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doi: 10.3934/dcdsb.2020205

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

Hailing Xuan , and  Xiaoliang Cheng

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

* Corresponding author: Hailing Xuan

Received  March 2020 Revised  April 2020 Published  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.

Keywords: Hemivariational inequality, history-dependent operators, numerical approximation, optimal order error estimate, damage, adhesion.
Mathematics Subject Classification: Primary:65M15, 74G30, 65N22;Secondary:74M15, 47J20.
Citation: Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulation of an adhesive contact problem with damage and long memory. Discrete & Continuous Dynamical Systems - B, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[15]

M. Frémond, Adhérence des solids, J. Mécan. Théor. Appl., 6 (1987), 383-407.   Google Scholar

[16]

M. Frémond and B. Nedjar, Damage, gradient of damage and principle of virtual work, Internat. J. Solids and Structures, 33 (1996), 1083-1103.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

M. Sofonea and T. Raafat, An electro-viscoelastic frictional contact problem with damage, Appl. Anal., 86 (2007), 503-518.  doi: 10.1080/00036810701286304.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[15]

M. Frémond, Adhérence des solids, J. Mécan. Théor. Appl., 6 (1987), 383-407.   Google Scholar

[16]

M. Frémond and B. Nedjar, Damage, gradient of damage and principle of virtual work, Internat. J. Solids and Structures, 33 (1996), 1083-1103.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

M. Sofonea and T. Raafat, An electro-viscoelastic frictional contact problem with damage, Appl. Anal., 86 (2007), 503-518.  doi: 10.1080/00036810701286304.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

Figure 1.  Reference configuration of the two-dimensional example
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Figure 2.  the deformed configuration at t = 0.125s, t = 0.5s, t = 0.75s and t = 1s
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Figure 3.  the damage field at t = 0.5s and t = 1s
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Figure 4.  the adhesion field at several times
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Figure 5.  the deformed configuration at several times
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Figure 6.  the adhesion field at several times
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Figure 7.  the deformed configuration at t = 0.5s and t = 1s
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Figure 8.  the damage field at t = 0.5s and t = 1s
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Figure 9.  the deformed configuration at $ \gamma_\nu = 1 $ and $ \gamma_\nu = 10 $, respectively
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