# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2021037
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## 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. Bartosz, X. L. Cheng, P. Kalita, Y. 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. Campo, J. 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. 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. [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. 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. [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. 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. [12] W. M. Han, M. 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. 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. [17] 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 Methods Appl., 13 (2020), 569-594.  doi: 10.4208/nmtma.oa-2019-0130.

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##### 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. Bartosz, X. L. Cheng, P. Kalita, Y. 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. Campo, J. 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. 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. [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. 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. [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. 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. [12] W. M. Han, M. 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. 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. [17] 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 Methods Appl., 13 (2020), 569-594.  doi: 10.4208/nmtma.oa-2019-0130.
Reference configuration of the two-dimensional example
Results with deformed configuration(left) and damage field(right)
Results with $a = 1, \ \mu = 0$ (left) and with $a = 1, \ \mu = 1$ (right)
Results with deformed configuration(left) and damage field(right)
numerical errors
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 0.194
 $h+k$ 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 0.194
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