We study a class of abstract hemivariational inequalities in a reflexive Banach space. For this class, using the theory of multivalued pseudomonotone mappings and a fixed-point argument, we provide a result on the existence and uniqueness of the solution. Next, we investigate a static frictional contact problem with unilateral constraints between a piezoelastic body and a conductive foundation. The contact, friction and electrical conductivity condition on the contact surface are described with the Clarke generalized subgradient multivalued boundary relations. We derive the variational formulation of the contact problem which is a coupled system of two hemivariational inequalities. Finally, for such system we apply our abstract result and prove its unique weak solvability.
IkedaT.Fundamentals of piezoelectricity. Oxford: Oxford University Press, 1990.
2.
NaniewiczZPanagiotopoulosPD.Mathematical theory of hemivariational inequalities and applications. New York: Marcel Dekker, 1995.
3.
PanagiotopoulosPD.Inequality problems in mechanics and applications. Boston: Birkhäuser, 1985.
4.
PanagiotopoulosPD.Hemivariational inequalities, applications in mechanics and engineering. Berlin: Springer–Verlag, 1993.
5.
BatraRCYangJS.Saint–Venant’s principle in linear piezoelectricity. J Elasticity1995; 38: 209–218.
6.
ChouguiNDrablaS.A quasistatic electro-viscoelastic contact problem with adhesion. Bull Malays Math Sci Soc2016; 39: 1439. doi:10.1007/s40840-015-0236-8.
7.
BenaissaHEssoufiEHFakharR.Existence results for unilateral contact problem with friction of thermo-electro-elasticity. Appl Math Mech Engl Ed2015. doi: 10.1007/s10483-015-1957-9.
8.
BenkhiraEHEssoufiEHFakharR.On convergence of the penalty method for a static unilateral contact problem with nonlocal friction in electro-elasticity. Eur J Appl Math2016; 27: 1–22.
9.
ChouguiNDrablaSHemiciN.Variational analysis of an electro-viscoelastic contact problem with friction and adhesion. J Korean Math Soc2016; 53: 161–185.
10.
EssoufiEHFakharRKokoJ.A decomposition method for a unilateral contact problem with Tresca friction arising in electro-elastostatics, Numer Func Anal Opt2015; 36: 1533–1558.
11.
HüeberSMateiAWohlmuthB.A contact problem for electro-elastic materials. Z Angew Math Mech2013; 93: 789–800.
12.
MigorskiSOchalASofoneaM.Analysis of a quasistatic contact problem for piezoelectric materials. J Math Anal Appl2011; 382: 701–713.
13.
MigorskiS.Hemivariational inequality for a frictional contact problem in elasto-piezoelectricity. Discrete Continuous Dyn Syst Ser B2006; 6: 1339–1356.
14.
MigorskiSOchalASofoneaM.Weak solvability of a piezoelectric contact problem. Eur J Appl Math2009; 20: 145–167.
15.
MigorskiSOchalASofoneaM.Nonlinear inclusions and hemivariational inequalities. Models and analysis of contact problems, Advances in Mechanics and Mathematics. Volume 26. New York: Springer, 2013.
16.
MigorskiS.Evolution hemivariational inequality for a class of dynamic viscoelastic nonmonotone frictional contact problems. Comput Math Appl2006; 52: 677–698.
17.
MigorskiSOchalASofoneaM.A dynamic frictional contact problem for piezoelectric materials. J Math Anal Appl2010; 361: 161–176.
18.
MigorskiSOchalASofoneaM.Analysis of a piezoelectric contact problem with subdifferential boundary condition. Roy Soc London Ser A2014; 144A: 1007–1025.
19.
MigorskiSOchalASofoneaM.History-dependent variational-hemivariational inequalities in contact mechanics. Nonlinear Anal Real World Appl2015; 22: 604–618.
20.
ClarkeFH.Optimization and nonsmooth analysis. New York: Wiley, Interscience, 1983.
LeVK.A range and existence theorem for pseudomonotone perturbations of maximal monotone operators. Proc Am Math Soc2011; 139: 1645–1658.
24.
BarbuVPrecupanuT.Convexity and optimization in Banach spaces. 4th ed.Dordrecht Heidelberg London New York: Springer, 2012.
25.
MigorskiSOchalASofoneaM.A class of variational-hemivariational inequalities in reflexive Banach spaces. J Elast2017; 127: 151–178.
26.
EckCJarušekJKrbečM.Unilateral contact problems: Variational methods and existence theorems. New York: Chapman/CRC Press, 2005.
27.
HanWSofoneaM.Quasistatic contact problems in viscoelasticity and viscoplasticity. Providence: Americal Mathematical Society, 2002.
28.
KikuchiNOdenJT.Contact problems in elasticity. A study in variational inequalities and finite element methods. Philadelphia: Society for Industrial and Applied Mathematics, 1988.
29.
LeXA.Dynamics of mechanical systems with Coulomb friction. Berlin: Springer–Verlag, 2003.
30.
ShillorMSofoneaMTelegaJJ.Models and Analysis of Quasistatic Contact, Lecture Notes in Physics. Volume 655. Berlin: Springer, 2004.
31.
SofoneaMMateiA.Mathematical models in contact mechanics, London mathematical society lecture note series. Volume 398. Cambridge: Cambridge University Press, 2012.
