Variational and Quasi-Variational Inequalities in Mechanics

ISBN-13: 9789048176199 / Angielski / Miękka / 2010 / 337 str.

Alexander S. Kravchuk; Pekka J. Neittaanmaki

Opis

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 sta- ments are physical or natural laws themselves. The essence of the variational approach for the solution of problems rel- ing to the determination of the real state of systems or processes consists in thecomparisonofclosestates.Theselectioncriteriafortheactualstatesmust be such that all the equations and conditions of the mathematical model are satis?ed. Historically, the ?rst variational theory was the Lagrange theory created to investigate the equilibrium of ?nite-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 in?nitesimally 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 dist- bances 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.

Recenzje

From the reviews:

"The main idea of the presented monograph is to deal with mathematical models connected with mechanical systems under unilateral constraints. ... Examples of analytical and numerical solutions are presented. Numerical solutions were obtained using the finite element and boundary element methods. ... The text contains a big amount of latest results achieved in mathematical modeling of contact problems in mechanics together with applications. It can be recommended both to graduate students and the researchers in applied mathematics and mechanics." (Igor Bock, Zentralblatt MATH, Vol. 1131 (9), 2008)

"The aim of this interesting book is the study of problems in the mathematical modelling of mechanical systems ... . The work is intended for a wide audience: this would include specialists in contact processes in structural and mechanical systems ... as well as those with a background in the mathematical sciences who seek a self-contained account of the mathematical theory of contact mechanics. The text is suitable for graduate students and researchers in applied mathematics, computational mathematics, and computational mechanics." (Ján Lovisek, Mathematical Reviews, Issue 2009 e)

Spis treści

1. Notation and Basics: 1.1. Notations and Conventions; 1.2. Functional spaces; 1.3. Bases and complete systems. Existence theorem; 1.4. Trace Theorem; 1.5. The laws of thermodynamics; 2. Variational Setting of Linear Steady-state Problems: 2.1. Problem of the equilibrium of system with a finite number of degrees of Freedom; 2.2. Equilibrium of the simplest continuous systems governed by ordinary differential Equations; 2.3. 3D and 2D problems on the equilibrium of linear elastic bodies; 3.4. Positive definiteness of the potential energy of linear systems; 3.Variational Theory for Nonlinear Smooth Systems: 3.1. Examples of nonlinear systems; 3.2. Differentiation of operators and functionals; 3.3. Existence and uniqueness theorems of the minimal point of a functional; 3.4. Condition for the potentiality of an operator; 3.5. Boundary value problems in the Hencky-Ilyushin theory of plasticity without discharge; 3.6. Problems in the elastic bodies theory with finite displacements and strain; 4. Unilateral Constraints and Non-Differentiable Functionals: 4.1. Introduction: systems with finite degrees of freedom; 4.2. Variational methods in contact problems for deformed bodies without friction; 4.3. Variational method in contact problem with friction; 5. The Transformation of Variational Principles: 5.1. Friedrichs Transformation; 5.2. Equilibrium, mixed and hybrid variational principles in the theory of elasticity; 5.3. The Young-Fenchel-Moreau duality transformation; 5.4. Applications of duality transformations in contact problems; 6. Non-Stationary Problems and Thermodynamics: 6.1. Traditional principles and methods; 6.2. Gurtin’s method; 6.3. Thermodynamics and mechanics of the deformed solids; 6.4. The variational theory of adhesion and crack initiation; 7. Solution Methods and Numerical implementation: 7.1. Frictionless contact problems: finite element method (FEM); 7.2. Friction contact problems: boundaryelement method (BEM); 8. Concluding Remarks: 8.1. Modelling. Identification problem. Optimization; 8.2. Development of the contact problems with friction, wear and adhesion; 8.3. Numerical implementation of the contact interaction phenomena; References; Index.