The first aim of this paper is to study a quasistatic frictionless contact problem for elastic materials, within the framework of the Mathematical Theory of Contact Mechanics. We model the behavior of the material with a constitutive law of the form

where $\boldsymbol{u}$ denotes the displacement field, $\boldsymbol{\sigma}$ represents the stress field, $\boldsymbol{\varepsilon}(\boldsymbol{u})$ is the linearized strain tensor and $\mathcal{F}$ is a fourth order tensor which describes the elastic properties of the material. The contact is modelled with a condition which involves normal compliance, memory term and infinite penetration. We prove the unique solvability of this model by using new arguments on historydependent variational inequalities presented in [6]. Also, we state and prove the dependence of the solution with respect to the data.

The second aim is to study the continuous dependence of the solution of a quasistatic frictionless contact problem for rate-type viscoplastic materials. We model the behavior of the material with a constitutive law of the form

Here $\mathcal{G}$ is a nonlinear constitutive function which describes the viscoplastic properties of the material. In (1.2) and everywhere in this paper the dot above a variable represents derivative with respect to the time variable $t$. The second part represents a continuation of [3] where a contact problem for viscoplastic materials of the form (1.2) was considered. The process was assumed to be quasistatic and the contact was modelled by using the normal compliance condition, finite penetration and memory term. The unique solvability of the solution was obtained. Also, the convergence of the solution of the problem with infinite penetration to the solution of the problem with finite penetration as the stiffness coefficient converges to infinity was proved. In the present paper we analyse the dependence of the solution of the viscoplastic contact problem in 3 with respect to the data.

The rest of the paper is structured as follows. In Section 2 we provide the notation we shall use as well as some preliminary material. In Section 3 we present the classical formulation of the first problem, list the assumption on the data and derive the variational formulation. Then we state and prove the unique weak solvability of the problem, Theorem 3.1, and a convergence result, Theorem 3.2. In Section 4 we introduce the classical formulation of the second problem and resume the results on its unique weak solvability obtained in 3. Then we state and prove a convergence result, Theorem 4.3,

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. $\mathbb{R}_{+}=[0,+\infty)$. We denote by $\mathbb{S}^{d}$ the space of second order symmetric tensors on $\mathbb{R}^{d}$. The inner product and norm on $\mathbb{R}^{d}$ and $\mathbb{S}^{d}$ are defined by

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 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

Also, we define the space

which is a Hilbert space endowed with the inner product

and the associated norm $\|\cdot\|_{Q_{1}}$. Here Div represents the divergence operator given by $\operatorname{Div}\boldsymbol{\sigma}=\left(\sigma_{ij,j}\right)$.

The assumption meas $\left(\Gamma_{1}\right)>0$ allows the use of Korn’s inequality which involves the completeness of the space ( $V,\|\cdot\|_{V}$ ). For an element $\boldsymbol{v}\in V$ we still write $\boldsymbol{v}$ for its trace 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 stress function $\boldsymbol{\sigma}$ we use the notation $\sigma_{\nu}$ and $\boldsymbol{\sigma}_{\tau}$ for the normal and the tangential components, i.e. $\sigma_{\nu}=(\boldsymbol{\sigma}\boldsymbol{\nu})\cdot\boldsymbol{\nu}$ and $\boldsymbol{\sigma}_{\tau}=\boldsymbol{\sigma}\boldsymbol{\nu}-\sigma_{\nu}\boldsymbol{\nu}$. For the convenience of the reader we recall the following Green’s formula:

We denote by $\mathbf{Q}_{\infty}$ the space of fourth order tensor fields given by

and $\mathbf{Q}_{\infty}$ is a real Banach space with the norm $\|\mathcal{E}\|_{\mathbf{Q}_{\infty}}=\sum_{1\leq i,j,k,l\leq d}\left\|\mathcal{E}_{ijkl}\right\|_{L^{\infty}(\Omega)}$. 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 on $X$. It is well known that $C\left(\mathbb{R}_{+};X\right)$ can be organized in a canonical way as a Fréchet space. Details can be found in (1) and 5, for instance. Here we restrict ourseleves to recall that the convergence of a sequence $\left(x_{m}\right)_{m}$ to the element $x$, in the space $C\left(\mathbb{R}_{+};X\right)$, can be described as follows:

Let $X$ be a real Hilbert space with inner product $(\cdot,\cdot)_{X}$ and associated norm $\|\cdot\|_{X}$. Let $K$ be a subset of $X$ and consider the operators $A:K\rightarrow X$, $\mathcal{R}:C\left(\mathbb{R}_{+};X\right)\rightarrow C\left(\mathbb{R}_{+};X\right)$ and function $f:\mathbb{R}_{+}\rightarrow X$. We are interested in the problem of finding a function $u\in C\left(\mathbb{R}_{+};X\right)$ such that $u(t)\in K$, for all $t\in\mathbb{R}_{+}$, and inequality below holds

In the study of (2.6) we assume that
(2.7) $K$ is a nonempty, closed, convex subset of $X$
and $A:K\rightarrow X$ is a strongly monotone Lipschitz continuous operator, i.e.
(2.8) $\left\{\begin{array}[]{l}\text{ (a) There exists }m>0\text{ such that }\\ \quad\left(Au_{1}-Au_{2},u_{1}-u_{2}\right)_{X}\geq m\left\|u_{1}-u_{2}\right\|_{X}^{2}\quad\forall u_{1},u_{2}\in K.\\ \text{ (b) There exists }M>0\text{ such that }\\ \left\|Au_{1}-Au_{2}\right\|_{X}\leq M\left\|u_{1}-u_{2}\right\|_{X}\quad\forall u_{1},u_{2}\in K.\end{array}\right.$

The operator $\mathcal{R}:C\left(\mathbb{R}_{+};X\right)\rightarrow C\left(\mathbb{R}_{+};X\right)$ satisfies
(2.9) $\left\{\begin{array}[]{l}\text{ For every }n\in\mathbb{N}^{*}\text{ there exists }r_{n}>0\text{ such that for all }t\in[0,n]\\ \left\|\mathcal{R}u_{1}(t)-\mathcal{R}u_{2}(t)\right\|_{X}\leq r_{n}\int_{0}^{t}\left\|u_{1}(s)-u_{2}(s)\right\|_{X}ds,\end{array}\right.$
for all $u_{1},u_{2}\in C\left(\mathbb{R}_{+};X\right)$, and, finally, we assume that
(2.10) $f\in C\left(\mathbb{R}_{+};X\right)$.

The next results, proved in [7], will be used in the rest of this paper.
THEOREM 2.1 Let $X$ be an Hilbert space and assume that (2.7)-(2.10) hold. Then, the inequality (2.6) has a unique solution $u\in C\left(\mathbb{R}_{+};K\right)$.

COROLLARY 2.2 Let $X$ be an Hilbert space and assume that (2.8)-(2.10) hold. Then there exists a unique function $u\in C\left(\mathbb{R}_{+};X\right)$ such that

To avoid any confusion, we note that here and below the notation $Au(t)$ and $\mathcal{R}u(t)$ are short hand notation for $A(u(t))$ and $(\mathcal{R}u)(t)$, for all $t\in\mathbb{R}_{+}$.
We end this section with a short description of the physical setting of the two contact problems.

An elastic body in the first problem and a viscoplastic body in the second problem 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 process is quasistatic and is studied in the interval of time $\mathbb{R}_{+}$.

The classical formulation of the first contact problem is the following.
Problem $\mathcal{P}_{1}$. 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 each $t\in\mathbb{R}_{+}$,

Here and below, in order to simplify the notation, we do not indicate explicitly the dependence of various functions on the variables $\boldsymbol{x}$ or $t$. Equation (3.1) represents the elastic constitutive law of the material. Equation (3.2) is the equation of equilibrium, conditions (3.3), (3.4) represent the displacement and traction boundary conditions, respectively, and condition (3.5) shows that the contact follows a normal compliance condition with memory term. At the moment $t$, the reaction of the foundation depends both on the current value of the penetration (represented by the term $\left.p\left(u_{\nu}(t)\right)\right)$ as well as on the history of the penetration (represented by the integral term). Finally, (3.6) is the frictionless condition.

We assume that the body forces and surface tractions have the regularity

Also, we assume that the normal compliance function $p$ verifies

and the surface memory function satisfies

Further details on the contact condition (3.5), normal compliance function $p$ and surface memory function $b$ can be found in 4 or 8 .

We turn now to the variational formulation of Problem $\mathcal{P}_{1}$. To this end, we assume in what follows that ( $\boldsymbol{u},\boldsymbol{\sigma}$ ) are sufficiently regular functions which satisfy (3.1)-(3.6). Let $\boldsymbol{v}\in V$ and $t\in\mathbb{R}_{+}$be given. We use Green’s formula (2.3), equation of equilibrium (3.2), we split the boundary integral over $\Gamma_{1},\Gamma_{2}$ and $\Gamma_{3}$ and, since $\boldsymbol{v}=\mathbf{0}$ on $\Gamma_{1}\times\mathbb{R}_{+}$and $\boldsymbol{\sigma}(t)\boldsymbol{\nu}=\boldsymbol{f}_{2}(t)$ on $\Gamma_{2}\times\mathbb{R}_{+}$, it follows that

Moreover, since $\boldsymbol{\sigma}(t)\boldsymbol{\nu}\cdot\boldsymbol{v}=\sigma_{\nu}(t)v_{\nu}+\boldsymbol{\sigma}_{\tau}(t)\cdot\boldsymbol{v}_{\tau}$ on $\Gamma_{3}$, condition (3.6) implies that $\int_{\Gamma_{3}}\boldsymbol{\sigma}(t)\boldsymbol{\nu}\cdot\boldsymbol{v}da=\int_{\Gamma_{3}}\sigma_{\nu}(t)v_{\nu}da$. We use the contact condition (3.5) to see that

We combine the above relations to deduce that

We use now (3.1) and (3.11) to derive the following variational formulation of the frictionless contact problem (3.1)-(3.6).

Problem $\mathcal{P}_{1}^{V}$. Find a displacement field $\boldsymbol{u}:\mathbb{R}_{+}\rightarrow V$ such that $\boldsymbol{u}(t)\in V$, for all $t\in\mathbb{R}_{+}$, and the equality below holds

Next, we prove the unique weak solvability of the variational problem $\mathcal{P}_{1}^{V}$. To this end we assume that the elasticity tensor $\mathcal{F}$ satisfies the following conditions.

We have the following existence and uniqueness result.
THEOREM 3.1 Assume that (3.13), (3.7)-(3.9) hold. Then, Problem $\mathcal{P}_{1}^{V}$ has a unique solution which satisfies $\boldsymbol{u}\in C\left(\mathbb{R}_{+};V\right)$.

Proof. We start with by providing an equivalent form to Problem $\mathcal{P}_{1}^{V}$. To this end, we use the Riesz representation Theorem to define the operators $P:V\rightarrow V,\mathcal{B}:C\left(\mathbb{R}_{+};V\right)\rightarrow C\left(\mathbb{R}_{+};V\right)$ and the function $\boldsymbol{f}:\mathbb{R}_{+}\rightarrow V$ by equalities

Then, it is easy to see that Problem $\mathcal{P}_{1}^{V}$ is equivalent to the problem of finding a function $\boldsymbol{u}:\mathbb{R}_{+}\rightarrow V$ such that for all $t\in\mathbb{R}_{+}$and $\boldsymbol{v}\in V$ the equality below holds

To solve the variational equation (3.17) we use Corollary 2.2 with $X=V$. To this end, we consider the operator $A:V\rightarrow V$ defined by

Then, for all $t\in\mathbb{R}_{+}$the equality (3.17) can be written as

Using (3.13), (3.8) and the definition of the operator $P$ we deduce that the operator $A$ is strongly monotone and Lipschitz continuous, i.e. it verifies (2.8).

Let $n\in\mathbb{N}^{*}$. Then, a simple calculation based on assumption (3.9) and inequality (2.2) shows that $\forall\boldsymbol{u}_{1},\boldsymbol{u}_{2}\in C\left(\mathbb{R}_{+};V\right),\forall t\in[0,n]$ the following inequality holds:

This inequality implies that the operator $\mathcal{B}$ given by (3.15) satisfies (2.9) with

Finally, using (3.7) and (3.16) we deduce that $\boldsymbol{f}\in C\left(\mathbb{R}_{+};V\right)$ and, therefore, (2.10) holds. It follows now from Corollary 2.2 that there exists a unique function $\boldsymbol{u}\in C\left(\mathbb{R}_{+};V\right)$ which satisfies the equation

And, using (3.15), (3.16) and (3.18) we deduce that there exists a unique solution $\boldsymbol{u}\in C\left(\mathbb{R}_{+};V\right)$ to the equality (3.12) for all $t\in\mathbb{R}_{+}$, which concludes the proof.

Let $\sigma$ be the function defined by (3.1). Then, it follows from (3.13) that $\boldsymbol{\sigma}\in C\left(\mathbb{R}_{+};Q\right)$. Moreover, it is easy to see that (3.11) holds for all $t\in\mathbb{R}_{+}$and, using standard arguments, it results from here that

Therefore, using the regularity $\boldsymbol{f}_{0}\in C\left(\mathbb{R}_{+};L^{2}(\Omega)^{d}\right)$ in (3.7) we deduce that $\operatorname{Div}\boldsymbol{\sigma}\in C\left(\mathbb{R}_{+};L^{2}(\Omega)^{d}\right)$ which implies that $\boldsymbol{\sigma}\in C\left(\mathbb{R}_{+};Q_{1}\right)$. A couple of functions ( $\boldsymbol{u},\boldsymbol{\sigma}$ ) which satisfies (3.1), (3.12) for all $t\in\mathbb{R}_{+}$is called a weak solution to the contact problem $\mathcal{P}_{1}$. We conclude that Theorem 3.1 provides the unique weak solvability of Problem $\mathcal{P}_{1}$. Moreover, the regularity of the weak solution is $\boldsymbol{u}\in C\left(\mathbb{R}_{+};V\right),\boldsymbol{\sigma}\in C\left(\mathbb{R}_{+};Q_{1}\right)$.

We study now the dependence of the solution of Problem $\mathcal{P}_{1}^{V}$ with respect to perturbations of the data. To this end, we assume in what follows that (3.13), (3.7)-(3.9) hold and we denote by $\boldsymbol{u}$ the solution of Problem $\mathcal{P}_{1}^{V}$ obtained in Theorem 3.1. For each $\rho>0$ let $\boldsymbol{f}_{0\rho},\boldsymbol{f}_{2\rho},p_{\rho}$ and $b_{\rho}$ be perturbations of $\boldsymbol{f}_{0},\boldsymbol{f}_{2},p$ and $b$ which satisfy conditions (3.7)-(3.9). We consider the following variational problem.

Problem $\mathcal{P}_{1\rho}^{V}$. Find a displacement field $\boldsymbol{u}_{\rho}:\mathbb{R}_{+}\rightarrow V$ such that $\boldsymbol{u}_{\rho}(t)\in V$, for all $t\in\mathbb{R}_{+}$, and the equality below holds for all $\boldsymbol{v}\in V$ :

Note that, here and below, $u_{\rho\nu}$ represents the normal component of the function $\boldsymbol{u}_{\rho}$.

It follows from Theorem 3.1 that, for each $\rho>0$ Problem $\mathcal{P}_{1\rho}^{V}$ has a unique solution $\boldsymbol{u}_{\rho}\in C\left(\mathbb{R}_{+};V\right)$. Consider now the following assumptions

We have the following convergence result.
THEOREM 3.2 Under assumptions (3.23)-(3.26) the solution $\boldsymbol{u}_{\rho}$ of Problem $\mathcal{P}_{1\rho}^{V}$ converges to the solution $\boldsymbol{u}$ of Problem $\mathcal{P}_{1}^{V}$, i.e.
(3.27) $\boldsymbol{u}_{\rho}\rightarrow\boldsymbol{u}\quad$ in $\quad C\left(\mathbb{R}_{+};V\right)\quad$ as $\quad\rho\rightarrow 0$.

Proof. Let $\rho>0$. We use the Riesz representation Theorem to define the operators $P_{\rho}:V\rightarrow V,\mathcal{B}_{\rho}:C\left(\mathbb{R}_{+};V\right)\rightarrow C\left(\mathbb{R}_{+};V\right)$ and the function $\boldsymbol{f}_{\rho}:\mathbb{R}_{+}\rightarrow V$ by equalities
$(3.28)\left(P_{\rho}\boldsymbol{u},\boldsymbol{v}\right)_{V}=\int_{\Gamma_{3}}p_{\rho}\left(u_{\nu}\right)v_{\nu}da\quad\forall\boldsymbol{u},\boldsymbol{v}\in V$,
$(3.29)\left(\mathcal{B}_{\rho}\boldsymbol{u}(t),\boldsymbol{v}\right)_{V}=\left(\int_{0}^{t}b_{\rho}(t-s)u_{\nu}^{+}(s)ds,v_{\nu}\right)_{L^{2}\left(\Gamma_{3}\right)}\forall\boldsymbol{u}\in C\left(\mathbb{R}_{+};V\right),\boldsymbol{v}\in V$,
$(3.30)\left(\boldsymbol{f}_{\rho}(t),\boldsymbol{v}\right)_{V}=\int_{\Omega}\boldsymbol{f}_{0\rho}(t)\cdot\boldsymbol{v}dx+\int_{\Gamma_{2}}\boldsymbol{f}_{2\rho}(t)\cdot\boldsymbol{v}da\quad\forall\boldsymbol{v}\in V,t\in\mathbb{R}_{+}$.
It follows from the proof of Theorem 3.1 that $\boldsymbol{u}$ is a solution of Problem $\mathcal{P}_{1}^{V}$ iff $\boldsymbol{u}$ solves equality (3.17), for all $t\in\mathbb{R}_{+}$. In a similar way, $\boldsymbol{u}_{\rho}$ is a solution of Problem $\mathcal{P}_{1\rho}^{V}$ iff $\boldsymbol{u}_{\rho}(t)\in V$, for all $t\in\mathbb{R}_{+}$and the following equality:

holds for all $\boldsymbol{v}\in V$.
Let $n\in\mathbb{N}^{*}$ and let $t\in[0,n]$. We take $\boldsymbol{v}=\boldsymbol{u}_{\rho}(t)-\boldsymbol{u}(t)$ in 3.31 and $\boldsymbol{v}=\boldsymbol{u}(t)-\boldsymbol{u}_{\rho}(t)$ in (3.17) and add the resulting equalities to obtain

Next, we use the definitions (3.28) and (3.14), the monotonicity of the function $p_{\rho}$ and assumption (3.26) to see that

Therefore, using the trace inequality (2.2), after some elementary calculus we find that

On the other hand the operator $\mathcal{B}_{\rho}$ verifies (2.9), i.e.

Using trace inequality we obtain

From (3.34) and (3.35) we conclude that