# American Institute of Mathematical Sciences

doi: 10.3934/eect.2020049

## On dynamic contact problem with generalized Coulomb friction, normal compliance and damage

 1 Pedagogical University of Krakow, Department of Mathematics, ul. 2, 30-084 Kraków, Poland 2 Jagiellonian University, Faculty of Mathematics and Computer Science, ul. Łojasiewicza 6, 30-348 Kraków, Poland

* Corresponding author

Dedicated to Professor Meir Shillor on the occasion of his 70th birthday

Received  October 2019 Published  March 2020

Fund Project: The research was supported by the H2020-MSCA-RISE-2018 Research and Innovation Staff Exchange Scheme Fellowship within the Project No. 823731 CONMECH. Work of PK was also supported by NCN (National Science Center) of the Republic of Poland by the projects no DEC-2017/25/B/ST1/00302 and UMO-2016/22/A/ST1/00077. The authors wish to thank Nikolaos S. Papageorgiou for useful comments

We formulate a dynamic problem which governs the displacement of a viscoelastic body which, on one hand, can come into frictional contact with a penetrable foundation, and, on the other hand, may undergo material damage. We formulate and prove the theorem on the existence and uniqueness of the weak solution to the formulated problem.

Citation: Leszek Gasiński, Piotr Kalita. On dynamic contact problem with generalized Coulomb friction, normal compliance and damage. Evolution Equations & Control Theory, doi: 10.3934/eect.2020049
##### References:
 [1] K. T. Andrews and M. Shillor, Thermomechanical behaviour of a damageable beam in contact with two stops, Appl. Anal., 85 (2006), 845-865.  doi: 10.1080/00036810600792857.  Google Scholar [2] K. T. Andrews, S. Anderson, R. S. R. Menike, M. Shillor, R. Swaminathan and J. Yuzwalk, Vibrations of a damageable string, Fluids and waves, Contemp. Math., Amer. Math. Soc., Providence, RI, 440 (2007), 1-13.  doi: 10.1090/conm/440/08474.  Google Scholar [3] V. Barbu, Optimal Control of Variational Inequalities, Research Notes in Mathematics, 100. Pitman, Boston, MA, 1984.  Google Scholar [4] E. Bonetti and G. Schimperna, Local existence for Frémond's model of damage in elastic materials, Continuum Mech. Therm., 16 (2004), 319-335.  doi: 10.1007/s00161-003-0152-2.  Google Scholar [5] P. G. Ciarlet, On Korn's inequality, Chin. Ann. Math. Ser. B, 31 (2010), 607-618.  doi: 10.1007/s11401-010-0606-3.  Google Scholar [6] J. C. Chipman, A. Roux, M. Shillor and M. Sofonea, A damageable spring, Machine Dyn. Res., 35 (2011), 82-96.   Google Scholar [7] F. H. Clarke, Optimization and Nonsmooth Analysis, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.  Google Scholar [8] Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic/Plenum Publishers, Boston, 2003. doi: 10.1007/978-1-4419-9158-4.  Google Scholar [9] Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer Academic Publishers, Boston, MA, 2003.  Google Scholar [10] M. Frémond, Non-Smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9.  Google Scholar [11] M. Frémond, K. L. Kuttler and M. Shillor, Existence and uniqueness of solutions for a one-dimensional damage model, J. Math. Anal. Appl., 229 (1999), 271-294.  doi: 10.1006/jmaa.1998.6160.  Google Scholar [12] L. Gasiński and P. Kalita, On quasi-static contact problem with generalized Coulomb friction, normal compliance and damage, Euro. J. Appl. Math., 27 (2016), 625-646.  doi: 10.1017/S0956792515000583.  Google Scholar [13] L. Gasiński and A. Ochal, Dynamic thermoviscoelastic problem with friction and damage, Nonlinear Anal. Real World Appl., 21 (2015), 63-75.  doi: 10.1016/j.nonrwa.2014.06.004.  Google Scholar [14] L. Gasiński, A. Ochal and M. Shillor, Variational-hemivariational approach to a quasistatic viscoelastic problem with normal compliance, friction and material damage, Z. Anal. Anwend., 34 (2015), 251-275.  doi: 10.4171/ZAA/1538.  Google Scholar [15] 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 [16] P. Kalita, Convergence of Rothe scheme for hemivariational inequalities of parabolic type, Int. J. Numer. Anal. Mod., 10 (2013), 445-465.   Google Scholar [17] P. Kalita, Semidiscrete variable time-step $\theta$-scheme for nonmonotone evolution inclusion, arXiv: 1402.3721v1. Google Scholar [18] K. L. Kuttler, J. Purcell and M. Shillor, Analysis and simulations of a contact problem for a nonlinear dynamic beam with a crack, Q. J. Mech. Appl. Math., 65 (2012), 1-25.  doi: 10.1093/qjmam/hbr018.  Google Scholar [19] K. L. Kuttler and M. Shillor, Quasistatic evolution of damage in an elastic body, Nonlinear Anal. Real World Appl., 7 (2006), 674-699.  doi: 10.1016/j.nonrwa.2005.03.026.  Google Scholar [20] 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 [21] 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 [22] 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 [23] T. Roubíček, Nonlinear Partial Differential Equations with Applications, Second edition, International Series of Numerical Mathematics, 153. Birkhäuser/Springer Basel AG, Basel, 2013. doi: 10.1007/978-3-0348-0513-1.  Google Scholar [24] M. Shillor, M. Sofonea and J. J. Telega, Models and Analysis of Quasistatic Contact, Lecture Notes in Physics, 655. Springer, Berlin, 2004. doi: 10.1007/b99799.  Google Scholar [25] M. Sofonea, C. Avramescu and A. Matei, A fixed point result with applications in the study of viscoplastic frictionless contact problems, Commun. Pure Appl. Ana., 7 (2008), 645-658.  doi: 10.3934/cpaa.2008.7.645.  Google Scholar [26] M. Sofonea, W. M. Han and M. Shillor, Analysis and Approximations of Contact Problems with Adhesion or Damage, Pure and Applied Mathematics, 276. Chapman & Hall/CRC, Boca Raton, FL, 2006.  Google Scholar [27] P. Szafraniec, Dynamic nonsmooth frictional contact problems with damage in thermoviscoelasticity, Math. Mech. Solids, 21 (2016), 525-538.  doi: 10.1177/1081286514527860.  Google Scholar

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##### References:
 [1] K. T. Andrews and M. Shillor, Thermomechanical behaviour of a damageable beam in contact with two stops, Appl. Anal., 85 (2006), 845-865.  doi: 10.1080/00036810600792857.  Google Scholar [2] K. T. Andrews, S. Anderson, R. S. R. Menike, M. Shillor, R. Swaminathan and J. Yuzwalk, Vibrations of a damageable string, Fluids and waves, Contemp. Math., Amer. Math. Soc., Providence, RI, 440 (2007), 1-13.  doi: 10.1090/conm/440/08474.  Google Scholar [3] V. Barbu, Optimal Control of Variational Inequalities, Research Notes in Mathematics, 100. Pitman, Boston, MA, 1984.  Google Scholar [4] E. Bonetti and G. Schimperna, Local existence for Frémond's model of damage in elastic materials, Continuum Mech. Therm., 16 (2004), 319-335.  doi: 10.1007/s00161-003-0152-2.  Google Scholar [5] P. G. Ciarlet, On Korn's inequality, Chin. Ann. Math. Ser. B, 31 (2010), 607-618.  doi: 10.1007/s11401-010-0606-3.  Google Scholar [6] J. C. Chipman, A. Roux, M. Shillor and M. Sofonea, A damageable spring, Machine Dyn. Res., 35 (2011), 82-96.   Google Scholar [7] F. H. Clarke, Optimization and Nonsmooth Analysis, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.  Google Scholar [8] Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic/Plenum Publishers, Boston, 2003. doi: 10.1007/978-1-4419-9158-4.  Google Scholar [9] Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer Academic Publishers, Boston, MA, 2003.  Google Scholar [10] M. Frémond, Non-Smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9.  Google Scholar [11] M. Frémond, K. L. Kuttler and M. Shillor, Existence and uniqueness of solutions for a one-dimensional damage model, J. Math. Anal. Appl., 229 (1999), 271-294.  doi: 10.1006/jmaa.1998.6160.  Google Scholar [12] L. Gasiński and P. Kalita, On quasi-static contact problem with generalized Coulomb friction, normal compliance and damage, Euro. J. Appl. Math., 27 (2016), 625-646.  doi: 10.1017/S0956792515000583.  Google Scholar [13] L. Gasiński and A. Ochal, Dynamic thermoviscoelastic problem with friction and damage, Nonlinear Anal. Real World Appl., 21 (2015), 63-75.  doi: 10.1016/j.nonrwa.2014.06.004.  Google Scholar [14] L. Gasiński, A. Ochal and M. Shillor, Variational-hemivariational approach to a quasistatic viscoelastic problem with normal compliance, friction and material damage, Z. Anal. Anwend., 34 (2015), 251-275.  doi: 10.4171/ZAA/1538.  Google Scholar [15] 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 [16] P. Kalita, Convergence of Rothe scheme for hemivariational inequalities of parabolic type, Int. J. Numer. Anal. Mod., 10 (2013), 445-465.   Google Scholar [17] P. Kalita, Semidiscrete variable time-step $\theta$-scheme for nonmonotone evolution inclusion, arXiv: 1402.3721v1. Google Scholar [18] K. L. Kuttler, J. Purcell and M. Shillor, Analysis and simulations of a contact problem for a nonlinear dynamic beam with a crack, Q. J. Mech. Appl. Math., 65 (2012), 1-25.  doi: 10.1093/qjmam/hbr018.  Google Scholar [19] K. L. Kuttler and M. Shillor, Quasistatic evolution of damage in an elastic body, Nonlinear Anal. Real World Appl., 7 (2006), 674-699.  doi: 10.1016/j.nonrwa.2005.03.026.  Google Scholar [20] 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 [21] 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 [22] 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 [23] T. Roubíček, Nonlinear Partial Differential Equations with Applications, Second edition, International Series of Numerical Mathematics, 153. Birkhäuser/Springer Basel AG, Basel, 2013. doi: 10.1007/978-3-0348-0513-1.  Google Scholar [24] M. Shillor, M. Sofonea and J. J. Telega, Models and Analysis of Quasistatic Contact, Lecture Notes in Physics, 655. Springer, Berlin, 2004. doi: 10.1007/b99799.  Google Scholar [25] M. Sofonea, C. Avramescu and A. Matei, A fixed point result with applications in the study of viscoplastic frictionless contact problems, Commun. Pure Appl. Ana., 7 (2008), 645-658.  doi: 10.3934/cpaa.2008.7.645.  Google Scholar [26] M. Sofonea, W. M. Han and M. Shillor, Analysis and Approximations of Contact Problems with Adhesion or Damage, Pure and Applied Mathematics, 276. Chapman & Hall/CRC, Boca Raton, FL, 2006.  Google Scholar [27] P. Szafraniec, Dynamic nonsmooth frictional contact problems with damage in thermoviscoelasticity, Math. Mech. Solids, 21 (2016), 525-538.  doi: 10.1177/1081286514527860.  Google Scholar
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