{0}\cdot (\mathbf {v} - \mathbf {u}(t))\,dx + \int \limits _{\varGamma _2}\mathbf {f}_2(t)\cdot (\mathbf {v} - \mathbf {u}(t))\,da\qquad \forall \,\mathbf {v}\in U.\nonumber \end{aligned}$$
(55)

Using now the definitions (47), (48) and (52) yields

$$\begin{aligned}&(A\mathbf {u}(t),\mathbf {v}-\mathbf {u}(t))_V+ (\varTheta \mathbf {u}(t)\cdot (\varvec{\varepsilon }(\mathbf {v})- \varvec{\varepsilon }(\mathbf {u}(t)))_Q\\&\quad + (\varPhi \mathbf {u}(t))\,(v^+_\nu -u^+_\nu (t)))_{L^2(\varGamma _3)} + (\mu \mathscr {R}(\mathbf {u}(t),\varTheta (\mathbf {u}(t)) \mathbf {n}^{*},\mathbf {v}_\tau - \mathbf {u}_\tau (t))_{L^2(\varGamma _3)^d} \nonumber \\&\qquad \ge (\mathbf {f}(t),\mathbf {v}-\mathbf {u}(t))_V\qquad \forall \,\mathbf {v}\in U. \end{aligned}$$

This inequality combined with the definitions (50) and (51) shows that the variational inequality (54) holds.

Conversely, assume that \((\mathbf {u},\varvec{\sigma }) \in C(\mathbb {R}_+; U\times \varSigma )\) is a couple of functions which satisfies (53) and (54) and let \(t\in \mathbb {R}_+\). Then, using the definitions (47), (48), (50)–(52) it follows that (55) holds. Moreover, recall that the regularity \(\varvec{\sigma } \in C(\mathbb {R}_+;\varSigma )\) implies that \(P_\varSigma \varvec{\sigma }(t)=\varvec{\sigma }(t)\) and, in addition, (53) yields \(\varvec{\sigma }(t)={\mathscr {E}}\varvec{\varepsilon }(\mathbf {u}(t)) +\varTheta \mathbf {u}(t)\). Substituting these equalities in (55) and using (49) we see that (42) holds. Finally, to conclude, we note that (41) is a direct consequence of (53) and the definition of the operator \(\varTheta \) in Lemma 5.

The interest in Lemma 5 arrises in the fact that it decouples the unknowns \(\mathbf {u}\) and \(\varvec{\sigma }\) in the system (41)–(42). The next step is to provide the unique solvability of the variational inequality (54) in which the unknown is the displacement field. To this end we need the following intermediate result on the operator \(\mathscr {R}\).

Lemma 6

There exists \(L_{\mathscr {R}}>0\) which depends only on \(\varOmega \), \(\varGamma _3\) and R, such that

$$\begin{aligned} \Vert {\mathscr {R}}(\mathbf {u}_1, \mathbf {z}_1)&-{\mathscr {R}}(\mathbf {u}_2, \mathbf {z}_2)\Vert _{L^{2}(\varGamma _3)} \le L_{\mathscr {R}}\left( \Vert \mathbf {u}_1 - \mathbf {u}_2\Vert _{V} + \Vert \mathbf {z}_1 - \mathbf {z}_2\Vert _{Q} \right) \\&\forall \, \mathbf {u}_1, \mathbf {u}_2 \in V, \ \mathbf {z}_1, \mathbf {z}_2 \in Q.\nonumber \end{aligned}$$
(56)

Proof

Let \(\mathbf {u}_1, \mathbf {u}_2 \in V, \, \mathbf {z}_1, \mathbf {z}_2 \in Q\). Then, by the definition (48) of the operator \(\mathscr {R}\) combined with inequality (17) we have

$$\begin{aligned}&\Vert {\mathscr {R}}(\mathbf {u}_1, \mathbf {z}_1) -{\mathscr {R}}(\mathbf {u}_2, \mathbf {z}_2)\Vert _{L^{2}(\varGamma _3)}\\&\qquad \le c_R \Vert P_\varSigma ({\mathscr {E}}\varvec{\varepsilon }(\mathbf {u}_1)+\mathbf {z}_1)-P_\varSigma ( {\mathscr {E}}\varvec{\varepsilon }(\mathbf {u}_2)+\mathbf {z}_2)\Vert _{Q_1}.\nonumber \end{aligned}$$
(57)

On the other hand, the definition of the set \(\varSigma \) and the nonexpansivity of the operator \(P_\varSigma \) yields

$$\begin{aligned}&\Vert P_\varSigma ({\mathscr {E}}\varvec{\varepsilon }(\mathbf {u}_1)+\mathbf {z}_1)-P_\varSigma ( {\mathscr {E}}\varvec{\varepsilon }(\mathbf {u}_2)+\mathbf {z}_2)\Vert _{Q_1}\\&\qquad \le \Vert {\mathscr {E}}\varvec{\varepsilon }(\mathbf {u}_1)-{\mathscr {E}}\varvec{\varepsilon }(\mathbf {u}_2)+\mathbf {z}_1 - \mathbf {z}_2\Vert _{Q}\nonumber \end{aligned}$$
(58)

We now combine inequalities (57) and (58) to see that

$$\begin{aligned} \Vert {\mathscr {R}}(\mathbf {u}_1, \mathbf {z}_1) -{\mathscr {R}}(\mathbf {u}_2, \mathbf {z}_2)\Vert _{L^{2}(\varGamma _3)}\le c_R \left( \Vert {\mathscr {E}}\varvec{\varepsilon }(\mathbf {u}_1) - {\mathscr {E}}\varvec{\varepsilon }(\mathbf {u}_2) + \mathbf {z}_1 - \mathbf {z}_2\Vert _{Q}\right) \end{aligned}$$
(59)

Lemma 6 is now a consequence of inequality (59) and assumption (26).

We proceed with the following existence and uniqueness result.

Lemma 7

The variational inequality (54) has a unique solution with regularity \(\mathbf {u} \in C(\mathbb {R}_+, U)\).

Proof

It is straightforward to see that inequality (54) represents a variational inequality of the form (7) in which \(X=V\), \(K=U\) and \(Y=Q \times L^{2}(\varGamma _3) \times Q\). Therefore, in order to prove its unique solvability, we check in what follows the assumptions of Theorem 2.

First, we note that assumption (1) is obviously satisfied. Next, we use the definition (47), assumptions (26), (29)(b) and inequality (16) to obtain that

$$\begin{aligned} \Vert A\mathbf {u} - A\mathbf {v}\Vert _V \le (L_{\mathscr {E}} + c_{0}^{2}L_{p})\Vert \mathbf {u} - \mathbf {v}\Vert _V \qquad \forall \, \mathbf {u}, \mathbf {v} \in V, \end{aligned}$$
(60)

where \(L_{\mathscr {E}}\) is a positive constant which depends on the elasticity operator \(\mathscr {E}\). On the other hand, from (26)(c) and (29)(c) the we deduce that

$$\begin{aligned} (A\mathbf {u} - A\mathbf {v}, \mathbf {u} - \mathbf {v})_V \ge m_{\mathscr {E}}\Vert \mathbf {u} - \mathbf {v}\Vert _V^{2}. \end{aligned}$$
(61)

We conclude from above that the operator A satisfies condition (2) with \(L=L_{\mathscr {E}} + c_{0}^{2}L_{p}\) and \(m= m_{\mathscr {E}}\).

Let \(\mathbf {w}=(\mathbf {x},y,\mathbf {z})\in Y\) and \(\mathbf {u}\in V\) be fixed. Then, using the properties of the traces it is easy to see that the function \(\mathbf {v}\mapsto j(\mathbf {w},\mathbf {u},\mathbf {v})\) is convex and continuous and, therefore, it satisfies condition (3)(a). We now consider the elements \(\mathbf {w}_1=(\mathbf {x}_1,y_1,\mathbf {z}_1)\), \(\mathbf {w}_2= (\mathbf {x}_2,y_2,\mathbf {z}_2)\in Y\), \(\mathbf {u}_1, \mathbf {u}_2, \mathbf {v}_1, \mathbf {v}_2 \in V\). Then, using inequality (56), assumption (31) and inequality (16) we find that

$$\begin{aligned}&j(\mathbf {w}_1, \mathbf {u}_1, \mathbf {v}_2) - j(\mathbf {w}_1, \mathbf {u}_1, \mathbf {v}_1) + j(\mathbf {w}_2, \mathbf {u}_2, \mathbf {v}_1) - j(\mathbf {w}_2, \mathbf {u}_2, \mathbf {v}_2)\\&\quad \le \Vert \mathbf {x}_1 - \mathbf {x}_2 \Vert _{Q}\Vert \mathbf {v}_1 - \mathbf {v}_2 \Vert _{V} + c_{0} \Vert y_1 - y_2 \Vert _{L^{2}(\varGamma _3)}\Vert \mathbf {v}_1 - \mathbf {v}_2 \Vert _{V} \\&\qquad + c_{0}L_{\mathscr {R}}\Vert \mu \Vert _{L^{\infty }(\varGamma _3)}\, \left( \Vert \mathbf {u}_1 - \mathbf {u}_2 \Vert _{V} + \Vert \mathbf {z}_1 - \mathbf {z}_2 \Vert _{Q}\right) \Vert \mathbf {v}_1 -\mathbf {v}_2 \Vert _{V} \\&\qquad \quad \le \alpha \,\Vert \mathbf {w}_{1} - \mathbf {w}_{2}\Vert _{Z}\Vert \mathbf {v}_1 -\mathbf {v}_2 \Vert _{V}+\beta \,\Vert \mathbf {u}_1 - \mathbf {u}_2 \Vert _{V}\Vert \mathbf {v}_1 -\mathbf {v}_2 \Vert _{V} \end{aligned}$$

where \(\alpha =2\max \,\{1, \,c_{0}, \,c_{0}L_{\mathscr {R}}\Vert \mu \Vert _{L^{\infty }(\varGamma _3)}\,\}\) and \(\beta = c_{0}L_{\mathscr {R}}\Vert \mu \Vert _{L^{\infty }(\varGamma _3)}\). It follows from here that j satisfies condition (3)(b). Let

$$\begin{aligned} \mu _0= \frac{m_{\mathscr {E}}}{c_{0}L_{\mathscr {R}}}, \end{aligned}$$
(62)

which, clearly, depends only on \(\varOmega , \varGamma _1, \varGamma _3\), R and \({\mathscr {E}}\). Then, it is easy to see that if the smallness assumption \(\Vert \mu \Vert _{L^{\infty }(\varGamma _3)} < \mu _0\) is satisfied we have \(\beta < m\) and, therefore, condition (6) holds.

Next, let \(\mathbf {u}, \mathbf {v} \in C(\mathbb R_+;V), n \in \mathbb {N}\) and let \(t \in [0, n]\). Then, using (49) and taking into account (30)(b) and (16) we obtain that

$$\begin{aligned}&\Vert \varPhi \mathbf {u}(t) - \varPhi \mathbf {v}(t)\Vert _{L^{2}(\varGamma _3)}=\left\| F\left( \int \limits _{0}^{t} u_{\nu }^{+}(s)ds\right) - F\left( \int \limits _{0}^{t} v_{\nu }^{+}(s)ds\right) \right\| _ {L^{2}(\varGamma _3)}\\&\qquad \le L_F\left\| \int \limits _{0}^{t} (u_{\nu }^{+}(s) - v_{\nu }^{+}(s))ds \right\| _ {L^{2}(\varGamma _3)} \le c_{0}L_{F}\,\int \limits _{0}^{t}\Vert \mathbf {u}(s) - \mathbf {v}(s)\Vert _V\,ds. \end{aligned}$$

Therefore, using this the definition (50) of the operator \(\mathscr {S}\) and (45) we have

$$\begin{aligned}&\Vert {\mathscr {S}}\mathbf {u}(t) - {\mathscr {S}}\mathbf {v}(t)\Vert _{Q \times L^{2}(\varGamma _3) \times Q}\le \left( 2\,\theta _n+ c_{0}L_{F}\,\right) \int \limits _{0}^{t}\Vert \mathbf {u}(s) - \mathbf {v}(s)\Vert _V\,ds. \end{aligned}$$

It follows from here that the operator \({\mathscr {S}}\) satisfies condition (4). Finally, we note that assumption (28) on the body forces and traction and definition (52) imply that \(\mathbf {f} \in C(\mathbb {R}_+; V)\).

We conclude from above that all the assumptions of Theorem 2 are satisfied. Therefore, we deduce that inequality (54) has a unique solution \(\mathbf {u} \in C(\mathbb {R}_+; U)\) which concludes the proof.

We now have all the ingredients to provide the proof of Theorem 3.

Proof (Proof of Theorem 3 )

[Proof of Theorem 3] Let \(\mathbf {u}\in C(\mathbb {R}_+;U)\) be the unique solution of inequality (54) obtained in Lemma 7 and let \(\varvec{\sigma }\) the function defined by (53). Then, using assumption (26) it follows that \(\varvec{\sigma }\in C(\mathbb {R}_+;Q)\). Let \( t \in \mathbb {R}_+\) be given. Arguments similar to those used in the proof of Lemma 7 show that

$$\begin{aligned}&\int \limits _{\varOmega }\varvec{\sigma }(t)\cdot (\varvec{\varepsilon }(\mathbf {v})-\varvec{\varepsilon }(\mathbf {u}(t)))\,dx + \int \limits _{\varGamma _3}p(u_\nu (t))(v_\nu -u_\nu (t))\,da \\&\quad + \int \limits _{\varGamma _3}F\left( \int \limits _{0}^{t} u_{\nu }^{+}(s)ds\right) \, (v^+_\nu -u^+_\nu (t))\,da\\&\qquad +\int \limits _{\varGamma _3}\mu |R(P_\varSigma (\varvec{\sigma }_{\nu }(t))|\,\mathbf {n}^{*}\cdot (\mathbf {v}_\tau - \mathbf {u}_\tau (t))\,da \nonumber \\&\qquad \quad \ge \int \limits _{\varOmega } \mathbf {f}_{0}\cdot (\mathbf {v} - \mathbf {u}(t))\,dx + \int \limits _{\varGamma _2}\mathbf {f}_2(t)\cdot (\mathbf {v} - \mathbf {u}(t))\,da\qquad \forall \, \mathbf {v}\in U.\nonumber \end{aligned}$$

Let \(\varvec{\varphi }\in C_0^\infty (\varOmega )^d\). We test in this inequality with \(\mathbf {v}=\mathbf {u}(t)\pm \varvec{\varphi }\) to deduce that

$$\int \limits _{\varOmega }\varvec{\sigma }(t)\cdot \varvec{\varepsilon }(\varvec{\varphi })\,dx = \int \limits _{\varOmega } \mathbf {f}_{0}\cdot \varvec{\varphi }\,dx $$

which implies that \(\text {Div}\varvec{\sigma }(t)+\mathbf {f}_0=\mathbf {0}\) in \(\varOmega \). It follows from here that \(\varvec{\sigma }(t)\in \varSigma \) and, moreover, \(\varvec{\sigma }\in C(\mathbb {R}_+;\varSigma )\).

We conclude from above that \((\mathbf {u},\varvec{\sigma })\) represents a couple of functions which satisfies (53)–(54) and, in addition, it has the regularity \((\mathbf {u},\varvec{\sigma })\in C(\mathbb {R}_+;U\times \varSigma )\). The existence part in Theorem 3 is now a direct consequence of Lemma 5. The uniqueness part follows from the uniqueness of the solution of the variational inequality (54), guaranteed by Lemma 7.