Analysis of a dynamic frictional contact problem for hyperviscoelastic material with non-convex energy density

Using the time approximation method we obtain the existence of a weak solution for the dynamic contact problem with damping and a non-convex stored elastic energy function. On the contact boundary we assume the normal compliance law and the generalization of the Coulomb friction law which allows for non-monotone dependence of the friction force on the tangential velocity. The existence result is accompanied by two numerical examples, one of them showing lack of uniqueness for the numerical solution.

[1]  J. J. Telega,et al.  Models and analysis of quasistatic contact , 2004 .

[2]  Barkawi Sahari,et al.  A review of constitutive models for rubber-like materials , 2010 .

[3]  J. C. Simo,et al.  The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics , 1992 .

[4]  P. Wriggers.  Computational contact mechanics , 2012 .

[5]  A. Bressan,et al.  Random extremal solutions of differential inclusions , 2016 .

[6]  E. Zeidler.  Nonlinear functional analysis and its applications , 1988 .

[7]  Etienne Emmrich,et al.  Evolution equations of second order with nonconvex potential and linear damping: existence via convergence of a full discretization , 2013 .

[8]  Piotr Kalita,et al.  Convergence of Rothe scheme for hemivariational inequalities of parabolic type , 2011, 1108.5469.

[9]  E. Rabinowicz.  The Nature of the Static and Kinetic Coefficients of Friction , 1951 .

[10]  Ioan R. Ionescu,et al.  Slip-dependent friction in dynamic elasticity , 2003 .

[11]  T. Roubíček.  Nonlinear partial differential equations with applications , 2005 .

[12]  Anna Ochal,et al.  Hemivariational inequality for viscoelastic contact problem with slip-dependent friction , 2005 .

[13]  Mircea Sofonea,et al.  A class of history-dependent variational-hemivariational inequalities , 2016 .

[14]  Mircea Sofonea,et al.  A Hyperelastic Dynamic Frictional Contact Model with Energy-Consistent Properties , 2015 .

[15]  F. Armero,et al.  On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics. Part II: second-order methods , 2001 .

[16]  Gero Friesecke,et al.  Implicit time discretization and global existence for a quasi-linear evolution equation with nonconvex energy , 1997 .

[17]  P. Tallec,et al.  Energy-controlling time integration methods for nonlinear elastodynamics and low-velocity impact , 2006 .

[18]  J. T. Oden,et al.  Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws , 1987 .

[19]  Jaroslav Haslinger,et al.  Finite Element Method for Hemivariational Inequalities , 1999 .

[20]  O. Yeoh.  Some Forms of the Strain Energy Function for Rubber , 1993 .

[21]  P. Kalita,et al.  A dynamic viscoelastic contact problem with normal compliance, finite penetration and nonmonotone slip rate dependent friction , 2015 .

[22]  M. Barboteu,et al.  A frictionless viscoelastodynamic contact problem with energy consistent properties: Numerical analysis and computational aspects , 2009 .

[23]  M. Boyce,et al.  A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials , 1993 .

[24]  Meir Shillor,et al.  Dynamic Contact with Normal Compliance Wear and Discontinuous Friction Coefficient , 2002, SIAM J. Math. Anal..

[25]  M. Barboteu,et al.  Formulation and analysis of two energy-consistent methods for nonlinear elastodynamic frictional contact problems , 2009 .

[26]  Zdzisław Denkowski,et al.  An Introduction to Nonlinear Analysis: Theory , 2013 .

[27]  J. Jarusek,et al.  On the solvability of dynamic elastic‐visco‐plastic contact problems , 2008 .

[28]  J. Aubin.  Set-valued analysis , 1990 .

[29]  Stanisław Migórski,et al.  Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction , 2005 .

[30]  J. Haslinger,et al.  Solution of Variational Inequalities in Mechanics , 1988 .

[31]  Carsten Carstensen,et al.  A theory of discretization for nonlinear evolution inequalities applied to parabolic Signorini problems , 1999 .

[32]  M. Sofonea,et al.  Nonlinear Inclusions and Hemivariational Inequalities , 2013 .

[33]  P. Panagiotopoulos,et al.  Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications , 1999 .

[34]  Mircea Sofonea,et al.  Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems , 2012 .

[35]  M. Cocou.  Existence of solutions of a dynamic Signorini's problem with nonlocal friction in viscoelasticity , 2002 .

[36]  Mikaël Barboteu,et al.  ANALYSIS OF A CONTACT PROBLEM WITH NORMAL COMPLIANCE, FINITE PENETRATION AND NONMONOTONE SLIP DEPENDENT FRICTION , 2014 .

[37]  L. Gasiński.  Evolution hemivariational inequality with hysteresis operator in higher order term , 2008 .

[38]  Evolution hemivariational inequalities with hysteresis , 2004 .

[39]  P. D. Panagiotopoulos,et al.  Mathematical Theory of Hemivariational Inequalities and Applications , 1994 .

[40]  Patrick Laborde,et al.  On the discretization of contact problems in elastodynamics , 2006 .

[41]  Meir Shillor,et al.  DYNAMIC CONTACT WITH SIGNORINI'S CONDITION AND SLIP RATE DEPENDENT FRICTION , 2004 .

[42]  Mircea Sofonea,et al.  History-dependent variational–hemivariational inequalities in contact mechanics ☆ , 2015 .

[43]  J. Ball,et al.  Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity , 1982 .

[44]  J. Moreau.  On Unilateral Constraints, Friction and Plasticity , 2011 .

[45]  A. Kamran,et al.  Mechanical Characterization and FE Modelling of a Hyperelastic Material , 2015 .

[46]  Ioan R. Ionescu,et al.  Dynamic contact problems with slip dependent friction in viscoelasticity , 2002 .

[47]  J. Nitsche.  On Korn's second inequality , 1981 .

[48]  Christof Eck,et al.  Unilateral Contact Problems: Variational Methods and Existence Theorems , 2005 .

[49]  Oscar Gonzalez,et al.  Exact energy and momentum conserving algorithms for general models in nonlinear elasticity , 2000 .

[50]  M. Cao,et al.  Existence of solutions for a dynamic Signorini's contact problem , 2006 .

[51]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[52]  On quasi-static contact problem with generalized Coulomb friction, normal compliance and damage† , 2015, European Journal of Applied Mathematics.

[53]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[54]  J. Lions,et al.  Inequalities in mechanics and physics , 1976 .

[55]  T. Laursen.  Computational Contact and Impact Mechanics , 2003 .