The aim of this paper is to study a frictionless contact problem for rate-type viscoplastic materials within the framework of the Mathematical Theory of Contact Mechanics. We model the the material’s behavior with a constitutive law of the form

where $\boldsymbol{u}$ denotes the displacement field, $\boldsymbol{\sigma}$ represents the stress tensor and $\varepsilon(\boldsymbol{u})$ is the linearized strain tensor. Here $\mathcal{E}$ is a linear operator which describes the elastic properties of the material and $\mathcal{G}$ is a nonlinear constitutive function which describes its viscoplastic behavior. In (1) and everywhere in this paper the dot above a variable represents the derivative with respect to the time variable $t$. Quasistatic frictionless contact problems for materials of the form (11) have been considered in [1,2,6, 9, 11] and the references therein. In [6, 9] the contact was modelled with both the Signorini and the normal compliance condition which describe a rigid and an elastic foundation, respectively. In 1, 11 the contact was modelled with normal compliance and unilateral constraint. This condition, introduced for the first time in [7], models an elastic-rigid behavior of the foundation.

The present paper represents a continuation of the short note [5]. There, a model which involves a contact condition with normal compliance, unilateral constraint and memory term was considered. This condition takes into account both the deformability, the rigidity, and the memory effects of the foundation. An existence and uniqueness result was proved and the contact process was studied on an unbounded interval of time which implies the use of the framework of Fréchet spaces of continuous functions, instead of that of the classical Banach spaces of continuous functions defined on a bounded interval of time. The aim of this work is to provide a penalization method in the study of the contact model in 5. Penalization methods in the study of contact problems were used by many authors, mainly for numerical reasons. The main ingredient of these methods arises in the fact that they remove the constraints by considering penalized problems defined on the whole space; these approximative problems have a unique solution which converges to the solution of the original problem, as the penalization parameter converges to zero.

The rest of the paper is structured as follows. In Section 2 we present the notation we shall use as well as some preliminary material. In Section 3 we describe the model of the contact process, list the assumptions on the data and derive the variational formulation of the problem. Then we state an existence and uniqueness result, Theorem 3.1, proved in [5]. In Section 4 we present the weak solvability of the penalized problem then we state and prove our main convergence 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 given $r\in\mathbb{R}$ we denote by $r^{+}$its positive part, i.e. $r=\max\{r,0\}$. Let $\Omega$ be a bounded domain $\Omega\subset\mathbb{R}^{d}(d=1,2,3)$ with a Lipschitz continuous boundary $\Gamma$ and let $\Gamma_{1}$ be a measurable part of $\Gamma$ such that meas $\left(\Gamma_{1}\right)>0$. We use the notation $\boldsymbol{x}=\left(x_{i}\right)$ for a typical point in $\Omega\cup\Gamma$ and we denote by $\boldsymbol{\nu}=\left(\nu_{i}\right)$ the outward unit normal at $\Gamma$. 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. An index that follows a comma represents the partial derivative with respect to the corresponding component of the spatial variable, e.g. $u_{i,j}=\partial u_{i}/\partial x_{j}$. We denote by $\mathbb{S}^{d}$ the space of second order symmetric tensors on $\mathbb{R}^{d}$ or, equivalently, the space of symmetric matrices of order $d$. The inner product and norm on $\mathbb{R}^{d}$ and $\mathbb{S}^{d}$ are defined by

In addition, we use standard notation for the Lebesgue and Sobolev spaces associated to $\Omega$ and $\Gamma$ and, moreover, we consider the spaces

These are real Hilbert spaces endowed with the inner products

and the associated norms $\|\cdot\|_{V}$ and $\|\cdot\|_{Q}$, respectively. Here $\varepsilon$ represents the deformation operator given by

Completeness of the space ( $V,\|\cdot\|_{V}$ ) follows from the assumption meas $\left(\Gamma_{1}\right)>$ 0 , which allows the use of Korn’s inequality.

For an element $\boldsymbol{v}\in V$ we still write $\boldsymbol{v}$ for the trace of $\boldsymbol{v}$ on the boundary and we denote by $v_{\nu}$ and $\boldsymbol{v}_{\tau}$ the normal and tangential components of $\boldsymbol{v}$ on $\Gamma$, given by $v_{\nu}=\boldsymbol{v}\cdot\boldsymbol{\nu},\boldsymbol{v}_{\tau}=\boldsymbol{v}-v_{\nu}\boldsymbol{\nu}$. Let $\Gamma_{3}$ be a measurable part of $\Gamma$. Then, by the Sobolev trace theorem, there exists a positive constant $c_{0}$ which depends on $\Omega,\Gamma_{1}$ and $\Gamma_{3}$ such that

Also, for a regular function $\boldsymbol{\sigma}\in Q$ we use the notation $\sigma_{\nu}$ and $\boldsymbol{\sigma}_{\tau}$ for the normal and the tangential traces, i.e. $\sigma_{\nu}=(\boldsymbol{\sigma}\boldsymbol{\nu})\cdot\boldsymbol{\nu}$ and $\boldsymbol{\sigma}_{\tau}=\boldsymbol{\sigma}\boldsymbol{\nu}-\sigma_{\nu}\boldsymbol{\nu}$. Moreover, we recall that the divergence operator is defined by the equality $\operatorname{Div}\boldsymbol{\sigma}=\left(\sigma_{ij,j}\right)$ and, finally, the following Green’s formula holds:

Finally, we consider the space of fourth order tensor fields

This is a real Banach space with the norm $\|\mathcal{E}\|_{\mathbf{Q}_{\infty}}=\max_{1\leq i,j,k,l\leq d}\left\|\mathcal{E}_{ijkl}\right\|_{L^{\infty}(\Omega)}$. Moreover, a simple calculation shows that

For each Banach 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 $C\left(\mathbb{R}_{+};X\right)$ 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 [4] and [8], for instance. Here we restrict ourseleves 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:

The physical setting is as follows. A viscoplastic body occupies a bounded domain $\Omega\subset\mathbb{R}^{d}(d=1,2,3)$ with a Lipschitz continuous boundary $\Gamma$, 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 and we study it in the interval of time $\mathbb{R}_{+}=[0,\infty)$. Then, the classical formulation of the contact problem we consider in this paper is the following.

Problem $\mathcal{P}$. 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 $t\in\mathbb{R}_{+}$,

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

Here and below, in order to simplify the notation, we do not indicate explicitly the dependence of various functions on the spatial variable $\boldsymbol{x}\in\Omega\cup\Gamma$. Equation (6) represents the viscoplastic constitutive law of the material already introduced in Section 1. Equation (7) is the equilibrium equation in which Div denotes the divergence operator for tensor valued functions. Conditions (8) and (9) are the displacement and traction boundary conditions, respectively, and condition (10) represents the contact condition with normal compliance, unilateral constraint and memory term, in which $\sigma_{\nu}$ denotes the normal stress, $u_{\nu}$ is the normal displacement, $g\geq 0$ and $p,b$ are given functions. This condition was first introduced in [5] and, in the case when $b$ vanishes, was used in [7, 10, for instance. Condition (11) shows that the tangential stress on the contact surface, denoted $\boldsymbol{\sigma}_{\tau}$, vanishes. We use it here since we assume that the contact process is frictionless. Finally, (12) represents the initial conditions in which $\boldsymbol{u}_{0}$ and $\boldsymbol{\sigma}_{0}$ denote the initial displacement and the initial stress field, respectively.

Next, we list the assumptions on the data, present the variational formulation of the problem $\mathcal{P}$ and then we state and prove its unique weak solvability. To this end, we assume that the elasticity tensor $\mathcal{E}$, the nonlinear constitutive function $\mathcal{G}$ and the normal compliance function $p$ satisfy the following conditions.

Moreover, the densities of body forces, surface tractions and the memory function are such that

Finally, the initial data verifies

We introduce the set of admissible displacements $U$ given by

Next, using the Riesz representation theorem we define the operators $P$ : $V\rightarrow V,\mathcal{B}:C\left(\mathbb{R}_{+},V\right)\rightarrow C\left(\mathbb{R}_{+},L^{2}\left(\Gamma_{3}\right)\right)$ and the function $f:\mathbb{R}_{+}\rightarrow V$ by equalities

In order to derive the variational formulation of the Problem $\mathcal{P}$ we introduce the operator $\mathcal{S}$ by the following lemma.

Lemma 3.1 Assume that (14) and (18) hold. Then, for each function $\boldsymbol{u}\in C\left(\mathbb{R}_{+};V\right)$ there exists a unique function $\mathcal{S}\boldsymbol{u}\in C\left(\mathbb{R}_{+};Q\right)$ such that

Moreover, the operator $\mathcal{S}:C\left(\mathbb{R}_{+};V\right)\rightarrow C\left(\mathbb{R}_{+};Q\right)$ satisfies the following condition: for every $n\in\mathbb{N}$ there exists $k_{n}>0$ such that, $\forall\boldsymbol{u}_{1},\boldsymbol{u}_{2}\in C\left(\mathbb{R}_{+};V\right),\forall t\in[0,n]$,

The variational formulation of Problem $\mathcal{P}$ is the following.
Problem $\mathcal{P}^{V}$. Find a displacement field $\boldsymbol{u}:\mathbb{R}_{+}\rightarrow U$ and a stress field $\boldsymbol{\sigma}:\mathbb{R}_{+}\rightarrow Q$ such that, for all $t\in\mathbb{R}_{+}$,

The proof of Lemma 3.1 including the variational formulation $\mathcal{P}^{V}$ were obtained in [5]. Note that (25) is a consequence of (6) and (12), while (26) can be easily obtained by using integrations by parts, (7)-(11) and notation (19)-(21). The unique weak solvability of Problem $\mathcal{P}$ follows from the following result.

Theorem 3.1 Assume that (13)-(18) hold. Then Problem $\mathcal{P}^{V}$ has a unique solution, which satisfies $\boldsymbol{u}\in C\left(\mathbb{R}_{+};U\right)$ and $\boldsymbol{\sigma}\in C\left(\mathbb{R}_{+};Q\right)$.

The proof of Theorem 3.1 was given in [5], based on an abstract result provided by 11, 12 .

In this section we introduce a penalized contact problem $\mathcal{P}_{\mu}$ and we prove that its unique weak solution converges to the weak solution of problem $\mathcal{P}$.

Let $q$ be a function which satisfies

Let $\mu>0$ and consider the function $p_{\mu}$ defined by

We deduce that the function $p_{\mu}$ satisfies condition (15), i.e.

This allows us to consider the operator $P_{\mu}:V\rightarrow V$ defined by

and we note that $P_{\mu}$ is a monotone Lipschitz continuous operator.
With these notation, we consider the following contact problem.
Problem $\mathcal{P}_{\mu}$. Find a displacement field $\boldsymbol{u}_{\mu}:\Omega\times\mathbb{R}_{+}\rightarrow\mathbb{R}^{d}$ and a stress field $\sigma_{\mu}:\Omega\times\mathbb{R}_{+}\rightarrow\mathbb{S}^{d}$ such that, for all $t\in\mathbb{R}_{+}$,

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

Note that here and below $u_{\mu\nu}$ represents the normal component of the displacement field $\boldsymbol{u}_{\mu}$ and $\sigma_{\mu\nu},\boldsymbol{\sigma}_{\mu\tau}$ represent the normal and tangential components of the stress tensor $\sigma_{\mu}$, respectively. The equations and boundary conditions in problem (31)-(37) have a similar interpretation as those in problem (6)-(12). The difference arises in the fact that here we replace the contact condition with normal compliance, memory term and unilateral constraint (10) with the contact condition with normal compliance and memory term (35). In this condition $\mu$ represents a penalization parameter which may be interpreted as a deformability coefficient of the foundation, and then $\frac{1}{\mu}$ is the surface stiffness coefficient.

Using notation (221), (21) and (30) by similar arguments as in the case of Problem $\mathcal{P}$ we obtain the following variational formulation of Problem $\mathcal{P}_{\mu}$.
Problem $\mathcal{P}_{\mu}^{V}$. Find a displacement field $\boldsymbol{u}_{\mu}:\mathbb{R}_{+}\rightarrow U$ and a stress field
$\boldsymbol{\sigma}_{\mu}:\mathbb{R}_{+}\rightarrow Q$ such that, for all $t\in\mathbb{R}_{+}$,

We have the following existence, uniqueness and convergence result.
Theorem 4.1 Assume that (13) - (18) and (29) hold. Then
a) For each $\mu>0$ there exists a unique solution $\boldsymbol{u}_{\mu}\in V$ to Problem $\mathcal{P}_{\mu}^{V}$.
b) The solution $\boldsymbol{u}_{\mu}$ of Problem $\mathcal{P}_{\mu}^{V}$ converges strongly to the solution $\boldsymbol{u}$ of Problem $\mathcal{P}^{V}$, that is

as $\mu\rightarrow 0$, for all $t\in\mathbb{R}_{+}$.
Note that the convergence (40) above is understood in the following sense: for all $t\in\mathbb{R}_{+}$and for every sequence $\left\{\mu_{n}\right\}\subset\mathbb{R}_{+}$converging to 0 as $n\rightarrow\infty$ we have $\boldsymbol{u}_{\mu_{n}}(t)\rightarrow\boldsymbol{u}(t)$ in $V,\boldsymbol{\sigma}_{\mu_{n}}(t)\rightarrow\boldsymbol{\sigma}(t)$ in $Q$ as $n\rightarrow\infty$.

The proof of Theorem 4.1 is carried out in several steps that we present in what follows. To this end we assume below that (13)-(18) and (29) hold. Let $\mu>0$. We consider the auxiliary problem of finding a displacement field $\widetilde{\boldsymbol{u}}_{\mu}:\mathbb{R}_{+}\rightarrow V$ such that, for all $t\in\mathbb{R}_{+}$,

This problem is an intermediate problem between (39) and (26), since here $\mathcal{S}\boldsymbol{u}(t),\mathcal{B}\boldsymbol{u}(t)$ are knowns, taken from the problem $\mathcal{P}^{V}$.

We have the following existence and uniqueness result.
Lemma 4.1 There exists a unique function $\widetilde{\boldsymbol{u}}_{\mu}\in C\left(\mathbb{R}_{+};V\right)$ which satisfies (41), for all $t\in\mathbb{R}_{+}$.

Proof. We define the operator $A_{\mu}:V\rightarrow V$ and the function $\widetilde{\boldsymbol{f}}:\mathbb{R}_{+}\rightarrow V$ by equalities

for all $\boldsymbol{u},\boldsymbol{v}\in V,t\in\mathbb{R}_{+}$. We note that (17), (16), (22) and (23) yield $\widetilde{\boldsymbol{f}}\in C\left(\mathbb{R}_{+};V\right)$.

Let $t\in\mathbb{R}_{+}$. Based on (42)-(43), it is easy to see that (41) is equivalent with the nonlinear variational inequality of the first kind

Next, by (13) and the properties of operator $P_{\mu}$ it follows that $A_{\mu}$ is a strongly monotone and Lipschitz continuous operator. Therefore, using standard arguments on variational inequalities we deduce that there exists a unique solution $\widetilde{\boldsymbol{u}}_{\mu}\in C\left(\mathbb{R}_{+};U\right)$ for (44), which concludes the proof.

We proceed with the following weak convergence result.
Lemma 4.2 As $\mu\rightarrow 0$,

for all $t\in\mathbb{R}_{+}$.
Proof. Let $t\in\mathbb{R}_{+}$. We take $\boldsymbol{v}=\mathbf{0}$ in (41) to obtain

On the other hand, the properties (29) yield $\left(P_{\mu}\widetilde{\boldsymbol{u}}_{\mu}(t),\widetilde{\boldsymbol{u}}_{\mu}(t)\right)_{V}\geq 0$, and from (45) we deduce that

From (13) we obtain that

Note that here and below $c$ is a constant which does not depend on $\mu$ and $t$ and whose value can change from line to line. This inequality shows that the sequence $\left\{\widetilde{\boldsymbol{u}}_{\mu}(t)\right\}_{\mu}\subset V$ is bounded. Hence, there exists a subsequence of the sequence $\left\{\widetilde{\boldsymbol{u}}_{\mu}(t)\right\}_{\mu}$, still denoted $\left\{\widetilde{\boldsymbol{u}}_{\mu}(t)\right\}_{\mu}$, and an element $\widetilde{\boldsymbol{u}}(t)\in V$ such that

Next we study the properties of the element $\widetilde{\boldsymbol{u}}(t)$. It follows from (45) that

and, since $\left\{\widetilde{\boldsymbol{u}}_{\mu}(t)\right\}_{\mu}$ is a bounded sequence in $V$, we deduce that

This implies that $\int_{\Gamma_{3}}p_{\mu}\left(\widetilde{u}_{\mu\nu}(t)\right)\widetilde{u}_{\mu\nu}(t)da\leq c$ and, since $\int_{\Gamma_{3}}p_{\mu}\left(\widetilde{u}_{\mu\nu}(t)\right)gda\geq$ 0, it follows that