# American Institute of Mathematical Sciences

May  2021, 26(5): 2781-2804. doi: 10.3934/dcdsb.2020205

## Numerical analysis and simulation of an adhesive contact problem with damage and long memory

 School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

* Corresponding author: Hailing Xuan

Received  March 2020 Revised  April 2020 Published  May 2021 Early access  June 2020

This paper studies an adhesive contact model which also takes into account the damage and long memory term. The deformable body is composed of a viscoelastic material and the process is taken as quasistatic. The damage of the material caused by the compression or the tension is involved in the constitutive law and the damage function is modelled through a nonlinear parabolic inclusion. Meanwhile, the adhesion process is modelled by a bonding field on the contact surface while the contact is described by a nonmonotone normal compliance condition. The variational formulation of the model is governed by a coupled system which consists of a history-dependent hemivariational inequality for the displacement field, a nonlinear parabolic variational inequality for the damage field and an ordinary differential equation for the adhesion field. We first consider a fully discrete scheme of this system and then focus on deriving error estimates for numerical solutions. Under appropriate solution regularity assumptions, an optimal order error estimate is derived. At the end of this paper, {we report some two-dimensional numerical simulation results} for the contact problem in order to provide numerical evidence of the theoretical results.

Citation: Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulation of an adhesive contact problem with damage and long memory. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2781-2804. doi: 10.3934/dcdsb.2020205
##### References:
 [1] T. Ammar, B. Benabderrahmane and S. Drabla, Frictional contact problems for electro-viscoelastic materials with long-term memory, damage, and adhesion, Electron. J. Diff. Equ., 2014 (2014), 21 pp. [2] T. A. Angelov, On a rolling problem with damage and wear, Mech. Res. Comm., 26 (1999), 281-286.  doi: 10.1016/S0093-6413(99)00025-7. [3] K. Atkinson and W. M. Han, Theortical Numerical Analysis: A Functional Analysis Framework, Third edition, Texts in Applied Mathematics, 39. Springer, Dordrecht, 2009. doi: 10.1007/978-1-4419-0458-4. [4] K. Bartosz, X. L. Cheng, P. Yu, Y. J. Kalita and C. Zheng, Rothe method for parabolic variational–hemivariational inequalities, J. Math. Anal. Appl., 423 (2015), 841-862.  doi: 10.1016/j.jmaa.2014.09.078. [5] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Elements Methods, Third edition, Texts in Applied Mathematics, 15. Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0. [6] M. Campo, J. R. Fernández, W. M. Han and M. Sofonea, A dynamic viscoelastic contact problem with normal compliance and damage, Finite Elem. Anal. Des., 42 (2005), 1-24.  doi: 10.1016/j.finel.2005.04.003. [7] M. Campo, J. R. Fernández and J. M. Viaño, Numerical analysis and simulations of a quasistatic frictional contact problem with damage in viscoelasticity, J. Comput. Appl. Math., 192 (2006), 30-39.  doi: 10.1016/j.cam.2005.04.046. [8] M. Campo, J. R. Fernández and T.-V. Hoarau-Mantel, Analysis of two frictional viscoplastic contact problems with damage, J. Comput. Appl. Math., 196 (2006), 180-197.  doi: 10.1016/j.cam.2005.08.025. [9] O. Chau, Numerical analysis of a thermal contact problem with adhesion, Comp. Appl. Math., 37 (2018), 5424-5455.  doi: 10.1007/s40314-018-0642-2. [10] O. Chau, J. R. Fernández, M. Shillor and M. Sofonea, Variational and numerical analysis of a quasistatic viscoelastic contact problem with adhesion, J. Comput. Appl. Math., 159 (2003), 431-465.  doi: 10.1016/S0377-0427(03)00547-8. [11] X. L. Cheng, Q. C. Xiao, S. Migórski and A. Ochal, Error estimate for quasistatic history-dependent contact model, Comput. Math. Appl., 77 (2019), 2943-2952.  doi: 10.1016/j.camwa.2018.08.058. [12] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathematics and its Applications, Vol. 4. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. [13] M. Cocu and R. Rocca, Existence results for unilateral quasistatic contact problem with friction and adhesion, Math. Model Numer. Anal., 34 (2000), 981-1001.  doi: 10.1051/m2an:2000112. [14] M. Frémond, Equilibre des structures qui adhèrent à leur support, C. R. Acad. Sci. Paris Série II Méc. Phys. Chim. Sci. Univers Sci. Terre, 295 (1982), 913-916. [15] M. Frémond, Adhérence des solids, J. Mécan. Théor. Appl., 6 (1987), 383-407. [16] M. Frémond and B. Nedjar, Damage, gradient of damage and principle of virtual work, Internat. J. Solids and Structures, 33 (1996), 1083-1103. [17] W. M. Han, S. Migórski and M. Sofonea, A class of variational-hemivariational inequalities with applications to frictional contact problems, SIAM J. Math. Anal., 46 (2014), 3891-3912.  doi: 10.1137/140963248. [18] W. M. Han, M. Shillor and M. Sofonea, Variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance, friction and damage, J. Comput. Appl. Math., 137 (2001), 377-398.  doi: 10.1016/S0377-0427(00)00707-X. [19] W. M. Han and M. Sofonea, Numerical analysis of hemivariational inequalities in contact mechanics, Acta Numer., 28 (2019), 175-286.  doi: 10.1017/S0962492919000023. [20] W. M. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, AMS/IP Studies in Advanced Mathematics, 30. American Mathematical Society, Providence, RI, International Press, omerville, MA, 2002. [21] W. M. Han, M. Sofonea and M. Barboteu, Numerical analysis of elliptic hemivariational inequalities, SIAM J. Numer. Anal., 55 (2017), 640-663.  doi: 10.1137/16M1072085. [22] L. Jianu, M. Shillor and M. Sofonea, A viscoelastic frictioness contact problem with adhesion, Appl. Anal., 80 (2001), 233-255.  doi: 10.1080/00036810108840990. [23] Y. X. Li and Z. H. Liu, A quasistatic contact problem for viscoelastic materials with friction and damage, Nonlinear Anal., 73 (2010), 2221-2229.  doi: 10.1016/j.na.2010.05.051. [24] S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, 26. Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5. [25] Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Monographs and Textbooks in Pure and Applied Mathematics, 188. Marcel Dekker, Inc., New York, 1995. [26] B. Nedjar, A theoretical and computational setting for a geometrically nonlinear gradient damage modelling framework, Comput. Mech., 30 (2002), 65-80.  doi: 10.1007/s00466-002-0368-1. [27] M. Sofonea, W. M. Han and M. Shillor, Analysis and Approximation of Contact Problems with Adhesion or Damage, Pure and Applied Mathematics (Boca Raton), 276. Chapman & Hall/CRC, Boca Raton, FL, 2006. [28] M. Sofonea and T. Raafat, An electro-viscoelastic frictional contact problem with damage, Appl. Anal., 86 (2007), 503-518.  doi: 10.1080/00036810701286304. [29] W. Xu, Z. P. Huang, W. M. Han, W. B. Chen and C. Wang, Numerical analysis of history-dependent hemivariational inequalities and applications to viscoelastic contact problems with normal penetration, Comput. Math. Appl., 77 (2019), 2596-2607.  doi: 10.1016/j.camwa.2018.12.038. [30] H. L. Xuan, X. L. Cheng, W. M. Han and Q. C. Xiao, Numerical analysis of a dynamic contact problems with history-dependent operators, Numer. Math-theory. Me., 13 (2020), 569-594.

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##### References:
 [1] T. Ammar, B. Benabderrahmane and S. Drabla, Frictional contact problems for electro-viscoelastic materials with long-term memory, damage, and adhesion, Electron. J. Diff. Equ., 2014 (2014), 21 pp. [2] T. A. Angelov, On a rolling problem with damage and wear, Mech. Res. Comm., 26 (1999), 281-286.  doi: 10.1016/S0093-6413(99)00025-7. [3] K. Atkinson and W. M. Han, Theortical Numerical Analysis: A Functional Analysis Framework, Third edition, Texts in Applied Mathematics, 39. Springer, Dordrecht, 2009. doi: 10.1007/978-1-4419-0458-4. [4] K. Bartosz, X. L. Cheng, P. Yu, Y. J. Kalita and C. Zheng, Rothe method for parabolic variational–hemivariational inequalities, J. Math. Anal. Appl., 423 (2015), 841-862.  doi: 10.1016/j.jmaa.2014.09.078. [5] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Elements Methods, Third edition, Texts in Applied Mathematics, 15. Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0. [6] M. Campo, J. R. Fernández, W. M. Han and M. Sofonea, A dynamic viscoelastic contact problem with normal compliance and damage, Finite Elem. Anal. Des., 42 (2005), 1-24.  doi: 10.1016/j.finel.2005.04.003. [7] M. Campo, J. R. Fernández and J. M. Viaño, Numerical analysis and simulations of a quasistatic frictional contact problem with damage in viscoelasticity, J. Comput. Appl. Math., 192 (2006), 30-39.  doi: 10.1016/j.cam.2005.04.046. [8] M. Campo, J. R. Fernández and T.-V. Hoarau-Mantel, Analysis of two frictional viscoplastic contact problems with damage, J. Comput. Appl. Math., 196 (2006), 180-197.  doi: 10.1016/j.cam.2005.08.025. [9] O. Chau, Numerical analysis of a thermal contact problem with adhesion, Comp. Appl. Math., 37 (2018), 5424-5455.  doi: 10.1007/s40314-018-0642-2. [10] O. Chau, J. R. Fernández, M. Shillor and M. Sofonea, Variational and numerical analysis of a quasistatic viscoelastic contact problem with adhesion, J. Comput. Appl. Math., 159 (2003), 431-465.  doi: 10.1016/S0377-0427(03)00547-8. [11] X. L. Cheng, Q. C. Xiao, S. Migórski and A. Ochal, Error estimate for quasistatic history-dependent contact model, Comput. Math. Appl., 77 (2019), 2943-2952.  doi: 10.1016/j.camwa.2018.08.058. [12] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathematics and its Applications, Vol. 4. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. [13] M. Cocu and R. Rocca, Existence results for unilateral quasistatic contact problem with friction and adhesion, Math. Model Numer. Anal., 34 (2000), 981-1001.  doi: 10.1051/m2an:2000112. [14] M. Frémond, Equilibre des structures qui adhèrent à leur support, C. R. Acad. Sci. Paris Série II Méc. Phys. Chim. Sci. Univers Sci. Terre, 295 (1982), 913-916. [15] M. Frémond, Adhérence des solids, J. Mécan. Théor. Appl., 6 (1987), 383-407. [16] M. Frémond and B. Nedjar, Damage, gradient of damage and principle of virtual work, Internat. J. Solids and Structures, 33 (1996), 1083-1103. [17] W. M. Han, S. Migórski and M. Sofonea, A class of variational-hemivariational inequalities with applications to frictional contact problems, SIAM J. Math. Anal., 46 (2014), 3891-3912.  doi: 10.1137/140963248. [18] W. M. Han, M. Shillor and M. Sofonea, Variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance, friction and damage, J. Comput. Appl. Math., 137 (2001), 377-398.  doi: 10.1016/S0377-0427(00)00707-X. [19] W. M. Han and M. Sofonea, Numerical analysis of hemivariational inequalities in contact mechanics, Acta Numer., 28 (2019), 175-286.  doi: 10.1017/S0962492919000023. [20] W. M. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, AMS/IP Studies in Advanced Mathematics, 30. American Mathematical Society, Providence, RI, International Press, omerville, MA, 2002. [21] W. M. Han, M. Sofonea and M. Barboteu, Numerical analysis of elliptic hemivariational inequalities, SIAM J. Numer. Anal., 55 (2017), 640-663.  doi: 10.1137/16M1072085. [22] L. Jianu, M. Shillor and M. Sofonea, A viscoelastic frictioness contact problem with adhesion, Appl. Anal., 80 (2001), 233-255.  doi: 10.1080/00036810108840990. [23] Y. X. Li and Z. H. Liu, A quasistatic contact problem for viscoelastic materials with friction and damage, Nonlinear Anal., 73 (2010), 2221-2229.  doi: 10.1016/j.na.2010.05.051. [24] S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, 26. Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5. [25] Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Monographs and Textbooks in Pure and Applied Mathematics, 188. Marcel Dekker, Inc., New York, 1995. [26] B. Nedjar, A theoretical and computational setting for a geometrically nonlinear gradient damage modelling framework, Comput. Mech., 30 (2002), 65-80.  doi: 10.1007/s00466-002-0368-1. [27] M. Sofonea, W. M. Han and M. Shillor, Analysis and Approximation of Contact Problems with Adhesion or Damage, Pure and Applied Mathematics (Boca Raton), 276. Chapman & Hall/CRC, Boca Raton, FL, 2006. [28] M. Sofonea and T. Raafat, An electro-viscoelastic frictional contact problem with damage, Appl. Anal., 86 (2007), 503-518.  doi: 10.1080/00036810701286304. [29] W. Xu, Z. P. Huang, W. M. Han, W. B. Chen and C. Wang, Numerical analysis of history-dependent hemivariational inequalities and applications to viscoelastic contact problems with normal penetration, Comput. Math. Appl., 77 (2019), 2596-2607.  doi: 10.1016/j.camwa.2018.12.038. [30] H. L. Xuan, X. L. Cheng, W. M. Han and Q. C. Xiao, Numerical analysis of a dynamic contact problems with history-dependent operators, Numer. Math-theory. Me., 13 (2020), 569-594.
Reference configuration of the two-dimensional example
the deformed configuration at t = 0.125s, t = 0.5s, t = 0.75s and t = 1s
the damage field at t = 0.5s and t = 1s
the adhesion field at several times
the deformed configuration at several times
the adhesion field at several times
the deformed configuration at t = 0.5s and t = 1s
the damage field at t = 0.5s and t = 1s
the deformed configuration at $\gamma_\nu = 1$ and $\gamma_\nu = 10$, respectively
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