Modeling and numerical analysis of a viscoelastic contact problem for locking materials with time-fractional derivatives

Jinxia Cen; Mustapha Bouallala; Abdesadik Bendarag

doi:10.1515/gmj-2025-2071

Article

Modeling and numerical analysis of a viscoelastic contact problem for locking materials with time-fractional derivatives

Jinxia Cen , Mustapha Bouallala and Abdesadik Bendarag

Published/Copyright: August 29, 2025

Published by

Become an author with De Gruyter Brill

Author Information Explore this Subject

From the journal Georgian Mathematical Journal

Abstract

This article explores a new contact problem involving viscoelastic materials locked with a rigid foundation. The constitutive relation in our study is constructed using a fractional Kelvin–Voigt model to describe displacement behavior. The contact is characterized by a variation of the Signorini condition, while friction is modeled by a nonlocal Coulomb’s friction law. We present the mathematical model for the viscoelastic process, provide the variational formulation, and establish the existence of a solution. We present fully discrete finite element schemes for the variational problem and derive error estimates for the approximations.

Keywords: Locking material; Caputo derivative fractional viscoelastic constitutive law; normal compliance; Coulomb’s friction; variational inequality; weak solution; finite element method; error estimate

MSC 2020: 74M15; 74S40; 47J20; 49J40; 35D30; 74S05

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12371312

Funding statement: This project has received funding from the Natural Science Foundation of Guangxi Grant Nos. 2021GXNSFFA196004 and 2024GXNSFBA010337, the National Natural Science Foundation of China Grant No. 12371312, the Natural Science Foundation of Chongqing Grant No. CSTB2024NSCQ-JQX0033. It is also supported by the Postdoctoral Fellowship Program of CPSF under Grant Number GZC20241534, the Systematic Project of Center for Applied Mathematics of Guangxi (Yulin Normal University) Nos. 2023CAM002 and 2023CAM003.

A Appendix

Now, let us revisit the established definitions in fractional calculus theory and nonlinear analysis, given in references [14, 16, 27, 22].

Definition A.1 (Riemann–Liouville fractional integral).

Let X be a Banach space and ( 0 , T ) a finite time interval. The Riemann–Liouville fractional integral of order α > 0 for a given function f ∈ L 1 ⁢ ( 0 , T ; X ) is defined by

I t α 0 ⁢ f ⁢ ( t ) = 1 Γ ⁢ ( α ) ⁢ ∫ 0 t ( t - s ) α - 1 ⁢ f ⁢ ( s ) ⁢ 𝑑 s for all ⁢ t ∈ ( 0 , T ) ,

where Γ ⁢ ( ⋅ ) stands for the Gamma function defined by

Γ ⁢ ( α ) = ∫ 0 ∞ t α - 1 ⁢ e - t ⁢ 𝑑 t .

To further elaborate on the definition, we introduce I t 0 0 = I , where I represents the identity operator. This implies that I t 0 0 ⁢ f ⁢ ( t ) = f ⁢ ( t ) for almost every t ∈ ( 0 , T ) .

Definition A.2 (Caputo derivative of order, 0 < α ≤ 1 ).

Let X be a Banach space, 0 < α ≤ 1 and ( 0 , T ) a finite time interval. For a given function f ∈ A ⁢ C ⁢ ( 0 , T ; W ) , the Caputo fractional derivative of f is defined by

D t α 0 C ⁢ f ⁢ ( t ) = I t 1 - α 0 ⁢ f ′ ⁢ ( t ) = 1 Γ ⁢ ( 1 - α ) ⁢ ∫ 0 t ( t - s ) - α ⁢ f ′ ⁢ ( s ) ⁢ 𝑑 s for all ⁢ t ∈ ( 0 , T ) .

The notation A ⁢ C ⁢ ( 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

D t 1 0 C ⁢ f ⁢ ( t ) = I ⁢ f ′ ⁢ ( t ) = f ′ ⁢ ( t ) for a.e. ⁢ t ∈ ( 0 , T ) .

Lemma A.3.

Let T > 0 be given. For a positive integer N, define k = T N . Assume that { g n } n = 1 N and { e n } n = 1 N are two sequences of nonnegative numbers satisfying

e n ≤ c ⁢ g n + c ⁢ ∑ j = 1 n k ⁢ e j , n = 1 , … , N ,

for a positive constant c independent of N or k. Then there exists a positive constant c, independent of N and k, such that

max 1 ≤ n ≤ N ⁡ e n ≤ c ⁢ max 1 ≤ n ≤ N ⁡ g n .

Acknowledgements

The authors wish to thank the two knowledgeable referees for their corrections and helpful remarks.

References

[1] Y. Ayyad, M. Barboteu and J. R. Fernández, A frictionless viscoelastodynamic contact problem with energy consistent properties: Numerical analysis and computational aspects, Comput. Methods Appl. Mech. Engrg. 198 (2009), no. 5–8, 669–679. 10.1016/j.cma.2008.10.004Search in Google Scholar

[2] M. Barboteu, W. Han and S. Migórski, On numerical approximation of a variational-hemivariational inequality modeling contact problems for locking materials, Comput. Math. Appl. 77 (2019), no. 11, 2894–2905. 10.1016/j.camwa.2018.08.004Search in Google Scholar

[3] M. Bouallala, Variational and error estimation for a frictionless contact problem in thermo-viscoelasticity with time fractional derivatives, Commun. Anal. Comput. 2 (2024), no. 1, 1–18. 10.3934/cac.2024001Search in Google Scholar

[4] M. Bouallala and E.-H. Essoufi, A thermo-viscoelastic fractional contact problem with normal compliance and Coulomb’s friction, J. Math. Phys. Anal. Geom. 17 (2021), no. 3, 280–294. 10.15407/mag17.03.280Search in Google Scholar

[5] M. Bouallala, E.-H. Essoufi, V. T. Nguyen and W. Pang, A time-fractional of a viscoelastic frictionless contact problem with normal compliance, European Phys. J. Spec. Topics 232 (2023), no. 14, 2549–2558. 10.1140/epjs/s11734-023-00962-xSearch in Google Scholar

[6] S. Bourichi and E.-H. Essoufi, Penalty method for an unilateral contact problem with Coulomb’s friction for locking materials, Int. J. Math. Model. Comput. 6 (2016), no. 1, 61–81. Search in Google Scholar

[7] F. Demengel, Displacements bounded deformation and measures stress, Annals of Superior School of Pise, 1972. Search in Google Scholar

[8] F. Demengel and P. Suquet, On locking materials, Acta Appl. Math. 6 (1986), no. 2, 185–211. 10.1007/BF00046725Search in Google Scholar

[9] M. Di Paola, R. Heuer and A. Pirrotta, Fractional visco-elastic Euler–Bernoulli beam, Internat. J. Solids Structures 50 (2013), no. 22–23, 3505–3510. 10.1016/j.ijsolstr.2013.06.010Search in Google Scholar

[10] C. Eck and J. Jarušek, Existence results for the static contact problem with Coulomb friction, Math. Models Methods Appl. Sci. 8 (1998), no. 3, 445–468. 10.1142/S0218202598000196Search in Google Scholar

[11] L. Essafi and M. Bouallala, Penalty method for unilateral contact problem with Coulomb’s friction in time-fractional derivatives, Demonstr. Math. 57 (2024), no. 1, Article ID 20240050. 10.1515/dema-2024-0050Search in Google Scholar

[12] E.-H. Essoufi and A. Zafrar, Dual methods for frictional contact problem with electroelastic-locking materials, Optimization 70 (2021), no. 7, 1581–1608. 10.1080/02331934.2020.1745794Search in Google Scholar

[13] Z. Faiz, O. Baiz and H. Benaissa, Penalization of a frictional thermoelastic contact problem with generalized temperature dependent conditions, Rend. Circ. Mat. Palermo (2) 74 (2025), no. 1, Paper No. 74. 10.1007/s12215-025-01195-8Search in Google Scholar

[14] W. Han and M. Sofonea, Evolutionary variational inequalities arising in viscoelastic contact problems, SIAM J. Numer. Anal. 38 (2000), no. 2, 556–579. 10.1137/S0036142998347309Search in Google Scholar

[15] C. Ionescu, A. Lopes, D. Copot, J. A. T. Machado and J. H. T. Bates, The role of fractional calculus in modeling biological phenomena: A review, Commun. Nonlinear Sci. Numer. Simul. 51 (2017), 141–159. 10.1016/j.cnsns.2017.04.001Search in Google Scholar

[16] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud. 204, Elsevier Science, Amsterdam, 2006. Search in Google Scholar

[17] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity—An Introduction to Mathematical Models, World Scientific, Hackensack, 2022. Search in Google Scholar

[18] J. A. C. Martins and J. T. Oden, Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws, Nonlinear Anal. 11 (1987), no. 3, 407–428. 10.1016/0362-546X(87)90055-1Search in Google Scholar

[19] S. Migórski and J. Ogorzały, A variational-hemivariational inequality in contact problem for locking materials and nonmonotone slip dependent friction, Acta Math. Sci. Ser. B (Engl. Ed.) 37 (2017), no. 6, 1639–1652. 10.1016/S0252-9602(17)30097-8Search in Google Scholar

[20] J. Nečas and I. Hlaváček, Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction, Stud. Appl. Mech. 3, Elsevier, Amsterdam, 1980. Search in Google Scholar

[21] J. T. Oden and E. B. Pires, Nonlocal and nonlinear friction laws and variational principles for contact problems in elasticity, Trans. ASME Ser. E. J. Appl. Mech. 50 (1983), no. 1, 67–76. 10.1115/1.3167019Search in Google Scholar

[22] I. Podlubny, Fractional Differential Equations, Math. Sci. Eng. 198, Academic Press, San Diego, 1999. Search in Google Scholar

[23] W. Prager, Elastic solids of limited compressibility, Proc. Int. Congr. Appl. Mech. Brussels 1956 (1956), 205–211. Search in Google Scholar

[24] W. Prager, On ideal locking materials, Trans. Soc. Rheology 1 (1957), no. 1, 169–175. 10.1122/1.548818Search in Google Scholar

[25] W. Prager, On elastic, perfectly locking materials, Applied Mechanics, Proceedings of the 11th International Congress of Applied Mechanics (Munich 1964), Springer, Berlin (1966), 538–544. 10.1007/978-3-662-29364-5_72Search in Google Scholar

[26] A. G. Radwan, Resonance and quality factor of the R ⁢ L α ⁢ C α fractional circuit, IEEE J. Emerging Select. Topics Circuits Syst. 3 (2013), no. 3, 377–385. 10.1109/JETCAS.2013.2272838Search in Google Scholar

[27] S. Shen, F. Liu, J. Chen, I. Turner and V. Anh, Numerical techniques for the variable order time fractional diffusion equation, Appl. Math. Comput. 218 (2012), no. 22, 10861–10870. 10.1016/j.amc.2012.04.047Search in Google Scholar

[28] M. Sofonea, A nonsmooth static frictionless contact problem with locking materials, Anal. Appl. (Singap.) 16 (2018), no. 6, 851–874. 10.1142/S0219530518500215Search in Google Scholar

[29] M. Sofonea, History-dependent inequalities for contact problems with locking materials, J. Elasticity 134 (2019), no. 2, 127–148. 10.1007/s10659-018-9684-3Search in Google Scholar

[30] M. Sofonea and S. Migórski, Variational-Hemivariational Inequalities with Applications, Monogr. Res. Notes Math., CRC Press, Boca Raton, 2018. 10.1201/9781315153261Search in Google Scholar

[31] S. Zeng, Z. Liu and S. Migorski, A class of fractional differential hemivariational inequalities with application to contact problem, Z. Angew. Math. Phys. 69 (2018), no. 2, Paper No. 36. 10.1007/s00033-018-0929-6Search in Google Scholar

[32] S. Zeng and S. Migórski, A class of time-fractional hemivariational inequalities with application to frictional contact problem, Commun. Nonlinear Sci. Numer. Simul. 56 (2018), 34–48. 10.1016/j.cnsns.2017.07.016Search in Google Scholar

[33] C. Zou, L. Zhang, X. Hu, Z. Wang, T. Wik and M. Pecht, A review of fractional-order techniques applied to lithium-ion batteries, lead-acid batteries, and supercapacitors, J. Power Sources 390 (2018), 286–296. 10.1016/j.jpowsour.2018.04.033Search in Google Scholar

Received: 2025-01-06

Revised: 2025-03-29

Accepted: 2025-05-14

Published Online: 2025-08-29

You are currently not able to access this content.

https://doi.org/10.1515/gmj-2025-2071

Keywords for this article

Locking material; Caputo derivative fractional viscoelastic constitutive law; normal compliance; Coulomb’s friction; variational inequality; weak solution; finite element method; error estimate