American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum
  • DCDS-B Home
  • This Issue
  • Next Article
    A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations
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

[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

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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  the deformed configuration at t = 0.125s, t = 0.5s, t = 0.75s and t = 1s
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  the damage field at t = 0.5s and t = 1s
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  the adhesion field at several times
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  the deformed configuration at several times
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  the adhesion field at several times
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  the deformed configuration at t = 0.5s and t = 1s
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  the damage field at t = 0.5s and t = 1s
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  the deformed configuration at $ \gamma_\nu = 1 $ and $ \gamma_\nu = 10 $, respectively
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Stanisław Migórski, Yi-bin Xiao, Jing Zhao. Fully history-dependent evolution hemivariational inequalities with constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020047
[2]

Zhenhai Liu, Van Thien Nguyen, Jen-Chih Yao, Shengda Zeng. History-dependent differential variational-hemivariational inequalities with applications to contact mechanics. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020044
[3]

Xiaoliang Cheng, Stanisław Migórski, Anna Ochal, Mircea Sofonea. Analysis of two quasistatic history-dependent contact models. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2425-2445. doi: 10.3934/dcdsb.2014.19.2425
[4]

Mircea Sofonea, Meir Shillor. A viscoplastic contact problem with a normal compliance with limited penetration condition and history-dependent stiffness coefficient. Communications on Pure & Applied Analysis, 2014, 13 (1) : 371-387. doi: 10.3934/cpaa.2014.13.371
[5]

Huan-Zhen Chen, Zhao-Jie Zhou, Hong Wang, Hong-Ying Man. An optimal-order error estimate for a family of characteristic-mixed methods to transient convection-diffusion problems. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 325-341. doi: 10.3934/dcdsb.2011.15.325
[6]

Stanisław Migórski. A note on optimal control problem for a hemivariational inequality modeling fluid flow. Conference Publications, 2013, 2013 (special) : 545-554. doi: 10.3934/proc.2013.2013.545
[7]

Leszek Gasiński. Optimal control problem of Bolza-type for evolution hemivariational inequality. Conference Publications, 2003, 2003 (Special) : 320-326. doi: 10.3934/proc.2003.2003.320
[8]

Changjie Fang, Weimin Han. Stability analysis and optimal control of A stationary Stokes hemivariational inequality. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020046
[9]

Patrick Henning, Mario Ohlberger. A-posteriori error estimate for a heterogeneous multiscale approximation of advection-diffusion problems with large expected drift. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1393-1420. doi: 10.3934/dcdss.2016056
[10]

Changjie Fang, Weimin Han. Well-posedness and optimal control of a hemivariational inequality for nonstationary Stokes fluid flow. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5369-5386. doi: 10.3934/dcds.2016036
[11]

Engu Satynarayana, Manas R. Sahoo, Manasa M. Higher order asymptotic for Burgers equation and Adhesion model. Communications on Pure & Applied Analysis, 2017, 16 (1) : 253-272. doi: 10.3934/cpaa.2017012
[12]

Svetlana Matculevich, Pekka Neittaanmäki, Sergey Repin. A posteriori error estimates for time-dependent reaction-diffusion problems based on the Payne--Weinberger inequality. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2659-2677. doi: 10.3934/dcds.2015.35.2659
[13]

Zijia Peng, Cuiming Ma, Zhonghui Liu. Existence for a quasistatic variational-hemivariational inequality. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020058
[14]

Gabriele Bonanno, Beatrice Di Bella. Fourth-order hemivariational inequalities. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 729-739. doi: 10.3934/dcdss.2012.5.729
[15]

Yusuke Murase, Atsushi Kadoya, Nobuyuki Kenmochi. Optimal control problems for quasi-variational inequalities and its numerical approximation. Conference Publications, 2011, 2011 (Special) : 1101-1110. doi: 10.3934/proc.2011.2011.1101
[16]

Mohamed Assellaou, Olivier Bokanowski, Hasnaa Zidani. Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3933-3964. doi: 10.3934/dcds.2015.35.3933
[17]

Stanislaw Migórski. Hemivariational inequality for a frictional contact problem in elasto-piezoelectricity. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1339-1356. doi: 10.3934/dcdsb.2006.6.1339
[18]

Lijing Xi, Yuying Zhou, Yisheng Huang. A class of quasilinear elliptic hemivariational inequality problems on unbounded domains. Journal of Industrial & Management Optimization, 2014, 10 (3) : 827-837. doi: 10.3934/jimo.2014.10.827
[19]

Zhenhai Liu, Stanislaw Migórski. Noncoercive damping in dynamic hemivariational inequality with application to problem of piezoelectricity. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 129-143. doi: 10.3934/dcdsb.2008.9.129
[20]

Rua Murray. Approximation error for invariant density calculations. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 535-557. doi: 10.3934/dcds.1998.4.535

2019 Impact Factor: 1.27

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences