A dynamic frictional contact problem between a viscoelastic body and a foundation is studied. The contact is modeled with normal damped response and a friction law. The constitutive law with long memory is assumed to be nonlinear. The existence result is proved using nonlinear monotone operators, fixed point argument, and extension procedure. Moreover, the exponential stability of the energy solution is established using the multiplier method.
ChenTHuangNJSofoneaM. A differential variational inequality in the study of contact problems with wear. Nonlinear Anal Real World Appl2022; 67: 103619.
17.
ChenTHuRSofoneaM. Analysis and control of an electro-elastic contact problem. Math Mech Solids2022; 27(5): 813–827.
18.
LiYChengXWangX. Optimal control of a quasistatic frictional contact problem with history-dependent operators. Int J Numer Anal Model2023; 20(1): 29–46.
19.
LiuYMigórskiSNguyenVT, et al. Existence and convergence results for an elastic frictional contact problem with nonmonotone subdifferential boundary conditions. Acta Appl Math2021; 41(4): 1151–1168.
20.
MateiA. A three-field variational formulation for a frictional contact problem with prescribed normal stress. Fractal Fractional2022; 6(11): 651.
21.
MateiAMicuS. Boundary optimal control for a frictional contact problem with normal compliance. Appl Math Optim2018; 78(2): 379–401.
22.
MateiAMicuS. Boundary optimal control for nonlinear antiplane problems. Nonlinear Anal Theory Methods Appl2011; 74(5): 1641–1652.
23.
CîndeaNMateiAMicuS, et al. Boundary optimal control for antiplane contact problems with power-law friction. Appl Math Comput 202; 386: 125448.
24.
MateiAMicuSNiţăC. Optimal control for antiplane frictional contact problems involving nonlinearly elastic materials of Hencky type. Math Mech Solids2018; 23(3): 308–328.
25.
SofoneaMXiaoY. Weak formulations of quasistatic frictional contact problems. Commun Nonlinear Sci Numer Simul2021; 101: 105888.
26.
SofoneaMShillorM. Tykhonov well-posedness and convergence results for contact problems with unilateral constraints. Technologies2021; 9(1): 1.
27.
XiaoYSofoneaM. On the optimal control of variational–hemivariational inequalities. J Math Anal Appl2019; 475(1): 364–384.
28.
MigórskiSZengB. A new class of history-dependent evolutionary variational–hemivariational inequalities with unilateral constraints. Appl Math Optim2021; 84: 2671–2697.
29.
HanWSofoneaM. Numerical analysis of hemivariational inequalities in contact mechanics. Acta Numerica2019; 28: 175–286.
30.
PengZGamorskiPMigórskiS. Boundary optimal control of a dynamic frictional contact problem. ZAMM J Appl Math Mech2020; 100(10): e201900144.
31.
MigórskiSHanWZengS. A new class of hyperbolic variational–hemivariational inequalities driven by non-linear evolution equations. Eur J Appl Math2021; 32(1): 59–88.
32.
RochdiMShillorMSofoneaM. A quasistatic contact problem with directional friction and damped response. Appl Anal1998; 68(3–4): 409–422.
33.
RenonNSofoneaMShillorM. Un modèle mathématique pour un problème de contact elasto-plastique avec écrouissage. Application à la scarification,- ( Actes du Dixieme Seminaire Franco-Polonais de Mecanique). Lille: Institut Polytechnique de Varsovie and Universite des Sciences et Technologies de Lille, 2002.
34.
SofoneaMXiaoYCoudercM. Optimization problems for a viscoelastic frictional contact problem with unilateral constraints. Nonlinear Anal Real World Appl2019; 50: 86–103.
35.
SofoneaMNiculescuCMateiA. An antiplane contact problem for viscoelastic materials with long-term memory. Math Model Anal2006; 11(2): 213–228.
36.
MateiACiurceaR. Weak solutions for contact problems involving viscoelastic materials with long memory. Mathematics and Mechanics of Solids2011; 16(4): 393–405.
37.
BarbuV. Nonlinear semigroups and differential equations in Banach spaces. Dordrecht: Springer, 1976.
38.
SegalI. Non-linear semi-groups. Ann Math1963; 78(2): 339–364.
39.
GeorgievVTodorovaG. Existence of a solution of the wave equation with nonlinear damping and source terms. J Diff Eq1994; 109(2): 295–308.
40.
VitillaroE. Global existence for the wave equation with nonlinear boundary damping and source terms. J Diff Eq2002; 186(1): 259–298.
