Variational analysis of a frictional contact problem with wear and damage
Abstract
We study a quasistatic problem describing the contact with friction and wear between a piezoelectric body and a moving foundation. The material is modeled by an electro-viscoelastic constitutive law with long memory and damage. The wear of the contact surface due to friction is taken into account and is described by the differential Archard condition. The contact is modeled with the normal compliance condition and the associated law of dry friction. We present a variational formulation of the problem and establish, under a smallness assumption on the data, the existence and uniqueness of the weak solution. The proof is based on arguments of parabolic evolutionary inequations, elliptic variational inequalities and Banach fixed point.
Keyword : quasistatic process, electro-viscoelastic materials, damage, normal compliance, friction, wear, existence and uniqueness, fixed point arguments, weak solution
How to Cite
Mesai Aoun, M. S., Selmani, . M., & Ahmed, A. A. (2021). Variational analysis of a frictional contact problem with wear and damage. Mathematical Modelling and Analysis, 26(2), 170-187. https://doi.org/10.3846/mma.2021.11942
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
K.T. Andrews, M. Shillor, S. Wright and A. Klarbring. A dynamic thermoviscoelastic contact problem with friction and wear. International journal of engineering science, 35(14):1291–1309, 1997. https://doi.org/10.1016/S0020-7225(97)87426-5
R.C. Batra and J.S. Yang. Saint-Venant’s principle in linear piezoelectricity. Journal of Elasticity, 38(2):209–218, 1995. https://doi.org/10.1007/BF00042498
P. Bisegna, F. Maceri and F. Lebon. The unilateral frictional contact of a piezoelectric body with a rigid support. In Contact mechanics, pp. 347–354. Springer Publishing, 2002. https://doi.org/10.1007/978-94-017-1154-8_37
M. Dalah and M. Sofonea. Antiplane frictional contact of electro-viscoelastic cylinders. Electronic Journal of Differential Equations (EJDE)[electronic only], 2007(161):1–14, 2007.
G. Duvant and J.L. Lions. Inequalities in mechanics and physics, volume 219. Springer Science & Business Media, 2012.
W. Han and M. Sofonea. Evolutionary variational inequalities arising in viscoelastic contact problems. SIAM Journal on Numerical Analysis, 38(2):556– 579, 2000. https://doi.org/10.1137/S0036142998347309
T. Ikeda. Fundamentals of piezoelectricity. Oxford University Press, Oxford, 1990.
L. Kaki and M. Denche. Variational analysis for some frictional contact problems. Boletim da Sociedade Paranaense de Matema´tica, 38(7):21–36, 2020. https://doi.org/10.5269/bspm.v38i7.44258
Z. Lerguet, M. Shillor and M. Sofonea. A frictional contact problem for an electro-viscoelastic body. Electronic Journal of Differential Equations (EJDE)[electronic only], 2007(170):1–16, 2007.
F. Maceri and P. Bisegna. The unilateral frictionless contact of a piezoelectric body with a rigid support. Mathematical and Computer Modelling, 28(4-8):19– 28, 1998. https://doi.org/10.1016/S0895-7177(98)00105-8
J.A. C Martins and J.T. Oden. Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws. Nonlinear Analysis TMA, 11(3):407–428,1987. https://doi.org/10.1016/0362-546X(87)90055-1
S. Migórski. Hemivariational inequality for a frictional contact problem in elastopiezoelectricity. Discrete & Continuous Dynamical Systems-B, 6(6):1339–13356, 2006. https://doi.org/10.3934/dcdsb.2006.6.1339
M. Rochdi, M. Shillor and M. Sofonea. Quasistatic viscoelastic contact with normal compliance and friction. Journal of Elasticity, 51(2):105–126, 1998. https://doi.org/10.1023/A:1007413119583
M. Selmani. A dynamic problem with adhesion and damage in electroviscoelasticity with long-term memory. JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only], 10(1):1–19, 2009.
M. Selmani. A frictional contact problem involving piezoelectric materials with long memory. Mediterranean Journal of Mathematics, 12(3):1177–1197, 2015. https://doi.org/10.1007/s00009-014-0430-1
M. Selmani et al. Frictional contact problem with wear for electro-viscoelastic materials with long memory. Bulletin of the Belgian Mathematical Society-Simon Stevin, 20(3):461–479, 2013. https://doi.org/10.36045/bbms/1378314510
M. Selmani and L. Selmani. A dynamic frictionless contact problem with adhesion and damage. Bulletin of the Polish Academy of Sciences. Mathematics, 55(1):17–34, 2007. https://doi.org/10.4064/ba55-1-3
M. Selmani and L. Selmani. A frictional contact problem with wear and damage for electro-viscoelastic materials. Applications of Mathematics, 55(2):89–109, 2010. https://doi.org/10.1007/s10492-010-0004-x
M. Shillor, M. Sofonea and J.J. Telega. Quasistatic viscoelastic contact with friction and wear diffusion. Quart. Appl. Math, 62(2):379–399, 2004. https://doi.org/10.1090/qam/2054605
M. Sofonea and R. Arhab. An electro-viscoelastic contact problem with adesion. Dynamics of Continuous Discrete and Impulsive Systems Series A, 14(4):577, 2007. https://doi.org/10.1080/00036810701286304
M. Sofonea and E. Essoufi. A piezoelectric contact problem with slip dependent coefficient of friction. Mathematical Modelling and Analysis, 9(3):229–242, 2004. https://doi.org/10.3846/13926292.2004.9637256
M. Sofonea and E. Essoufi. Quasistatic frictional contact of viscoelastic piezoelectric body. Ad . Math. Sci. Appl, 14(3):613–631, 2004.
M. Sofonea, W. Han and M. Shillor. Analysis and approximation of contact problems with adhesion or damage. CRC Press, 2006. https://doi.org/10.1201/9781420034837
M. Sofonea, F. Pătrulescu and Y. Souleiman. Analysis of a contact problem with wear and unilateral constraint. Applicable Analysis, 95(11):2590–2607, 2016. https://doi.org/10.1080/00036811.2015.1102892
M. Sofonea and Y. Souleiman. Analysis of a sliding frictional contact problem with unilateral constraint. Mathematics and Mechanics of Solids, 22(3):324–342, 2015. https://doi.org/10.1177/1081286515591304
M. Sofonea and Y. Souleiman. A viscoelastic sliding contact problem with normal compliance, unilateral constraint and memory term. Mediterranean Journal of Mathematics, 13:2863–2886, 2016. https://doi.org/10.1007/s00009-015-0661-9
B. Souraya and A.A. Abdelaziz. Analysis of a dynamic contact problem for electro-viscoelastic materials. Milan Journal of Mathematics, 86(1):105–124, 2018. https://doi.org/10.1007/s00032-018-0282-4
N. Strömberg. Continuum thermodynamics of contact, friction and wear. Ph.D. Thesis. Linkoping University, Sweeden, 1995.
N. Strömberg, L. Johansson and A. Klarbring. Derivation and analysis of a generalized standard model for contact, friction and wear. International Journal of Solids and Structures, 33(13):1817–1836, 1996. https://doi.org/10.1016/0020-7683(95)00140-9
R.C. Batra and J.S. Yang. Saint-Venant’s principle in linear piezoelectricity. Journal of Elasticity, 38(2):209–218, 1995. https://doi.org/10.1007/BF00042498
P. Bisegna, F. Maceri and F. Lebon. The unilateral frictional contact of a piezoelectric body with a rigid support. In Contact mechanics, pp. 347–354. Springer Publishing, 2002. https://doi.org/10.1007/978-94-017-1154-8_37
M. Dalah and M. Sofonea. Antiplane frictional contact of electro-viscoelastic cylinders. Electronic Journal of Differential Equations (EJDE)[electronic only], 2007(161):1–14, 2007.
G. Duvant and J.L. Lions. Inequalities in mechanics and physics, volume 219. Springer Science & Business Media, 2012.
W. Han and M. Sofonea. Evolutionary variational inequalities arising in viscoelastic contact problems. SIAM Journal on Numerical Analysis, 38(2):556– 579, 2000. https://doi.org/10.1137/S0036142998347309
T. Ikeda. Fundamentals of piezoelectricity. Oxford University Press, Oxford, 1990.
L. Kaki and M. Denche. Variational analysis for some frictional contact problems. Boletim da Sociedade Paranaense de Matema´tica, 38(7):21–36, 2020. https://doi.org/10.5269/bspm.v38i7.44258
Z. Lerguet, M. Shillor and M. Sofonea. A frictional contact problem for an electro-viscoelastic body. Electronic Journal of Differential Equations (EJDE)[electronic only], 2007(170):1–16, 2007.
F. Maceri and P. Bisegna. The unilateral frictionless contact of a piezoelectric body with a rigid support. Mathematical and Computer Modelling, 28(4-8):19– 28, 1998. https://doi.org/10.1016/S0895-7177(98)00105-8
J.A. C Martins and J.T. Oden. Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws. Nonlinear Analysis TMA, 11(3):407–428,1987. https://doi.org/10.1016/0362-546X(87)90055-1
S. Migórski. Hemivariational inequality for a frictional contact problem in elastopiezoelectricity. Discrete & Continuous Dynamical Systems-B, 6(6):1339–13356, 2006. https://doi.org/10.3934/dcdsb.2006.6.1339
M. Rochdi, M. Shillor and M. Sofonea. Quasistatic viscoelastic contact with normal compliance and friction. Journal of Elasticity, 51(2):105–126, 1998. https://doi.org/10.1023/A:1007413119583
M. Selmani. A dynamic problem with adhesion and damage in electroviscoelasticity with long-term memory. JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only], 10(1):1–19, 2009.
M. Selmani. A frictional contact problem involving piezoelectric materials with long memory. Mediterranean Journal of Mathematics, 12(3):1177–1197, 2015. https://doi.org/10.1007/s00009-014-0430-1
M. Selmani et al. Frictional contact problem with wear for electro-viscoelastic materials with long memory. Bulletin of the Belgian Mathematical Society-Simon Stevin, 20(3):461–479, 2013. https://doi.org/10.36045/bbms/1378314510
M. Selmani and L. Selmani. A dynamic frictionless contact problem with adhesion and damage. Bulletin of the Polish Academy of Sciences. Mathematics, 55(1):17–34, 2007. https://doi.org/10.4064/ba55-1-3
M. Selmani and L. Selmani. A frictional contact problem with wear and damage for electro-viscoelastic materials. Applications of Mathematics, 55(2):89–109, 2010. https://doi.org/10.1007/s10492-010-0004-x
M. Shillor, M. Sofonea and J.J. Telega. Quasistatic viscoelastic contact with friction and wear diffusion. Quart. Appl. Math, 62(2):379–399, 2004. https://doi.org/10.1090/qam/2054605
M. Sofonea and R. Arhab. An electro-viscoelastic contact problem with adesion. Dynamics of Continuous Discrete and Impulsive Systems Series A, 14(4):577, 2007. https://doi.org/10.1080/00036810701286304
M. Sofonea and E. Essoufi. A piezoelectric contact problem with slip dependent coefficient of friction. Mathematical Modelling and Analysis, 9(3):229–242, 2004. https://doi.org/10.3846/13926292.2004.9637256
M. Sofonea and E. Essoufi. Quasistatic frictional contact of viscoelastic piezoelectric body. Ad . Math. Sci. Appl, 14(3):613–631, 2004.
M. Sofonea, W. Han and M. Shillor. Analysis and approximation of contact problems with adhesion or damage. CRC Press, 2006. https://doi.org/10.1201/9781420034837
M. Sofonea, F. Pătrulescu and Y. Souleiman. Analysis of a contact problem with wear and unilateral constraint. Applicable Analysis, 95(11):2590–2607, 2016. https://doi.org/10.1080/00036811.2015.1102892
M. Sofonea and Y. Souleiman. Analysis of a sliding frictional contact problem with unilateral constraint. Mathematics and Mechanics of Solids, 22(3):324–342, 2015. https://doi.org/10.1177/1081286515591304
M. Sofonea and Y. Souleiman. A viscoelastic sliding contact problem with normal compliance, unilateral constraint and memory term. Mediterranean Journal of Mathematics, 13:2863–2886, 2016. https://doi.org/10.1007/s00009-015-0661-9
B. Souraya and A.A. Abdelaziz. Analysis of a dynamic contact problem for electro-viscoelastic materials. Milan Journal of Mathematics, 86(1):105–124, 2018. https://doi.org/10.1007/s00032-018-0282-4
N. Strömberg. Continuum thermodynamics of contact, friction and wear. Ph.D. Thesis. Linkoping University, Sweeden, 1995.
N. Strömberg, L. Johansson and A. Klarbring. Derivation and analysis of a generalized standard model for contact, friction and wear. International Journal of Solids and Structures, 33(13):1817–1836, 1996. https://doi.org/10.1016/0020-7683(95)00140-9