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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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
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