# A frictional contact problem with wear diffusion

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## Abstract

This paper constructs and analyzes a model for the dynamic frictional contact between a viscoelastic body and a moving foundation. The contact involves wear of the contacting surface and the diffusion of the wear debris. The relationships between the stresses and displacements on the contact boundary are modeled by the normal compliance law and a version of the Coulomb law of dry friction. The rate of wear of the contact surface is described by the differential form of the Archard law. The effects of the diffusion of the wear particles that cannot leave the contact surface on the surface are taken into account. The novelty of this work is that the contact surface is a manifold and, consequently, the diffusion of the debris takes place on a curved surface. The interest in the model is related to the wear of mechanical joints and orthopedic biomechanics where the wear debris are trapped, they diffuse and often cause the degradation of the properties of joint prosthesis and various implants. The model is in the form of a differential inclusion for the mechanical contact and the diffusion equation for the wear debris on the contacting surface. The existence of a weak solution is proved by using a truncation argument and the Kakutani–Ky Fan–Glicksberg fixed point theorem.

## Keywords

Viscoelastic material Coulomb friction Archard wear Diffusion on manifold Variational inequality## Mathematics Subject Classification

47J20 47J22 74M10 74M15 70K75## 1 Introduction

This work studies a nonlinear dynamical model for the process of contact between a viscoelastic body and a reactive foundation when wear debris is generated and diffuses on the contact surface. The model includes subdifferential friction boundary condition, and considerably extends the model and the results in [20], which were announced in [19] and further developed in [9, 10]. Additional information and details can be found in [21]. The research in [20] was motivated, in part, by biomechanical applications. Indeed, such problems arise in artificial joints after arthroplasty (knee, hip, shoulder, elbow, etc.) where debris is produced by articulating parts of the prosthesis and is transported to the bone-implant interface. The debris causes the deterioration of the interface, and is believed to be an important factor leading to prosthesis loosening (see, e.g., [17, 18] and references therein). Thus, there is a considerable interest in modeling such complex contact problems arising in implanted joints. This pertains to both cement-less (the so-called “press-fit”) and cemented implants.

We present a mathematical model for the dynamics of such problems. The contact process is assumed to include friction and wear between a viscoelastic body and a reactive foundation. Contact is described with a generalized compliance condition and friction with a general subdifferential law. We assume that the wear generation process takes place only on a part of the contact surface, and the wear rate is described by a generalized differential Archard condition that allows for the diffusion of the debris on the whole of the contact surface. This is the main novelty in the model. Such phenomena of wear diffusion can be found in many engineering settings, but in mathematical publications on contact and wear, it is tacitly assumed that the wear debris is removed from the surface once they are formed, which is the case some cases, such as car engines where the oil transports the debris away. The only mathematical works (that we are aware of) in which the wear debris remains on the surface and its diffusion is taken into account are [19, 20], but there the contact surface was assumed to be planar. However, in most cases in applications, and those in joint replacements, the surfaces are curved. Moreover, in [19, 20] the authors considered a quasistatic process and a moving foundation.

The novelty of this paper lies in that the process is assumed to be dynamic, the contact surface is a manifold and so we use of surface gradients and the Laplace–Beltrami operator instead of the linear diffusion equation. Also, we use a general nonmonotone subdifferential conditions to model friction, which is an extension of the classical formulation as a variational inequality with a subdifferential in the sense of convex analysis. In addition, the method of proof is new and very different from the usual one based on the use of results for variational inclusions.

The model for the processes consists of two coupled equations: the first is the dynamic equation of motion of a viscoelastic body and it contains a frictional multivalued term. The second one describes the diffusion of the wear debris on the contact surface of the body. Our key result is the theorem on the existence of a weak solution to the problem. In contrast to [9, 10] (where the debris diffusion is modeled, but the contact surface is assumed to be flat) we do not use the Banach fixed point argument, but we base our approach on the Kakutani–Ky Fan–Glicksberg theorem that allows us to remove of the limitations on the constants present in the model at the cost of getting only existence, and not the uniqueness of a solution. In such a way we present a new way to obtain existence results for contact problems with friction and wear diffusion.

We remark here that we do not take into account adhesion effects in the model, and in many contact problems, one should also take into account the process of adhesion that is coupled with friction and wear diffusion. For instance, clinical practice shows that adhesion plays an important role at the bone-implant interface, and for further details we refer to [17, 18] and the references therein.

The main mathematical difficulties of this paper lie in the formulation of the wear diffusion not on a subset of \({\mathbb {R}}^2\), but on a 2D manifold in \({\mathbb {R}}^3\). Similar setup in context of modeling the chemical processes of surfactant adsorption and desorption was considered in [8]. Also, due to the fact we do not impose any smallness condition on the constants in the model, we cannot use the Banach fixed point argument (such as it is done in [14]) that also asserts the solution uniqueness. In our approach, we do not need any assumptions on the smallness of the data, but we obtain only the existence of a weak solution. Due to the rather general assumptions on the nonlinearities appearing in the problem, we are forced to use a truncation, and we first obtain the solutions to the truncated problem. We then obtain the necessary a priori estimates and remove this restriction by passing to the limit with truncation parameter.

The paper is organized as follows. Section 2 describes the ‘classical model’ for the process. We also describe shortly the equation for the wear diffusion on the contact manifold. Section 3 lists the assumptions on the problem data and derives its variational formulation, Problem \(P_{V}\). It is a system coupling an evolutionary differential inclusion for the displacements with a diffusion equation on the curved contact surface for the wear. Our main result, Theorem 3.2, states that under certain reasonable assumptions on the setting and problem data, there exists a solution of the variational problem, which is a weak solution for the ‘classical’ model. The proof of the main existence result is done in Sect. 4, and is based on the approach described above. Finally, Sect. 4 concludes with a short discussion and some open problems for further study.

## 2 The model

We consider a viscoelastic body that occupies a bounded domain \(\Omega \subseteq {\mathbb {R}}^{d}\), \(d=2,3\) that is acted upon by volume forces and surface tractions. Although the case \({\mathbb {R}}^{2}\) is of interest mathematically, in this case the contact surface is a curve and there doesn’t seem to be applied interest in such a case, so we have \(d=3\) in mind. As a result, the body may come in frictional contact with a foundation and, consequently, a part of the contacting surface may undergo wear. The wear particles or debris produced in this process remain on the contact surface and undergo diffusion. Thus, grooves and surface damage occur causing changes in the shape and properties of the contacting surface. We construct a mathematical model for the evolution of the mechanical state of the body during the time interval [0, *T*], where \(0<T<+\infty \). The unknowns in the problem are the displacements and the surface wear function. We refer to [20] for a more thorough discussion and additional details of the process. The main novelty here is that the contact surface is curved, while there and in [9, 10] the contact surface was assumed to be flat, and moreover, here the process is dynamic.

We let \(\Gamma \) denote the boundary of \(\Omega \) that is assumed to be Lipschitz continuous. We assume that \(\Gamma \) consists of three pairwise disjoint sets: \({\overline{\Gamma }}_D\) where the body is held fixed and \(\mu _{d-1}(\Gamma _D)>0\); \({\overline{\Gamma }}_N\) where surface tractions act; and \({\overline{\Gamma }}_C\) that is the potential contact surface, where friction and wear take place. The set \(\Gamma _C\) is assumed to be a \(C^2\) manifold with smooth boundary \(\partial \Gamma _C\). We note here that the assumption \(\mu _{d-1}(\Gamma _D)>0\) is not essential, but it allows to avoid certain technical difficulties, such as the lack of the Korn inequality. We use the notation \(\Omega _T=\Omega \times (0,T)\), \(\Gamma _T=\Gamma \times (0,T)\), and similarly for \(\Gamma _{DT}, \Gamma _{NT}\) and \(\Gamma _{CT}\).

The body is held clamped on \(\Gamma _{D}\) and so the displacement field vanishes there. A volume force of density \(f_0\) acts in \(\Omega _T\) and surface tractions of density \(f_N\) are applied on \(\Gamma _{NT}\). An initial gap function *g* can exist between the potential contact surface \(\Gamma _{C}\) and the foundation and it is measured along the outward normal \(\nu \).

- (H1)
\(a_{ijkl},b_{ijkl}\in L^\infty (\Omega )\),

- (H2)
\(a_{ijkl}=a_{jikl}=a_{klij},\ b_{ijkl}=b_{jikl}=b_{klij}\) for \(i,j,k,l=1,\ldots ,d\),

- (H3)
\(a_{ijkl}\xi _{ij}\xi _{kl} \ge \alpha |\xi |^2,\ b_{ijkl}\xi _{ij}\xi _{kl} \ge 0\) for \(\alpha >0\) and all symmetric matrices \((\xi _{ij})_{i,j=1}^d\).

*u*satisfies the momentum law

*S*be a smooth surface in \({\mathbb {R}}^d\), if

*G*is a smooth function defined in a neighborhood of

*S*, the

*surface or tangent gradient*on

*S*is defined as

*G*on

*S*, recalling that \(\nu \) denotes the unit outer normal vector to

*S*. Thus, the surface gradient at \(x \in S\) is the projection of the gradient at

*x*onto the tangent plane to

*S*at

*x*. Note, that for the above definition of \( \nabla _S G\) to make sense, we need to extend

*G*from

*S*to an open neighborhood in \({\mathbb {R}}^d\), however, such an extension always exists for smooth

*S*and the value of \( \nabla _S G\) does not depend on the choice of the extension (see, e.g., [6]). If we denote the components of the

*surface gradient*by

*surface divergence*of the surface gradient, i.e.,

*S*has a smooth boundary \(\Gamma _S= \partial S\) and denote by \(\nu _S\) the unit outer normal to

*S*on \(\Gamma _S\). Then, Green’s formula on

*S*is given by (see, e.g., [7])

*S*. In our setting, \(S=\Gamma _C\) and \(\Gamma _S=\partial \Gamma _C\). For the sake of somewhat simplified notation we use \( \nabla _\Gamma \) for the gradient and \(\Delta _\Gamma \) for the Laplace–Beltrami operator on \(\Gamma _C\). We use the notion of the Sobolev space \(H^1(S)\) of functions on the manifold

*S*, i.e., functions in \(L^2(S)\) that have their surface gradients belongs to \(L^2(S)^d\), see [2] for the definition and properties of these functions on manifolds without boundary and [1] for manifolds with smooth boundary, which is the case here.

- (H4)
\(h_w:\mathbb {R}^d\times \mathbb {R}^d\rightarrow \mathbb {R}\) is continuous and for some \(C_w>0\), and for every \(u,v\in {\mathbb {R}}^d, \theta \in {\mathbb {R}}\), \(|h_w(u,v)|\leqslant C_w(1+|u|^2+|v|^2)\).

- (H5)
\(h_\nu :{\mathbb {R}}^d \rightarrow {\mathbb {R}}\) is continuous and \(|h_\nu (u)|\leqslant C_\nu (1+|u|)\) for every \(u,v\in {\mathbb {R}}^d, \theta \in {\mathbb {R}}\), for some \(C_\nu >0\).

*j*is a locally Lipschitz function and \(\partial j\) stands for its Clarke subdifferential (see Sect. 3 for details). We suppose that \(h_\tau \) and \(j_\tau \) satisfy

- (H6)
\(h_\tau :{\mathbb {R}}^d\times {\mathbb {R}}^d\times {\mathbb {R}}\rightarrow {\mathbb {R}}_+\) is a continuous function and \(|h_\tau (u,v,\theta )|\leqslant C_\tau (1+|u| + |v|+|\theta |)\) for every \(u,v\in {\mathbb {R}}^d, \theta \in {\mathbb {R}}\), for some \(C_\tau >0\) and

- (H7)
\(j_{\tau }:\Gamma _{C}\times {\mathbb {R}}^d\longrightarrow {\mathbb {R}}\) is a function such that \(j_{\tau }(\cdot ,\xi )\) is measurable on \(\Gamma _{C}\) for every \(\xi \in {\mathbb {R}}^d\), \(j_{\tau }(x,\cdot )\) is locally Lipschitz on \( {\mathbb {R}}^d\) for a.e. \(x\in \Gamma _{C}\) and moreover \(\zeta \cdot \xi \geqslant 0\) for \(\zeta \in \partial j_\tau (x,\xi )\) for all \(\xi \in {\mathbb {R}}^n\) a.e \(x\in \Gamma _C\).

- (H8)
there exist \(c_{1\tau }>0\) such that \(\Vert \partial j_{\tau }(x,\xi )\Vert \leqslant c_{1\tau }\) for every \(\xi \in {\mathbb {R}}^d\) and a.e. \(x\in \Gamma _{C}\).

*r*|, i.e.,

## 3 Variational formulation

We turn to the variational formulation of problem (2.1)–(2.9). To that end, we first introduce the concepts that are needed below, and then the variational formulation. In what follows, \(i,j=1,\ldots , d\) everywhere, the summation convention over repeated indices is used, and an index following a comma indicates a partial derivative.

For a reflexive Banach space *E*, we denote by \({\langle \cdot , \cdot \rangle }_{E^* \times E}\) the duality pairing between the dual space \(E^*\) and *E*. If *E* is a Hilbert space, then the scalar product in *E* is denoted by \((\cdot ,\cdot )_E\). Throughout this paper, we denote by *C* a generic positive constant that depends on the problem data and may change its value form line to line. By \(|\cdot |\) we denote the Euclidean norm in \({\mathbb {R}}^d\) or \({\mathbb {S}}^d\), the space of symmetric \(d\times d\) matrices.

*v*instead of \(\gamma v\).

*N*is any set of measure zero. It is possible to generalize the notion of the Clarke subdifferential to functionals defined on Banach spaces, cf., [4, 5, 14], but for our purposed it is sufficient to consider this definition on \({\mathbb {R}}^d\).

### Problem 3.1

*H*(

*h*) as a collection of the hypotheses \(H(h_w), H(h_\nu )\) and \(H(h_\tau )\).

We are now able to state the main theorem of this paper.

### Theorem 3.2

Assume that \(u_0\in V, v_0\in H\), \(\theta _0\in L^2(\Gamma _C)\), \(f_0\in L^2(0,T;V^*), \ f_2\in L^2(0,T;L^2(\Gamma _N)^d)\), and \(\kappa >0\). Under hypotheses (H1)–(H8) there exists a solution to Problem 3.1.

We conclude that the model (2.1)–(2.9) has a weak or variational solution. The uniqueness of the solution remains an unresolved question.

## 4 Proof of Theorem 3.2

In this section, we prove the existence theorem. The idea of the proof is as follows. First, we decouple the coupled Problem 3.1 by replacing the coupling terms with given functions and introduce truncation operators. We obtain the existence of solutions for the decoupled and truncated problems independently. Then, we apply the Kakutani–Ky Fan–Glicksberg fixed point theorem to show the existence result for the original problem. Finally, we pass to the limit with the truncation parameter. In the proof we always assume (H1)–(H8), and that \(u_0\in V, v_0\in H\), \(\theta _0\in L^2(\Gamma _C)\), \(f_0\in L^2(0,T;V^*), \ f_2\in L^2(0,T;L^2(\Gamma _N)^d)\), and \(\kappa >0\), so we do not repeat these assumptions in the auxiliary lemmas below.

We start by recalling the fixed point theorem.

### Theorem 4.1

(Kakutani–Ky Fan–Glicksberg) Let \(S\subset E\) be a nonempty, compact, and convex set, where *E* is a locally convex Hausdorff topological vector space. Let the set-valued function \(\varphi :S\rightarrow 2^S\) have nonempty, convex values, and let \(\text {Gr}(\varphi ) = \{ (x,y)\in S\, |\ y\in \varphi (x) \} \) be a closed set in the product topology of \(E\times E\). Then, the set \(\{x\in S\mid x\in \varphi (x)\}\) of fixed points of \(\varphi \) is nonempty and compact.

Next, for \(l>0\), we define truncation operators \(N_l:{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d\) and \(M_l:{\mathbb {R}}\rightarrow {\mathbb {R}}\) by

\(\begin{array}{ll} \qquad N_l(x)={\left\{ \begin{array}{ll} x, &{}\quad |x|\leqslant l, \\ \frac{x}{|x|} l, &{}\quad |x|>l. \end{array}\right. } &{}\quad M_l(x)={\left\{ \begin{array}{ll} x, &{}\quad |x|\leqslant l, \\ \frac{x}{|x|} l, &{}\quad |x|>l. \end{array}\right. } \end{array}\)

The following lemma is straightforward to show, and we present the proof for the sake of completeness.

### Lemma 4.2

Truncation operators \(N_l\) and \(M_l\) are Lipschitz continuous with a constant 1.

### Proof

Now we fix \(l>0\) (large), choose the functions \({\overline{v}}\in {\mathcal {W}}, \ \overline{\xi } \in L^2(0,T;L^2(\Gamma _C)^d)\) and \(\overline{\theta } \in {\mathcal {W}}_\Gamma \), and let \({\overline{u}}\) given by (3.5) using \({\overline{v}}\). Consider now the following two auxiliary problems.

### Problem 4.3

### Problem 4.4

We note that by using the given functions and the truncations, the two problems are uncoupled.

### Lemma 4.5

There exists a unique solution to Problem 4.3

### Proof

For the proof of Lemma we refer to [14]. \(\square \)

### Lemma 4.6

There exists a unique solution to Problem 4.4.

### Proof

For the proof, we refer to classical results on parabolic problems, see, e.g., [13]. \(\square \)

In the next step, we introduce the following coupled, but still truncated problem.

### Problem 4.7

We now show the existence of a solution to Problem 4.7 by using Lemmas 4.5 and 4.6 and the fixed point theorem, Theorem 4.1.

In what follows, we check all the assumption of Theorem 4.1, and summarize the steps in the lemmas. First, we derive the necessary a-priori estimates.

### Lemma 4.8

*v*and \(\theta \) be the solutions of Problems 4.3 and 4.4, respectively. Then, the following estimates hold:

*T*,

*f*, the constants present in (H1)–(H8), and

*l*.

### Proof

*t*) for \(t\in (0,T)\) and choosing appropriate value of \(\varepsilon \) yields

*t*) for \(t\in (0,T)\) we obtain

Next, we define the space \(Z= {\mathcal {W}}\times {\mathcal {W}}_\Gamma \times L^2(0,T;L^2(\Gamma _C)^d)\) and consider the solution operator \(\Lambda :Z \rightarrow 2^Z\), which assigns to a triple \(({\overline{v}},\overline{\theta },\overline{\xi })\) a triple \((v,\theta ,\xi )\), where *v* and \(\theta \) are the solutions of Problems 4.3 and 4.4, respectively, and \(\xi \) is a \(L^2\)-measurable selection out of \(\partial j({\overline{v}}_\tau )\). We have the following lemma.

### Lemma 4.9

### Proof

### Lemma 4.10

\(\Lambda \) has nonempty and convex values.

### Proof

The result follows from the convexity in the definition of the Clarke subdifferential, see, e.g., [4], the existence of \(\xi \) given in Lemma 4.8, and the existence and uniqueness of *v* and \(\theta \) established in Lemmas 4.5 and 4.6. \(\square \)

### Lemma 4.11

\(Gr(\Lambda )\) is sequentially closed in \((w-Z)\times (w-Z)\) topology.

### Proof

We choose three sequences such that \({\overline{v}}_n\rightarrow {\overline{v}}\) weakly in \({\mathcal {W}}\), \(\overline{\theta }_n\rightarrow \overline{\theta }\) weakly in \({\mathcal {W}}_\Gamma \) and \(\overline{\xi }_n \rightarrow \overline{\xi }\) weakly in \(L^2(0,T;L^2(\Gamma _C)^d)\). Define \(v_n, \theta _n\) and \(\xi _n\) as, respectively, the solutions of Problems 4.3 and 4.4 corresponding to \({\overline{v}}_n, \overline{\theta }_n, \overline{\xi }_n\), and the \(L^2\) selection of \(\partial j_\tau ({\overline{v}}_{n\tau }(x,t))\). Assume that \(v_n\rightarrow v\) weakly in \({\mathcal {W}}\), \(\theta _n\rightarrow \theta \) weakly in \({\mathcal {W}}_\Gamma \) and \(\xi _n \rightarrow \xi \) weakly in \(L^2(0,T;L^2(\Gamma _C)^d)\). We need to show that *v* and \(\theta \) are the solutions of Problems 4.3 and 4.4 that correspond to \({\overline{v}}, \overline{\theta }\) and \(\overline{\xi }\), and that \(\xi \) is the \(L^2\) selection of \(\partial j_\tau ({\overline{v}}_{\tau }(x,t))\).

*v*and \(\theta \) are the solutions of Problems 4.3 and 4.4 corresponding to \({\overline{v}}, \overline{\theta }\) and \(\overline{\eta }\), we need to write (4.1) and (4.2) for \(v_n\) and \(\theta _n\), and then pass to the limit \(n \rightarrow \infty \). It is clear that \(\theta _n(0) \rightarrow \theta (0)\) weakly in \(L^2(\Gamma _C)\) and \(v_n(0) \rightarrow v(0)\) weakly in

*H*, which implies that \(\theta \) and

*v*satisfy the same initial conditions as \(\theta _n, v_n\). Moreover, the following hold,

*A*and

*G*, we obtain that

The next step is essentially the last one.

### Lemma 4.12

The operator \(\Lambda \) has a fixed point.

### Proof

Consider \(\Lambda |_{B(R_1,R_2,R_3)}\), where \(B(R_1,R_2,R_3)\) is given by Lemma 4.9. It follows from the lemma that \(\Lambda (B(R_1,R_2,R_3))\subset 2^{B(R_1,R_2,R_3)}\). Then, Lemma 4.10 shows that this mapping has nonempty and convex values. From Lemma 4.11 we deduce that \(Gr(\Lambda |_{B(R_1,R_2,R_3)})\) is sequentially closed in the \((w-Z)\times (w-Z)\) topology. Since the topology is weak, we need the following argument to show that this set is closed. But, \(Gr(\Lambda |_{B(R_1,R_2,R_3)})\subset B(R_1,R_2,R_3)\times B(R_1,R_2,R_3)\), which is bounded, closed and convex in the reflexive space \(Z\times Z\); therefore, \(Gr(\Lambda |_{B(R_1,R_2,R_3)})\) is sequentially compact, and so \((w-Z)\times (w-Z)\) is compact and \((w-Z)\times (w-Z)\) is closed. Taking into account Lemma 4.10, the assertion of the lemma follows now directly from Theorem 4.1. \(\square \)

We have shown that all the assumptions of the fixed point theorem, Theorem 4.1, hold true and that establishes the following theorem, which guarantees the existence of a solution of the truncated problem.

### Theorem 4.13

There exists a solution to Problem 4.7.

The last step in the proof of our main theorem is to show that we can remove the truncation operators from Problem 4.7.

### Proof of Theorem 3.2

*l*. To that end, we choose \(\eta =v(t)\) in (4.3) and using again the Cauchy inequality with \(\varepsilon >0\), we obtain

*C*is independent of

*l*. This estimate is crucial for the proof of the theorem.

Thus, the model has at least one solution. The question of uniqueness remains unresolved, but in view of the complexity of the system and its nonlinearities, it is unlikely. Indeed, the uniqueness of solution to Problem 3.1 does not follow from the presented argument, as it does in a case of the Banach fixed point theorem. Moreover, we suspect that uniqueness would require additional smallness assumptions on the data and stronger assumption on the functions \(h_\tau ,h_\nu ,h_w\).

As has been already mentioned, establishing an existence theorem for a purely elastic model is of considerable mathematical interest.

## Notes

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