VARIATIONAL AND QUASI-VARIATIONAL INEQUALITIES IN MECHANICS

Transcription

1 VARIATIONAL AND QUASI-VARIATIONAL INEQUALITIES IN MECHANICS

2 SOLID MECHANICS AND ITS APPLICATIONS Volume 147 Series Editor: G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity. For a list of related mechanics titles, see final pages.

3 Variational and Quasi-Variational Inequalities in Mechanics by ALEXANDER S. KRAVCHUK Moscow State University, Moscow, Russia and PEKKA J. NEITTAANMÄKI University of Jyväskyl ä, Jyväskylä, Finland

4 A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN (HB) ISBN (e-book) Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Printed on acid-free paper All Rights Reserved 2007 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

5 Preface The variational method is a powerful tool to investigate states and processes in technical devices, nature, living organisms, systems, and economics. The power of the variational method consists in the fact that many of its statements are physical or natural laws themselves. The essence of the variational approach for the solution of problems relating to the determination of the real state of systems or processes consists in the comparison of close states. The selection criteria for the actual states must be such that all the equations and conditions of the mathematical model are satisfied. Historically, the first variational theory was the Lagrange theory created to investigate the equilibrium of finite-dimensional mechanical systems under holonomic bilateral constraints (bonds). The selection criterion proposed by Lagrange is the admissible displacement principle. In accordance with this principle, the work of the prescribed forces (supposed to be constant) on infinitesimally small, kinematically admissible (virtual) displacements is zero. It is known that equating the virtual work performed for potential systems to zero is equivalent to the stationarity conditions for the total energy of the system. The transition from bilateral constraints to unilateral ones was performed by O. L. Fourier. Fourier demonstrated that the virtual work on small disturbances of a stable equilibrium state of a mechanical system under unilateral constraints must be positive (or, at least, nonnegative). Therefore, for such a system the corresponding mathematical model is reduced to an inequality and the problem becomes nonlinear. The dynamic theory of systems under unilateral constraints was proposed by M. V. Ostrogradski and completed by J. R. Mayer and E. Zermelo. The Ostrogradski method is an algorithm for the integration of the equations of motion. According to this algorithm, only the bonds which reduce to equality must be taken into account, because the strict inequality constraints do not influence the motion. The selection of such bonds is performed using a special method in our work as well.

6 vi Preface Later, this approach was generalized to continuous systems, i.e., as a problem of continuum mechanics. At first such a problem was considered by Ostrogradski. The first research into the unilateral problem in the mechanics of solids was performed by A. Signorini in The Signorini problem consists of finding the equilibrium of a deformed solid in a smooth rigid shell. The aim of this book is to consider a wide set of problems arising in the mathematical modeling of mechanical systems under unilateral constraints. Most attention is devoted to the interaction of deformed solids. In these investigations elastic and nonelastic deformations and friction and adhesion phenomena are taken into account. All the necessary mathematical tools are given: local boundary value problem formulations, construction of variational equations and inequalities, the transition to minimization problems, existence and uniqueness theorems, and variational transformations (Friedrichs and Young Fenchel Moreau) to dual and saddle-point search problems. Important new results concern contact problems with friction. The Coulomb friction law (and some others), in which relative sliding velocities appear, is considered. The corresponding quasi-variational inequality is constructed as well as an appropriate iterative method for its solution. Convergence is demonstrated. Outlines of the variational approach to nonstationary and dissipative systems and to the construction of the governing equation are also given. Examples of analytical and numerical solutions are presented. Numerical solutions were obtained with the finite element and boundary element methods (BEMs) with all the necessary definitions and theorems. For the variational principles and variational methods of the mechanics, the deformed solids are classical tools for the mathematical modeling of processes in technical devices, constructions, and nature. Much has been published on the subject. The foundation of the methods developed in the first half of the past century can be found in [CH53]. More recent results can be found in [LL50, Mik64b, Was68, Rek77, HHNL88, You69] and others. The application of computers and numerical methods to the solution of engineering problems gave a powerful stimulus to the development of variational approaches. Classical methods were revised and new methods were created for extremal problems under unilateral constraints. Such problems arise in mathematical programming, optimization and control, see, e.g., [Roc70, Lio68, Ban83a, FM68, Fle81, NST06, HN96] and many others. Important theoretical results were, simultaneously, obtained for these new problems in physical sciences, mechanics, biology, and other areas. The contributions of French and Italian mathematicians (J.-L. Lions, G. Duvaut, G. Stampacchia, J. Céa, D. Kinderlehrer, R. Glowinski) must be noted in this connection, see [DL72, KS80, Céa64, Glo84, GLT81]. Considerable successes were attained in the work of P. D. Panagiotopoulos [AP92, Pan85] devoted to the contact problems in the mechanics of solids. Note that the contact problems in the classical theory of elasticity first were solved analytically, with the potential theory method [Her95], complex

7 Preface vii variables methods [Mus53] and others. For a survey of these methods and the analytical solutions, see [Gla80]. One can conclude from the analysis of the contents of these (and other) books and articles that few publications exist on problems such as variational formulations of contact with friction, algorithms for their solution and convergence theorems, contact with adhesion, and numerical solutions to technical problems. In addition, the modern state of the theory and applications of solutions to unilateral problems permits the inclusion of these topics in academic courses. Therefore, it is needed a textbook which contains all the necessary mathematical tools. These arguments motivated the creation of this book. The book contains eight chapters. Bearing in mind the possible use of the book as a textbook, the authors proceed from simple examples to general theory. We give the notations and mathematical tools in Chapter 1. In Chapter 2, the equilibrium of linear systems is considered for finite-dimensional systems, continuous systems, and to linear elastic bodies. Chapter 3 is devoted to nonlinear smooth systems without unilateral constraints. It considers the differentiation of functionals and operators, extremal conditions, existence and uniqueness theorems for minimization problems, and operator potentiality conditions. Two examples from the mechanics of solids are considered: boundary value problems (BVP) for the Hencky Ilyushin theory of plasticity without discharge and BVP for nonlinear elastic bodies with finite displacements and strains. Problems with unilateral constraints are investigated in Chapter 4. Contact interaction of deformed bodies with smooth contact surfaces and with finite friction are considered. It is demonstrated that the local frictionless contact problem is equivalent to the minimization of a functional via a variational inequality. This statement is generalized to a system of deformed bodies with new results on the influence of the different forms of the impenetrability condition. Generalizations on nonlinear governing equations are proposed, including processes with finite displacements and strains. Laws of Coulomb-type governing friction include the relative sliding velocity of particles on contact surfaces. Admissible displacement and velocity fields are constructed with the Ostrogradski method. It is found that the corresponding variational formulation is a quasi-variational inequality. An iterative method for the solution of this inequality is proposed. An a priori solution estimate is given as well as the foundation of the transition from velocities to displacements in the governing law. Chapter 5 is devoted to transformations of variational problems with unilateral constraints. Following the presentation method, the simplest problems without unilateral constraints are considered first. For such problems (BVP for an ordinary differential equation and BVP for the Poisson equation) the classical Friedrichs transformation is adequate and permits solutions to the dual problems as well as to the saddle-point problems. This method permits us to find the equilibrium and the mixed and hybrid variational principles [BF91].

8 viii Preface It is well known that the generalization of the Friedrichs transformation to problems with unilateral constraints was made by Young, Fenchel, and Moreau. This transformation was applied to contact problems and enables one to obtain a set of variational problems including some new variational principles. In Chapter 6, the variational principles and methods for nonstationary problems are considered, including variational principles for dissipative systems and the Gurtin method for the dynamic elasticity problem and for viscoelastic BVP. Here, a description of contact interaction with adhesion is given. Chapter 7 is devoted to solutions method, algorithms, and their numerical implementation. The finite element and BEMs are presented. The solution of some model problems as well as important technical ones are given for 2D and 3D formulations. In conclusion (Chapter 8), we outline some modern research directions in the variational theory of unilateral problems, including relations of this theory with identification and optimization problems, history and recent results obtained in the theory of contact problems with friction, wear phenomena, large displacements and deformations, adhesion which plays an important role in micromechanics and nanomechanics of deformed solids. A survey is given on the numerical solution of unilateral problems as well, using the finite element and BEMs. It must be emphasized that the variational approach has been developed rapidly in the last few years and has been successfully used in medicine, biology, economics, heat conduction modeling, tomography, and many others branches of science and technology. The starting point of our work on unilateral problems was a collaboration with Professors R. Glowinski and P. A. Raviart. We are happy to acknowledge their contributions. We thank all our colleagues for fruitful discussions on many aspects of unilateral problems, in particular, Professors J. Haslinger, B. E. Pobedrya, N. V. Banichuk and the Full Member of the Russian Academy of Sciences I. G. Goryacheva. We appreciate the considerable contributions of the editor of our book, Professor G. M. L. Gladwell, for his numerous remarks and suggestions on the content of the book and exposition of the results. We express our sincere thanks to M.-L. Rantalainen for her excellent technical assistance and J. Räbinä for graphical illustration. Part of the research was supported by grants RFFI, No , , by Academy of Finland, and by Tekes (Finnish Funding Agency for Technology and Innovation) MASI programme. Jyväskylä, January 2007 Alexander Kravchuk Pekka Neittaanmäki

9 Contents 1 Notations and Basics Notations and conventions Space of independent variables Tensors and vectors Summation notation Algebraic operation Differential operator notation Functional spaces Open, closed, and compact sets in R n Metric spaces Sets in a metric space Normed linear spaces Inner product spaces, Hilbert spaces, and Lebesgue spaces Generalized derivatives, Sobolev spaces Some embedding theorems in a Sobolev space Dual (conjugate) spaces and weak convergence. Quotient space Bases and complete systems. Existence theorem Bases and complete systems Existence of a solution of a set of linear equations Tracetheorem Laws of thermodynamics Basic definitions First law of thermodynamics Second law of thermodynamics Variational Setting of Linear Steady-state Problems Problem of the equilibrium of systems with a finite number of degrees of freedom

10 x Contents 2.2 Equilibrium of the simplest continuous systems governed by ordinary differential equations Second-order problems Problems for fourth-order equations D and 2D problems on the equilibrium of linear elasticbodies Strain Stresses Strain stress relation (the generalized Hooke law) Formulation of boundary value problems D problems of the linear theory of elasticity Transition to the variational formulation Positive definiteness of the potential energy oflinearsystems Uniqueness of the minimum point Positive definiteness inequality for scalar functions Applications to linear elasticity problems Variational Theory for Nonlinear Smooth Systems Examplesofnonlinearsystems Differentiation of operators and functionals Existence and uniqueness theorems of the minimal point of a functional Condition for the potentiality of an operator Boundary value problems in the Hencky Ilyushin theory of plasticity without discharge Problems in the elastic bodies theory with finite displacements andstrain Vectors and strains in systems of curvilinear coordinates Strains and stress Equilibrium (motion) equations Governing equations Principle of virtual work Unilateral Constraints and Nondifferentiable Functionals Introduction: systems with finite degrees of freedom Example Comment of the development of the variational inequalities method Variational methods in contact problems for deformed bodies without friction Contact problems for a beam and membrane Contact between an elastic body and a rigid stamp

11 Contents xi Contact of a system of deformed bodies Geometrically nonlinear theory of elasticity Comments Variational method in contact problem with friction Generalities Quasi-variational inequality for the contact problem with friction Interpretation Solution of the quasi-variational inequality A priori estimate of the dynamic problem solution Dynamic contact of an elastic solid and a system of rigid moving indenters Local potential method Proportional processes Further generalization Comments Transformation of Variational Principles Friedrichs transformation Introduction Boundary value problem for the ordinary differential equation Dirichlet problem for the Poisson equation Equilibrium, mixed and hybrid variational principles inthetheoryofelasticity Castigliano principle and the Reissner principle Hybrid principles Kinematic constraints in the domain (Herrmann principle) Young Fenchel Moreau duality transformation Definition of duality transformation General definition of subdifferential and subgradient Method of solving minimization (maximization) problems Transition to the saddle-point problem Special cases Applications of duality transformations in contact problems Disturbance of the Arrow Hurwitz form in the problem (P) Disturbance of the Castigliano form Combined disturbance (Lagrangians) Generalization for the deformation theory of plasticity Contact problem for several elastic bodies Comments

12 xii Contents 6 Nonstationary Problems and Thermodynamics Traditional principles and methods Differential variational principles Integral principles Study of dissipative systems with the finite number of degrees of freedom Continuous conservative systems Example of a dissipative continuous system Gurtinmethod Wave equation Heat conduction equation Dynamic problem of the linear theory of elasticity Theory of linear viscoelasticity Thermodynamics and mechanics of the deformed solids Generalities Extremum principles for dissipation and entropy Extremum principles in the theory of plastic flow Theory of normal dissipative mechanisms Variational theory of adhesion and crack initiation Theory of adhesive joints formation and destruction Corollaries I and II of the laws of thermodynamics State equations Johnson Kendall Roberts theory of adhesion (JKRtheory) Examples in the JKR theory Models of accumulation of damages on the surface Model of the viscous crack motion Solution Methods and Numerical Implementation Frictionless contact problems: finite element method Generalities: continuous problem Finite element method: examples Friction contact problems: boundary element method Boundary element method Numerical examples to 2D contact problems Solution to 3D contact problems with the Boussinesq andcerrutiformulae Concluding Remarks Modeling, and identification problem, and optimization Development of the contact problems with friction, wear, and adhesion

13 Contents xiii 8.3 Numerical implementation of the contact interaction phenomena References Index...325