The theory of variational inequalities plays an important role in the study of nonlinear boundary value problems arising in mechanics, physics and engineering science. At the heart of this theory is the intrinsic inclusion of free boundaries in an elegant mathematical formulation. General results on the analysis of the variational inequalities, including existence and uniqueness results, can be found in [1, 2, 11, 13, 17, 22, for instance. Details concerning the numerical analysis of variational inequalities, including solution algorithms and error estimates, can be found in [6, 10]. References in the study of mathematical and numerical analysis of variational inequalities arising in hardening plasticity include [7, 8).

Phenomena of contact between deformable bodies abound in industry and everyday life. For this reason, considerable progress has been achieved recently in modelling, mathematical analysis and numerical simulations of various contact processes and, as a result, a general mathematical theory of contact mechanics is currently emerging. It is concerned with the mathematical structures which underlie general contact problems with different constitutive laws, i.e. materials, varied geometries and different contact conditions. To this end, it uses various mathematical concepts which include both variational and hemivariational inequalities and multivalued inclusions. An early attempt to study frictional contact problems within the framework of variational inequalities was made in [4]. An excellent reference on analysis and numerical approximations of contact problems involving elastic materials with or without friction is 12]. The variational analysis of various contact problems can be found in the monographs [5, 9, 10, 12, 16, 17, 20. The state of the art in the field can be found in the proceedings [14, 18, 24] and in the special issue [19], as well.

Existence, uniqueness and regularity results in the study of a new class of variational inequalities were proved in [21. There, the first trait of novelty lies in the fact that, unlike the results obtained in literature, the variational inequalities considered were defined on an unbounded interval of time. The second novelty was related to their special structure, which involves two nondifferentiable convex functionals, one of them depending on the history of the solution. This class of variational inequalities represents a general framework in which a large number of quasistatic contact problems, associated with various constitutive laws and frictional contact conditions, can be cast, as exemplified in [22].

Our intention in this current paper is to present a penalization method in the study of the variational inequalities introduced in [21 and to apply it to a new model of contact. Penalization methods in the study of elliptic variational inequalities were used by many authors, mainly for numerical reasons. Details can be found in [6] and the references therein. The main ingredient of these methods arises from the fact that they remove the constraints by considering penalized problems defined on the whole space; these approximative problems have unique solutions which converge to
the solutions of the original problems, as the penalization parameter converges to zero.

The rest of the paper is structured as follows. In Section 2 we present a new mathematical model of contact which is of applied interest and which motivates the abstract study we present in this paper. In Section 3 we state the abstract problem and recall its unique solvability, obtained in [21]. Then we state the penalized problems and prove our main result, Theorem 3.2. The proof of this theorem is given in Section 4. Further, we illustrate the use of the abstract results in the study of the contact model introduced in Section 2. To this end, in Section 5 we list the assumptions on the data and derive the variational formulation. Then we state and prove Theorem 5.1 which concerns the unique weak solvability of the model. Next, in Section 6, we use our abstract penalization method. Our main result in this section is given by Theorem 6.1 which states the existence of a unique weak solution of the penalized contact problems and its convergence to the weak solution of the original contact model. Finally, in Section 7, we present some concluding remarks.

The physical setting is as follows. A viscoelastic body occupies a bounded domain $\Omega\subset\mathbb{R}^{d}(d=1,2,3)$ with a Lipschitz continuous boundary $\Gamma$ which is divided into three measurable parts $\Gamma_{1},\Gamma_{2}$ and $\Gamma_{3}$, such that meas $\left(\Gamma_{1}\right)>0$. The body is subject to the action of body forces of density $\boldsymbol{f}_{0}$. We also assume that it is fixed on $\Gamma_{1}$ and surface tractions of density $\boldsymbol{f}_{2}$ act on $\Gamma_{2}$. On $\Gamma_{3}$, the body is in frictionless contact with a deformable obstacle, the so-called foundation. We assume that the contact process is quasistatic, we study it in the interval of time $\mathbb{R}_{+}=[0,\infty)$, and we denote by $\boldsymbol{\nu}$ and $\mathbb{S}^{d}$ the outward unit normal at $\Gamma$ and the space of second order symmetric tensors on $\mathbb{R}^{d}$, respectively. Then, the classical formulation of the contact problem we consider in the rest of this paper is the following.

Problem $\mathcal{Q}$. Find a displacement field $\boldsymbol{u}:\Omega\times\mathbb{R}_{+}\rightarrow\mathbb{R}^{d}$ and a stress field $\boldsymbol{\sigma}$ : $\Omega\times\mathbb{R}_{+}\rightarrow\mathbb{S}^{d}$ such that

for all $t\in\mathbb{R}_{+}$, and there exists $\xi:\Omega\times\mathbb{R}_{+}\rightarrow\mathbb{R}$ which satisfies

for all $t\in\mathbb{R}_{+}$.
Here and below, in order to simplify the notation, we do not indicate explicitly the dependence of various functions on the spatial variable $\boldsymbol{x}$. Equation (2.1) represents the viscoelastic constitutive law with long memory in which $\mathcal{A}$ is the elasticity operator, $\mathcal{B}$ represents the relaxation tensor and $\varepsilon(\boldsymbol{u})$ denotes the linearized strain tensor. Equation (2.2) represents the equation of equilibrium in which Div denotes the divergence operator for tensor valued functions. Conditions (2.3) and (2.4) are the displacement boundary condition and the traction boundary condition, respectively. Condition (2.5) is the frictionless condition and it shows that the tangential stress on the contact surface, denoted by $\boldsymbol{\sigma}_{\tau}$, vanishes. More details on the equations and conditions (2.1)-(2.5) can be found in 22.

We now describe the contact condition (2.6) in which our main interest lies and which represents the main novelty of the model. Here $\sigma_{\nu}$ denotes the normal stress, $u_{\nu}$ is the normal displacement and $u_{\nu}^{+}$may be interpreted as the penetration of the body’s surface asperities and those of the foundation. Moreover, $p$ is a Lipschitz continuous increasing function which vanishes for a negative argument, $F$ is a positive function and $g>0$. This condition can be derived in the following way. Let $t\in\mathbb{R}_{+}$be given. First, we assume that the penetration is limited by the bound $g$ and, therefore, the normal displacement satisfies the inequality

Next, we assume that the normal stress has an additive decomposition of the form

in which the function $\sigma_{\nu}^{D}(t)$ describes the deformability of the foundation and the functions $\sigma_{\nu}^{R}(t),\sigma_{\nu}^{M}(t)$ describe the rigidity and the memory properties of the foundation, respectively. We assume that $\sigma_{\nu}^{D}(t)$ satisfies a normal compliance contact condition, that is

The part $\sigma_{\nu}^{R}(t)$ of the normal stress satisfies the Signorini condition in the form with a gap function, i.e.

Finally, the function $\sigma_{\nu}^{M}(t)$ satisfies the condition

We combine (2.8), (2.9) and write $-\sigma_{\nu}^{M}(t)=\xi(t)$ to see that

Then we substitute equality (2.12) in (2.10) and use (2.7), (2.11) to obtain the contact condition (2.6).

We now present additional details of the contact condition (2.6). The inequalities and equalities below in this section are valid at an arbitrary point $\boldsymbol{x}\in\Gamma_{3}$. First, we recall that (2.6) describes a condition with unilateral constraint, since inequality (2.7) holds at each moment of time. Next, assume that at a given moment $t$ there is separation between the body and the foundation, i.e. $u_{\nu}(t)<0$. Then, since $p\left(u_{\nu}(t)\right)=0$, (2.6) shows that $\sigma_{\nu}(t)=0$, i.e. the reaction of the foundation vanishes. Note that the same behaviour of the normal stress is described both in the classical normal compliance condition and in the Signorini contact condition, when there is separation. Assume now that at the moment $t$ there is penetration which did not reach the bound $g$, i.e. $0<u_{\nu}(t)<g$. Then (2.6) yields

This equality shows that, at the moment $t$, the reaction of the foundation depends on the penetration and represents a normal compliance-type condition. Note that (2.6) also shows that if at the moment $t$ we have penetration which satisfies $0<u_{\nu}(t)<g$ then $-\sigma_{\nu}(t)\geq F$. Indeed, if $0<u_{\nu}(t)<g$ then (2.13) holds and this implies that $-\sigma_{\nu}(t)\geq F$. We conclude from above that if $-\sigma_{\nu}(t)<F$ then there is no penetration and, therefore, $F$ represents a yield limit of the normal pressure, under which the penetration is not possible. This kind of behaviour characterizes a rigidelastic foundation.

In conclusion, condition (2.6) shows that when there is separation between the body’s surface and the foundation then the normal stress vanishes; the penetration arises only if the normal stress reaches the critical value $F$; when there is penetration the contact follows a normal compliance condition of the form (2.13) but up to the limit $g$ and then, when this limit is reached, the contact follows a Signorini-type unilateral condition with the gap $g$. For this reason we refer to this condition as to a multivalued normal compliance contact condition with unilateral constraint. It can be interpreted physically as follows. The foundation is assumed to be made of a hard material covered by a thin layer of a soft material with thickness $g$. The soft material has a rigid-elastic behaviour, i.e. is deformable, allows penetration, but only if the normal stress arrives to the yield value $F$; the contact with this layer is modelled
with normal compliance, as shown in equality (2.13). The hard material is perfectly rigid and, therefore, it does not allow penetration; the contact with this material is modelled with the Signorini contact condition.

Two questions arise in the study of the unilateral contact problem $\mathcal{Q}$. The first one concerns its unique solvability; the second one concerns the approach of the solution by the solution of a contact model with normal compliance without unilateral constraint. The answers to the questions above are provided by the variational analysis of this contact problem, presented in Section 5 and 6. This analysis is carried out based on the abstract existence, uniqueness and convergence result that we present in the next section.

Everywhere below we use the notation $\mathbb{N}^{*}$ for the set of positive integers and $\mathbb{R}_{+}=[0,\infty)$. For each normed space $X$ we use the notation $C\left(\mathbb{R}_{+};X\right)$ for the space of continuous functions defined on $\mathbb{R}_{+}$with values in $X$. For a subset $K\subset X$ we still use the symbol $C\left(\mathbb{R}_{+};K\right)$ for the set of continuous functions defined on $\mathbb{R}_{+}$with values in $K$. It is well known that, if $X$ is a Banach space, then $C\left(\mathbb{R}_{+};X\right)$ can be organized in a canonical way as a Fréchet space, i.e. as a complete metric space in which the corresponding topology is induced by a countable family of seminorms. Details can be found in 3 and [15], for instance. Here we only need to recall that the convergence of a sequence $\left(x_{k}\right)_{k}$ to the element $x$, in the space $C\left(\mathbb{R}_{+};X\right)$, can be described as follows:

Consider now a real Hilbert space $X$ with inner product $(\cdot,\cdot)_{X}$ and associated norm $\|\cdot\|_{X}$. Also, let $K$ be a subset of $X$, let $A:X\rightarrow X,\mathcal{S}:C\left(\mathbb{R}_{+};X\right)\rightarrow C\left(\mathbb{R}_{+};X\right)$ be two operators, and let $j:X\rightarrow\mathbb{R},f:\mathbb{R}_{+}\rightarrow X$ be two functions. We assume in what follows that

and $A$ is strongly monotone and Lipschitz continuous operator, i.e.

Moreover, we assume that the operator $\mathcal{S}$ satisfies the following condition:

Following the terminology in [21, 22] we refer to an operator $\mathcal{S}$ which satisfies (3.4) as a history-dependent operator. Finally, we suppose that

With the data above, we consider the following problem.
Problem $\mathcal{P}$. Find a function $u:\mathbb{R}_{+}\rightarrow X$ such that, for all $t\in\mathbb{R}_{+}$, the inequality below holds:

Following [21, 22] we refer to (3.7) as a history-dependent variational inequality. It represents the framework in which the variational formulation of a large number of contact problems can be cast, with the appropriate choice of spaces and operators. Details can be found in [9, 21, 22, 23] and the references therein. The solvability of Problem $\mathcal{P}$ is provided by the following existence and uniqueness result, proved in [21.

Theorem 3.1 Let $X$ be a Hilbert space and assume that (3.2)-(3.6) hold. Then, Problem $\mathcal{P}$ has a unique solution $u\in C\left(\mathbb{R}_{+};K\right)$.

In order to formulate the penalized problems associated to Problem $\mathcal{P}$ we consider an operator $G:X\rightarrow X$ which satisfies the following conditions:

Note that conditions (3.8) (a) and (b) show that $G$ is a monotone Lipschtz continuous operator. Also, note that such an operator $G$ always exists. For example consider the operator $G:X\rightarrow X$ defined by

where $P_{K}:X\rightarrow K$ represents the projection operator onto $K$. Then, using the properties of the projections, it is easy to see that the operator $G$ satisfies condition (3.8).

Next, for each $\mu>0$ we consider the following problem.
Problem $\mathcal{P}_{\mu}$. Find a function $u_{\mu}:\mathbb{R}_{+}\rightarrow X$ such that, for all $t\in\mathbb{R}_{+}$, the inequality below holds:

Note that, in contrast to Problem $\mathcal{P}$, in Problem $\mathcal{P}_{\mu}$ the constraint $u(t)\in K$ is removed and is replaced with an additional term which contains the penalization parameter $\mu$. For this reason, we refer to Problem $\mathcal{P}_{\mu}$ as a penalized problem associated to Problem $\mathcal{P}$.

We have the following existence, uniqueness and convergence result, which represents the main result of this section.

Theorem 3.2 Let $X$ be a Hilbert space and assume that (3.2)-(3.6), (3.8) hold. Then:

1. 1.

   For each $\mu>0$ Problem $\mathcal{P}_{\mu}$ has a unique solution which satisfies $u_{\mu}\in C\left(\mathbb{R}_{+};X\right)$.

   Report issue for preceding element
2. 2.

   The solution $u_{\mu}$ of Problem $\mathcal{P}_{\mu}$ converges to the solution $u$ of Problem $\mathcal{P}$, that is

   Report issue for preceding element

for each $t\in\mathbb{R}_{+}$.
Note that the convergence (3.10) above is understood in the following sense: for each $t\in\mathbb{R}_{+}$and for every sequence $\left\{\mu_{n}\right\}\subset\mathbb{R}_{+}$converging to 0 as $n\rightarrow\infty$ we have $u_{\mu_{n}}(t)\rightarrow u(t)$ as $n\rightarrow\infty$.

The proof of Theorem 3.2 will be carried out in several steps that we present in what follows. To this end, below in this section we assume that (3.2)-(3.6), (3.8) hold and
we denote by $c$ a positive constant which may depend on $t,A,\mathcal{S},j,f$ and $u$, but is independent of $\mu$, and whose value may change from line to line. The following lemma shows the unique solvability of the nonlinear inequality (3.9).

Lemma 4.1 For each $\mu>0$ there exists a unique function $u_{\mu}\in C\left(\mathbb{R}_{+};X\right)$ which satisfies the inequality (3.9) for all $t\in\mathbb{R}_{+}$.

Proof. Let $\mu>0$. Using (3.3) and (3.8) it is easy to show that the operator

is a strongly monotone Lipschitz continuous operator on $X$. Lemma 4.1 is now a consequence of Theorem 3.1 used with $K=X$.

Next, we consider the following intermediate problem.
Problem $\widetilde{\mathcal{P}}_{\mu}$. Find a function $\widetilde{u}_{\mu}:\mathbb{R}_{+}\rightarrow X$ such that, for all $t\in\mathbb{R}_{+}$, the inequality below holds:

Note that inequality (3.9) is a history-dependent variational inequality, since the operator $\mathcal{S}$ is applied to the unknown $u_{\mu}$. In contrast, the variational inequality (4.1) is a time-dependent variational inequality, since here $\mathcal{S}u$ is a given function. The following lemma shows the unique solvability of the nonlinear inequality (4.1).

Lemma 4.2 For each $\mu>0$ there exists a unique function $\widetilde{u}_{\mu}\in C\left(\mathbb{R}_{+};X\right)$ which satisfies the inequality (4.1), for all $t\in\mathbb{R}_{+}$.

Proof. The proof is obtained by similar arguments to those used in the proof of Lemma 4.1.

Next we investigate the properties of the sequence $\left\{\widetilde{u}_{\mu}(t)\right\}$ for a fixed $t\in\mathbb{R}_{+}$.

Lemma 4.3 For each $t\in\mathbb{R}_{+}$there exists a subsequence of the sequence $\left\{\widetilde{u}_{\mu}(t)\right\}$, again denoted $\left\{\widetilde{u}_{\mu}(t)\right\}$, which converges weakly to $u(t)$, i.e.

Proof. Let $t\in\mathbb{R}_{+},\mu>0$ and let $v_{0}\in K$. We use (4.1) to obtain

and, therefore,

We use (3.5) to see that there exist $\omega\in X$ and $\alpha\in\mathbb{R}$, which do not depend on $t$, such that

and, therefore,

Then, we combine (4.3), (3.3), (3.8) (c) and (4.4) to find that

We use now (4.5), the elementary inequality

and the triangle inequality

As a result we deduce that there exists $c>0$ which depends on $v_{0}$ but does not depend on $\mu$ such that

Inequality (4.6) shows that the sequence $\left\{\widetilde{u}_{\mu}(t)\right\}$ is bounded in $X$. Therefore, it follows that there exists a subsequence of the sequence $\left\{\widetilde{u}_{\mu}(t)\right\}$, again denoted $\left\{\widetilde{u}_{\mu}(t)\right\}$ and an element $\widetilde{u}(t)\in X$ such that

Next, we investigate the properties of the element $\widetilde{u}(t)\in X$. First of all, we show that $\widetilde{u}(t)\in K$. To this end, we use (4.1) to deduce that

We now write

then we use the Lipschitz continuity of the operator $A$ and inequality (4.4) to obtain that

We combine now this inequality and (4.6) to see that there exists a positive constant $c$ which depends on $t,A,f,\mathcal{S},j,u$ and $v$, but is independent on $\mu$, such that

We take now $v=\widetilde{u}(t)$ in (4.9), then we pass to the upper limit as $\mu\rightarrow 0$ in the resulting inequality to obtain

Therefore, using assumption (3.8) (a), (b) the convergence (4.7) and standard arguments on pseudomonotone operators (see Proposition 1.23 in [22, for instance) we deduce that

On the other hand, the inequality (4.9) implies that

We combine the inequalities (4.10) and (4.11) to see that

and, taking $v=\widetilde{u}(t)-G\widetilde{u}(t)$ in this inequality yields $\|G\widetilde{u}(t)\|_{X}^{2}\leq 0$. We conclude that $G\widetilde{u}(t)=0_{X}$ and, using assumption (3.8)(d) it follows that

Next, from inequality (4.1) and assumption (3.8) (c) we find that

We now take $v=\widetilde{u}(t)\in K$ in (4.13) and obtain

then we pass to the upper limit as $\mu\rightarrow 0$ in this inequality and use the weak convergence (4.7) and the assumption (3.5). As a result we obtain

and, using again the argument on pseudomonotonicity employed in the proof of Lemma 4.3, it follows that

On the other hand, passing to the lower limit as $\mu\rightarrow 0$ in (4.13) and using (4.7) yields

We combine now the inequalities (4.15) and (4.16) to see that

Next, we take $v=u(t)$ in (4.17) and $v=\widetilde{u}(t)$ in (3.7). Then, adding the resulting inequalities and using the strong monotonicity of the operator $A$ we obtain that

which concludes the proof.
The next step is provided by the following weak convergence result.

Lemma 4.4 For each $t\in\mathbb{R}_{+}$the whole sequence $\left\{\widetilde{u}_{\mu}(t)\right\}$ converges weakly in $X$ to $u(t)$ as $\mu\rightarrow 0$.

Proof. Let $t\in\mathbb{R}_{+}$. A carefully examination of the proof of Lemma 4.3 shows that any weak convergent subsequence of the sequence $\left\{\widetilde{u}_{\mu}(t)\right\}\subset X$ converges weakly to $u(t)$, where, recall, $u(t)$ is the element of $X$ which solves the variational inequality (3.7) at the moment $t$. This inequality has a unique solution and, moreover, estimate (4.6) shows that the sequence $\left\{\widetilde{u}_{\mu}(t)\right\}$ is bounded in $X$. Lemma 4.4 is now a consequence of a standard compactness argument.

We proceed with the following strong convergence result.
Lemma 4.5 For each $t\in\mathbb{R}_{+}$the sequence $\left\{\widetilde{u}_{\mu}(t)\right\}$ converges strongly in $X$ to $u(t)$, that is

Proof. Let $\mu>0$ and $t\in\mathbb{R}_{+}$. We take $v=\widetilde{u}(t)$ in (4.15) to see that

then we combine this inequality with (4.14) to obtain that

Finally, we use (4.18) to find that

On the other hand, from the weak convergence of the sequence $\left\{\widetilde{u}_{\mu}(t)\right\}$ to $u(t)$, guaranteed by Lemma 4.4, it follows that

Next, from the strong monotonicity of the operator $A$ we have

The strong convergence (4.19) is now a consequence of (4.20)-(4.22).
The last step is provided by the following strong convergence result.
Lemma 4.6 For each $t\in\mathbb{R}_{+}$the sequence $\left\{u_{\mu}(t)\right\}$ converges strongly in $X$ to $u(t)$, that is

Proof. Let $t\in\mathbb{R}_{+}$and $n\in\mathbb{N}^{*}$ be such that $t\in[0,n]$. Let also $\mu>0$. We take $v=u_{\mu}(t)$ in (4.1) and $v=\widetilde{u}_{\mu}(t)$ in (3.9). Then, adding the resulting inequalities we deduce that