The mathematical modelling of contact phenomena is rather complex and, usually, leads to strongly nonlinear boundary value problems. The reason arise in the fact that, as shown in [ $6,7,13,23,24,29,31$ ], accurate mathematical models need to take into consideration the additional phenomena involved in contact processes. These phenomena are the friction, the heat generation, the wear and the adhesion of contacting surfaces, among others. Wear

is defined as the material loss or change in surface texture occurring when two surfaces of mechanical components contact each other. As the contact process evolves, the contacting surfaces evolve too, via their wear. Wear in sliding systems is often very slow but it is persisting, continuous and cumulative. Its characterization represents one of the basic tasks in the study of machine elements. Indeed, in the process of design of machine elements and tools operating in contact conditions, engineers need to know areas of contact, contact stresses, and they need to predict wear of rubbing elements.

Wear of contact surfaces represents a complex phenomenon. Following [10,25], it is customary to distinguish among the following wear types: adhesive, abrasive, contact fatigue, freeting, oxidation, corrosion and erosion. In terms of the severity of wear on the wearing surfaces, two broad types of wear phenomena have been mentionned in [1]: severe wear and mild wear. Severe wear is characterized by high wear rates, extensive plastic deformation, transfer of material to the harder counter face, and flake-like metallic wear debris. Mild wear, by contrast, is characterized by low wear rates, minimal plastic deformation, formation of a surface film protecting against metal-to-metal contact, and oxide wear debris.

Due to its crucial role in various technological and biomechanical processes, the wear phenomenon subjects of numerous experimental and theoretical studies. For instance, the evolution of wear gaps in fretting problems was studied numerically in [35], by using the finite element method. Numerical simulations of wear shapes due to pitting phenomena for various operating conditions have been investigated in [9], by using arguments of fracture mechanics. A thermoelastic wheel-rail contact problem with wear has been studied in [4]. Numerical methods for wear problems with application to implanted knee joints has been developed in [28]. An original analytical approach to wear was performed in [10]. General models for frictional contact with wear could be find in [36, 37] as well as in the survey [38]. The mathematical analysis of various models of frictional contact with wear, including existence and uniqueness results of the weak solution, was carried out in [11, 12, 26-29].

A new mathematical model which describes the equilibrium of an elastic body in frictional contact with a moving foundation was recently considered in [34]. There, the contact was modeled with a normal compliance condition with unilateral constraint associated to a sliding version of Coulomb’s law of dry friction, and the wear of the foundation was described with a version of Archard’s law. A variational formulation of the problem was derived, in a form of the system which couples a time-dependent equation for the stress field, a time-dependent variational inequality for the displacement field and an integral equation for the wear function. The unique weak solvability of the model was proved, by using arguments on time-dependent variational inequalities and fixed point. This result was completed with a convergence result which shows that the solution of a penalized frictional contact problem with wear converges to the solution of the contact model, as the penalization parameter converges to zero.

The current paper represents a continuation of [34] and contains two main novelties. The first one concerns the mathematical model since, in contrast with [34], we consider here that the deformable body is viscoplastic and we model its behavior with a viscoplastic constitutive law with internal state variable. The analysis of this model could be carried out by using arguments similar to those used in [34], with a different choice of spaces and operators. Nevertheless, we choose to present here a different approach, which consists the second novelty of this paper. Thus, in contrast with [34], we derive a mixed variational formulation of the problem in which the unknowns are the stress field, the displacement field, the internal state variable, the wear function and the Lagrange multiplier, then we prove its unique solvability by using a recent abstract existence result in the study of mixed variational problems, proved in [32]. Mixed variational problems involving Lagrange multipliers have been
used both in analysis and mechanics, in the study of minimization problems. They provide a useful framework in which a large number of problems involving unilateral constraints can be cast and can be solved numerically. Their study is based on arguments on duality, saddle points theory and fixed point. The literature in the field is extensive, see for instance $[3,8,14,18]$ and the references therein. The analysis of various mixed variational problems associated to contact models can be found in [15, 16, 19-21], for instance.

The rest of the manuscript is structured as follows. In Sect. 2 we present the notation and some preliminary material, including a new abstract result, Thereom 2.2. In Sect. 3 we introduce the model of sliding frictional contact with wear and list the assumption on the data. Then, in Sect. 4 we derive its mixed variational formulation. In Sect. 5 we state and prove our main existence and uniqueness result, Theorem 5.1, which provides the unique solvability of the viscoplastic contact problem with wear. Finally, in Sect. 6 we present relevant particular cases of our contact model and we comment on the corresponding existence and uniqueness results. We also provide a comparison between the fixed point method used in [34] and the Lagrange multiplier method used in the current paper. At the best of our knowledge, these two methods represent the main functional methods used in the study of contact problems with unilateral constraints.

Everywhere in this paper we use the notation $\mathbb{R}_{+}$for the set of positive real numbers and $\mathbb{N}$ for the set of positive integers. Given two sets $X$ and $Y$ we use the notation $X\times Y$ for their cartesian product and, ( $x,y$ ) will represent a typical point of the set $X\times Y$. All the vector spaces considered below are real vector spaces and, for a vector space $X$, we use the notation $0_{X}$ for the zero element of $X$. In addition, if ( $X,\|\cdot\|_{X}$ ) and ( $Y,\|\cdot\|_{Y}$ ) are normed spaces, then $\|\cdot\|_{X\times Y}$ represents the norm of the space $X\times Y$ given by

We use similar notation for the product of more than two sets or spaces. For a normed 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 $X$ and, 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 $K$.

Now, assume that $\left(X,\|\cdot\|_{X}\right)$ and $\left(Y,\|\cdot\|_{Y}\right)$ are normed spaces and $\mathcal{S}:C\left(\mathbb{R}_{+};X\right)\rightarrow C\left(\mathbb{R}_{+};Y\right)$. Then, we recall that the operator $\mathcal{S}$ is called a history-dependent operator if the following property holds:

Note that in (2.1) and everywhere below the notation $\mathcal{S}\eta(t)$ represents the value of the function $\mathcal{S}\eta$ at the point $t$, i.e. $\mathcal{S}\eta(t)=(\mathcal{S}\eta)(t)$. The notion of history-dependent operator was introduced in [30] and used in a number of papers, see for instance [31] and the references therein. Such kind of operators arise both in Functional Analysis, Theory of Partial Differential Equations and Solid Mechanics, as well. One of their main properties is given by the following fixed point result.

Theorem 2.1 Let ( $X,\|\cdot\|_{X}$ ) be a Banach space and let $\mathcal{S}:C\left(\mathbb{R}_{+};X\right)\rightarrow C\left(\mathbb{R}_{+};X\right)$ be a history-dependent operator. Then the operator $\mathcal{S}$ has a unique fixed point $\eta^{*}\in C\left(\mathbb{R}_{+};X\right)$.

Note that Theorem 2.1 represents a particular case of a more general result proved in [33]. Its proof is based on the fact that, if $X$ is a Banach space, then $C\left(\mathbb{R}_{+};X\right)$ 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.

We turn now to an abstract result which represents a consequence of Theorem 2.1 and which will be used twice in Sect. 5 of this manuscript. Thus, we assume in what follows that $\left(X,\|\cdot\|_{X}\right)$ is a normed space, ( $Y,\|\cdot\|_{Y}$ ) is a Banach space and $A:X\rightarrow Y$ and $G:\mathbb{R}_{+}\times X\times Y\rightarrow Y$ are given operators, which satisfy the following conditions:

Theorem 2.2 Let ( $X,\|\cdot\|_{X}$ ) be a normed space, ( $Y,\|\cdot\|_{Y}$ ) a Banach space and assume that (2.2)-(2.3) hold. Then, there exists an operator $\mathcal{S}:C\left(\mathbb{R}_{+};X\right)\rightarrow C\left(\mathbb{R}_{+};Y\right)$ such that for all functions $x\in C\left(\mathbb{R}_{+};X\right)$ and $y\in C\left(\mathbb{R}_{+};Y\right)$, equality

holds if and only if

Moreover, the operator $\mathcal{S}:C\left(\mathbb{R}_{+};X\right)\rightarrow C\left(\mathbb{R}_{+};Y\right)$ is a history-dependent operator.
Proof Let $x\in C\left(\mathbb{R}_{+};X\right)$ and consider the operator $\Lambda:C\left(\mathbb{R}_{+};Y\right)\rightarrow C\left(\mathbb{R}_{+};Y\right)$ defined by

for all $\tau\in C\left(\mathbb{R}_{+};Y\right)$ and $t\in\mathbb{R}_{+}$. Note that, using the assumptions (2.2)-(2.3), it follows that the operator $\Lambda$ is well defined. Moreover, it depends on $x$ but, for simplicity, we do not indicate explicitly this dependence.

Let $\tau_{1},\tau_{2}\in C\left(\mathbb{R}_{+};Y\right)$ and let $t\in\mathbb{R}_{+}$. Then, using definition (2.6) and assumption (2.3), we deduce that

This inequality combined with Theorem 2.1 shows that the operator $\Lambda$ has a unique fixed point in $C\left(\mathbb{R}_{+};Y\right)$, denoted $\mathcal{S}x$. Moreover, combining (2.6) with equality $\Lambda(\mathcal{S}x)=\mathcal{S}x$ we deduce that $\mathcal{S}x$ is the unique element of the space $C\left(\mathbb{R}_{+};Y\right)$, which satisfies

This implies the equivalence between equalities (2.4) and (2.5), for all functions $x\in C\left(\mathbb{R}_{+};X\right)$ and $y\in C\left(\mathbb{R}_{+};Y\right)$.

To proceed, let $x_{1},x_{2}\in C\left(\mathbb{R}_{+};X\right),n\in\mathbb{N}$ and $t\in[0,n]$. Then, using (2.8) and taking into account (2.2) and (2.3), we obtain that

Using now the Gronwall argument we deduce that

This inequality shows that (2.1) holds with $s_{n}=L_{G}\left(L_{A}+1\right)e^{nL_{G}}$, which concludes the proof.

Note that Theorem 2.2 is important since it underlies the history-dependence feature of the solution of the implicit integral equation (2.4). It will be usefull in the study of viscoplastic constitutive laws, as explained in Sect. 5.

Next, we recall an existence and uniqueness result for mixed variational problems. To this end, let ( $X,(\cdot,\cdot)_{X},\|\cdot\|_{X}$ ) and ( $Y,(\cdot,\cdot)_{Y},\|\cdot\|_{Y}$ ) be two real Hilbert spaces and we consider two operators $A:X\rightarrow X,\tilde{\mathcal{S}}:C\left(\mathbb{R}_{+};X\right)\rightarrow C\left(\mathbb{R}_{+};X\right)$, a bilinear form $b:X\times Y\rightarrow\mathbb{R}$, two functions $f,h:\mathbb{R}_{+}\rightarrow X$ and a set $\Lambda\subset Y$. We assume that the following conditions hold:

$\Lambda$ is a closed convex unbounded subset of $Y$ that contains $0_{Y}$.

With these data we introduce the following evolutionary problem.
Problem 2.3 Find the functions $u:\mathbb{R}_{+}\rightarrow X$ and $\lambda:\mathbb{R}_{+}\rightarrow\Lambda$ such that

for all $t\in\mathbb{R}_{+}$
The unique solvability of Problem 2.3 is provided in the next theorem.
Theorem 2.4 Assume (2.10)-(2.14). There exists $d_{0}>0$ which depends only on $A$ and $b$ such that, if $\tilde{d}_{n}<d_{0}$ for all positive integers $n$, then Problem 2.3 has a unique solution ( $u,\lambda$ ). Moreover, the solution satisfies $u\in C\left(\mathbb{R}_{+};X\right)$ and $\lambda\in C\left(\mathbb{R}_{+};\Lambda\right)$.

Theorem 2.4 was obtained in [32]. Its proof is based on results on generalized saddle point problems and various estimates, combined with a fixed point argument. The smalness assumption $\tilde{d}_{n}<d_{0}$ in the statement of Theorem 2.4 is needed when using the fixed point argument which follows from the Banach contractions principle.

We end this section with further notation and preliminaries related to the contact model we are interested in. We denote by $\mathbb{S}^{d}(d=1,2,3)$ 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 $\|\kappa\|$ for the Euclidean norm of the element $\kappa\in\mathbb{R}^{m}$, where $m\in\mathbb{N}$, and $\mathbf{0}$ for the zero element of the spaces $\mathbb{R}^{d},\mathbb{S}^{d}$ and $\mathbb{R}^{m}$.

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 $\Omega\cup\Gamma$ and we denote by $\boldsymbol{v}=\left(v_{i}\right)$ the outward unit normal at $\Gamma$. Also, 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 $v_{\nu}=\boldsymbol{v}\cdot\boldsymbol{v},\boldsymbol{v}_{\tau}=\boldsymbol{v}-v_{\nu}\boldsymbol{v}$. 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

As in [32] we consider the space

where $\left.\boldsymbol{v}\right|_{\Gamma_{3}}$ denotes the restriction of the trace of the element $\boldsymbol{v}\in V$ to $\Gamma_{3}$. We recall that $W\subset H^{1/2}\left(\Gamma_{3};\mathbb{R}^{d}\right)$, where $H^{1/2}\left(\Gamma_{3};\mathbb{R}^{d}\right)$ is the space of the restriction on $\Gamma_{3}$ of traces on $\Gamma$ of functions of $H^{1}(\Omega)^{d}$. We denote by $D$ the dual of the space $W$, and by $\langle\cdot,\cdot\rangle_{\Gamma_{3}}$ the duality pairing between $D$ and $W$. Nevertheless, for simplicity, we write $\langle\boldsymbol{\mu},\boldsymbol{v}\rangle_{\Gamma_{3}}$ instead of $\left\langle\boldsymbol{\mu},\left.\boldsymbol{v}\right|_{\Gamma_{3}}\right\rangle_{\Gamma_{3}}$, when $\boldsymbol{\mu}\in D$ and $\boldsymbol{v}\in V$.

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. $\sigma_{\nu}=(\boldsymbol{\sigma}\boldsymbol{v})\cdot\boldsymbol{v}$ and $\boldsymbol{\sigma}_{\tau}=\boldsymbol{\sigma}\boldsymbol{v}-\sigma_{\nu}\boldsymbol{v}$. 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

This inequality will be used in several places, in Sect. 5.

The physical setting is similar to that considered in [34] and can be resumed as follows. A viscoplastic body occupies a bounded domain $\Omega\subset\mathbb{R}^{d}$ with a Lipschitz continuous boundary $\Gamma$, divided into three measurable parts $\Gamma_{1},\Gamma_{2}$ and $\Gamma_{3}$ such that meas $\left(\Gamma_{1}\right)>0$ and, in addition, $\Gamma_{3}$ is plane. The body is subject to the action of body forces of density $\boldsymbol{f}_{0}$. It is fixed on $\Gamma_{1}$ and time-dependent surfaces tractions of density $\boldsymbol{f}_{2}$ act on $\Gamma_{2}$. On $\Gamma_{3}$, the body is in sliding frictional contact with a moving obstacle, the so-called foundation, which is made of a hard material covered by a layer of soft material of thickness $g$. The friction implies the wear of the foundation that we model it with a surface variable, the wear function. Then, the classical formulation of the contact problem is the following.

Problem $\mathcal{P}$ Find a stress field $\boldsymbol{\sigma}:\Omega\times\mathbb{R}_{+}\rightarrow\mathbb{S}^{d}$, a displacement field $\boldsymbol{u}:\Omega\times\mathbb{R}_{+}\rightarrow\mathbb{R}^{d}$, an internal state variable $\kappa:\Omega\times\mathbb{R}_{+}\rightarrow\mathbb{R}^{m}$ and a wear function $w:\Gamma_{3}\times\mathbb{R}_{+}\rightarrow\mathbb{R}$ such that

We now provide a brief description of the equations and conditions in Problem $\mathcal{P}$. Here and below, in order to simplify the notation, we do not indicate explicitly the dependence of various functions on the spatial variable $\boldsymbol{x}$.

First, (3.1) and (3.2) represent the rate-type viscoplastic constitutive law with internal state variable in which we assume that elasticity tensor $\mathcal{E}$ and the constitutive functions $\mathcal{G}$ and $\boldsymbol{G}$ satisfy the following conditions:

Constitutive equations of the form (3.1)-(3.2) could describe both elasticity, plasticity, creep, relaxation, hardening and softening phenomena. For this reason they have been considered in the literature in order to model the behavior of real materials like rubbers, metals, pastes, rocks and so on. Various results and mechanical interpretation concerning constitutive laws of this form may be found in [5] and [17], for instance. Here we restrict ourselves to provide three clasical examples of such equations, with our without internal state variables.

The first example is one-dimensional and does not involve internal state variable. It is of the form

with

Here where $E>0$ is the Young modulus, $k_{1},k_{2}>0$ are viscosity constants, $f$ and $g$ are Lipschitz continuous functions with $g(\varepsilon)<f(\varepsilon)$, and $F_{1},F_{2}:\mathbb{R}_{+}\rightarrow\mathbb{R}$ are increasing functions with $F_{1}(0)=F_{2}(0)=0$. Note that the domain of elastic behavior of the material is characterized by the inequalities $g(\varepsilon)\leq\sigma\leq f(\varepsilon)$. Plastic deformations occur only for $\sigma>f(\varepsilon)$ in extension or for $\sigma<g(\varepsilon)$ in compression. Therefore, since the yield limit (in extension and in compression) depends on the deformation, we conclude that the model (3.14), (3.15) represents a model with hardening.

A second example of an elastic-viscoplastic constitutive law without internal state variable is Perzyna’s law given by

Here $\mathcal{E}$ is a fourth order tensor satisfying (3.11), $\mathcal{E}^{-1}$ denotes its inverse, $\delta>0$ is a viscosity constant, $\mathcal{K}$ is a nonempty, closed, convex set in the space $\mathbb{S}^{d}$ of symmetric tensors and $\mathcal{P}_{\mathcal{K}}$ represents the projection operator. Notice that in this case the function $\mathcal{G}$ does not depend on $\boldsymbol{\varepsilon}$ and is given by

Since $\sigma=\mathcal{P}_{\mathcal{K}}\sigma$ iff $\sigma\in\mathcal{K}$, from (3.16) we see that viscoplastic deformations occur only for the stress tensors $\sigma$ outside the set $\mathcal{K}$. Thus, the set $\mathcal{K}$ represents the domain of elastic behavior of the material. It is usually defined by

where $\mathcal{F}:\mathbb{S}^{d}\rightarrow\mathbb{R}$ is a convex function such that $\mathcal{F}(\mathbf{0})<0$. The function $\mathcal{F}$ is called the yield function and the equation $\mathcal{F}(\boldsymbol{\sigma})=0$ represents the yield condition.

A concrete example of an elastic-viscoplastic constitutive law of the form (3.1), (3.2) is given by the Perzyna’s law with internal state variable,

Here $\mathcal{P}_{\mathcal{K}(\boldsymbol{\kappa})}$ represents the projection mapping on the von Mises convex set $\mathcal{K}(\boldsymbol{\kappa})$ defined by equality

$\boldsymbol{\sigma}^{D}$ being the deviator of $\boldsymbol{\sigma}$, and $\omega:\mathbb{R}\rightarrow\mathbb{R}$ is a given positive function. Note that, as explained in [13], the variable $\kappa$ given by (3.19) represents the irreversible equivalent strain.

Equation (3.3) is the equilibrium equation and we use it here since we assume that the process is quasistatic. Conditions (3.4) and (3.5) are the displacement boundary condition and traction boundary condition, respectively. We assume that the densities of body forces and surface tractions are such that

Conditions (3.6)-(3.8) were introduced and justified in [34] and, for this reason, we do not present here in detail. We restrict ourselves to mention that (3.6) represents the contact condition in which the normal compliance function $p$ satisfies
(a) $p:\Gamma_{3}\times\mathbb{R}\rightarrow\mathbb{R}_{+}$.
(b) There exists $L_{p}>0$ such that

(d) The mapping $\boldsymbol{x}\mapsto p(\boldsymbol{x},r)$ is measurable on $\Gamma_{3}$, for any $r\in\mathbb{R}$.
(e) $p(\boldsymbol{x},r)=0$ for all $r\leq 0$, a.e. $\boldsymbol{x}\in\Gamma_{3}$.

This condition was derived by assuming an additive decomposition of the normal stress into two components which satisfy the Signorini condition in the form with a gap function and the normal compliance contact condition with wear, respectively.

Condition (3.7) represents a sliding version of the classical Coulomb law of dry friction. Here $\eta$ represents the friction coefficient, $\boldsymbol{n}^{*}$ is the unitary vector defined by

where $\boldsymbol{v}^{*}$ is the velocity of the foundation, supposed to be a non vanishing time-dependent function in the plane of $\Gamma_{3}$. This condition was derived under the assumption that the velocity of the foundation $\boldsymbol{v}^{*}(t)$ is large in comparison with the tangential velocity $\dot{\boldsymbol{u}}_{\tau}(t)$. Here, we assume that the coefficient of friction and velocity of the foundation verify two following conditions:

The differential equation (3.8) represents a version of Archard’s law which governs the evolution of the wear function and, again, it was derived under the assumption that the velocity of the foundation $\boldsymbol{v}^{*}(t)$ is large in comparison with the tangential velocity $\dot{\boldsymbol{u}}_{\tau}(t)$. Here

$k$ being the wear coefficient, assumed to be such that

Condition (3.9) represents the initial condition for the wear function and shows that at the initial moment the materials involved in the process are new. Next, (3.10) represent the initial conditions for the rest of the unknowns 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 assume in what follows that these initial data have the regularity

Finally, we assume that

where, we recall, $\tilde{\theta}_{v}=\tilde{\boldsymbol{\theta}}\cdot\boldsymbol{v}$. This assumption concerns only the geometry of the problem and was already used in [2], for instance. It is needed in order to derive a mixed variational formulation to Problem $\mathcal{P}$.