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December  2020, 9(4): 981-993. doi: 10.3934/eect.2020060

## A dynamic viscoelastic problem with friction and rate-depending contact interactions

 Aix Marseille Univ, CNRS, Centrale Marseille, LMA UMR 7031, Marseille, France, Laboratoire de Mécanique et d'Acoustique, 4 Impasse Nikola Tesla, CS 40006, 13453 Marseille Cedex 13, France

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

Received  December 2019 Revised  January 2020 Published  December 2020 Early access  May 2020

The aim of this work is to study a dynamic problem that constitutes a unified approach to describe some rate-depending interactions between the boundaries of two viscoelastic bodies, including relaxed unilateral contact, pointwise friction or adhesion conditions. The classical formulation of the problem is presented and two variational formulations are given as three and four-field evolution implicit equations. Based on some approximation results and an equivalent fixed point problem for a multivalued function, we prove the existence of solutions to these variational evolution problems.

Citation: Marius Cocou. A dynamic viscoelastic problem with friction and rate-depending contact interactions. Evolution Equations and Control Theory, 2020, 9 (4) : 981-993. doi: 10.3934/eect.2020060
##### References:
 [1] L.-E. Andersson, Existence results for quasistatic contact problems with Coulomb friction, Appl. Math. Optim., 42 (2000), 169-202.  doi: 10.1007/s002450010009. [2] P. Boieri, F. Gastaldi and D. Kinderlehrer, Existence, uniqueness, and regularity results for the two-body contact problem, Appl. Math. Optim., 15 (1987), 251-277.  doi: 10.1007/BF01442654. [3] O. Chau, W. Han and M. Sofonea, A dynamic frictional contact problem with normal damped response, Acta Appl. Math., 71 (2002), 159-178.  doi: 10.1023/A:1014501802247. [4] O. Chau, M. Shillor and M. Sofonea, Dynamic frictionless contact with adhesion, Z. Angew. Math. Phys., 55 (2004), 32-47.  doi: 10.1007/s00033-003-1089-9. [5] M. Cocou, Existence of solutions of a dynamic Signorini's problem with nonlocal friction in viscoelasticity, Z. Angew. Math. Phys., 53 (2002), 1099-1109.  doi: 10.1007/PL00012615. [6] M. Cocou, A class of implicit evolution inequalities and applications to dynamic contact problems, Ann. Univ. Buchar. Math. Ser., 4(LXII) (2013), 167-178. [7] M. Cocou, A class of dynamic contact problems with Coulomb friction in viscoelasticity, Nonlinear Anal. Real World Appl., 22 (2015), 508-519.  doi: 10.1016/j.nonrwa.2014.08.012. [8] M. Cocou, A variational inequality and applications to quasistatic problems with Coulomb friction, Appl. Anal., 97 (2018), 1357-1371.  doi: 10.1080/00036811.2017.1376249. [9] M. Cocou and R. Rocca, Existence results for unilateral quasistatic contact problems with friction and adhesion, M2AN Math. Model. Numer. Anal., 34 (2000), 981-1001.  doi: 10.1051/m2an:2000112. [10] M. Cocou and G. Scarella, Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body, Z. Angew. Math. Phys., 57 (2006), 523-546.  doi: 10.1007/s00033-005-0013-x. [11] M. Cocou, M. Schryve and M. Raous, A dynamic unilateral contact problem with adhesion and friction in viscoelasticity, Z. Angew. Math. Phys., 61 (2010), 721-743.  doi: 10.1007/s00033-009-0027-x. [12] C. Eck, J. Jarušek and M. Krbec, Unilateral contact problems. Variational methods and existence theorems, in Pure and Applied Mathematics (Boca Raton), 270, Chapman & Hall/CRC, Boca Raton, FL, 2005. doi: 10.1201/9781420027365. [13] K. Fan, Fixed point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 121-126.  doi: 10.1073/pnas.38.2.121. [14] W. Han and M. Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity, in AMS/IP Studies in Advanced Mathematics, 30, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2002. [15] J. Jarušek, Dynamic contact problems with given friction for viscoelastic bodies, Czechoslovak Math. J., 46 (1996), 475-487. [16] N. Kikuchi and J. T. Oden, Contact problems in elasticity: A study of variational inequalities and finite element methods, in SIAM Studies in Applied Mathematics, 8, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. doi: 10.1137/1.9781611970845. [17] K. L. Kuttler, Dynamic friction contact problems for general normal and friction laws, Nonlinear Anal., 28 (1997), 559-575.  doi: 10.1016/0362-546X(95)00170-Z. [18] K. L. Kuttler and M. Shillor, Dynamic bilateral contact with discontinuous friction coefficient, Nonlinear Anal., 45 (2001), 309-327.  doi: 10.1016/S0362-546X(99)00345-4. [19] K. L. Kuttler and M. Shillor, Dynamic contact with Signorini's condition and slip rate dependent friction, Electron. J. Differential Equations, 2004 (2004), 1-21. [20] K. L. Kuttler, M. Shillor and J. R. Fernández, Existence and regularity for dynamic viscoelastic adhesive contact with damage, Appl. Math. Optim., 53 (2006), 31-66.  doi: 10.1007/s00245-005-0837-y. [21] J. A. C. Martins and J. T. Oden, Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws, Nonlinear Anal., 11 (1987), 407-428.  doi: 10.1016/0362-546X(87)90055-1. [22] S. Migórski and A. Ochal, A unified approach to dynamic contact problems in viscoelasticity, J. Elasticity, 83 (2006), 247-275.  doi: 10.1007/s10659-005-9034-0. [23] S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities, Springer, New York, 2013. [24] Z. Naniewicz and P. D. Panagiotopoulos, Mathematical theory of hemivariational inequalities and applications, in Monographs and Textbooks in Pure and Applied Mathematics, 188, Marcel Dekker, Inc., New York, 1995. [25] P. D. Panagiotopoulos, Hemivariational Inequalities. Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-51677-1. [26] P. J. Rabier and O. V. Savin, Fixed points of multi-valued maps and static Coulomb friction problems, J. Elasticity, 58 (2000), 155-176.  doi: 10.1023/A:1007619416280. [27] M. Raous, L. Cangémi and M. Cocou, A consistent model coupling adhesion, friction, and unilateral contact, Comput. Methods Appl. Mech. Engrg., 177 (1999), 383-399.  doi: 10.1016/S0045-7825(98)00389-2. [28] R. Rocca and M. Cocou, Existence and approximation of a solution to quasistatic Signorini problem with local friction, Internat. J. Engrg. Sci., 39 (2001), 1233-1255.  doi: 10.1016/S0020-7225(00)00089-6. [29] R. Rocca and M. Cocou, Numerical analysis of quasistatic unilateral contact problems with local friction, SIAM J. Numer. Anal., 39 (2001), 1324-1342.  doi: 10.1137/S0036142900382600. [30] M. Shillor, M. Sofonea and J. J. Telega, Models and analysis of quasistatic contact, in Lecture Notes in Physics, 655, Springer, Berlin, Heidelberg, 2004. [31] J. Simon, Compact sets in the space $L^{p}(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360. [32] M. Sofonea, W. Han and M. Shillor, Analysis and approximation of contact problems with adhesion or damage, in Pure and Applied Mathematics (Boca Raton), 276, Chapman & Hall/CRC, Boca Raton, FL, 2006.

show all references

##### References:
 [1] L.-E. Andersson, Existence results for quasistatic contact problems with Coulomb friction, Appl. Math. Optim., 42 (2000), 169-202.  doi: 10.1007/s002450010009. [2] P. Boieri, F. Gastaldi and D. Kinderlehrer, Existence, uniqueness, and regularity results for the two-body contact problem, Appl. Math. Optim., 15 (1987), 251-277.  doi: 10.1007/BF01442654. [3] O. Chau, W. Han and M. Sofonea, A dynamic frictional contact problem with normal damped response, Acta Appl. Math., 71 (2002), 159-178.  doi: 10.1023/A:1014501802247. [4] O. Chau, M. Shillor and M. Sofonea, Dynamic frictionless contact with adhesion, Z. Angew. Math. Phys., 55 (2004), 32-47.  doi: 10.1007/s00033-003-1089-9. [5] M. Cocou, Existence of solutions of a dynamic Signorini's problem with nonlocal friction in viscoelasticity, Z. Angew. Math. Phys., 53 (2002), 1099-1109.  doi: 10.1007/PL00012615. [6] M. Cocou, A class of implicit evolution inequalities and applications to dynamic contact problems, Ann. Univ. Buchar. Math. Ser., 4(LXII) (2013), 167-178. [7] M. Cocou, A class of dynamic contact problems with Coulomb friction in viscoelasticity, Nonlinear Anal. Real World Appl., 22 (2015), 508-519.  doi: 10.1016/j.nonrwa.2014.08.012. [8] M. Cocou, A variational inequality and applications to quasistatic problems with Coulomb friction, Appl. Anal., 97 (2018), 1357-1371.  doi: 10.1080/00036811.2017.1376249. [9] M. Cocou and R. Rocca, Existence results for unilateral quasistatic contact problems with friction and adhesion, M2AN Math. Model. Numer. Anal., 34 (2000), 981-1001.  doi: 10.1051/m2an:2000112. [10] M. Cocou and G. Scarella, Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body, Z. Angew. Math. Phys., 57 (2006), 523-546.  doi: 10.1007/s00033-005-0013-x. [11] M. Cocou, M. Schryve and M. Raous, A dynamic unilateral contact problem with adhesion and friction in viscoelasticity, Z. Angew. Math. Phys., 61 (2010), 721-743.  doi: 10.1007/s00033-009-0027-x. [12] C. Eck, J. Jarušek and M. Krbec, Unilateral contact problems. Variational methods and existence theorems, in Pure and Applied Mathematics (Boca Raton), 270, Chapman & Hall/CRC, Boca Raton, FL, 2005. doi: 10.1201/9781420027365. [13] K. Fan, Fixed point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 121-126.  doi: 10.1073/pnas.38.2.121. [14] W. Han and M. Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity, in AMS/IP Studies in Advanced Mathematics, 30, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2002. [15] J. Jarušek, Dynamic contact problems with given friction for viscoelastic bodies, Czechoslovak Math. J., 46 (1996), 475-487. [16] N. Kikuchi and J. T. Oden, Contact problems in elasticity: A study of variational inequalities and finite element methods, in SIAM Studies in Applied Mathematics, 8, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. doi: 10.1137/1.9781611970845. [17] K. L. Kuttler, Dynamic friction contact problems for general normal and friction laws, Nonlinear Anal., 28 (1997), 559-575.  doi: 10.1016/0362-546X(95)00170-Z. [18] K. L. Kuttler and M. Shillor, Dynamic bilateral contact with discontinuous friction coefficient, Nonlinear Anal., 45 (2001), 309-327.  doi: 10.1016/S0362-546X(99)00345-4. [19] K. L. Kuttler and M. Shillor, Dynamic contact with Signorini's condition and slip rate dependent friction, Electron. J. Differential Equations, 2004 (2004), 1-21. [20] K. L. Kuttler, M. Shillor and J. R. Fernández, Existence and regularity for dynamic viscoelastic adhesive contact with damage, Appl. Math. Optim., 53 (2006), 31-66.  doi: 10.1007/s00245-005-0837-y. [21] J. A. C. Martins and J. T. Oden, Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws, Nonlinear Anal., 11 (1987), 407-428.  doi: 10.1016/0362-546X(87)90055-1. [22] S. Migórski and A. Ochal, A unified approach to dynamic contact problems in viscoelasticity, J. Elasticity, 83 (2006), 247-275.  doi: 10.1007/s10659-005-9034-0. [23] S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities, Springer, New York, 2013. [24] Z. Naniewicz and P. D. Panagiotopoulos, Mathematical theory of hemivariational inequalities and applications, in Monographs and Textbooks in Pure and Applied Mathematics, 188, Marcel Dekker, Inc., New York, 1995. [25] P. D. Panagiotopoulos, Hemivariational Inequalities. Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-51677-1. [26] P. J. Rabier and O. V. Savin, Fixed points of multi-valued maps and static Coulomb friction problems, J. Elasticity, 58 (2000), 155-176.  doi: 10.1023/A:1007619416280. [27] M. Raous, L. Cangémi and M. Cocou, A consistent model coupling adhesion, friction, and unilateral contact, Comput. Methods Appl. Mech. Engrg., 177 (1999), 383-399.  doi: 10.1016/S0045-7825(98)00389-2. [28] R. Rocca and M. Cocou, Existence and approximation of a solution to quasistatic Signorini problem with local friction, Internat. J. Engrg. Sci., 39 (2001), 1233-1255.  doi: 10.1016/S0020-7225(00)00089-6. [29] R. Rocca and M. Cocou, Numerical analysis of quasistatic unilateral contact problems with local friction, SIAM J. Numer. Anal., 39 (2001), 1324-1342.  doi: 10.1137/S0036142900382600. [30] M. Shillor, M. Sofonea and J. J. Telega, Models and analysis of quasistatic contact, in Lecture Notes in Physics, 655, Springer, Berlin, Heidelberg, 2004. [31] J. Simon, Compact sets in the space $L^{p}(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360. [32] M. Sofonea, W. Han and M. Shillor, Analysis and approximation of contact problems with adhesion or damage, in Pure and Applied Mathematics (Boca Raton), 276, Chapman & Hall/CRC, Boca Raton, FL, 2006.
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