A time-fractional of a viscoelastic frictionless contact problem with normal compliance

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Eur. Phys. J. Spec. Top. https://doi.org/10.1140/epjs/s11734-023-00962-x

THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS

Regular Article

A time-fractional of a viscoelastic frictionless contact problem with normal compliance Mustapha Bouallala1,2,a , E. L.-Hassan Essoufi2,b , Van Thien Nguyen3,c , and Wei Pang4,d 1 2 3 4

Department of Mathematics and Computer Science, Polydisciplinary Faculty, Modeling and Combinatorics Laboratory, Cadi Ayyad University, B.P. 4162, Safi, Morocco Faculty of Science and Technology, Hassan 1st University Settat Laboratory Mathematics, Computer Science and Engineering Sciences (MISI), 26000 Settat, Morocco Department of Mathematics, FPT University, Education zone, Hoa Lac High Tech Park, Km 29 Thang Long Highway, Thach That ward, Hanoi, Vietnam Center for Applied Mathematics of Guangxi, and Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin 537000, Guangxi, People’s Republic of China Received 15 March 2023 / Accepted 9 August 2023 © The Author(s), under exclusive licence to EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2023 Abstract In this paper, we propose a new model of dynamic frictionless contact problem between a viscoelastic body and a rigid foundation. The constitutive relation is modeled with the fractional Kelvin-Voight law. The contact is described with the normal compliance condition. We derive a weak formulation, and we prove the existence of its weak solution. The proofs are based on the abstract of monotone operator, Caputo derivative, Galerkin method and Banach fixed point theorem.

1 Introduction The theory of monotone operators is a branch of functional analysis that studies the properties of monotone operators in vector spaces. These operators play a crucial role in various areas of applied mathematics, including partial differential equations, nonlinear analysis, and convex optimization. Classic references on the theory of monotone operators include [23–31]. The Caputo derivative is a generalization of the classical derivative, introduced by Italian mathematician Michele Caputo in the 1960s [32]. It is used to describe non-local derivative phenomena such as anomalous diffusion, fractal deformation or non-local behavior in various scientific fields such as physics, engineering and applied sciences, see [9, 13, 18, 21, 22, 36–40, 43]. The Galerkin method is named after mathematician Boris Galerkin, who developed it in the 1920s. Galerkin’s method is a numerical approach used for solving partial differential equations (PDEs). It is based on the variational formulation of the PDE and allows for an approximate solution to be obtained by projecting the PDE onto a finite subspace of test functions [33, 34]. This method finds extensive application in fluid and a

e-mail: [email protected] e-mail: e.h.essoufi@gmail.com c e-mail: [email protected] d e-mail: [email protected] (corresponding author) b

0123456789().: V,-vol

solid mechanics. Banach fixed point theory deals with the existence and properties of fixed points for contracting operators in complete spaces [35, 41, 42]. Fractional calculus has been utilized in biological models to capture long-term memory properties observed in biological systems. One notable example is immunological memory, which enables the immune system to recognize and remember pathogens it has encountered in the past. This memory feature facilitates a quicker and more efficient response to subsequent infections or invasions by the same pathogen, resulting in an enhanced immune defense (see [45, 46]). The analysis and dynamic behavior of a novel dengue model, which encompasses phenomena such as dengue infection transmission with vaccination, treatment, and reinfection, necessitates the application of fractional calculus as described in [47]. We recall that the application of the Caputo and Atangana-Baleanu fractional order derivative to study the complex interactions between CD4+ T cells and HIV viruses is quite recent, as it was introduced in [47]. In the mechanical modeling of gums and rubbers, fractional calculus holds significant importance, particularly when dealing with materials that possess the ability to remember past deformations and exhibit viscoelastic behavior. Indeed, fractional derivation is naturally introduced into them, for fractional Kelvin-Voigt constitutive laws and fractional Maxwell models can be seen in [9, 18, 19].

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The authors of [20, 22] have studied a class of hemivariational inequalities involving the time-fractional order integral operator, using the fractional KelvinVoigt law in which the contact is described by the Clarke subdifferential of a nonconvex and nonsmooth functional. A similar model with a frictionless contact problem can be found in [8]. Very recently, in 2021, Bouallala et al. [3] have initiated the study of a new model of a fractional contact problem with normal compliance and Coulomb’s friction in thermoviscoelasticity. Dynamic contact problems for viscoelastic materials have been studied in [1, 6, 10, 11] and for piezoelectric materials in [1, 14, 16]. Mathematical model which describes the dynamic frictional contact between a thermo-viscoelastic body and a conductive foundation [2]. In 2002 Han et al. [7] presented the solvability results of dynamic and quasistatic of adhesive contact for elastic and viscoelastic material. An excellent reference to dynamic friction contact problems by a general subdifferential condition on the velocity can be found in the proceedings in [4]. The trait of novelty of this paper is that here we study of a dynamic frictionless contact problem for a viscoelastic body with a rigid foundation. We use, below, the Kelvin-Voight constitutive law with time fractional α σ(t) = Bε(C 0 Dt u(t)) + Aε(u(t)),

C 0

 Dtα u(t) − Div σ(t) = f0 (t).

(1.1)

(1.2)

The contact is modeled with the normal compliance. We derive a variational formulation, and We establish the existence of a weak solution to the model. The outline of this paper is as follows: The second section provides a description of the model for the dynamic frictionless contact process between a viscoelastic body and a rigid foundation with timefractional behavior. In the third section, we present the notations, we list the assumptions on the problem’s data, we derive the variational formulation and we start our main result. The proof of the existence result is presented in the fourth section, where it is carried out in several steps. The paper concludes in the fifth section with a summary of the results and observations obtained. Finally, in the Appendix, we recall some necessary definitions and results that are useful for the proof of the main result.

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Below is the physical formulation of the fractional viscoelastic contact problem. A viscoelastic material is under consideration, occupying the domain Ω ⊂ Rd , d = 1, 2, 3, in its reference configuration. It is assumed that the domain Ω is bounded with a smooth boundary denoted ∂Ω = Γ. This boundary is divided into three disjoint measurable parts: ΓD , ΓN , and ΓC , with the condition meas(ΓD ) > 0. The time interval considered is [0, T ], where T > 0. The body is clamped on ΓD × (0, T ). A volume force with density f0 acts in Ω × (0, T ), while a surface traction with density f1 acts on ΓN × (0, T ). On ΓC × (0, T ) the body can come into contact with an obstacle refereed to as the foundation. Only the outward normal vector ν is measured in this context. Throughout the following discussions, we denote Sd as the space of second-order symmetric tensors on Rd . The symbols “·” and · represent the inner product and the Euclidean norm, respectively, on Sd and Rd . This means that for any u, v ∈ Rd and σ, τ ∈ Sd , we have the following definitions. 1

u · v =ui vi , vRd = (v, v) 2

and 1 2

where α ∈ (0, 1] is the material constant and t ∈ [0, T ]. The physical framework of this study is established upon the following considerations: the generalization of behavior laws from rheological models employed in linear viscoelasticity, which incorporate springs and dampers [48]. Furthermore, fractional models allow for the description of viscoelastic behavior across a broad frequency range using only a few parameters [49, 50]. In addition, the second novelty of this work is the modeling of the equation of motion with fractional time as follows: α ρC 0 Dt

2 Mechanical equations

σ · τ =σij τij , τ Sd = (τ , τ ) . In our notation, vν and vτ represent the normal and tangential components of v on the boundary Γ. These components are defined as vν = v · ν, and vτ = v − vν ν, respectively, see [44]. Similarly, we define the normal and tangential components of tensor field σ by σν = (σν) · ν and στ = σν − σν ν. We denote by u : Ω×]0, T [→ Rd the displacement field, σ = (σij ) : Ω×]0, T [→ Sd the stress tensor and ε(u) = (εij (u)) = 12 (ui.j + uj.i ) the linearized strain tensor. The mechanical problem formulation of the fractional contact problem can be expressed as follows: Problem (P): Find a displacement field u : Ω×]0, T [−→ Rd and a stress tensor σ : Ω×]0, T [−→ Sd such that α σ(t) = Bε(C 0 Dt u(t)) + Aε(u(t)) in Ω × (0, T ), (2.1) 2α ρC 0 Dt u(t) − Div σ(t) = f0 (t) in Ω × (0, T ),

(2.2)

u = 0 on ΓD × (0, T ),

(2.3)

σ(t)ν = f1 (t) on ΓN × (0, T ),

(2.4)

C α 0 Dt u(0,

(2.5)

u(0, x) = u0 ,

x) = w0 in Ω,

− σν (u(t)) = pν (uν (t)), στ = 0 on ΓC × (0, T ). (2.6)

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Equation (2.1) represents the time fractional KelvinVoigt viscoelastic constitutive law of the Caputo type as described in [20]. In this equation, B = (bijkl ) and A = (aijkl ) refer to the viscosity tensor (fourth-order) and elastic tensor, respectively. Relation (2.2) is the equation of motion with time-fractional where ρ represents the mass density, here and below Div denotes the divergence operator for the tensor, i.e., Div σ = (σij, j ). Furthermore, relations (2.3)–(2.4) are the mechanical boundary conditions. The initial condition is described by equation (2.5). Finally, equation (2.6) represents the normal compliance frictionless contact condition, where pν is a prescribed function. When pν is positive, uν represents the penetration of the surface asperities into the foundation.

Vad ={v ∈ V , vν ≤ 0 on ΓC }, endowed with the inner product and norm given by: 1

(u, v)V = (ε(u), ε(v))H , vV = (v, v)V2 , for all v ∈ V.

Since meas(ΓD ) > 0, Korn’s inequality holds: There exists cK > 0 depending only on Ω and ΓD such that: ε(v)H ≥ cK vH1 , for all v ∈ V.

(3.2)

In addition, by the Sobolev trace theorem, there exists a constant cS > 0 depending only on Ω, ΓD , and ΓC such that: v[L2 (ΓC )]d ≤ cS vV , for all v ∈ V.

3 Weak formulation and main result In this section, we derive a weak formulation of Problem (P) and investigate its solvability. Everywhere in what follows, we use the classical notation for the Lp and Sobolev spaces associated with Ω and Γ. Let X be a Banach space, and T a positive real number. Then, for k = 1, 2, . . ., we also use the classical notation for Lp (0, T ; X), C (0, T ; X ), and W k, p (0, T ; X). We now introduce the functional spaces. 

d  d H = L2 (Ω) , H1 = H 1 (Ω) ,   H = τ = (τij ) | τij = τji ∈ L2 (Ω), i, j = 1, ..., d , and

(3.3)

Next, for all u, v in V , we define the following operators: a : V × V → R, a(u, v) := (Aε(u), ε(v))H , b : V × V → R, b(u, v) := (Bε(u), ε(v))H , (3.4) and the mappings j : V × V → R given by:  pν (uν (t))vν da.

j(u(t), v) :=

(3.5)

ΓC

By virtue of the Riesz’s representation theorem, we can find an element f (t) ∈ V such that:

H1 = {σ ∈ H | Div σ ∈ H}.



These are real Hilbert spaces endowed with the following inner products:

 f0 (t) · vdx +

(f (t), v)V = Ω

f1 (t) · vda, for all v ∈ V. ΓN

(3.6)

 (u, v)H =

ρui vi dx,

In our analysis of the mechanical problem defined by equations (2.1) to (2.6), we make the following hypothesis:

Ω

(u, v)H1 =(u, v)H + (ε(u) + ε(v))H ,  (σ, τ )H = σij τij dx,

(HP1)

Ω

(i)

(σ, τ )H1 =(σ, τ )H + (Div σ + Div τ )H , and the associated norms |·|H , |·|H1 , |·|H , and |·|H1 . When σ is a regular function, the following Green’stype formula holds:

aijkl =ajikl = alkij ∈ L∞ (Ω), bijkl =bjikl = blkij ∈ L∞ (Ω).

 (σ, ε(v))H + (Div σ, v)H =

σν · vda, for all v ∈ H1 . Γ

(3.1)

Moreover, keeping in mind (2.3)–(2.4), we introduce the following space and set of admissible displacements: V ={v ∈ H1 , v = 0 on ΓD }, and

The operators a and b are chosen to be bilinear and satisfy the usual property of symmetry.

(ii)

The operators a and b satisfy the properties of ellipticity a(u, u) ≥ ma u2V , b(u, u) ≥ mb u2V , for all u ∈ V , where ma and mb are positive constants.

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(HP2)

The operators a and b satisfy the usual property of boundedness. There exist positive constants Ma and Mb such that for all u, v in V , we have: |a(u, v )|≤ Ma uV vV , |b(u, v )|≤ Mb uV vV .

(HP3)

(HP4)

(i)

The force, traction, and initial conditions will be assumed as follows: f0 ∈   C 0, T ; L2 (Ω2 )d , f1 ∈ C 0, T ; L2 (ΓN )d and u0 ∈ Vad . (ii) The mass density satisfies ρ ∈ L∞ (Ω), and there exists ρ∗ > 0 such that ρ(x) ≥ ρ∗ a.e. x ∈ R. The normal compliance function pν satisfies the following hypothesis: (i) (ii) (iii) (iv)

pν : ΓC × R → R+ ; x → pν (x, u) is measurable on ΓC , for all u ∈ R; x → pν (x, u) = 0 for u ≤ 0, a.e. x ∈ ΓC ; There exists Lν > 0 such that

≤ Lν |u − v|, for all u, v ∈ R+ .

Let V ⊂ H ⊂ V ∗ be an evolution triple of spaces, where V is a reflexive and separable Banach space, H is a separable Hilbert space, and the embedding V → H is dense and continuous. The embedding operator between V and H is denoted by ι and is assumed to be compact. The dual space to V is denoted by V ∗ , and the dual mapping ι : H → V of ι is also linear, continuous and compact. Finally, by employing the standard procedure based on Green’s formula (3.1), we obtain the following fractional formulation of Problem (P): Problem (PV): Find a displacement field u : Ω×]0, T [−→ Rd and a stress tensor σ : Ω×]0, T [−→ Sd for v ∈ V and α ∈ (0, 1] such that

C 0

Dt2α u(t), v

C 0

 Dtα u(t) + Aε(u(t)),

V

+b



C 0

Dtα u(t), v

C α 0 Dt u(0,

C 0

Dtα (w(t)), v

 V

(4.1)

+ b(w(t), v) + a(0 Itα w(t) + u0 , v)

+ j(0 Itα w(t) + u0 , v) = (f (t), v)V . u(0, x) = u0 ,

C α 0 Dt u(0,

x) = w0 .

(4.2) (4.3)

x) = w0 .

(z(t), v)V = a(0 Itα w(t) + u0 , v) + j(0 Itα w(t) + u0 , v). (4.4) Then, we can express problem (4.1)–(4.2) as follows: Problem (PV1): Find a displacement field w ∈ V and a stress tensor σ ∈ H a.e. t ∈]0, T [, v ∈ V and α ∈ (0, 1] such that 

σz (t) = Bε(wz (t)) + Aε( 0 Itα wz (t) + u0 ), C α 0 Dt wz (t),

(4.5)



v

V

+ b(wz (t), v) + (z(t), v)V = (f (t), v)V .

(4.6)

We have the following result Lemma 4.1 For all v ∈ V and t ∈ (0, T ), Problem (PV1) has at least one wz ∈ W 1, 2 (0, T ; V )  solution  2 2 d and σz ∈ L 0, T ; L Ω, S .

(3.8)

Proof Using Riesz’s representation theorem, we define the operator fz as follows:

(3.9)

Our main result on existence, which will be proven in the next section, can be stated as follows: Theorem 3.1 Assuming that assumptions (HP 1) to (HP 4) hold, we can conclude that Problem (PV) has at least one solution (u, σ) that satisfies the following conditions:    u ∈ W 2, 2 (0, T ; V ), and σ ∈ L2 0, T ; L2 Ω, Sd . (3.10)

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σ(t) = Bε(w(t)) + Aε(0 Itα w(t) + u0 ),

(3.7)



+ a(u(t), v) + j(u(t), v) = (f (t), v)V , u(0, x) = u0 ,

The proof of Theorem 3.1 is conducted in several steps, relying on various techniques such as variational inequalities, the Galerkin method, the compactness method, and the Banach fixed point theorem. α In the first step, let us denote w(t) = C 0 Dt u(t) for a.e. t ∈ (0, T ). Applying Proposition 6.1 (iii), we obtain u(t) = α 0 It w(t) + u0 . Therefore, we can reformulate problem (3.7)–(3.8) as follows: Find w ∈ V and σ ∈ H such that

In the second step, let z ∈ L2 (0, T ; V ) be given by:

|pν (., u) − pν (., v )|

σ(t) = Bε

4 Proof of main result

(fz (t), v)V = (f (t), v)V − (z(t), v)V .

(4.7)

Equation (4.6) can be expressed as follows: C 0

Dtα wz (t), v

 V

+ b(wz (t), v) = (fz (t), v)V . (4.8)

Now, we begin the Faedo-Galerkin method. For this purpose, we assume that the functions βk , for k = 1, ..., m, are eigenfunctions of −Δ such that (βk )1≤k≤m form a Hilbertian basis of H 1 (Ω).

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Fix a positive integer m and consider the search for a function wzm : (0, T ) → H 1 (Ω)d in the following form:

wzm :=

m

dim (t)βi .

C α 0 Dt

1 2 wzm (t)V 2

≤

C 0

 Dtα wzm (t), wzm (t) V

+ b(wzm (t), wzm (t))V = (fz (t), wzm (t))V .

(4.9)

(4.16)

i=1

We denote Fm the vector space generated by β1 , β2 , ..., βm . Whence wzm ∈ Fm and wzm → wz in V . An approximate problem: For each integer m ≥ 1, our approximated problem takes the form: α 2 Find wzm ∈ L2 (0, T ; Fm ) such that C 0 Dt wzm ∈ L (0, T ; Fm ) and C

α 0 Dt wzm (t), βk

 V

+ b(wzm (t), βk ) = (fz (t), βk )V . (4.10)

Assuming that wzm has the structure given by (4.9), we observe that: C 0

Dtα wzm (t), βk

 V

=

C α k 0 Dt dm (t),

(4.11)

By applying Young’s inequality and using assumption (HP1), we obtain: 2

mb wzm (t)V ≤ b(wzm (t), wzm (t)),

(4.17)

and |(fz (t), wzm (t))| ≤

1  2 2 fz (t)V + wzm (t)V , 2 2 (4.18)

for all  > 0. Thus  2 2 2 C α 1 w D (t) + c1 wzm (t)V ≤ c2 fz (t)V . zm 0 t V 2 (4.19) Using Proposition 6.1(ii), we can deduce the following estimate:  t 2c1 2 2 wzm (t)V + (t − s)α−1 wzm (s)V ds Γ(α) 0

 2 2 (4.20) ≤ c2 fz (t)V + w0 V .

b(wzm (t), βk ) = Bdkm (t),

(4.12)

(fz (t), βk )V = fzk (t).

(4.13)

Then, we have that T∗ = +∞. Fix any v ∈ V , with v≤ 1, and write v = v1 + v2 , m where v1 ∈ spam{βk }k=1 and (v2 , βk ) = 0, (k = 1, ..., m). m Since, the functions {βk }k=1 are orthogonal in V

(4.14)

v1 V ≤ vV ≤ 1.

Then, (4.10) can be written as follows C α k 0 Dt dm (t)

  = h t, dkm (t) ,

  where h t, dkm (t) = fzk (t) − Bdkm (t). Next, assumption (HP2) implies that:

By relation (4.10), we deduce that C 0



 k

  

h t, dm (t) − h t, dkm (t) ≤ Mb dkm (t) − dim (t) . 1 2 1 2 (4.15) Then, by applying the standard methods in fractional calculus (see, [13, Proposition 4.6]), there exists a unique solvability continuous functions dm (t) =  d1m (t), d2m (t), ..., dm m (t) on [0, T∗ ) satisfying (4.14). Here and in the following, c1 and c2 represent positive generic constants whose values may vary from line to line. Estimates: Multiply equation (4.10) by dim (t), summing for i = 1, ..., m, and utilizing the fact that wzm → 12 |wzm |2V is a convex functional, we obtain:

(4.21)

Dtα wzm (t), v1

 V

+ b(wzm (t), v1 )V = (fz (t), v1 )V . (4.22)

and due to the continuity of b, we have: |b(wzm (t), v1 )| ≤ Mb wzm (t)V ,

(4.23)

|(fz (t), v1 )| ≤ fz (t)V .

(4.24)

Thus, we can conclude that:  C α 0 Dt wz (t) ∗ ≤ c1 + c2 (wz (t) + fz (t) ). m m V V V (4.25) From (4.20), there exists a positive constant c such that C α  0 Dt wz (t) 2 ≤ c. m L (0, T ;V ∗ )

(4.26)

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Passage to the limit: By utilizing the previous estimates and applying the compactness result stated in Theorem (6.1) for Caputo’s derivative, we can deduce the existence of a subsequence wzml and wz ∈ L2 (0, T ; V ) such that: (4.27)

|j(0 Itα wz (t1 ) + u0 , v) − j(0 Itα wz (t2 ) + u0 , v)| Lν cS T α wz (t1 ) − wz (t2 )V vV . ≤ Γ(α + 1) (4.34)

weakly in L2 (0, T ; V ∗ ). (4.28)

By combining the previous inequalities, we can conclude that there exists a positive constant c that depends on Ma , Lν , cS , α, and T . This constant satisfies the following inequality:

wzml → wz strongly in L2 (0, T ; V ), and C α 0 Dt wzml

→

C α 0 Dt wz

Based on (3.3), (3.5), and hypothesis (HP4), we can derive the following result:

|Λz(t1 ) − Λz(t2 )|V ≤ c|wz (t1 ) − wz (t2 )|V ,

Then



b wzml (t), v → b(wz (t), v) in R,

 (4.29)   C α α → C 0 Dt wzml , v 0 Dt wz , v V in R, V

and α 0 It wzml

→

α 0 It wz

(4.35)

and by the regularity of wz , we conclude that Λz ∈ C(0, T ; V ). Now, let z1 , z2 ∈ L2 (0, T ; V ) and t ∈ [0, T ]. Similar to (4.35), we can express the following relation: Λz1 (t) − Λz2 (t)V ≤ c|wz1 (t) − wz2 (t)|V .

weakly in L2 (0, T ; V ).

(4.36)

(4.30) On the other hand, using (4.8), we can deduce that

Therefore, a weak solution wz ∈ W 1, 2 (0, T ; V ) exists for equation (4.6). Now, considering (4.4), (HP1), and (HP3), it can be  concluded that σz ∈ L2 0, T ; L2 Ω, Sd . Hence, the proof of Lemma 4.1 is now complete.  In the last step, for z ∈ L2 (0, T ; V ), we use the solution wz obtained in Lemma 4.1. Additionally, we consider the operator Λ : C(0, T ; V ) → C(0, T ; V ) defined by.

C 0

Dtα wz1 (t) −

C α 0 Dt wz2 (t),

 wz1 (t) − wz2 (t) V

+ b(wz1 (t) − wz2 (t), wz1 (t) − wz2 (t)) + (z1 (t) − z2 (t), wz1 (t) − wz2 (t))V = 0.

(4.37)

The definition given by (6.2) implies  C α 0 Dt wz1 (t) −



 C α 0 Dt wz2 (t)

V

≤

(Λz(t), v)V := a(0 Itα wz (t) + u0 , v) + j(0 Itα wz (t) + u0 , v).

T 1−α w˙ z1 (t) − w˙ z2 (t)V . Γ(α)

(4.38)

(4.31) The combination of (HP1)(ii) and (4.37)–(4.38), imply that

We obtain the following result. Lemma 4.2 For all z ∈ L2 (0, T ; V ), α ∈ (0, 1], the operator Λz :]0, T [→ V is continuous. Moreover, there exists a unique element z ∗ ∈ L2 (0, T ; V ) such that Λz ∗ = z ∗ .

2

mb wz1 (t) − wz2 (t)V T 1−α w˙ z1 (t) − w˙ z2 (t)V wz1 (t) − wz2 (t)V Γ(α) + z1 (t) − z2 (t)V wz1 (t) − wz2 (t)V . (4.39) ≤

Consider z ∈ L2 (0, T ; V ) and t1 , t2 ∈]0, T [. It can be deduced from (4.30) that (Λz(t1 ) − Λz(t2 ), v) = a(0 Itα wz (t1 ) − 0 Itα wz (t2 ), + j(0 Itα wz (t1 ) + u0 , v) − j(0 Itα wz (t2 ) + u0 , v).

v)

(4.32) By applying Definition 6.2 and considering (HP2), we deduce that |a(0 Itα wz (t1 ) − 0 Itα wz (t2 ), v)| Ma T α wz (t1 ) − wz (t2 )V vV . (4.33) ≤ Γ(α + 1)

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 Since wzi (t) = have

0

t

w˙ zi (s)d + wzi (0), for i = 1, 2, we 

wz1 (t) − wz2 (t)V ≤

0

t

w˙ z1 (s) − w˙ z2 (s)V ds. (4.40)

By integrating (4.39) over the interval [0, T ] and considering the initial condition wz1 (0) = wz2 = w(0), along with (4.40), we obtain the following expression:

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t

wz1 (s) − wz2 (s)2V ds ≤

mb 0



+ 0

t

T 1−α wz1 (t) − wz2 (t)2V Γ(α)

z1 (s) − z2 (s)V wz1 (s) − wz2 (s)V ds.

(4.41)

1 2 Taking into account the inequality xy ≤ γx2 + 4γ y , where γ > 0, and applying Gronwall’s inequality, we can conclude that  t 2 2 wz1 (t) − wz2 (t)V ≤ z1 (s) − z2 (s)V ds. 0

(4.42)

Now, combining this inequality with (4.36), we can deduce that  t 2 2 Λz1 (t) − Λz2 (t)V ≤ c z1 (s) − z2 (s)V ds. 0

(4.43)

and assumptions on the data. The proof is based on the Caputo’s fractional derivative, the Galerkin method, the compactness method, and the Banach fixed point theorem. The difficulty in solving this type of problem lies in the coupling of viscoelastic physics in fractional time and the non-linearity of the boundary conditions. This work can be considered as a foundation for studying other problems with piezoelectric or thermoviscoelastic behavior, considering various types of contact and friction. The work presented here encompasses various extensions and perspectives, including the following notable aspects: (1)

Modeling new contact problems based on industry projects such as energy production. (2) Study contact models using optimization tools and the associated optimal control problems. (3) Use numerical methods based on convex optimization, such as the projecting conjugate gradient and augmented Lagrangian..., as well as techniques from artificial intelligence

For all t ∈ [0, T ], iterating this inequality n times yields the following result: Λ z1 (t) − Λ n

≤

n

2 z2 (t)L2 (0, T ;V )

(cT )n 2 z1 (t) − z2 (t)L2 (0, T ;V ) . n!

(4.44)

This implies that for sufficiently large n, Λn becomes a contraction in the Banach space L2 (0, T ; V ). Consequently, Λ has a unique fixed point. We are now prepared to establish the proof of Theorem (3.1). Proof of Theorem 3.1 Let z ∗ ∈ L2 (0, T ; V ) be the fixed point of the operator Λ and denote w∗ the solution of (4.6), for z = z ∗ . Define σ ∗ (t) = Bε(w∗ (t)) + Aε( 0 Itα w∗ (t) + u0 ). Using (4.31) and taking into account Λ(z ∗) = z ∗ , we  find (w∗ , σ ∗ ) ∈ W 1, 2 (0, T ; V ) × L2 0, T ; L2 Ω, Sd . Hence, we infer that u∗ ∈ W 2, 2 (0, T ; V ) given by u∗ (t) = 0 Itα w∗ (t) + u0 for a.e. t ∈ (0, T ). Finally, we deduce that (u, σ) constitutes a solution to Problem (PV). This completes the proof. 

5 Conclusion In this paper, we examine a frictionless normal compliance contact problem involving a viscoelastic body and a foundation. The constitutive law of the material follows the Kelvin-Voigt model, incorporating fractional time derivatives using the fractional Caputo operator. The variational formulation of this problem refers to a nonlinear dynamical system that exists in fractional time. A result establishing the existence of a weak solution has been obtained under appropriate conditions

Acknowledgements This project has received funding from the Natural Science Foundation of Guangxi Grant Nos. 2021GXNSFFA196004 and GKAD21220144, NNSF of China Grant Nos. 12001478 and 12371312, the Startup Project of Doctor Scientific Research of Yulin Normal University No. G2023ZK13, the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No. 823731 CONMECH, and the project cooperation between Guangxi Normal University and Yulin Normal University. Data availability Data sharing does not apply to this article as no datasets were generated or analyzed during the current study.

Appendix In this paragraph, we recall some results about the fractional calculus and nonlinear analysis that can find it as soon as the following references [5, 12, 15, 17, 18]. Definition 6.1 (Riemann-Liouville fractional integral ) Let X be a Banach space and (0, T ) be a finite time interval. The Riemann-Liouville fractional integral of order α > 0 for a given function f ∈ L1 (0, T ; X) is defined by α 0 It f (t)

= Jα (f (t)) =

1 Γ(α)



t

0

(t − s)α−1 f (s)ds, ∀t ∈ (0, T ),

where Γ(.) stands for the Gamma function defined by  Γ(α) =

∞

tα−1 e−t dt.

0

123

Eur. Phys. J. Spec. Top.

To complement the definition, we set 0 It0 = I, where I is the identity operator, which means that 0 It0 f (t) = f (t) for a.e. t ∈ (0, T ). Definition 6.2 (Caputo derivative of order 0 < α ≤ 1) Let X be a Banach space, 0 < α ≤ 1 and (0, T ) be a finite time interval. For a given function f ∈ AC(0, T ; X), the Caputo fractional derivative of f is defined by =0 It1−α f  (t)  t 1 = (t − s)−α f  (s)ds, ∀t ∈ (0, T ). Γ(1 − α) 0

ii)

for y ∈ AC(0, T ; X) and α ∈ (0, 1], we have α C α 0 It 0 Dt y(t)

iii)

= y(t) − y(0) for a.e. t ∈ (0, T ),

for y ∈ L1 (0, T ; X), we have C α α 0 Dt 0 It y(t)

= y(t) for a.e. t ∈ (0, T ).

C α 0 Dt f (t)

The notation AC (0, T ; X ) refers to the space of all absolutely continuous functions from (0, T ) into X . It is obvious that if α = 1, the Caputo derivative reduces to the classical first-order derivative, that is, we have C 1 0 Dt f (t)

= If  (t) = f  (t), for a.e. t ∈ (0, T ).

Additionally, we recall the compactness criteria, which is analogous to the Aubin-Lions Lemma, for the existence of weak solutions to time-fractional PDEs as presented in [15]. Theorem 6.1 Let T > 0, α ∈ (0, 1) and p ∈ [1, ∞). let B0 , B , B1 be a Banach space. B0 → B compactly and B → B1 continuously. Suppose W ⊂ L1loc (0, T ; B0 ) satisfies: (i)

There exists r1 ∈ [1, ∞) and c1 > 0 such that ∀u ∈ W .  t   1 sup Jα urB10 = sup (t − s)α−1 t∈(0, T ) t∈(0, T ) Γ(α) 0 urB10 (s)ds ≤ c1 ;

(ii) (iii)

(6.1)

There exists p1 ∈ (p, ∞], W is bounded in Lp1 (0, T , B); There exists r2 ∈ [1, ∞) and c2 > 0 such that ∀u ∈ W , there is an assignment of initial value u0 for u so that the weak Caputo derivative satisfies: C α  0 Dt u r ≤ c2 , (6.2) L 1 (0, T ;B ) 1

then, W is relatively compact in Lp (0, T , B). Finally, we have obtained these results on the composition of fractional order Caputo derivative and integral operators Proposition 6.1 Let X be a Banach space and α, β > 0. Then, the following statements hold i)

for y ∈ L1 (0, T ; X), we have 0Itα 0 Itβ y(t) =0 Itα+β y(t) for a.e. t ∈ (0, T ),

123

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