doi: 10.3934/mcrf.2021037
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Numerical analysis and simulations of a frictional contact problem with damage and memory

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

* Corresponding author: Hailing Xuan

Received  November 2020 Revised  March 2021 Early access July 2021

In this paper, we study a frictional contact model which takes into account the damage and the memory. The deformable body consists of a viscoelastic material and the process is assumed to be quasistatic. The mechanical damage of the material which caused by the tension or the compression is included in the constitutive law and the damage function is modelled by a nonlinear parabolic inclusion. Then the variational formulation of the model is governed by a coupled system consisting of a history-dependent hemivariational inequality and a nonlinear parabolic variational inequality. We introduce and study a fully discrete scheme of the problem and derive error estimates for numerical solutions. Under appropriate solution regularity assumptions, an optimal order error estimate is derived for the linear finite element method. Several numerical experiments for the contact problem are given for providing numerical evidence of the theoretical results.

Citation: Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulations of a frictional contact problem with damage and memory. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2021037
References:
[1]

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.

[2]

K. Atkinson and W. M. Han, Theortical Numerical Analysis. A Functional Analysis Framework, Third edition, Texts in Applied Mathematics, Vol. 39, Springer, Dordrecht, 2009. doi: 10.1007/978-1-4419-0458-4.

[3]

K. BartoszX. L. ChengP. KalitaY. J. Yu 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.

[4]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Elements Methods, Third edition, Texts in Applied Mathematics, Vol. 15, Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0.

[5]

M. CampoJ. R. Fernández and A. Rodrguez-arós, A quasistatic contact problem with normal compliance and damage involving viscoelastic materials with long memory, Appl. Numer. Math., 58 (2008), 1274-1290.  doi: 10.1016/j.apnum.2007.07.003.

[6]

M. CampoJ. 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.

[7]

X. L. Cheng and Q. C. Xiao, Numerical analysis of an adhesive contact problem with long memory, East Asian J. Appl. Math., 10 (2020), 72-88. 

[8]

X. L. ChengQ. C. XiaoS. 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.

[9]

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.

[10]

M. Frémond and B. Nedjar, Damage, gradient of damage and principle of virtual power, Internat. J. Solids and Structures, 33 (1996), 1083-1103.  doi: 10.1016/0020-7683(95)00074-7.

[11]

W. M. HanM. 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.

[12]

W. M. HanM. Sofonea and D. Danan, Numerical analysis of stationary variational-hemivariational inequalities, Numer. Math., 139 (2018), 563-592.  doi: 10.1007/s00211-018-0951-9.

[13]

W. M. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, AMS/IP Studies in Advanced Mathematics, Vol. 30, American Mathematical Society, Providence, RI, International Press, Somerville, MA, 2002. doi: 10.1090/amsip/030.

[14]

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.

[15]

M. Sofonea, W. M. Han and M. Shillor, Analysis and Approximation of Contact Problems with Adhesion or Damage, Pure and Applied Mathematics (Boca Raton), Vol. 276, Chapman & Hall/CRC, Boca Raton, FL, 2006.

[16]

W. XuZ. P. HuangW. M. HanW. 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.

[17]

H. L. XuanX. L. ChengW. M. Han and Q. C. Xiao, Numerical analysis of a dynamic contact problems with history-dependent operators, Numer. Math. Theory Methods Appl., 13 (2020), 569-594.  doi: 10.4208/nmtma.oa-2019-0130.

show all references

References:
[1]

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.

[2]

K. Atkinson and W. M. Han, Theortical Numerical Analysis. A Functional Analysis Framework, Third edition, Texts in Applied Mathematics, Vol. 39, Springer, Dordrecht, 2009. doi: 10.1007/978-1-4419-0458-4.

[3]

K. BartoszX. L. ChengP. KalitaY. J. Yu 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.

[4]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Elements Methods, Third edition, Texts in Applied Mathematics, Vol. 15, Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0.

[5]

M. CampoJ. R. Fernández and A. Rodrguez-arós, A quasistatic contact problem with normal compliance and damage involving viscoelastic materials with long memory, Appl. Numer. Math., 58 (2008), 1274-1290.  doi: 10.1016/j.apnum.2007.07.003.

[6]

M. CampoJ. 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.

[7]

X. L. Cheng and Q. C. Xiao, Numerical analysis of an adhesive contact problem with long memory, East Asian J. Appl. Math., 10 (2020), 72-88. 

[8]

X. L. ChengQ. C. XiaoS. 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.

[9]

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.

[10]

M. Frémond and B. Nedjar, Damage, gradient of damage and principle of virtual power, Internat. J. Solids and Structures, 33 (1996), 1083-1103.  doi: 10.1016/0020-7683(95)00074-7.

[11]

W. M. HanM. 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.

[12]

W. M. HanM. Sofonea and D. Danan, Numerical analysis of stationary variational-hemivariational inequalities, Numer. Math., 139 (2018), 563-592.  doi: 10.1007/s00211-018-0951-9.

[13]

W. M. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, AMS/IP Studies in Advanced Mathematics, Vol. 30, American Mathematical Society, Providence, RI, International Press, Somerville, MA, 2002. doi: 10.1090/amsip/030.

[14]

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.

[15]

M. Sofonea, W. M. Han and M. Shillor, Analysis and Approximation of Contact Problems with Adhesion or Damage, Pure and Applied Mathematics (Boca Raton), Vol. 276, Chapman & Hall/CRC, Boca Raton, FL, 2006.

[16]

W. XuZ. P. HuangW. M. HanW. 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.

[17]

H. L. XuanX. L. ChengW. M. Han and Q. C. Xiao, Numerical analysis of a dynamic contact problems with history-dependent operators, Numer. Math. Theory Methods Appl., 13 (2020), 569-594.  doi: 10.4208/nmtma.oa-2019-0130.

Figure 1.  Reference configuration of the two-dimensional example
Figure 2.  Results with deformed configuration(left) and damage field(right)
Figure 3.  Results with $ a = 1, \ \mu = 0 $ (left) and with $ a = 1, \ \mu = 1 $ (right)
Figure 4.  Results with deformed configuration(left) and damage field(right)
Figure 5.  numerical errors
Table 1.  numerical errors
$ h+k $ 0.046875 0.09375 0.1875 0.375
$ \Vert {\boldsymbol{u }}- {\boldsymbol{u }}^{hk}\Vert_V+\Vert \zeta-\zeta^{hk}\Vert_{Z_0} $ $ 0.324\times 10^{-1} $ $ 0.631\times 10^{-1} $ 0.1082 0.1940
$ h+k $ 0.046875 0.09375 0.1875 0.375
$ \Vert {\boldsymbol{u }}- {\boldsymbol{u }}^{hk}\Vert_V+\Vert \zeta-\zeta^{hk}\Vert_{Z_0} $ $ 0.324\times 10^{-1} $ $ 0.631\times 10^{-1} $ 0.1082 0.1940
[1]

Furi Guo, Jinrong Wang, Jiangfeng Han. Impulsive hemivariational inequality for a class of history-dependent quasistatic frictional contact problems. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021057

[2]

Stanisław Migórski, Yi-bin Xiao, Jing Zhao. Fully history-dependent evolution hemivariational inequalities with constraints. Evolution Equations and Control Theory, 2020, 9 (4) : 1089-1114. doi: 10.3934/eect.2020047

[3]

Zhenhai Liu, Van Thien Nguyen, Jen-Chih Yao, Shengda Zeng. History-dependent differential variational-hemivariational inequalities with applications to contact mechanics. Evolution Equations and Control Theory, 2020, 9 (4) : 1073-1087. doi: 10.3934/eect.2020044

[4]

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

[5]

Soumia Saïdi, Fatima Fennour. Second-order problems involving time-dependent subdifferential operators and application to control. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022019

[6]

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

[7]

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 and Continuous Dynamical Systems - B, 2011, 15 (2) : 325-341. doi: 10.3934/dcdsb.2011.15.325

[8]

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

[9]

Changjie Fang, Weimin Han. Stability analysis and optimal control of a stationary Stokes hemivariational inequality. Evolution Equations and Control Theory, 2020, 9 (4) : 995-1008. doi: 10.3934/eect.2020046

[10]

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

[11]

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

[12]

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

[13]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5217-5226. doi: 10.3934/dcdsb.2020340

[14]

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

[15]

Masahiro Kubo. Quasi-subdifferential operators and evolution equations. Conference Publications, 2013, 2013 (special) : 447-456. doi: 10.3934/proc.2013.2013.447

[16]

Zijia Peng, Cuiming Ma, Zhonghui Liu. Existence for a quasistatic variational-hemivariational inequality. Evolution Equations and Control Theory, 2020, 9 (4) : 1153-1165. doi: 10.3934/eect.2020058

[17]

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

[18]

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

[19]

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

[20]

Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu. Periodic solutions for time-dependent subdifferential evolution inclusions. Evolution Equations and Control Theory, 2017, 6 (2) : 277-297. doi: 10.3934/eect.2017015

2021 Impact Factor: 1.141

Article outline

Figures and Tables

[Back to Top]