Quasistatic Elastic Contact with Adhesion

Abstract

The aim of this paper is the variational study of the contact with adhesion between an elastic material and a rigid foundation in the quasistatic process where the deformations are supposed to be small. The behavior of this material is modelled by a nonlinear elastic law and the contact is modelled with Signorini's conditions and adhesion. The evolution of bonding field is described by a nonlinear differential equation. We derive a variational formulation of the mechanical problem, and we prove the existence and uniqueness of the weak solution using a theorem on variational inequalities, the theorem of Cauchy-Lipschitz, a lemma of Gronwall, as well as the fixed point of Banach.

1. Introduction

The phenomena of contact with or without friction between deformable bodies or between deformable and rigid bodies abound in industry and everyday life. The contact of the tires with the roads, the shoe with disc of break, pistons with skirts are current examples. Because of the importance of contact process in structural and mechanical systems, a considerable effort has been made in its mathematical modeling, mathematical analysis, and numerical simulations.

Process of adhesion is important in many industrial settings where parts, nonmetallic, are glued together. Recently, composite materials reached prominence, since they are very strong and light, and therefore, of considerable importance in aviation, space exploration, and in the automotive industry. Composite materials may undergo delamination under stress, a process in which different layers debond and move relatively to each other. To model the process when bonding is not permanent and debonding may take place, there is a need to add adhesion to the description of the contact process. For these reasons, adhesive contact between bodies when a glue is added to prevent the surfaces from relative motion, has recently received increased attention in the mathematical literature. In this paper we introduce an internal variable of surface, known as bonding field and denoted in this paper by 𝛽, which describes the fractional density of active bonds on the contact surface. When 𝛽=1 at a point of contact surface, the adhesion is complete and all the bonds are active; when 𝛽=0 all the bonds are inactive, severed, and there is no adhesion; when 0<𝛽<1 the adhesion is partial and only a fraction of the bonds is active. The problems of contact with adhesion were studied by several authors. Significant results on these problems can be found in [1–7] and references therein.

Here, the novelty consists in the introduction of the bonding field into the contact between a material elastic and a rigid foundation where the process is quasistatic and the material is modelled by a nonlinear elastic law. The main contribution of this study lies in the proof of existence and uniqueness of the weak solution of the mechanical problem.

This work is organized as follows. In Section 2 we present some notations and preliminaries. In Section 3 we state the mechanical models of elastic contact with adhesion, list the assumptions on the data of the mechanical problem and deduce its variational formulation. In Section 4 we state and prove the existence of a unique weak solution to the mechanical problem; the proof is based on arguments of evolutionary equations and Banach fixed point.

2. Notations and Preliminaries

In this section, we specify the standard notations used and we remind of some definitions and necessary results for the study of this mechanical problem.

We denote by 𝑆^𝑁 the space of second order symmetric tensors on ℝ^𝑁 (𝑁=2,3) while “⋅” and |⋅| represent the inner product and the Euclidean norm on ℝ^𝑁 and 𝑆^𝑁, respectively. Thus, for every 𝑢,𝑣∈ℝ^𝑁, and 𝜎,𝜏∈𝑆^𝑁 we have𝑢⋅𝑣=𝑢_𝑖𝑣_𝑖,|𝑢|=(𝑢⋅𝑢)^1/2,𝜎⋅𝜏=𝜎_𝑖𝑗𝜏_𝑖𝑗,|𝜎|=(𝜎⋅𝜎)^1/2. (2.1)

Here and below, the indices 𝑖, 𝑗 run between 1, 𝑁 and the summation convention over repeated indices is adopted.

Let Ω⊂ℝ^𝑁 be a bounded domain with a Lipschitz boundary Γ and let 𝜈 denote the unit outer normal on Γ. Moreover, we use also the spaces𝐻=𝑢=𝑢_𝑖/𝑢_𝑖∈𝐿²(Ω),ℋ=𝜎=𝜎_𝑖𝑗/𝜎_𝑖𝑗=𝜎_𝑗𝑖∈𝐿²(Ω,𝐻₁)={𝑢∈𝐻/𝜀(𝑢)∈ℋ},ℋ₁={𝜎∈ℋ/Div𝜎∈𝐻}, (2.2) where 𝜀∶𝐻₁→ℋ, Div∶ℋ→𝐻 are the deformation and the divergence operators, respectively, defined by𝜀(𝑢)=𝜀_𝑖𝑗(𝑢),𝜀_𝑖𝑗(𝑢)=12𝜕_𝑗𝑢_𝑖+𝜕_𝑖𝑢_𝑗,Div𝜎=𝜕_𝑗𝜎_𝑖𝑗. (2.3)

The spaces 𝐻, ℋ, 𝐻₁, and ℋ₁ are real Hilbert spaces endowed with the canonical inner products given by⟨𝑢,𝑣⟩_𝐻=_Ω𝑢_𝑖𝑣_𝑖⟨⟩𝑑𝑥,∀𝑢,𝑣∈𝐻,𝜎,𝜏_ℋ=_Ω𝜎_𝑖𝑗𝜏_𝑖𝑗⟨⟩𝑑𝑥,∀𝜎,𝜏∈ℋ,𝑢,𝑣_𝐻1=⟨𝑢,𝑣⟩_𝐻+⟨𝜀(𝑢),𝜀(𝑣)⟩_ℋ⟨⟩,∀𝑢,𝑣∈𝐻,𝜎,𝜏_ℋ1=⟨𝜎,𝜏⟩_ℋ+⟨Div𝜎,Div𝜏⟩_𝐻,∀𝜎,𝜏∈ℋ. (2.4)

The associated norms are denoted by |⋅|_𝐻, |⋅|_ℋ, |⋅|_𝐻1, and |⋅|_ℋ1, respectively.

Since the boundary Γ is Lipschitz continuous, the unit outward normal vector 𝜈 on the boundary is defined almost everywhere for every vector field 𝑢∈𝐻₁, we also use the notation 𝑢 for the trace of 𝑢 on Γ and we denote by 𝑢_𝜈 and 𝑢_𝜏 the normal and tangential components of 𝑢 on the boundary Γ given by𝑢_𝜈=𝑢⋅𝜈,𝑢_𝜏=𝑢−𝑢_𝜈𝜈. (2.5)

For a regular (say 𝐶¹) stress field 𝜎, the application of its trace on the boundary to 𝜈 is the Cauchy stress vector 𝜎𝜈. We define, similarly, the normal and tangential components of the stress on the boundary Γ by𝜎_𝜈=(𝜎𝜈)⋅𝜈,𝜎_𝜏=𝜎𝜈−𝜎_𝜈𝜈. (2.6)

And we recall that the following Green's formula holds⟨𝜎,𝜀(𝑢)⟩_ℋ+⟨Div𝜎,𝑢⟩_𝐻=_Γ𝜎𝜈𝑢𝑑𝑠,∀𝑢∈𝐻₁. (2.7)

Let Γ₁ be a measurable part of Γ such that meas Γ₁>0 and let 𝑉 be the closed subset of 𝐻₁ defined by𝑉=𝑣∈𝐻₁/𝑣=0onΓ₁. (2.8) Since measΓ₁>0, then Korn's inequality holds, and thus there exists a constant 𝑐>0, depending only on Ω and Γ₁ such that|𝜀(𝑢)|_ℋ≥𝑐|𝑢|_𝐻1∀𝑢∈𝑉. (2.9)

A proof of Korn's inequality may be found in [8, page 79].

We define the inner product over the space 𝑉 by⟨𝑢,𝑣⟩_𝑉=⟨𝜀(𝑢),𝜀(𝑣)⟩_ℋ,∀𝑢,𝑣∈𝑉. (2.10)

It follows from Korn's inequality that |⋅|_𝑉 and |⋅|_𝐻1are equivalent norms on 𝑉. Therefore (𝑉,|⋅|_𝑉) is a real Hilbert space. Moreover, by the Sobolev trace theorem, there exists a positive constant 𝑐>0, depending only on Ω, Γ₁ and Γ₃ such that|𝑣|_𝐿²_(Γ3)^𝑁≤𝑐|𝑣|_𝑉,∀𝑣∈𝑉. (2.11)

Finally, we shall use the notation 𝒬 for the set defined by𝒬=𝛽∈𝐶0,𝑇;𝐿²Γ₃∶0≤𝛽(𝑡)≤1∀𝑡∈[0,𝑇]a.e.onΓ₃,𝛽(0)=𝛽₀. (2.12)

We end this preliminary with the following version of the classical theorem of Cauchy-Lipschitz which can be found in [9, page 60].

Theorem 2.1. Assume that (𝑋,|⋅|_𝑋) is a real Banach space. Let 𝐹(𝑡,⋅)∶𝑋→𝑋 be an operator defined almost everywhere on ]0,𝑇[, satisfying the following conditions:(1)|𝐹(𝑡,𝑢)−𝐹(t,𝑣)|_𝑋≤𝐿_𝐹|𝑢−𝑣|_𝑋∀𝑢,𝑣∈𝑋 a.e. 𝑡∈]0,𝑇[, for some 𝐿_𝐹,(2)𝑡↦𝐹(𝑡,𝑢)∈𝐿^𝑝([0,𝑇];𝑋),∀𝑢∈𝑋, and some 𝑝≥1.Then, for every 𝑢₀∈𝑋, there exists a unique function 𝑢∈𝑊^1,𝑝([0,𝑇];𝑋) such that 𝑢̇(𝑡)((𝑡][,𝑢(0)=𝐹𝑡,𝑢))a.e.𝑡∈0,𝑇=𝑢₀. (2.13)

This theorem will be used in Section 4 to prove the theorem of existence and uniqueness of weak solution of the mechanical problem.

3. Problem Statement and Variational Formulation

We consider an elastic material which occupies a bounded domain Ω⊂ℝ^𝑁 (𝑁=2,3) and assume that its boundary Γ is regular and partitioned into three disjoint measurable parts Γ₁, Γ₂, and Γ₃ such that meas Γ₁>0. Let [0,𝑇] be the time interval of interest, where 𝑇>0. The material is clamped on Γ₁×]0,𝑇[ and therefore the displacement field vanishes there. We also assume that a volume force of density 𝑓₀ acts in Ω×]0,𝑇[ and that a surface traction of density 𝑓₂ acts on Γ₂×]0,𝑇[. On Γ₃×]0,𝑇[ the material may come in contact with arigid foundation. Moreover, the process is quasistatic and the evolution of the bonding field is described by a nonlinear differential equation.

Under these conditions, the formulation of the mechanical problem is the following.

Problem P. Find a displacement field 𝑢∶Ω×[0,𝑇]→ℝ^𝑁, a stress field 𝜎∶Ω×[0,𝑇]→𝑆^𝑁 and a bonding field 𝛽∶Γ₃×[0,𝑇]→[0,1] such that 𝜎(𝑡)(𝜀(𝑢(𝑡][,(𝑡)=𝐹)))inΩ×0,𝑇Div𝜎+𝑓₀(𝑡)=0inΩ×]0,𝑇[,𝑢=0,onΓ₁×]0,𝑇[,𝜎𝜈=𝑓₂,onΓ₂×][𝑢0,𝑇,_𝜈≤0,𝜎_𝜈+𝛾_𝜈𝑅𝑢_𝜈𝛽²≤0,𝜎_𝜈+𝑅𝑢_𝜈𝛽²𝑢_𝜈=0,onΓ₃×][𝜎0,𝑇,_𝜏=0,onΓ₃×][̇𝛾0,𝑇,𝛽=−_𝜈𝛽−𝑅𝑢_𝜈+²−𝜖_𝑎+,onΓ₃×][𝛽(0)0,𝑇,=0,onΓ₃. (3.1)(3.2)(3.3)(3.4)(3.5)(3.6)(3.7)(3.8)

Here (3.1) is the nonlinear elastic constitutive law. Equation (3.2) represents the equilibrium (3.3) and (3.4) are the displacement-traction boundary conditions, respectively. Conditions (3.5) represent the Signorini conditions with adhesion where 𝛾_𝜈 is a given adhesion coefficient and 𝑅 is the truncation operator defined by𝑅(s)=|𝑠|𝐿,if𝑠≥𝐿,𝑠,if<𝐿,−𝐿,if𝑠≤−𝐿. (3.9) Here 𝐿>0 being the characteristic length of the bond, beyond which it does not offer additional traction. The introduction of the operator 𝑅, defined below, is motivated by the mathematical argument but is not restrictive in terms of application, since no restriction on the size of the parameter 𝐿 is made in the sequel. Thus, by choosing 𝐿 very large, we can assume that 𝑅(𝑢_𝜈)=𝑢_𝜈 and, therefore, from (3.5) we recover the contact conditions:𝑢_𝜈≤0,𝜎_𝜈+𝛾_𝜈𝑢_𝜈𝛽²≤0,𝜎_𝜈+𝑢_𝜈𝛽²𝑢_𝜈=0,onΓ₃×]0,𝑇[, (3.10)

it follows from (3.5) that there is no penetration between the material and the foundation, since 𝑢_𝜈≤0 during the process. Also, note that when the bonding vanishes, then the contact conditions (3.5) become the classical Signironi contact conditions with zero gap function, that is,𝑢_𝜈≤0,𝜎_𝜈≤0,𝜎_𝜏=0,𝜎_𝜈𝑢_𝜈=0,onΓ₃×]0,𝑇[. (3.11)

Condition (3.6) represents the frictionless contact and shows that the tangential stress vanishes on the contact surface during the process. Equation (3.7) describes the evolution of the bonding field with given material parameters 𝛾_𝜈 and 𝜖_𝑎. Also, the data 𝛽₀ in (3.8) is the given initial bonding field.

Assumptions. For the variational study of the mechanical problem, we assume that the operator 𝐹 satisfies the following conditions:𝐹∶Ω×𝑆^𝑁⟶𝑆^𝑁suchthat∃𝑚>0suchthat𝐹𝑥,𝜀₁−𝐹𝑥,𝜀₂⋅𝜀₁−𝜀₂≥𝑚𝜀₁−𝜀₂²a.e.𝑥∈Ω∀𝜀₁,𝜀₂∈𝑆^𝑁,∃𝐿>0suchthat𝐹𝑥,𝜀₁−𝐹𝑥,𝜀₂≤𝐿𝜀₁−𝜀₂a.e.𝑥∈Ω∀𝜀₁,𝜀₂∈𝑆^𝑁,Themapping𝑥⟼𝐹(𝑥,𝜀)isLebesguemeasurablea.e.𝑥∈Ω,∀𝜀∈𝑆^𝑁,Themapping𝑥⟼𝐹(𝑥,0)∈ℋ. (3.12a)(3.12b)(3.12c)(3.12d)

We suppose that the adhesion coefficients satisfy𝛾_𝜈∈𝐿^∞Γ₃,𝜖_𝑎∈𝐿²Γ₃,𝛾_𝜈,𝜖_𝑎≥0a.e.onΓ₃. (3.13) And the body forces and surface traction have the regularity:𝑓₀∈𝑊^1,∞(0,𝑇;𝐻),𝑓₂∈𝑊^1,∞0,𝑇;𝐿²Γ₂^𝑁. (3.14) The initial data satisfy𝛽₀∈𝒬. (3.15) We use the convex subset of admissible displacements defined by𝑈=𝑢∈𝑉suchthat𝑢_𝜈≤0onΓ₃. (3.16) It follows from (3.14) and Riesz-Frechet representation theorem that there exists a unique function 𝑓∶[0,𝑇]→𝑉 such that:𝑓(𝑡),𝑣_𝑉=𝑓₀(𝑡),𝑣_𝐻+𝑓₂(𝑡),𝑣_{𝐿2(Γ2)𝑁}∀𝑣∈𝑉,𝑡∈]0,𝑇[. (3.17) Moreover, we note that the conditions (3.14) imply𝑓∈𝑊^1,∞(0,𝑇;𝑉). (3.18) Finally, we define the adhesion functional 𝑗∶𝐿^∞(Γ₃)×𝑉×𝑉→ℝ by𝑗𝛽,𝑢,𝑣=−_Γ3𝛾_𝜈𝛽²−𝑅𝑢_𝜈+𝑣_𝜈𝑑𝑠. (3.19)

3.1. Variational Formulation

By applying Green's formula, and using the equilibrium equation and the boundary conditions, we easily deduce the following variational formulation of the mechanical Problem P.

Problem PV. Find a displacement 𝑢∶[0,𝑇]→𝑉, and a bonding field 𝛽∶[0,𝑇]→𝐿^∞(Γ₃) such that: 𝑢(𝑡)∈𝑈,⟨𝐹(𝜀(𝑢(𝑡))),𝜀(𝑣−𝑢(𝑡))⟩_ℋ+𝑗𝛽(𝑡),𝑢(𝑡),𝑣−𝑢(𝑡)≥𝑓(𝑡),𝑣−𝑢(𝑡)_𝑉][,𝛽̇(𝑡)𝛾∀𝑣∈𝑈,𝑡∈0,𝑇=−_𝜈𝛽(𝑡)−𝑅𝑢_𝜈(𝑡)₊²−𝜖_𝑎+][,𝛽(0)a.e.𝑡∈0,𝑇=𝛽₀. (3.20)(3.21)(3.22)

Note that the variational Problem PV is formulated in terms of displacement and bonding field, since the stress field was eliminated. However, if the solution (𝑢,𝛽) of these variational problems is known, the corresponding stress field 𝜎 can be easily obtained by using the nonlinear elastic law (3.1).

Below in this subsection 𝛽, 𝛽₁, and 𝛽₂ denote various elements of 𝒬, and 𝑢, 𝑢₁, 𝑢₂, and 𝑣 represent elements of 𝑉, and 𝑐>0 represent constants which may depend on Ω, Γ₁, Γ₃, 𝛾_𝜈, and 𝐿. Note that the adhesion functional 𝑗 is linear with respect to the last argument and therefore𝑗𝛽,𝑢,−𝑣=−𝑗𝛽,𝑢,𝑣. (3.23) Next using (3.19), as well as the properties (3.9) of the truncation operator 𝑅, we find𝑗𝛽₁,𝑢₁,𝑢₂−𝑢₁+𝑗𝛽₂,𝑢₂,𝑢₁−𝑢₂≤𝑐_Γ3𝛽₁−𝛽₂𝑢₁−𝑢₂𝑑𝑠. (3.24)

And by the Sobolev theorem trace we obtain𝑗𝛽₁,𝑢₁,𝑢₂−𝑢₁+𝑗𝛽₂,𝑢₂,𝑢₁−𝑢₂≤𝑐𝛽₁−𝛽_2𝐿2(Γ3)𝑢₁−𝑢_2𝑉. (3.25) We now take 𝛽=𝛽₁=𝛽₂ in (3.25) to deduce𝑗𝛽,𝑢₁,𝑢₂−𝑢₁+𝑗𝛽,𝑢₂,𝑢₁−𝑢₂≤0. (3.26) Similar computations, based on the Lipschitz continuity of 𝑅, show that the following inequality also holds𝑗𝛽,𝑢₁,𝑣−𝑗𝛽,𝑢₂,𝑣≤𝑐𝑢₁−𝑢_2𝑉|𝑣|_𝑉. (3.27) We take 𝑢₁=𝑣 and 𝑢₂=0 in (3.26) then we use equality 𝑅(0)=0 and (3.23) to obtain𝑗𝛽,𝑣,𝑣≥0. (3.28) The inequalities (3.25)–(3.28) will be used in various places to prove the theorem of existence and uniqueness of the weak solution.

4. Existence and Uniqueness of Weak Solution

Theorem 4.1. Assume that (3.12a)–(3.15) hold. Then there exists a unique solution (𝑢,𝛽) to Problem PV and it satisfies 𝑢∈𝑊^1,∞(0,𝑇;𝑉),𝛽∈𝑊^1,∞0,𝑇;𝐿^∞Γ₃. (4.1)

A triple (𝑢,𝜎,𝛽) which satisfies (3.1) and (3.20)–(3.22) is called weak solution of the mechanical Problem P. We conclude that under the stated assumptions, the mechanical problem has a unique weak solution. The regularity of the weak solution in terms of stress is given by𝜎∈𝑊^1,∞0,𝑇;ℋ₁. (4.2) Indeed, taking 𝑣=𝜑∈𝐷^∞(Ω) in (3.20) and using (3.1), (3.17) we find: Div𝜎(𝑡)+𝑓₀(𝑡)=0. (4.3) Now (3.14) and (4.3) imply that Div𝜎∈𝑊^1,∞(0,𝑇;𝐻), which in its turn implies 𝜎∈𝑊^1,∞(0,𝑇;ℋ₁).

The proof of Theorem 4.1 will be carried out in several steps. In the first step we consider the following problem in which 𝛽∈𝒬, is given.

Problem 1. Find a displacement 𝑢_𝛽∶[0,𝑇]→𝑉 such that 𝑢_𝛽(𝑡)∈𝑈,𝐹𝜀𝑢_𝛽(𝑡),𝜀𝑣−𝑢_𝛽(𝑡)_ℋ+𝑗𝛽(𝑡),𝑢_𝛽(𝑡),𝑣−𝑢_𝛽(𝑡)≥𝑓(𝑡),𝑣−𝑢_𝛽(𝑡)_𝑉∀𝑣∈𝑈,𝑡∈]0,𝑇[. (4.4) We have the following result.

Lemma 4.2. There exists a unique solution to Problem 1 which satisfies 𝑢_𝛽∈𝐶(0,𝑇;𝑉). (4.5)

Proof. Let 𝑡∈[0,𝑇]. We consider the operator 𝐴_𝑡∶𝑉→𝑉 defined by 𝐴_𝑡𝑢,𝑣_𝑉=⟨𝐹(𝜀(𝑢(𝑡))),𝜀(𝑣)⟩_ℋ+𝑗𝛽(𝑡),𝑢(𝑡),𝑣∀𝑢,𝑣∈𝑉. (4.6)
Let 𝑢₁,𝑢₂∈𝑉. We have 𝐴_𝑡𝑢₁−𝐴_𝑡𝑢₂,𝑢₁−𝑢_2𝑉=𝐹𝜀𝑢₁−𝐹𝜀𝑢₂,𝜀𝑢₁−𝑢_2ℋ−𝑗𝛽,𝑢₁,𝑢₂−𝑢₁−𝑗𝛽,𝑢₂,𝑢₁−𝑢₂. (4.7)
We use (3.26), (3.12a), Korn's inequality and the equivalence of the two norms |⋅|_𝐻1 and |⋅|_𝑉 to show that 𝐴_𝑡𝑢₁−𝐴_𝑡𝑢₂,𝑢₁−𝑢_2𝑉≥𝑐𝑢₁−𝑢₂²_𝑉. (4.8)
The operator 𝐴_𝑡 is, therefore, strongly monotone.
Let 𝑢₁,𝑢₂,𝑣∈𝑉. We have 𝐴_𝑡𝑢₁−𝐴_𝑡𝑢₂,𝑣_𝑉≤𝐹𝜀𝑢₁−𝐹𝜀𝑢₂,𝜀(𝑣)_ℋ+𝑗𝛽,𝑢₁,𝑣−𝑗𝛽,𝑢₂,𝑣. (4.9)
Using Cauchy-Schwartz's inequality, (3.12b) and (3.27), one obtains 𝐴_𝑡𝑢₁−𝐴_𝑡𝑢_2𝑉≤𝑐𝑢₁−𝑢_2𝑉. (4.10)
Therefore, 𝐴_𝑡 is a continuous Lipschitz operator. Since 𝑈 is a nonempty convex closed subset of 𝑉, it follows from the standard results on elliptic variational inequalities that there exists a unique element 𝑢_𝛽, such that 𝑢_𝛽∈𝑈,𝐴_𝑡𝑢_𝛽,𝑣−𝑢_𝛽≥𝑓,𝑣−𝑢_𝛽𝑉∀𝑣∈𝑈. (4.11)
Using (4.6) we get 𝑢_𝛽(𝑡)∈𝑈,𝐹𝜀𝑢_𝛽(𝑡),𝜀𝑣−𝑢_𝛽(𝑡)_ℋ+𝑗𝛽(𝑡),𝑢_𝛽(𝑡),𝑣−𝑢_𝛽(𝑡)≥𝑓(𝑡),𝑣−𝑢_𝛽(𝑡)_𝑉∀𝑣∈𝑈,𝑡∈]0,𝑇[. (4.12)
We now show that 𝑢_𝛽∈𝐶(0,𝑇;𝑉).
Let 𝑡₁, 𝑡₂∈[0,𝑇]. Denote 𝑢_𝛽(𝑡_𝑖), 𝛽(𝑡_𝑖), and 𝑓(𝑡_𝑖) by 𝑢_𝑖, 𝛽_𝑖 and 𝑓_𝑖 (for 𝑖=1,2), respectively. Then we have 𝐹𝜀𝑢₁−𝐹𝜀𝑢₂,𝜀𝑢₁−𝑢_2ℋ≤𝑓₁−𝑓₂,𝑢₁−𝑢_2𝑉+𝑗𝛽₁,𝑢₁,𝑢₂−𝑢₁+𝑗𝛽₂,𝑢₂,𝑢₁−𝑢₂. (4.13)
Using (3.12a) we obtain 𝑚𝜀𝑢₁−𝑢₂²_ℋ≤𝑓₁−𝑓₂,𝑢₁−𝑢_2𝑉+𝑗𝛽₁,𝑢₁,𝑢₂−𝑢₁+𝑗𝛽₂,𝑢₂,𝑢₁−𝑢₂. (4.14)
Using Korn's inequality and the fact that |⋅|_𝑉, |⋅|_𝐻1 are equivalent norms on 𝑉, Cauchy-Schwartz's inequality and (3.25) we get 𝑢₁−𝑢_2𝑉≤𝑐𝑓₁−𝑓_2𝑉+𝛽₁−𝛽_2𝐿2(Γ3). (4.15)
From the previous inequality, the fact that 𝛽∈𝒬 and the regularity of the function 𝑓 given by (3.18), it follows that 𝑢_𝛽∈𝐶(0,𝑇;𝑉).

In the second step we use the displacement field 𝑢_𝛽, obtained in Lemma 4.2 and we consider the following auxiliary problem.

Problem 2. Find a bonding field 𝜃_𝛽∶[0,𝑇]→𝐿^∞(Γ₃) such that 𝜃̇_𝛽(𝑡)=−𝛾_𝜈𝜃_𝛽(𝑡)−𝑅𝑢_𝛽𝜈(𝑡)₊²−𝜖_𝑎+][,𝜃a.e.𝑡∈0,𝑇_𝛽(0)=𝛽₀. (4.16)(4.17) We have the following result.

Lemma 4.3. There exists a unique solution to Problem 2 which satisfies 𝜃_𝛽∈𝑊^1,∞0,𝑇;𝐿^∞Γ₃∩𝒬. (4.18)

Proof. Consider the mapping 𝐹_𝛽∶[0,𝑇]×𝐿^∞(Γ₃)→𝐿^∞(Γ₃) defined by 𝐹_𝛽𝑡,𝜃_𝛽=−𝛾_𝜈𝜃_𝛽(𝑡)−𝑅𝑢_𝛽𝜈(𝑡)₊²−𝜖_𝑎+. (4.19)
It follows from the properties of the truncation operator 𝑅, that 𝐹_𝛽 is Lipschitz continuous with respect to the second argument, uniformly in time. Moreover, for any 𝜃_𝛽∈𝐿^∞(Γ₃), the mapping
𝑡↦𝐹_𝛽(𝑡,𝜃_𝛽) belongs to 𝐿^∞(0,𝑇;𝐿^∞(Γ₃)). Then from Theorem 2.1, we deduce the existence of a unique function 𝜃_𝛽∈𝑊^1,∞(0,𝑇;𝐿^∞(Γ₃)), which satisfies (4.16)-(4.17). The regularity 𝜃_𝛽∈𝒬, follows from (4.16)-(4.17) and assumption 0≤𝛽₀≤1 a.e. on Γ₃. Indeed, (4.16) implies that for a.e. 𝑥∈Γ₃, the function 𝑡→𝜃_𝛽(𝑥,𝑡) is decreasing and its derivative vanishes when 𝛾_𝜈𝜃_𝛽(𝑡)[(−𝑅(𝑢_𝛽𝜈(𝑡)))₊]²≤𝜖_𝑎. Combining these properties with the inequality 0≤𝛽₀≤1 we deduce that 0≤𝜃_𝛽(𝑡)≤1, for all 𝑡∈[0,𝑡], a.e. on Γ₃, which shows that 𝜃_𝛽∈𝒬.

In the third step we denote by 𝑢_𝛽 the solution of Problem 1 and 𝜃_𝛽 the solution of Problem 2, for every 𝛽∈𝒬. Moreover, we define the operator Λ∶𝒬→𝒬 byΛ𝛽=𝜃_𝛽. (4.20)

Lemma 4.4. The operator Λ has a unique fixed point 𝛽^∗.

Proof. We show that for a positive integer 𝑝 the mapping Λ^𝑝 is a contraction on 𝒬. For this, suppose that 𝛽_𝑖 are two functions of 𝒬 and we denote by 𝑢_𝑖, 𝜃_𝑖 the functions obtained in Lemmas 4.2 and 4.3, respectively, for 𝛽=𝛽_𝑖, (𝑖=1,2).
Let 𝑡∈[0,𝑇]. We use (4.4) and (3.25) to deduce that 𝐹𝜀𝑢₁(𝑡)−𝐹𝜀𝑢₂(𝑡),𝜀𝑢₁(𝑡)−𝑢₂(𝑡)_ℋ≤𝑐𝛽₁(𝑡)−𝛽₂(𝑡)_𝐿2(Γ3)𝑢₁(𝑡)−𝑢₂(𝑡)_𝑉. (4.21)
Using the fact that 𝐹 is a strongly monotone, Korn's inequality and |⋅|_𝑉, |⋅|_𝐻1are equivalent norms on 𝑉, we get 𝑢₁(𝑡)−𝑢₂(𝑡)_𝑉≤𝑐𝛽₁(𝑡)−𝛽₂(𝑡)_𝐿2(Γ3), (4.22) which implies ^𝑡₀𝑢₁(𝑠)−𝑢₂(𝑠)_𝑉𝑑𝑠≤𝑐^𝑡₀𝛽₁(𝑠)−𝛽₂(𝑠)_𝐿2(Γ3)𝑑𝑠. (4.23)
On the other hand, it follows from (4.16) and (4.17) that 𝜃_𝑖(𝑡)=𝛽₀−^𝑡₀𝛾_𝜈𝜃_𝑖(𝑠)−𝑅𝑢_𝑖𝜈(𝑠)₊²−𝜖_𝑎+𝑑𝑠𝑖=1,2. (4.24)
And then 𝜃₁(𝑡)−𝜃₂(𝑡)_𝐿2(Γ3)≤𝑐^𝑡₀𝜃₁(𝑠)−𝑅𝑢_1𝜈(𝑠)₊²−𝜃₂(𝑠)−𝑅𝑢_2𝜈(𝑠)₊²_𝐿2(Γ3)𝑑𝑠. (4.25)
Using the definition (3.9) and writing 𝜃₁=𝜃₁−𝜃₂+𝜃₂, we get 𝜃₁(𝑡)−𝜃₂(𝑡)_𝐿2(Γ3)≤𝑐^𝑡₀𝜃₁(𝑠)−𝜃₂(𝑠)_𝐿2(Γ3)𝑑𝑠+𝑐^𝑡₀𝑢_1𝜈(𝑠)−𝑢_2𝜈(𝑠)_𝐿2(Γ3)𝑑𝑠. (4.26)
By Gronwall's inequality and the Sobolev trace theorem, it follows that: 𝜃₁(𝑡)−𝜃₂(𝑡)_𝐿2(Γ3)≤𝑐^𝑡₀𝑢₁(𝑠)−𝑢₂(𝑠)_𝑉𝑑𝑠. (4.27)
Using (4.20) and (4.27), we obtain Λ𝛽₁(𝑡)−Λ𝛽₂(𝑡)_𝐿2(Γ3)≤𝑐^𝑡₀𝑢₁(𝑠)−𝑢₂(𝑠)_𝑉𝑑𝑠. (4.28)
We now combine (4.23) and (4.28) to see that Λ𝛽₁(𝑡)−Λ𝛽₂(𝑡)_𝐿2(Γ3)≤𝑐^𝑡₀𝛽₁(𝑠)−𝛽₂(𝑠)_𝐿2(Γ3)𝑑𝑠. (4.29)
And reiterating this inequality 𝑝 times, yields Λ^𝑝𝛽₁−Λ^𝑝𝛽_{2𝐶(0,𝑇;𝐿2(Γ3))}≤𝑐^𝑝𝑇^𝑝𝛽𝑝!¹−𝛽_{2𝐶(0,𝑇;𝐿2(Γ3))}∀𝑝∈𝑁. (4.30)
Recall that 𝒬 is a nonempty closed set in the Banach space 𝐶(0,𝑇;𝐿²(Γ₃)) and note that inequality (4.30) shows that for 𝑝 sufficiently large Λ^𝑝∶𝒬→𝒬, then Λ has a unique fixed point 𝛽^∗∈𝒬.

Now, the proof of the Theorem 4.1, is a consequence of the previous lemmas.

Proof. Existence. Let 𝛽^∗∈𝒬 be the fixed point of Λ and 𝑢^∗ be the solution of Problem 1 for 𝛽=𝛽^∗, that is, 𝑢^∗=𝑢_𝛽^∗. Arguments similar to those used in proof of (4.22) lead to 𝑢^∗𝑡₁−𝑢^∗𝑡_2𝑉≤𝑐𝛽^∗𝑡₁−𝛽^∗𝑡_2𝐿2(Γ3)∀𝑡₁,𝑡₂∈[0,𝑇]. (4.31)
Since 𝛽^∗=𝜃_𝛽^∗ it follows from Lemma 4.3 that 𝛽^∗∈𝑊^1,∞(0,𝑇;𝐿^∞(Γ₃)) and therefore (4.31) implies that 𝑢^∗∈𝑊^1,∞(0,𝑇;𝑉). From (4.4), (4.16) and (4.17) we conclude that (𝑢^∗,𝛽^∗) is a solution of the Problem PV which satisfies (4.1).
Uniqueness. The uniqueness of the solution is a consequence of the fixed point of the operator Λ defined by (4.20). Indeed, let (𝑢,𝛽) be a solution of Problem PV which satisfies (4.1). Since 𝛽∈𝒬, it follows from (3.20) that 𝑢 is solution to Problem 1, but the Lemma 4.2 implies that this problem has a unique solution denoted 𝑢_𝛽, we get 𝑢=𝑢_𝛽. (4.32)
We put 𝑢=𝑢_𝛽 in (3.21) and using the initial condition (3.22) then we can see that 𝛽 is a solution to Problem 2, but the Lemma 4.3 implies that this last problem has a unique solution denoted 𝜃_𝛽, we get 𝛽=𝜃_𝛽. (4.33)
We use now (4.20) and (4.33) to see that Λ𝛽=𝛽, that is, 𝛽 is a fixed point of the operator Λ. It follows now from Lemma 4.4 that 𝛽=𝛽^∗. (4.34)
The uniqueness of the solution is now a consequence of (4.32) and (4.34).