Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives

1 Introduction

The study of the behavior of many materials, such as polymers, reveals a low-frequency dependence of their damping properties over a wide frequency range. This weak frequency dependence is challenging to describe within the framework of classical viscoelastic models (integer-order derivatives) and without an excessive number of parameters. The challenge is overcome by employing fractional-order operators instead of integer-order operators in the constitutive laws. This substitution leads to a decreased number of parameters required to characterize the material’s properties.

Time-fractional-order viscoelastic models find extensive applications in various fields such as mechanics, chemistry, and engineering [1–3]. For the specific viscoelastic materials, we can find the fractional constitutive model in previous studies [4–6].

The theory of fractional differential equations has been studied in several works, including [7–10] and more recently [11,12]. Models utilizing fractional derivatives and their corresponding modeling can be found in [5,13–15]. The investigation of differential hemivariational inequalities in Banach spaces was initiated [16,17]. In previous studies [18,19], the authors explore another application of partial differential equations (PDEs) involving fractional-order derivatives, specifically the synchronization of fractional-order stochastic systems in finite-dimensional spaces. Numerical illustrations are provided to validate the theoretical findings.

In 2018, Zeng et al. [17,20] addressed a novel category of generalized differential hemivariational inequalities. Their study incorporated the temporal fractional-order derivative operator while considering a frictional contact problem. They employed Rothe’s method to establish the existence of a weak solution to the contact problem. More recently, in the investigation by Bouallala and Essoufi [21], a fractional contact problem with normal compliance and Coulomb’s friction was examined. This research concentrated on the interplay between a thermo-viscoelastic body and a thermally conductive foundation. A more recent contribution in [22] initiated the exploration of a novel frictionless dynamic contact problem model for a viscoelastic body with normal compliance, incorporating the Kelvin-Voigt constitutive law with a time-fractional component.

The study of a contact problem with friction, considering fractional-order derivatives using the penalization method, presents several motivations. Theoretically, it allows for the generalization of classical models and stimulates research and the development of new methods and theories in applied mathematics and numerical analysis. Practically, important motivations include applications in engineering, more accurate predictions of material behavior, and the optimization of industrial processes.

Furthermore, given that the fractional-order derivative generalizes the constitutive laws of rheological models commonly used in linear viscoelasticity, including springs and dampers. The objective of this document is to analyze a frictional contact problem involving a viscoelastic material and a rigid foundation. In this framework, we express the constitutive equation using the fractional Kelvin-Voigt law. The process is quasi-static, the contact is unilateral, and friction is modeled by a version of Coulomb’s law. We introduce a weak formulation of the model, involving a variational inequality for the displacement field with a fractional derivative component in time.

The novelties of this study are as follows: first, the utilization of a time-fractional Kelvin-Voigt constitutive law for the viscosity tensor is as follows:

where α ∈ ( 0 , 1 ] and t ∈ [ 0 , T ] .

The second novelty lies in tackling a novel problem by penalizing the contact law and regularizing friction. This approach transforms the problem into a variational equation that includes a fractional-order derivative in time. Consequently, this penalized formulation manifests as a nonlinear quasi-variational equation involving fractional-order derivatives.

The third innovation of this study is the combination of two major methods, Rothe and finite difference-Galerkin, which are highly effective in the resolution of PDEs, particularly those involving fractional derivatives. Additionally, the former is used for a quasi-variational inequality and the latter for a variational equation, both of which are nonlinear and involve fractional-order derivatives.

We demonstrate the existence of a penalized solution using the Galerkin method for such equations. Among the difficulties encountered in this study are the estimation of certain non-linear terms and the convergence of the two problems. These challenges stem from the combination of fractional-order derivatives and non-linear boundary conditions.

The remaining sections of this article are organized as follows: In Section 2, we present the model of the contact quasistatic process of a viscoelastic body with the fractional Kelvin-Voigt law. We introduce some preliminary material and list assumptions on the problem data. Additionally, we derive the variational formulation of the problem and present the main results concerning the existence of a weak solution. In Section 3, we prove the existence of a weak solution by leveraging the Rothe method and a surjectivity theorem for multivalued pseudomonotone operators. In Section 4, we discuss the existence of a penalty problem and investigate the convergence of the solution as the penalty parameter approaches zero. The proof is based on utilizing the fractional Caputo derivative, implementing the Galerkin method, and employing the compactness method for the Caputo derivative. Finally, in Appendix, we provide a summary of relevant results, including known definitions and properties in nonlinear analysis and fractional calculus.

2 Problem statement and variational formulation

We assume that a viscoelastic body occupies a regular domain Ω of R d , d = 2 , 3 , which will be supposed bounded with a smooth boundary Γ = ∂ Ω . This boundary is divided into three open disjoint parts Γ D , Γ N , and Γ C , such that meas ( Γ D ) > 0 . The interval is denoted as [ 0 , T ] , where T represents a fixed positive value that defines the upper bound of the time interval.

The body is assumed to be clamped in Γ D × ( 0 , T ) and is subjected to a volume force f 1 in Ω × ( 0 , T ) . A density of traction force f N acts on Γ N × ( 0 , T ) . The normalized gap between Γ C and a rigid foundation is denoted by g (Figure 1).

Figure 1

Domain in the initial configuration.

In the following, we use S d to denote the space of second-order symmetric tensors on R d , while “⋅” and ∣⋅∣ will represent the inner product and the Euclidean norm on S d and R d , respectively.

We denote by u : Ω × ( 0 , T ) → R d the displacement field, σ = ( σ i j ) : Ω × ( 0 , T ) → S d the stress tensor, ε ( u ) = ( ε i j ( u ) ) the linearized strain tensor given by ε i j ( u ) = 1 2 ( u i , j + u j , i ) , and div and Div denote the divergence operator for vector-valued and tensor-valued functions, respectively. Specifically, D i v σ = ( σ i j , j ) and div ξ = ( ξ j , j ) .

We represent the normal and tangential components of the displacement field u on Γ as follows:

The normal and tangential components of the stress field σ on the boundary are defined as follows:

respectively, where ν denote the outward normal vector on Γ .

The classical formulation of the fractional contact problem can be expressed in the following manner:

• Problem (P): Find a displacement field u : Ω × ( 0 , T ) ⟶ R d and a stress field σ : Ω × ( 0 , T ) ⟶ S d such that

Equation (1) corresponds to the Caputo-type time-fractional Kelvin-Voigt viscoelastic constitutive law, as described in [23]. Here, ℬ = ( ℬ i j k l ) and V = ( V i j k l ) denote the elastic tensor and viscosity tensor, respectively, both of which are fourth-order tensors. Equation (2) represents the stress equilibrium condition. The relations (3) and (4) represent the mechanical boundary conditions. Additionally, the initial condition is described by equation (5). Relation (6) captures the frictional contact on Γ C with Signorini’s conditions. Furthermore, equation (7) represents Coulomb’s friction, where μ denotes the coefficient of friction.

In the context of a real Banach space X and 1 ≤ p ≤ ∞ , we adopt the conventional notation to represent the spaces L p ( 0 , T ; X ) , C ( 0 , T ; X ) , and W k , p ( 0 , T ; X ) , where k = 1 , 2 , … .

To establish the variational formulation of Problem (P), we will utilize the function spaces:

and

Endowed with the following inner products:

with the associated norm ‖ · ‖ H , ‖ · ‖ H 1 , ‖ · ‖ ℋ , and ‖ · ‖ ℋ 1 .

Taking into account (3), we introduce the following space:

endowed with the inner products and norm given by

The set of admissible displacements is defined as follows:

Given that meas ( Γ D ) > 0 , Korn’s inequality holds.

where c K > 0 is a constant, which depends only on Ω and Γ D .

Furthermore, according to Sobolev’s trace theorem, there exists a positive constant c d that depends solely on Ω and Γ C , such that

For simplicity, let us denote the following bilinear and symmetric operators:

Applying Riesz’s representation theorem, we define the element f ( t ) ∈ V as follows:

Also, we define the mapping j : V × V → R by

In the study of mechanical Problem (P), we impose the following assumptions:

1. 1. The viscosity tensor V : Ω × S d → S d and the elasticity ℬ : Ω × S d → S d exhibit the standard property of symmetry:

      V i j k l = V j i k l = V l k i j ∈ L ∞ ( Ω ) , ℬ i j k l = ℬ j i k l = ℬ l k i j ∈ L ∞ ( Ω ) .

   2. The forms a , b satisfy the property of ellipticity

      a ( u , u ) ≥ m a ‖ u ‖ V 2 , and b ( u , u ) ≥ m b ‖ u ‖ V 2 ,

      where m a , m b > 0 for all u ∈ V .

   3. The operators a and b adhere to the conventional property of boundedness.

      ∣ a ( u , v ) ∣ ≤ M a ‖ u ‖ V . ‖ v ‖ V , and ∣ b ( u , v ) ∣ ≤ M b ‖ u ‖ V . ‖ v ‖ V ,

      where M a , M b > 0 , for all u , v ∈ V .

2. 1. The forces and tractions satisfy the following conditions:

      f 1 ∈ L 2 ( 0 , T ; L 2 ( Ω ) ) , and f N ∈ L 2 ( 0 , T ; L 2 ( Γ N ) d ) .

   2. The gap function and the initial condition fulfill the following conditions:

      g > 0 , g ∈ L ∞ ( Γ C ) , and u 0 ∈ V a d .

3. The coefficient of friction μ : Γ C × R + → R + satisfies

   1. There exists L μ > 0 , for all x , y ∈ R +

      ∣ μ ( ⋅ , x 1 ) − μ ( ⋅ , x 2 ) ∣ < L μ ∣ x 1 − x 2 ∣ a.e. on Γ C .

   2. The mapping z ↦ μ ( z , x ) is measurable on Γ C , for all x ∈ R + .

   3. The mapping z ↦ μ ( z , x ) is μ * -bounded a.e. on Γ C , where

      μ * = sup t ∈ [ 0 , T ] ‖ μ ( t ) ‖ L ∞ ( Γ C ) .

4. The mapping j satisfies

   1. j is measurable on Γ C .

   2. j is locally Lipschitz on Γ C .

   3. j is a proper convex and l.s.c on V .

   4. There exists c j > 0 such that

      ‖ ∂ j ( v ) ‖ V * ≤ c j ( 1 + ‖ v ‖ V ) .

As V is dense in H , the inclusion mapping from ( V , ‖ ⋅ ‖ V ) to ( H , ‖ ⋅ ‖ H ) is continuous and dense. Consequently, we identify H with its dual space H * , and we express V ⊂ H ≡ H * ⊂ V * , with V * being the dual space of V .

By employing a standard procedure relying on Green’s formula, we derive the subsequent variational formulation of (1)–(7):

• Variational Problem (PV) : Find a displacement field u : Ω × ( 0 , T ) ⟶ R d and a stress field σ : Ω × ( 0 , T ) ⟶ S d for all v ∈ V such that

3 Existence result

In this section, we present and demonstrate the existence of the result

The demonstration of Theorem 3.1 hinges upon the application of logic involving the nonlinear operator, the Caputo derivative, and the Rothe method.

Given y ( t ) = D t α 0 C u ( t ) and u ( t ) = I t α 0 y ( t ) + u 0 , inequality (12) can be restated as follows:

Now, let N ∈ N be a fixed integer, and Δ t = δ = T N . We proceed to examine the following approximation of the fractional integral operator I t n α 0 y ( t ) by:

where t k = k δ . Additionally, we define the functional f δ k as follows:

for i = 1 , … , N .

By applying the Rothe method to equation (14), we derive the ensuing fractional Rothe problem:

Fractional Rothe Problem (FRP): Find { y δ k } ⊂ V for k = 1 , … , N such that

where

The subsequent result is as follows: