Research paper
Weak formulations of quasistatic frictional contact problems

https://doi.org/10.1016/j.cnsns.2021.105888Get rights and content

Highlights

  • different variational formulations for quasistatic frictional contact problems can be derived.

  • the displacement field and the strain field play symmetric roles in the structure of the mathematical models of contact.

  • sweeping process arguments can be succesfully used in the study of contact problems.

Abstract

We consider a general mathematical model which describes the quasistatic contact of a deformable body with an obstacle, the so-called foundation. The material’s behaviour is modeled with a visco-elastic-type constitutive law and the contact is described with a general interface law associated to a version of Coulomb’s law of dry friction. We list the assumptions on the data and provide relevant examples of constitutive laws and boundary conditions. Then, we derive two different variational formulations of the model in which the unknowns are the displacement and the strain field, respectively. We prove the equivalence of these formulations. Finally, we use recent arguments of sweeping process in order to obtain the existence of a unique weak solution to the contact model.

Introduction

Contact phenomena arise in industry and everyday life. They are modeled by strongly nonlinear boundary value problems which, usually, do not have classical solutions. For this reason, in the last decades, a considerable effort has been done in the study of variational analysis of different contact models. The literature in the field is extensive. It includes the books [9], [10], [12], [25], [26] and, more recently [7], [27], [28]. There, various models of contact have been considered together with their variational formulations. Then, existence and uniqueness results have been obtained by using various functional arguments, including arguments of monotonicity, convexity, nonsmooth analysis, multivalued analysis and fixed point. The numerical analysis of various models of contact can be found in [12], [13], [15], for instance.

The notion of variational formulation for a contact problem varies from author to author and even from paper to paper. For contact models which have a convex structure, most of the formulations considered in the literature are in a form of a variational inequality in which the unknown is either the displacement or the velocity field. References in the field include [7], [12], [25], [27]. The contact models formulated in terms of locally Lipschitz functions lead to hemivariational inequalities. References in the field are [26], [28], for instance. Weak formulations of contact problems in which the unknown is the stress field are also called dual formulations. Usually, such formulations lead to variational inequalities or inclusions. Some examples have been considered in [27], [29] for the convex case and in [14], [28] for the nonconvex case. There, existence, uniqueness and equivalence results have been obtained.

Sweeping processes are differential inclusions governed by the normal cones of a family of convex moving sets. Introduced in early seventy’s in the pioneering works of Moreau [18], [19], [20], sweeping processes have been intensively studied in the last decades, as illustrated in [8], [11], [16], [17] and, more recently, in [6], [21], [24], [30]. Arguments of sweeping process in the variational analysis of mathematical models of contact have been considered in [1], [2], [22], [23]. There, abstract existence and uniqueness results for various classes of sweeping processes have been obtained by using the properties of history-dependent operators. Then, these results have been used in the study of frictionless or frictional contact models with viscoelastic materials. The sweeping process considered in [1], [2] was formulated in terms of displacement while the sweeping process considered in [22], [23] was formulated in terms of the strain field.

The aim of this current paper is twofold. The first one is to establish two different variational formulations for quasistatic frictional contact problems with viscoelastic materials and to prove their equivalence. Thus, we consider a general class of contact problems with a convex structure for which we provide a first variational formulation in which the unknown is the displacement field and a sweeping process formulation in which the unknown is the strain field. Deriving these formulations and proving their equivalence show that the displacement field and the strain field play symmetric roles in the structure of the mathematical models of contact, which represents the first trait of novelty of this paper. Our second aim is to deduce existence and uniqueness results for the corresponding contact problems and, to this end, we use a sweeping process argument. At the best of our knowledge this represents the second trait of novelty of this paper.

The rest of the paper is organized as follows. In Section 2 we present preliminary material needed in the rest of the paper. In Section 3 we introduce the general contact model considered and list the assumptions on the data. Then we provide examples of constitutive laws and boundary conditions which satisfy these assumptions. Section 4 is devoted to the weak formulations of the contact models while Section 5 provides their equivalence. The unique weak solvability of the models is presented in Section 6. We end this paper with Section 7 in which we present some concluding remarks.

Section snippets

Preliminaries

In this section we introduce some notation and preliminary material. The notation we introduce here will be used everywhere in the next sections, associated to particular choices of spaces and operators. All the function spaces we consider in this paper are real spaces, even if we do not mention it explicitly.

Function spaces in Contact Mechanics. Everywhere below d{2,3} and Sd stands for the space of second order symmetric tensors on Rd. Moreover ``·, · and 0 represent the inner product,

The contact model

In this section we introduce a general mathematical model describing the mechanical state of a deformable body that occupies the domain Ω, in the time interval of interest I. The body is fixed on the part Γ1 of its boundary, is acted upon by traction forces on Γ2 and is in potential contact on Γ3 with an obstacle, the so-called foundation. The model we consider contains as particular cases several models studied in the literature. It is based on the following mechanical assumptions: the

Two weak formulations

In this section we derive two variational formulation for Problem P. To this end, besides assumptions (3.8), (3.9), (3.10) and (3.11) discussed in the previous section, we assume that the density of applied forces and the initial displacement have the regularityf0C(I;L2(Ω)d).f2C(I;L2(Γ2)d).u0V.We shall keep these assumptions everywhere in this section, even if we do not mention it explicitly. Now, we consider the functions j:L2(Γ3)×VR and f:IV defined byj(θ,v)=Γ3θvνda+Γ3Fb(θ)vτdaforallθ

An equivalence result

We start this section with the following preliminary result.

Lemma 5.1

Assume (3.11), (4.1), (4.2) and let ω, σQ, θL2(Γ3), tI such thatωNΣ(θ,t)(σ).Then, there exists a unique element uV such that ω=ε(u) and, moreover,(σ,ε(v)ε(u))+j(θ,v)j(θ,u)(f(t),vu)VforallvV.

Proof

First, we note that inclusion (5.1) implies thatσΣ(θ,t),(τσ,ω)Q0forallτΣ(θ,t).

Let zɛ(V) where, here and below, M represents the orthogonal of the set M in Q. Then (z,ɛ(v))Q=0 for all vV, which implies that σ±zΣ(θ,t). Therefore,

Existence and uniqueness results

We start this section with the following preliminary result which completes the statement of Lemma 4.1.

Lemma 6.1

Assume (3.11), (4.1), (4.2). Then, the multivalued mapping Σ:L2(Γ3)×I2Q satisfies assumption (K) on the spaces X=Q and Y=L2(Γ3).

Proof

Assume that θ1,θ2L2(Γ3), t1,t2I and zQ and denoteσ1=PΣ(θ1,t1)z,σ2=PΣ(θ2,t2)z.We use (2.17) to see that zPΣ(θ1,t1)zNΣ(θ1,t1)(PΣ(θ1,t1)z) and, therefore, (6.1) implies that zσ1NΣ(θ1,t1)σ1. Next, Lemma 5.1 implies that there exists a unique element u1V such thatσ

Conclusion

In this paper we considered a general frictional contact problem for viscoelastic materials. The constitutive law we used includes as particular cases various constitutive laws used in the literature, as the Kelvin-Voigt constitutive law, for instance. The contact condition presented here is very general, too, and includes as particular cases the normal compliance condition in a form with a gap function and the normal damped response condition. Friction was described with a version of Coulomb’s

CRediT authorship contribution statement

Mircea Sofonea: Conceptualization, Methodology, Writing - original draft. Yi-bin Xiao: Writing - review & editing.

Acknowledgements

This work has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No. 823731 CONMECH. It also was supported by the National Natural Science Foundation of China (72033002, 11771067, 11971003), the Applied Basic Project of Sichuan Province (2019YJ0204) and the Fundamental Research Funds for the Central Universities (ZYGX2019J095).

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