Phenomena of contact between deformable bodies abound in industry and everyday life. Contact of braking pads with wheels, tires with roads, pistons with skirts are just few simple examples. Common industrial processes such as metal forming, metal extrusion, involve contact evolutions. Owing to their inherent complexity, contact
phenomena lead to mathematical models expressed in terms of strongly nonlinear elliptic or evolutionary boundary value problems.

An early attempt to study frictional contact problems within the framework of variational inequalities was made in [3]. An excellent reference on analysis and numerical approximations of contact problems involving elastic materials with or without friction is [7]. The variational analysis of various contact problems, including existence and uniqueness results, can be found in the monographs [4, 6, 15, 22. The state of the art in the field can also be found in the proceedings [10, 18, 29] and in the special issue [21], as well.

To construct a mathematical model which describes a specific contact process we need to precise the material’s behavior and the contact conditions, among others. In this paper we assume that the material is viscoelastic and we describe its behavior with a constitutive law with long memory of the form

Here $\boldsymbol{u}$ denotes the displacement field, $\boldsymbol{\sigma}$ represents the stress, $\boldsymbol{\varepsilon}(\boldsymbol{u})$ is the linearized strain tensor and, finally, $\mathcal{A}$ and $\mathcal{B}$ are the elasticity operator and the relaxation tensor, respectively. Results and mechanical interpretations in the study of viscoelastic materials of the form (1.1) can be found in [3, 17, 28, for instance. The analysis of various contact problems which such kind of materials was provided in [19, 20, 24. There, the unique solvability of the problems was proved by using existence and uniqueness results for evolutionary variational inequalities involving a Volterra-type integral term; fully discrete schemes for the numerical approximation of the models were considered and error estimates were derived; finally, the schemes were implemented on a computer code and numerical simulations were presented. The analysis of models of antiplane frictional contact problems with viscoelastic materials of the form (1.1), including existence, uniqueness and convergence results, was performed in 25.

We turn now to describe some representative contact conditions used in the literature and, to this end, we denote by $u_{\nu}$ and $\sigma_{\nu}$ the normal displacement and the normal stress on the contact surface, respectively.

The so-called normal compliance contact condition describes a deformable foundation. It assigns a reactive normal pressure that depends on the interpenetration of the asperities on the body’s surface and those of the foundation. A general expression for this condition is

where $p$ is a nonnegative prescribed function which vanishes for negative argument. Indeed, when $u_{\nu}<0$ there is no contact and the normal pressure vanishes. When there is contact then $u_{\nu}$ is positive and represents a measure of the interpenetration of the asperities. Then, condition (1.2) shows that the foundation exerts a pressure on the body, which depends on the penetration.

A commonly used example of the normal compliance function $p$ is

Here the constant $c_{\nu}>0$ is the surface stiffness coefficient and $r^{+}=\max\{r,0\}$ denotes the positive part of $r$. A second example is provided by the truncated normal compliance function

where $\alpha$ is a positive coefficient related to the wear and hardness of the surface. In this case the contact condition (1.2) means that when the penetration is too large, i.e., when it exceeds $\alpha$, the obstacle offers no additional resistance to penetration.

The normal compliance contact condition was first introduced in [14] and since then used in many publications, see, e.g., [7, 8, 9, 11] and references therein. The term normal compliance was first introduced in [8, 9]. An idealization of the normal compliance, which is used often in engineering literature, and can also be found in mathematical publications, is the Signorini contact condition, in which the foundation is assumed to be perfectly rigid. It is obtained, formally, from the normal compliance condition (1.2), (1.3), in the limit when the surface stiffness coefficient becomes infinite, i.e., $c_{\nu}\rightarrow\infty$, and thus interpenetration is not allowed. This leads to the idea of regarding contact with a rigid support as a limiting case of contact with deformable support, whose resistance to compression increases. The Signorini contact condition can be stated in the following complementarity form:

This condition was first introduced in 23 and then used in many papers, see, e.g., [22] for further details and references. Assume now that there is an initial gap $g>0$ between the body and the foundation. Then the Signorini contact condition in a form with a gap function is given by

In various situations the reaction of the foundation at the moment $t$ depends on the history of the penetration and, therefore, it cannot be determinate as a function of the current value $u_{\nu}(t)$. In this case one can assume that the normal stress satisfies a condition of the form

in which $b$ represents a given function, the so-called surface memory function. Contact conditions of the form (1.7) have a simple physical interpretation if there are no cycles of contact and separation during the time interval of interest. For instance, assume in what follows that $b$ is a positive function. Moreover, assume that in the time interval
$[0;t]$ there is only penetration (i.e. $u_{\nu}(s)\geq 0$ for all $s\in[0,t]$ ). Then (1.7) shows that the reaction of the foundation at $t$ is towards the body (since $\sigma_{\nu}(t)\leq 0$ ). Also, if in the time interval $[0;t]$ there is separation (i.e. $u_{\nu}(s)<0$ for all $s\in[0,t]$ ) then there is no reaction at the moment $t$ (since $\sigma_{\nu}(t)=0$ ). Now, assume a situation in which $u_{\nu}$ is positive in time interval $\left[0,t_{0}\right]$ and negative on the time interval $\left[t_{0},t\right]$. Then, following (1.7) we have

since the integral on the remaining interval $\left[t_{0},t\right]$ vanishes. Assume, in addition, that the support of the function $b$ is included in the interval $[0,\delta]$ with $\delta>0$. Two possibilities arise. First, if $t-t_{0}>\delta$ it follows that $b(t-s)=0$ for all $s\in\left[0,t_{0}\right]$ and (1.7) shows the normal stress $\sigma_{\nu}(t)$ vanishes. Second, if $t-t_{0}\leq\delta$ (1.7) implies that $\sigma_{\nu}(t)\leq 0$ i.e. a residual pression exists at the moment $t$ on the body’s surface. We interpret this as a memory effect in which the foundation prevents the separation, moves towards the body and exerts a pression on a short interval of time of length $\delta$. Various other mechanical interpretation of the condition (1.7) could be obtained if $b$ is assumed to be a negative function or if this condition is associated to the normal compliance condition (1.2), as shown in Section 3 below. Note that conditions of the form (1.7) were considered in [13] in the study of a lumped model with contact and friction.

In the present paper we study a quasistatic frictionless contact problem for viscoelastic materials of the form (1.1). The novelty consists in the fact that the contact condition we use describes a deformable foundation which becomes rigid when the penetration reaches a critical bound and which developes memory effects. This contact condition includes as particular cases both the normal compliance condition (1.2), the Signorini condition (1.6) and the history-dependent condition (1.7). Considering such condition leads to a new and nonstandard mathematical model which, in a variational formulation, is given by a history-dependent variational inequality for the displacement field. We prove the unique weak solvability of the problem then we establish two convergence results.

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. In Section 4 we list the assumptions on the data and derive the variational formulation of the problem. Then we state and prove our main existence and uniqueness result, Theorem 4.1. In Section 5 we state and prove our first converge result, Theorem 5.1. It states the continuous dependence of the solution with respect to the data. Finally, in Section 6 we state and prove our second converge result, Theorem 6.1. It states that the weak solution of the problem with normal compliance, memory term and unilateral constraint can be approached by the weak solution of a problem with normal compliance and memory term, as the stiffness coefficient of the foundation converges to infinity.

Everywhere in this paper we use the notation $\mathbb{N}^{*}$ for the set of positive integers and $\mathbb{R}_{+}$will represent the set of non negative real numbers, i.e. $\mathbb{R}_{+}=[0,+\infty)$. We denote by $\mathbb{S}^{d}$ the space of second order symmetric tensors on $\mathbb{R}^{d}$ 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

Let $\Omega\subset\mathbb{R}^{d}(d=1,2,3)$ be a bounded domain with Lipschitz continuous boundary $\Gamma$ and let $\Gamma_{1},\Gamma_{2}$, and $\Gamma_{3}$ be three measurable parts 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 $\Omega$ 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 use standard notation for the Lebesgue and Sobolev spaces associated to $\Omega$ and $\Gamma$. In particular, we recall that the inner products on the Hilbert spaces $L^{2}(\Omega)^{d}$ and $L^{2}\left(\Gamma_{3}\right)^{d}$ are given by

and the associated norms will be denoted by $\|\cdot\|_{L^{2}(\Omega)^{d}}$ and $\|\cdot\|_{L^{2}\left(\Gamma_{2}\right)^{d}}$, respectively. Moreover, we consider the spaces

These are real Hilbert spaces endowed with the inner products

and the associated norms $\|\cdot\|_{V},\|\cdot\|_{Q}$ and $\|\cdot\|_{Q_{1}}$, respectively. Here $\boldsymbol{\varepsilon}$ and Div are the deformation and divergence operators 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 $V$ and we denote by $v_{\nu}$ and $\boldsymbol{v}_{\tau}$ the normal and tangential components of $\boldsymbol{v}$ on $\Gamma$ given by $v_{\nu}=\boldsymbol{v}\cdot\boldsymbol{\nu},\boldsymbol{v}_{\tau}=\boldsymbol{v}-v_{\nu}\boldsymbol{\nu}$. By the Sobolev trace theorem, there exists a positive constant $c_{0}$, depending on $\Omega$, $\Gamma_{1}$, and $\Gamma_{3}$, such that

For an regular function $\boldsymbol{\sigma}:\Omega\cup\Gamma\rightarrow\mathbb{S}^{d}$ we denote by $\sigma_{\nu}$ and $\boldsymbol{\sigma}_{\tau}$ the normal and the tangential components of the vector $\boldsymbol{\sigma}\boldsymbol{\nu}$ on $\Gamma$, respectively, and we recall that $\sigma_{\nu}=\boldsymbol{\sigma}\boldsymbol{\nu}\cdot\boldsymbol{\nu},\boldsymbol{\sigma}_{\tau}=\boldsymbol{\sigma}-\sigma_{\nu}\boldsymbol{\nu}$. Moreover, the following Green’s formula holds:

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

We note that $\mathbf{Q}_{\infty}$ is a real Banach space with the norm

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 continuously 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, 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 22 and [12], 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:

Equivalence (2.4) will be used several times in Section 5 of the paper.
Consider now a real Hilbert space $X$ with inner product $(\cdot,\cdot)_{X}$ and associated norm $\|\cdot\|_{X}$. Also, assume given a set $K\subset X$, the operators $A:K\rightarrow X,\mathcal{S}$ : $C\left(\mathbb{R}_{+};X\right)\rightarrow C\left(\mathbb{R}_{+};X\right)$ and a function $f:\mathbb{R}_{+}\rightarrow X$ such that:
$K$ is a closed, convex, nonempty subset of $X$.

We have following result, which represents a particular case of a more general existence and uniqueness result proved in 26 .

Theorem 2.1 Assume that (2.5)-(2.8) hold. Then there exists a unique function $u\in C\left(\mathbb{R}_{+};X\right)$ such that, for all $t\in\mathbb{R}_{+}$, the inequality below holds:

Following the terminology introduced in [26] we refer to (2.9) as a history-dependent variational inequality. To avoid any confusion, we note that here and below the notation $Au(t)$ and $\mathcal{S}u(t)$ are short hand notation for $A(u(t))$ and $(\mathcal{S}u)(t)$, i.e. $Au(t)=A(u(t))$ and $\mathcal{S}u(t)=(\mathcal{S}u)(t)$, for all $t\in\mathbb{R}_{+}$.

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 $\Gamma$, 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 the body is fixed on $\Gamma_{1}$ and surfaces tractions of density $\boldsymbol{f}_{2}$ act on $\Gamma_{2}$. On $\Gamma_{3}$, the body is in frictionless contact with a obstacle, the so-called foundation. We assume that the foundation is deformable and, therefore, the penetration is allowed. Nevertheless, when the penetration reaches a given bound $g$, the foundation becomes rigid. And, finally, there are memory effects during the contact process. The process is quasistatic, and it is studied in the interval of time $\mathbb{R}_{+}=[0,\infty)$. With these assumption, the classical formulation of the problem 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 $t\in\mathbb{R}_{+}$,

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 (3.1) represents the viscoelastic constitutive law of the material introduced in Section 1 and equation (3.2) is the equilibrium equation. Conditions (3.3) and (3.4) are the displacement and traction boundary conditions, respectively, and condition (3.6) 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.

We now describe the contact condition (3.5) in which our main interest is. Here $\sigma_{\nu}$ denotes the normal stress, $u_{\nu}$ is the normal displacement and $p$ is a Lipschitz continuous increasing function which vanishes for a negative argument. Moreover, $b$ is the surface memory function and $g>0$ is a given bound for the normal displacement. This condition can be derived in the following way. First, we assume that the penetration is limited by the bound $g$ and, therefore, at each time moment $t\in\mathbb{R}_{+}$, the normal displacement satisfies the inequality

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

in which the functions $\sigma_{\nu}^{D},\sigma_{\nu}^{R}$ and $\sigma_{\nu}^{M}$ describe the deformability, the rigidity and the memory properties of the foundation, at each $t\in\mathbb{R}_{+}$. We assume that the function $\sigma_{\nu}^{D}$ satisfies the normal compliance contact condition (1.2), that is

The part $\sigma_{\nu}^{R}$ of the normal stress satisfies the Signorini condition in the form with a gap function (1.6), i.e.

And, finally, the function $\sigma_{\nu}^{M}$ satisfies the memory condition (1.7), that is

We combine equalities (3.8), (3.10) and (3.11) to see that

Then we substitute equality (3.12) in (3.10) and use inequality (3.7) to obtain the contact condition (3.5).

Not that (3.5) describes a condition with unilateral constraint, since inequality (3.7) holds at each time moment. Assume now that at a given moment $t$ there is penetration which did not reach the bound $g$, i.e. $0<u_{\nu}(t)<g$. Then (3.5) yields

This equality shows that at the moment $t$, the reaction of the foundation depend 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 in (3.13)). When $b$ is a positive function the reaction of the foundation is larger than that given by the term $p\left(u_{\nu}(t)\right)$ and we conclude that equality (3.13) models the hardening phenomenon of the surface. When $b$ is a negative function the reaction of the foundation is smaller than that given by the term $p\left(u_{\nu}(t)\right)$ and we conclude that equality (3.13) models the softening phenomenon of the surface. Hardening and softening of contact surfaces represent an important phenomenon which appear in various industrial applications applications, see for instance [16] and references therein.

In conclusion, condition (3.5) shows that the contact follows a normal compliance condition with memory term of the form (3.13) 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 normal compliance contact condition with memory term and 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 viscoelastic behaviour, i.e. is deformable, allows penetration and presents memory effects; the contact with this layer is modelled with normal compliance and memory term. 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. To resume, the foundation has a rigid-viscoelastic behavior; its viscoelastic behavior is given by the layer of the soft material while its rigid behavior is given by the hard material.

In the particular case when $b=0$ the contact condition (3.5) was introduced in [5], in the study of a dynamic frictionless contact problem with elastic-visco-plastic
materials. Then, it was used in [1] and [27] in the study of various quasistatic contact problems. Also, note that when $g>0,p=0$ and $b=0$ condition (3.5) becomes the Signorini contact condition in a form with a gap function, (1.7). And, finally, if $b=0$ and $g\rightarrow\infty$, we recover the normal compliance contact condition with a zero gap function, (1.2).

To derive the variational formulation of the problem $\mathcal{P}$ we list the assumptions on the problem data. First, we assume that the elasticity operator $\mathcal{A}$ and the relaxation tensor $\mathcal{B}$ satisfy the following conditions.

The normal compliance and the surface memory function satisfy the conditions

Finally, we assume that the densities of body forces and surface tractions have the regularity

and, moreover, we introduce the set of admissible displacements fields defined by

Assume in what follows that ( $\boldsymbol{u},\boldsymbol{\sigma}$ ) are sufficiently regular functions which satisfy (3.1)-(3.6) and let $\boldsymbol{v}\in U$ and $t>0$ be given. We use the Green formula (2.2) and the equilibrium equation (3.2) to obtain

We split the boundary integral over $\Gamma_{1},\Gamma_{2}$ and $\Gamma_{3}$ and, since $\boldsymbol{v}-\boldsymbol{u}(t)=\mathbf{0}$ on $\Gamma_{1}$ and $\boldsymbol{\sigma}(t)\boldsymbol{\nu}=\boldsymbol{f}_{2}(t)$ on $\Gamma_{2}$, we obtain

Moreover, since

condition (3.6) implies that

We write now

then we use the contact conditions (3.5) and the definition (4.6) of the set $U$ to see that

and, therefore,

We combine now equalities (4.7), (4.8) then we use inequality (4.9) to deduce that

In addition, we note that the boundary condition (3.3), the first inequality in (3.5) and notation (4.6) imply that $\boldsymbol{u}(t)\in U$. Therefore, using the constitutive law (3.1) and inequality (4.10) we derive the following variational formulation of Problem $\mathcal{P}$.

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

In the study of the problems $\mathcal{P}^{V}$ we have the following existence and uniqueness result.

Theorem 4.1 Assume that (4.1) -(4.5) hold. Then, Problem $\mathcal{P}^{V}$ has a unique solution which satisfies $\boldsymbol{u}\in C\left(\mathbb{R}_{+};V\right)$.

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

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

To solve the variational inequality (4.15) we use Theorem 2.1 with $X=V$ and $K=U$. To this end we consider the operator $A:V\rightarrow V$ defined by

It is easy to see that condition (2.5) holds. Next, we use (4.1), (4.3) and (2.1) to see that the operator $A$ verifies condition (2.6). Let $n\in\mathbb{N}^{*}$. Then, a simple calculation based on assumptions (4.2), (4.4) and inequalities (2.1), (2.3) shows that