Hemivariational Inequality Approach Research Articles

A class of differential hemivariational inequalities constrained on nonconvex star-shaped sets

The purpose of this paper is to investigate a class of nonconvex-constrained differential hemivariational inequalities consisting of nonlinear evolution equations and evolutionary hemivariational inequalities. The admissible set of constraints is closed and star-shaped with respect to a certain ball in a reflexive Banach space. We construct an auxiliary inclusion problem and obtain the existence results by applying a surjectivity theorem for multivalued pseudomonotone operators and the properties of Clarke subgradient operator. Moreover, the existence of a solution to the original problem is established by hemivariational inequality approach and a penalization method in which a small parameter does not have to tend to zero. Finally, an application of the main results is provided.

Hemivariational Inequality Approach to Evolutionary Constrained Problems on Star-Shaped Sets

In this paper, we consider a nonconvex evolutionary constrained problem for a star-shaped set. The problem is a generalization of the classical evolution variational inequality of parabolic type. We provide an existence result; the proof is based on the hemivariational inequality approach, a surjectivity theorem for multivalued pseudomonotone operators in reflexive Banach spaces, and a penalization method. The admissible set of constraints is closed and star-shaped with respect to a certain ball; this allows one to use a discontinuity property of the generalized Clarke subdifferential of the distance function. An application of our results to a heat conduction problem with nonconvex constraints is provided.

Obstacle Problems with Cohesion: A Hemivariational Inequality Approach and Its Efficient Numerical Solution

Motivated by an obstacle problem for a membrane subject to cohesion forces, constrained minimization problems involving a nonconvex and nondifferentiable objective functional representing the total potential energy are considered. The associated first-order optimality system leads to a hemivariational inequality, which can also be interpreted as a special complementarity problem in function space. Besides an analytical investigation of first-order optimality, a primal-dual active set solver is introduced. It is associated to a limit case of a semismooth Newton method for a regularized version of the underlying problem class. For the numerical algorithms studied in this paper, global as well as local convergence properties are derived and verified numerically.

Hemivariational inequality approach to the dynamic viscoelastic contact problem with nonmonotone normal compliance and slip-dependent friction

We examine a mathematical model which describes dynamic viscoelastic contact problems with nonmonotone normal compliance condition and the slip displacement dependent friction. First, we derive a weak formulation of the model in the form of a hemivariational inequality. Then we embed the hemivariational inequality into a class of second-order evolution inclusions for which we provide a result on the existence of a solution. We conclude with examples of the subdifferential boundary conditions for contact with normal compliance and the slip dependent friction.

Hemivariational inequality approach to the dynamic viscoelastic sliding contact problem with wear

The dynamic contact problem of sliding contact with wear between a viscoelastic body and a foundation is formulated as a hemivariational inequality. The contact is modeled with a general normal damped response condition which is nonmonotone, possibly multivalued and has the subdifferential form. We establish the existence of weak solutions by using a surjectivity result for a class of pseudomonotone operators. We also deliver conditions under which the solution of the evolution inclusion associated with the hemivariational inequality is unique.

A review of the theory of static and quasi-static frictional contact problems in elasticity

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Andersson Lars-Erik and Klarbring Anders 2001A review of the theory of static and quasi-static frictional contact problems in elasticityPhil. Trans. R. Soc. A.3592519–2539http://doi.org/10.1098/rsta.2001.0908SectionRestricted accessA review of the theory of static and quasi-static frictional contact problems in elasticity Lars-Erik Andersson Lars-Erik Andersson Department of Mathematics, SE-581 83 Linköping, Sweden Google Scholar Find this author on PubMed Search for more papers by this author and Anders Klarbring Anders Klarbring Department of Mechanical Engineering, Linköping University, SE-581 83 Linköping,Sweden (; ) Google Scholar Find this author on PubMed Search for more papers by this author Lars-Erik Andersson Lars-Erik Andersson Department of Mathematics, SE-581 83 Linköping, Sweden Google Scholar Find this author on PubMed Search for more papers by this author and Anders Klarbring Anders Klarbring Department of Mechanical Engineering, Linköping University, SE-581 83 Linköping,Sweden (; ) Google Scholar Find this author on PubMed Search for more papers by this author Published:15 December 2001https://doi.org/10.1098/rsta.2001.0908AbstractThis paper gives a review of mathematical results of existence and uniqueness of solutions to problems of linear elasticity involving friction. Static, steady sliding and quasi-static problems are discussed. Both the case of a continuum body and that of a space-discrete one are treated. The intention is to describe the state of the art for these problems. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Vodička R, Mantič V and Roubíček T (2017) Quasistatic normal-compliance contact problem of visco-elastic bodies with Coulomb friction implemented by QP and SGBEM, Journal of Computational and Applied Mathematics, 10.1016/j.cam.2016.10.010, 315, (249-272), Online publication date: 1-May-2017. Bartosz K and Sofonea M (2016) Modeling and analysis of a contact problem for a viscoelastic rod, Zeitschrift für angewandte Mathematik und Physik, 10.1007/s00033-016-0718-z, 67:5, Online publication date: 1-Oct-2016. Mielke A and Roubíček T (2015) Applications in continuum mechanics and physics of solids Rate-Independent Systems, 10.1007/978-1-4939-2706-7_4, (235-458), . Hu Z, Wang Z, Zhang S, Zhang Y and Zhou H (2014)(2014) A combined reordering procedure for preconditioned GMRES applied to solving equations using Lagrange multiplier method, Engineering Computations, 10.1108/EC-07-2013-0184, 31:7, (1283-1304), Online publication date: 30-Sep-2014., Online publication date: 30-Sep-2014. Hu Z, Zhang S, Zhang Y, Zhou H and Li D (2014)(2014) An efficient preconditioned Krylov subspace method for large-scale finite element equations with MPC using Lagrange multiplier method, Engineering Computations, 10.1108/EC-03-2013-0077, 31:7, (1169-1197), Online publication date: 30-Sep-2014., Online publication date: 30-Sep-2014. Chen Q, Gunzburger M and Perego M (2013) Well-Posedness Results for a Nonlinear Stokes Problem Arising in Glaciology, SIAM Journal on Mathematical Analysis, 10.1137/110848694, 45:5, (2710-2733), Online publication date: 1-Jan-2013. Zozulya V (2011) Variational formulation and nonsmooth optimization algorithms in elastodynamic contact problems for cracked body, Computer Methods in Applied Mechanics and Engineering, 10.1016/j.cma.2010.07.005, 200:5-8, (525-539), Online publication date: 1-Jan-2011. (2011) Numerical Analysis of Cable Networks Nonsmooth Mechanics and Convex Optimization, 10.1201/b10839-13, (205-228) Hintermüller M, Kovtunenko V and Kunisch K (2011) Obstacle Problems with Cohesion: A Hemivariational Inequality Approach and Its Efficient Numerical Solution, SIAM Journal on Optimization, 10.1137/10078299, 21:2, (491-516), Online publication date: 1-Apr-2011. Angelov T (2009) Modelling and Solvability of a Rigid-Plastic Rolling Problem, Mathematics and Mechanics of Solids, 10.1177/1081286509105148, 15:6, (639-654), Online publication date: 1-Aug-2010. Angelov T (2009) Variational and numerical approach to a steady-state rolling problem, Journal of Engineering Mathematics, 10.1007/s10665-009-9284-0, 66:4, (311-323), Online publication date: 1-Apr-2010. Ahn Y (2010) Discontinuity of quasi-static solution in the two-node Coulomb frictional system, International Journal of Solids and Structures, 10.1016/j.ijsolstr.2010.06.007, 47:21, (2866-2871), Online publication date: 1-Oct-2010. Argatov I (2009) Averaging of a finely laminated elastic medium with roughness or adhesion on the contact surfaces of the layers, Journal of Applied Mathematics and Mechanics, 10.1016/j.jappmathmech.2010.01.014, 73:6, (734-746), Online publication date: 1-Jan-2009. Hild P (2009) Two results on solution uniqueness and multiplicity for the linear elastic friction problem with normal compliance, Nonlinear Analysis: Theory, Methods & Applications, 10.1016/j.na.2009.04.061, 71:11, (5560-5571), Online publication date: 1-Dec-2009. Mielke A A Mathematical Framework for Generalized Standard Materials in the Rate-Independent Case Multifield Problems in Solid and Fluid Mechanics, 10.1007/978-3-540-34961-7_12, (399-428) Mielke A (2005) Evolution of Rate-Independent Systems Handbook of Differential Equations Evolutionary Equations, 10.1016/S1874-5717(06)80009-5, (461-559), . Huang S and Schimmels J (2005) The Motion of a Compliantly Suspended Rigid Body Constrained by Multipoint Frictional Contact, Journal of Dynamic Systems, Measurement, and Control, 10.1115/1.2098878, 127:4, (623-632), Online publication date: 1-Dec-2005. Huang S and Schimmels J (2016) Spatial Compliant Motion of a Rigid Body Constrained by a Frictional Contact, The International Journal of Robotics Research, 10.1177/02783649030229004, 22:9, (733-756), Online publication date: 1-Sep-2003. This Issue15 December 2001Volume 359Issue 1789Theme Issue ‘Non-smooth mechanics’ compiled by F. G. Pfeiffer Article InformationDOI:https://doi.org/10.1098/rsta.2001.0908Published by:Royal SocietyPrint ISSN:1364-503XOnline ISSN:1471-2962History: Published online15/12/2001Published in print15/12/2001 License: Citations and impact Keywordsquasi-static problemexistenceunilateral contactlinear elasticityuniquenessCoulomb friction

A Bifurcation Theory for a Nonconvex Unilateral Laminated Plate Problem Formulated as a Hemivariational Inequality Involving a Potential Operator

AbstractThe present paper concerns the hemivariational inequality approach to the laminated plate theory under abstract subdifferential conditions. The mechanical problem is formulated as a nonlinear eigenvalue problem for hemivariational inequalities and a corresponding bifurcation theory is given.

New models for a class of adhesive grippers. The hemivariational inequality approach

New models for a class of adhesive grippers. The hemivariational inequality approach

New models for a class of adhesive grippers. The hemivariational inequality approach

A hemivariational inequality model for adhesive grasping problems is proposed and studied in this paper. The unilateral frictionless and frictional contact effects between the fingertips and the grasping object that lead to linear complementarity problems with singular matrices for the study of static equilibrium of the gripper-object system are generalized here to cover adhesive multifingered grippers. Adhesive effects are modelled by appropriately defined, generally nonconvex, yield sets in the space of contact stresses, friction stresses, gaps or frictional slips and their combinations. The hemivariational inequality problem that arises may involve copositive plus, symmetric matrices and nonempty closed sets for the frictionless gripper problem and copositive plus, nonsymmetric matrices with starshaped sets for the frictional case. Solvability conditions that guarantee the existence of a solution to the gripper problem are given. They specify the conditions which are required to hold between external forces, fingertip mechanical behavior and finger placement in order to solve the gripper problem.

Noncoercive hemivariational inequality approach to constrained problems for star-shaped admissible sets

The hemivariational inequality approach is used in order to establish the existence of solutions to a large class of noncoercive constrained problems in a reflexive Banach space, in which the set of all admissible elements is not convex but fulfills some star-shaped property.

On a class of Hemivariational inequalities involving hemicontinuous monotone operators. Application in unilateral mechanics

The hemivariational inequality approach of P.D. Panagiotopoulos [18]-[22] is used so as to establish the existence of solutions to a class of noncoercive and nonconvex constrained problems in a reflexive Banach space.

On the Hemivariational Inequality Approach to Nonconvex Constrained Problems in the Theory of von Kármán Plates

AbstractIn dieser Arbeit werden nichtkonvexe Randwertprobleme behandelt, die im Rahmen der von Kármánschen Plattentheorie zu Hemivariationsungleichungen führen.

Hemivariational inequality approach to constrained problems for star-shaped admissible sets

The hemivariational inequality approach is applied to establish the existence of solutions to a large class of nonconvex constrained problems in a reflexive Banach space. The admissible sets are supposed to be star-shaped with respect to a ball. Due to a discontinuity property of the Clarke directional differential related to the corresponding distance functions, the proposed method permits one to attain the solution without passing to zero with the penalization parameter. Some applications to nonconvex constrained variational problems illustrate the theory.

A hemivariational inequality approach to the rock interface problem

In the present paper a method is presented for the analysis of rock joints and interfaces. The mechanical response of rock joints and interfaces involves phenomena of nonclassic nature as debonding and slip along the interfaces, which can be completely described by nonmonotone possibly multivalued (e.g. sawtooth) phenomenological constitutive laws, the so-called complete laws, which result from the respective experimental stress-stain diagrams. Due to the lack of monotonicity of these laws, the rock interface problem cannot be studied by means of the classical variational methods and therefore, the need for a generalization of the classical variational theories to cover also such constitutive laws arises. This can be achieved by means of the notion of generalized gradient which was already applied to the solution of a large number of structural analysis problems involving nonconvex and nondifferentiable energy functions. First the interface behaviour is modelled through nonconvex superpotentials describing the debonding and/or stick/slip phenomena. Then the hemivariational inequalities are formulated and some methods for the numerical treatment are proposed.

Interlayer Slip and Delamination Effect: A Hemivariational Inequality Approach

In the present paper we formulate hemivariational inequalities describing the interlayer slip and delamination effect in elastic structures. After the formulation of the interface conditions which are of a nonmonotone multivalued nature, we derive hemivariational inequalities for the static and dynamic problems. Further some methods for the numerical treatment of the problems are given (regularisation method and substationarity-point method). Two numerical examples concerning simple adhesively connected elastic structures are given to illustrate the theory.

A hemivariational inequality approach to the unilateral contact problem and substationarity principles

The present paper deals with contact problems of structures for which the contact region is not a priori known. For the boundary conditions describing the contact both convex and nonconvex potential functions are considered. The first lead to variational inequalities and minimum energy principles and the second to a new type of variational forms, called hemivariational inequalities, and substationary-energy principles. Static and dynamic contact problems with friction and/or brittle fracture effects are considered. Applications illustrate the theory.