Phenomena of contact between deformable bodies or between deformable and rigid bodies abound in industry and everyday life. Contact of braking pads with wheels, tires with roads, pistons with skirts are just a few simple examples. Common industrial processes such as metal forming, metal extrusion, involve contact evolutions and, for this reason, a considerable effort has been developed in their modelling, mathematical analysis and numerical solution. Owing to their inherent complexity, contact phenomena lead to nonlinear and nonsmooth mathematical problems.

The aim of this paper is to study two frictionless contact problems for rate-type viscoplastic 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 and $\boldsymbol{\varepsilon}(\boldsymbol{u})$ is the linearized strain tensor. Here $\mathcal{E}$ is a fourth order tensor which describes the elastic properties of the material and $\mathcal{G}$ is a constitutive function which describes its viscoplastic behavior. In (1.1) and everywhere in this paper the dot above a variable represents derivative with respect to the time variable $t$.

Various results, examples and mechanical interpretations in the study of viscoplastic materials of the form (1.1) can be found in [3, 5] and references therein. Displacement-traction boundary value problems with such materials were considered in [5], both in the dynamic and quasistatic case. Quasistatic frictionless and frictional contact problems for materials of the form (1.1) were studied in various papers, see 4 and 13 for a survey. There, various models of contact were stated and they variational analysis, including existence and uniqueness results, was provided. The numerical analysis of the corresponding models can be found in [4] and the references therein. In all the above papers the process of contact was studied in a finite interval of time.

In [14] a quasistatic frictionless contact problem for viscoplastic materials of the form (1.1) was considered. The process was assumed to be quasistatic and the contact was modelled by using the normal compliance condition with infinite penetration. The unique solvability of the solution was obtained by using a fixed point argument. In contrast, in [2] was considered a problem with normal compliance and finite penetration and was proved the unique solvability of the models using new arguments on history-dependent variational inequalities presented in [15]. The present paper represents a continuation of [2] and we consider the problem with normal compliance, finite penetration and memory term. The same contact condition for viscoelastic materials was used in [16]. Also, we state and prove 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. And, finally, we provide numerical simulations which illustrate this convergence.

The rest of the paper is structured as follows. In Section 2 we introduce the notations we shall use as well as some preliminary material. In Section 3 we present the classical formulation of the two contact problems. In Section 4 we list the assumptions on the data and derive the variational formulation of the problems. Then we state and prove the unique weak solvability of each model. In Section 5 we state and prove a converge result, Theorem 5.1. Next, in Section 6 we present the numerical solution of the contact problem with normal compliance restricted by unilateral constraints and memory term. And, finally, in Section 7 some numerical simulations are presented including a numerical validation of the 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 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 $\Omega$ 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 $\boldsymbol{\varepsilon}$ 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 $V$ and we denote by $v_{\nu}$ and $\boldsymbol{v}_{\tau}$ the normal and tangential components of $\boldsymbol{v}$ on $\Gamma$ given by $v_{\nu}=\boldsymbol{v}\cdot\boldsymbol{\nu},\boldsymbol{v}_{\tau}=\boldsymbol{v}-v_{\nu}\boldsymbol{\nu}$. 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 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:

For a normed space ( $X,\|\cdot\|_{X}$ ) we use the notation $C\left(\mathbb{R}_{+};X\right)$ for the space of continuously functions defined on $\mathbb{R}_{+}$with values on $X$, and $C^{1}\left(\mathbb{R}_{+};X\right)$ for the space of continuous differentiable functions defined on $\mathbb{R}_{+}$with values on $X$. For a subset $K\subset X$ we still use the symbols $C\left(\mathbb{R}_{+};K\right)$ and $C^{1}\left(\mathbb{R}_{+};K\right)$ for the set of continuous and continuously differentiable functions defined on $\mathbb{R}_{+}$with values on $K$, respectively.

Let $X$ be a real Hilbert space with inner product $(\cdot,\cdot)_{X}$ and associated norm $\|\cdot\|_{X}$. 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:

We proceed with the following existence and uniqueness result in the study of nonlinear equations involving monotone operators.

Theorem 2.1 Let $X$ be a Hilbert space and let $A:X\rightarrow X$ be a strongly monotone Lipschitz continuous operator. Then, for each $f\in X$ there exists a unique element $u\in X$ such that $Au=f$.

The following result, proved in [15, will be used in Section 4 of this paper.
Theorem 2.2 Assume that (2.3)-(2.6) hold. Then there exists a unique function $u\in C\left(\mathbb{R}_{+};K\right)$ such that for all $t\in\mathbb{R}_{+}$, the inequality below holds:

We have the following consequence of Theorem 2.2.
Corollary 2.3 Let $X$ be a Hilbert space and assume that (2.3)-(2.6) hold. Then there exists a unique function $u\in C\left(\mathbb{R}_{+};X\right)$ such that

Following the terminology introduced in [15] we refer to (2.7) as a history-dependent quasivariational 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 $t\in\mathbb{R}_{+}$.

In this section we present the two problems which describe the frictionless contact process and present the assumption on the data. 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 problem is quasistatic, and we study the contact process in the interval of time $\mathbb{R}_{+}=[0,\infty)$.

In the first problem, unlike in [2], the contact is modelled with normal compliance and memory term in such a way that the penetration is not limited. Under these conditions, the classical formulation of the 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

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 viscoplastic 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. Finally, (3.7) represents the initial
conditions in which $\boldsymbol{u}_{0}$ and $\boldsymbol{\sigma}_{0}$ denote the initial displacement and the initial stress field, respectively.

The function $p$ is Lipschitz continuous, increasing and vanishes for a negative argument, i.e.

In the second problem the contact is again modelled with normal compliance and memory term but in such a way that the penetration is limited and associated to a unilateral constraint. The classical formulation of the problem is the following.

Problem $\mathcal{P}_{2}$. 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

Here $g\geq 0$ is given and $p$ is a function which satisfies (3.8). Conditions (3.9)(3.12) and (3.14) - (3.15) have the same interpretation as in the contact problem $\mathcal{P}_{1}$.

We now present the new contact condition (3.13), condition (3.5) can be presented using similar arguments. It 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 $t\in\mathbb{R}_{+}$. Also, we assume that the function $\sigma_{\nu}^{D}$ satisfies the normal compliance contact condition

Condition (3.18) combined with assumption (3.8) shows that when there is separation between the body and the obstacle (i.e. when $u_{\nu}<0$ ), then the reaction of the foundation vanishes (since $\sigma_{\nu}=0$ ); also, when there is penetration (i.e. when $u_{\nu}\geq$ 0 ), then the reaction of the foundation is towards the body (since $\sigma_{\nu}\leq 0$ ) and it is increasing with the penetration (since $p$ is an increasing function). Finally, we note that in this condition the penetration is not restricted and the normal stress is uniquely determined by the normal displacement.

Condition (3.18) was first introduced in [11, 12] in the study of dynamic contact problems with elastic and viscoelastic materials. The term normal compliance for this condition was first used in (8, 9. A first example of normal compliance function $p$ which satisfies condition (3.8) is

where $r^{+}=\max\{r,0\}$ and $c_{\nu}$ is a positive constant. In this case condition (3.18) shows that the reaction of the foundation is proportional to the penetration and, therefore, (3.8), (3.18) model the contact with a linearly elastic foundation. A second example of normal compliance function $p$ which satisfies condition (3.8) is given by

where $\alpha$ is a positive coefficient related to the wear and hardness of the surface and, again, $c_{\nu}>0$. In this case the contact condition (3.18) means that when the penetration is too large, i.e. when it exceeds $\alpha$, the obstacle backs off and offers no additional resistance to the penetration. We conclude that in this case the foundation has an elastic-perfectly plastic behavior.

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

And, finally, the function $\sigma_{\nu}^{M}$ satisfies the memory condition

in which $b$ represents a given function, the so-called surface memory function. Contact conditions of the form (3.21) 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 (3.21) 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 (3.21) 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 (3.21) shows the normal stress $\sigma_{\nu}(t)$ vanishes. Second, if $t-t_{0}\leq\delta$ (3.21) 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 (3.21) could be obtained if $b$ is assumed to be a negative function.

We combine equalities (3.17), (3.18) and (3.21) to see that

Then we substitute equality (3.22) in (3.20) and use inequality (3.16) to obtain the contact condition (3.13).

In this section we list the assumptions on the data, derive the variational formulations of the problems $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ and then we state and prove their unique weak solvability. To this end we assume that the elasticity tensor $\mathcal{E}$ and the constitutive function $\mathcal{G}$ satisfy the following conditions.

The surface memory function satisfies

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

In the study of Problem $\mathcal{P}_{1}$ we assume that the initial data satisfy

and, finally, in the study of Problem $\mathcal{P}_{2}$ we assume that

where $U$ denotes the set of admissible displacements defined by

In the rest of the section we denote by $c$ a positive generic constant that may depend on time and whose value may change from line to line. Also, we use the symbol " $\longrightarrow$ " to denote the weak convergence in the Hilbert space $V$.

We turn now to the variational formulation of the problems $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$. To this end, we use Riesz’s representation Theorem to define the operator $P:V\rightarrow V$ and the function $\boldsymbol{f}:\mathbb{R}_{+}\rightarrow V$ by equalities

Assume in what follows that ( $\boldsymbol{u},\boldsymbol{\sigma}$ ) are sufficiently regular functions which satisfy (3.1)-(3.7) and let $t>0$ be given. We integrate equation (3.1) with the initial conditions (3.7) to obtain

Then we use the Green formula (2.2), the equilibrium equation (3.2), the boundary conditions (3.3)-(3.6) and notation (4.8)-(4.9) to see that

We present the following existence and uniqueness result proved in 2 .
Lemma 4.1 Assume that (4.2) and (4.5) hold. Then, for each function $\boldsymbol{u}\in C\left(\mathbb{R}_{+};V\right)$ there exists a unique function $\mathcal{S}_{1}\boldsymbol{u}\in C\left(\mathbb{R}_{+};Q\right)$ such that

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

Using the operator $\mathcal{S}_{1}:C\left(\mathbb{R}_{+};V\right)\rightarrow C\left(\mathbb{R}_{+};Q\right)$ defined in Lemma 4.1 we deduce that (4.10) and (4.11) are equivalent with

We use again Riesz’s representation Theorem to define the operator $\mathcal{S}:C\left(\mathbb{R}_{+};V\right)\rightarrow C\left(\mathbb{R}_{+};V\right)$ by equality

and we obtain the following variational formulation of the Problem $\mathcal{P}_{1}$.
Problem $\mathcal{P}_{1}^{V}$. Find a displacement field $\boldsymbol{u}:\mathbb{R}_{+}\rightarrow V$ and a stress field $\boldsymbol{\sigma}:\mathbb{R}_{+}\rightarrow Q$, such that

hold, for all $t\in\mathbb{R}_{+}$.
In the study of the problem $\mathcal{P}_{1}^{V}$ we have the following existence and uniqueness result.

Theorem 4.2 Assume that (3.8) and (4.1)-(4.5) hold. Then, Problem $\mathcal{P}_{1}^{V}$ has a unique solution, which satisfies

Proof. We define the operator $A:V\rightarrow V$ by equality

With this notation we consider the problem of finding a function $\boldsymbol{u}:\mathbb{R}_{+}\rightarrow V$ such that, for all $t\in\mathbb{R}_{+}$, the following equality holds

To solve (4.19) we employ Corollary 2.3 with $X=V$. We use (4.1), (3.8) and (2.1) to see that the operator $A$ verifies condition (2.4), i.e. it is strongly monotone and Lipschitz continuous. In addition, using (4.3) we note that the operator $\mathcal{S}$ satisfies condition (2.5) with

for every $n\in\mathbb{N}$.
Finally, using (4.4) and (4.9) we deduce that $\boldsymbol{f}$ has the regularity expressed in (2.6). It follows now from Corollary 2.3 that there exists a unique function $\boldsymbol{u}\in C\left(\mathbb{R}_{+};V\right)$ which solves the equality (4.19), for any $t\in\mathbb{R}_{+}$.

Based on the results above we deduce the existence of a unique function $\boldsymbol{u}\in C\left(\mathbb{R}_{+};V\right)$ which satisfies (4.16) for any $t\in\mathbb{R}_{+}$. Let $\boldsymbol{\sigma}$ be defined by (4.15). Then it follows that the couple ( $\boldsymbol{u},\boldsymbol{\sigma}$ ) is the unique couple of functions with regularity (4.17) which satisfies (4.15)-(4.16).

Assume now that ( $\boldsymbol{u},\boldsymbol{\sigma}$ ) are sufficiently regular functions which satisfy (3.9)-(3.15) and, again, let $t>0$ be given. Then, using similar arguments as above we obtain the following variational formulation of Problem $\mathcal{P}_{2}$.
Problem $\mathcal{P}_{2}^{V}$. Find a displacement field $\boldsymbol{u}:\mathbb{R}_{+}\rightarrow V$ and a stress field $\boldsymbol{\sigma}:\mathbb{R}_{+}\rightarrow Q$, such that

hold, for all $t\in\mathbb{R}_{+}$.
In the study of the problem $\mathcal{P}_{2}^{V}$ we have the following existence and uniqueness result.

Theorem 4.3 Assume that (3.8), (4.1) -(4.4) and (4.6) hold. Then, Problem $\mathcal{P}_{2}^{V}$ has a unique solution, which satisfies

Proof. We use the Theorem 2.2 with $X=V$ and $K=U$ and arguments similar to those used in the proof of Theorem 4.2.

Everywhere in this section we assume that the function $p$ satisfies condition (3.8) and let $q$ be a function which satisfies

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

Using assumption (5.1) it follows that the function $p_{\mu}$ satisfies condition (3.8), i.e.

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

and, moreover, we note that $P_{\mu}$ is a monotone, Lipschitz continuous operator.
With these notation, we consider the following contact problem.

Problem $\mathcal{P}_{1\mu}$. 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