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 material’s behavior with a constitutive law of the form

where $\boldsymbol{u}$ denotes the displacement field, $\boldsymbol{\sigma}$ represents the stress tensor, $\boldsymbol{\varepsilon}(\boldsymbol{u})$ is the linearized strain tensor and $\boldsymbol{\kappa}$ denotes an internal state variable. 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 (1.1) and everywhere in this paper the dot above a variable represents the derivative with respect to the time variable $t$. Following [3, 7], the internal state variable $\boldsymbol{\kappa}$ is a vector-valued function whose evolution is governed by the differential equation

in which $\boldsymbol{G}$ is a nonlinear constitutive function with values in $\mathbb{R}^{m},m$ being a positive integer.

Various results, examples and mechanical interpretations in the study of viscoplastic materials of the form (1.1), (1.2) can be found in [3, 7, and the references therein. Quasistatic contact problems for such materials have been considered in 5, 6) and the references therein. There, the contact was assumed to be frictionless and was modelled with normal compliance; the unique weak solvability of the corresponding problems was proved by using arguments of nonlinear equations with monotone operators and fixed point; semi-discrete and fully discrete scheme were considered, error estimates and convergence results were proved and numerical simulation in the study of two-dimensional test problems were presented. The normal compliance contact condition was first introduced in [14] and since then used in many publications, see, e.g., 9, 10, 11, 13] and references therein. The term normal compliance was first introduced in [10, 11.

In the particular case without internal state variable the constitutive equation (1.1) reads

and was used in the literature in order to model the behaviour of various materials like rubbers, rocks, metals, pastes and polymers. Quasistatic frictionless contact problems for materials of the form (1.3) have been considered in [1, 6, 15, 18] and the references therein, under various contact conditions. In [6, 15] both the Signorini and the normal compliance condition were used which, recall, describe a contact with a rigid and elastic foundation, respectively. In [1, 18] the contact was modelled with normal compliance and unilateral constraint condition. This condition, introduced for the first time in [8], models an elastic-rigid behavior of the foundation.

With respect to the papers above mentioned, the current paper has three traits of novelties that we describe in what follows. First, the model we consider involves a contact condition with normal compliance, unilateral constraint and memory term. This condition takes into account both the deformability, the rigidity, and the memory effects of the foundation. Second, in contrast with the short note [4], we model the behavior of the material with a viscoplastic constitutive law with internal state variable. And, finally, we study the contact process 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 three ingredients above lead to a new and interesting mathematical model. The aim of this work is to prove the unique weak solvability of this model and to study the dependence of the weak solution with respect to the data.

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 converge result, Theorem 5.1. It states the continuous dependence of the solution with respect to the data.

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

Also, we use the notation $\|\boldsymbol{\kappa}\|$ for the Euclidean norm of the element $\boldsymbol{\kappa}\in\mathbb{R}^{m}$. 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 $\boldsymbol{\varepsilon}$ 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 $\boldsymbol{v}$ 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 denote by $\mathbf{Q}_{\infty}$ the space of fourth order tensor fields given by

and we recall 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 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 [2] and [12], 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 following fixed-point result will be used in Section 4 of the paper.
Theorem 2.1 Let ( $X,\|\cdot\|_{X}$ ) be a real Banach space and let $\Lambda:C\left(\mathbb{R}_{+};X\right)\rightarrow C\left(\mathbb{R}_{+};X\right)$ be a nonlinear operator with the following property: there exists $c>0$ such that

for all $u,v\in C\left(\mathbb{R}_{+};X\right)$ and for all $t\in\mathbb{R}_{+}$. Then the operator $\Lambda$ has a unique fixed point $\eta^{*}\in C\left(\mathbb{R}_{+};X\right)$.

Theorem 2.1 represents a simplified version of Corollary 2.5 in [16. We underline that in (2.5) and below, the notation $\Lambda\eta(t)$ represents the value of the function $\Lambda\eta$ at the point $t$, i.e. $\Lambda\eta(t)=(\Lambda\eta)(t)$.

Consider now a real Hilbert space $X$ with inner product $(\cdot,\cdot)_{X}$ and associated norm $\|\cdot\|_{X}$ as well as a normed space $Y$ with norm $\|\cdot\|_{Y}$. Let $K$ be a subset of $X$ and consider the operators $A:K\rightarrow X,\mathcal{R}:C\left(\mathbb{R}_{+};X\right)\rightarrow C\left(\mathbb{R}_{+};Y\right)$ as well as the functions $\varphi:Y\times K\rightarrow\mathbb{R},f:\mathbb{R}_{+}\rightarrow X$ such that:
$K$ is a nonempty closed convex subset of $X$.

The following result, proved in [17], will be used in Section 4 of this paper.
Theorem 2.2 Assume that (2.6)-(2.10) 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:

Following the terminology introduced in [17] we refer to an operator which satisfies condition (2.8) as a history-dependent operator. Moreover, (2.11) represents a historydependent quasivariational inequality.

Finally, assume that $X$ and $Y$ represent two real Hilbert spaces with the inner products $(\cdot,\cdot)_{X}$ and $(\cdot,\cdot)_{Y}$, and associated norms $\|\cdot\|_{X}$ and $\|\cdot\|_{Y}$, respectively. Then, we denote by $X\times Y$ the product of these spaces. We recall that $X\times Y$ is a real Hilbert space with the canonical inner product $(\cdot,\cdot)_{X\times Y}$ defined by

The associated norm of the space $X\times Y$, denoted $\|\cdot\|_{X\times Y}$, satisfies the inequality

This inequality will be used several times in Sections 4 and 5 of this manuscript.

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 $\boldsymbol{u}:\Omega\times\mathbb{R}_{+}\rightarrow\mathbb{R}^{d}$, a stress field $\boldsymbol{\sigma}:\Omega\times\mathbb{R}_{+}\rightarrow\mathbb{S}^{d}$ and an internal state variable $\boldsymbol{\kappa}:\Omega\times\mathbb{R}_{+}\rightarrow\mathbb{R}^{m}$ such that

for all $t\in\mathbb{R}_{+}$, there exists $\xi:\Gamma_{3}\times\mathbb{R}_{+}\rightarrow\mathbb{R}$ which satisfies

for all $t\in\mathbb{R}_{+}$and, moreover,

Here and below, in order to simplify the notation, we do not indicate explicitly the dependence of various functions on the spatial variable $\boldsymbol{x}$. Equations (3.1), (3.2) represent the rate-type viscoplastic constitutive law with internal state variable introduced in Section (1. Equation (3.3) represents the equation of equilibrium in which Div denotes the divergence operator for tensor valued functions. Conditions (3.4) and (3.5) are the displacement boundary condition and the traction boundary condition, respectively. Condition (3.6) is the frictionless condition and it shows that the tangential stress on the contact surface vanishes. Finally, (3.8) represents the initial conditions in which $\boldsymbol{u}_{0},\boldsymbol{\sigma}_{0},\boldsymbol{\kappa}_{0}$ denote the initial displacement, the initial stress field and the initial state variable, respectively.

We now describe the contact condition (3.7) in which our main interest is. 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, $b$ is a positive function and $g>0$. This condition can be derived in the following way. Let $t\in\mathbb{R}_{+}$be a given time moment. 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 $\sigma_{\nu}^{D}(t),\sigma_{\nu}^{R}(t)$ and $\sigma_{\nu}^{M}(t)$ describe the deformability, the rigidity and the memory properties of the foundation. 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.

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

on $\Gamma_{3}$. We combine (3.10), (3.11) and denote $-\sigma_{\nu}^{M}(t)=\xi(t)$ to see that

Then we substitute equality (3.14) in (3.12) and use (3.9), (3.13) to obtain the contact condition (3.6).

We now present additional details of the contact condition (3.7). The inequalities and equalities below in this section are valid in an arbitrary point $\boldsymbol{x}\in\Gamma_{3}$. First, we recall that (3.7) describes a condition with unilateral constraint, since inequality (3.9) holds at each time moment. Next, assume that at a given moment $t$ there is penetration which did not reach the bound $g$, i.e. $0<u_{\nu}(t)<g$. Then (3.7) 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 $p\left(u_{\nu}(t)\right)$ ) as well as on the history of the penetration (represented by the integral term in (3.15)). Assume now that at a given moment $t$ there is separation between the body and the foundation, i.e. $u_{\nu}(t)<0$. Then, since $p\left(u_{\nu}(t)\right)=0$, (3.7) shows that $\sigma_{\nu}(t)=0$,
i.e. the reaction of the foundation vanishes. Note that the same behavior of the normal stress is described both in the classical normal compliance condition and in the Signorini contact condition, when separation arises.

In conclusion, condition (3.7) shows that when there is separation then the normal stress vanishes; when there is penetration the contact follows a normal compliance condition with memory term of the form (3.15) 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 unilateral constraint and memory term. 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, as shown in equality (3.15). 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 this section we list the assumptions on the data, derive 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 $\mathcal{E}$ and the constitutive functions $\mathcal{G}$ and $\boldsymbol{G}$ satisfy the following conditions.

The densities of body forces and surface tractions are such that

and the normal compliance function $p$ satisfies

Also, the surface memory function and the initial data verify

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

Assume in what follows that ( $\boldsymbol{u},\boldsymbol{\sigma},\boldsymbol{\kappa}$ ) are sufficiently regular functions which satisfy (3.1)-(3.8) and let $\boldsymbol{v}\in U$ and $t>0$ be given. First, we integrate equations (3.1), (3.2) with the initial conditions (3.8) to obtain

and

Next, we use Green formula (2.2) and the equilibrium equation (3.3) to see that

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

Moreover, since

taking into account the frictionless condition (3.6) we obtain

We write now

then we use the contact conditions (3.7) and the definition (4.8) of the set $U$ to see that

We use (3.7), again, and the hypothesis (4.6) on function $b$ to deduce that

Then we add the inequalities (4.12) and (4.13) and integrate the result on $\Gamma_{3}$ to find that

Finally, we combine (4.11) and (4.14) to deduce that

We gather the results above to obtain the following variational formulation of Problem $\mathcal{P}$.

Problem $\mathcal{P}^{V}$. Find a displacement field $\boldsymbol{u}:\mathbb{R}_{+}\rightarrow U$, a stress field $\boldsymbol{\sigma}:\mathbb{R}_{+}\rightarrow Q$ and an internal state variable $\boldsymbol{\kappa}:\mathbb{R}_{+}\rightarrow L^{2}(\Omega)^{m}$ such that (4.9), (4.10) and (4.15) hold, for all $t\in\mathbb{R}_{+}$.

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

Theorem 4.1 Assume that (4.1) -(4.7) hold. Then, Problem $\mathcal{P}^{V}$ has a unique solution which satisfies