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The mathematical theory of contact mechanics is a growing field in engineering and scientific computing. This book is intended as a unified and readily accessible source for mathematicians, applied mathematicians, mechanicians, engineers and scientists, as well as advanced students. The first part describes models of the processes involved like friction, heat generation and thermal effects, wear, adhesion and damage. The second part presents many mathematical models of practical interest and demonstrates the close interaction and cross-fertilization between contact mechanics and the theory of variational inequalities. The last part reviews further results, gives many references to current research and discusses open problems and future developments. The book can be read by mechanical engineers interested in applications. In addition, some theorems and their proofs are given as examples for the mathematical tools used in the models.
Índice: Function spaces and their properties; Introduction to finite difference and finite element approximations; Variational inequalities; Constitutive relations in solid mechanics; Background on variational and numerical analysis in contact mechanics; Contact problems in elasticity; Bilateral contact with slip dependent friction; Frictional contact with normal compliance; Frictional contact with normal damped response; Other viscoelastic contact problems; Frictionless contact with dissipative potential; Frictionless contact between two viscoplastic bodies; Bilateral contact with Tresca's friction law; Other viscoelastic contact problems; Bibliography; Index.
This work establishes a mathematical existence theory for solutions of some quasi-static models in contact mechanics with dry friction. The models are finite dimensional and friction is modeled according to Coulomb's law. The main focus is on the geometric non-linearity which is due to the curved obstacle surface.
This book introduces readers to a mathematical theory of contact problems involving deformable bodies. It covers mechanical modeling, mathematical formulations, variational analysis, and the numerical solution of the associated formulations. The authors give a complete treatment of some contact problems by presenting arguments and results in modeling, analysis, and numerical simulations. Variational analysis of the models includes existence and uniqueness results of weak solutions,as well as results of continuous dependence of the solution on the data and parameters. Also discussed are links between different mechanical models.
We formulate and study two mathematical models of a thermoforming process involving a membrane and a mould as implicit obstacle problems. In particular, the membrane-mould coupling is determined by the thermal displacement of the mould that depends in turn on the membrane through the contact region. The two models considered are a stationary (or elliptic) model and an evolutionary (or quasistatic) one. For the first model, we prove the existence of weak solutions by solving an elliptic quasi-variational inequality coupled to elliptic equations. By exploring the fine properties of the variation of the contact set under non-degenerate data, we give sufficient conditions for the existence of regular solutions, and under certain contraction conditions, also a uniqueness result. We apply these results to a series of semi-discretised problems that arise as approximations of regular solutions for the evolutionary or quasistatic problem. Here, under certain conditions, we are able to prove existence for the evolutionary problem and for a special case, also the uniqueness of time-dependent solutions.
Modelling of frictional devices is one of the greatest remaining problems in structural mechanics. The dissipative properties of frictional interfaces are central in design and analysis of built-up structures such as engines, aerospace structures and bladed-disk assemblies. These interfaces exhibit complex, nonlinear behavior and are the main source of uncertainly in finite element models for built-up structures. It is shown that the frictional interfaces are the main cause of damping and challenges in predicting the dynamic characteristics (stiffness, mode shapes, ...) of a systems containing interfaces. Improved methods for modeling joints have been presented recently, however, numerical computation of the dynamic behavior of a system containing interfaces can be excessively expensive due to the nonlinearity introduced by the interfaces. The resulting simulations are still quite expensive, often impractical for a system that contains many joints, and none of the simulations has been thoroughly validated against measurements. Especially, when a dynamic simulation is implemented, the computational cost becomes completely prohibitive since the response must span many cycles of oscillation. Indeed in most practical applications, the inertial terms can be neglected in a local sense (i.e. near the joint) since accelerations in those regions are small relative to elastic forces, which makes the implementation of quasi-static methods logical. The primary contribution of this work is the development and validation of novel quasi-static methods for modeling and analyzing the dynamic response of structures that contain frictional interfaces. The work seeks to answer the following questions: Are quasi-static methods effective to predict the dynamic characteristics of a nonlinear system? Are the results obtained using quasi-static methods verifiable against measurements? Are they applicable to complex and realistic systems containing many joints? How much efficiency is provided by using quasi-static methods instead of dynamic methods? What are the limitation of using quasi-static methods to be implemented as an alternative to the dynamic methods? Are the proposed quasi-static methods applicable for a nonlinear system subjected to harmonic forces? If quantitative predictions of quantities such as the nonlinear Frequency Response Functions are desired, at what level can the proposed quasi-static methods be used effectively? To address these concerns, several quasi-static methods are proposed in this thesis for different systems involving interfaces subjected to harmonic or random dynamic forces. The results obtained are verified against experimental measurements, the advantages of the proposed methods as well as their limitations are investigated, and the computational cost of these methods are compared with the alternative methods. Examples are presented in which the proposed quasi-static methods are demonstrated as feasible, efficient methods in order to predict the dynamic features of built-up systems including frictional devices.
This carefully edited book offers a state-of-the-art overview on formulation, mathematical analysis and numerical solution procedures of contact problems. The contributions collected in this volume summarize the lectures presented by leading scientists in the area of contact mechanics, during the 4th Contact Mechanics International Symposium (CMIS) held in Hannover, Germany, 2005.
This monograph presents an original method to unify the mathematical theories of well-posed problems and contact mechanics. The author uses a new concept called the Tykhonov triple to develop a well-posedness theory in which every convergence result can be interpreted as a well-posedness result. This will be useful for studying a wide class of nonlinear problems, including fixed-point problems, inequality problems, and optimal control problems. Another unique feature of the manuscript is the unitary treatment of mathematical models of contact, for which new variational formulations and convergence results are presented. Well-Posed Nonlinear Problems will be a valuable resource for PhD students and researchers studying contact problems. It will also be accessible to interested researchers in related fields, such as physics, mechanics, engineering, and operations research.
An invaluable instrument for gaining a wide-ranging perspective on the latest developments in mathematical aspects of scientific computing, discovering new applications and the most recent developments in long-standing applications. Provides an insight into the state of the art of Numerical Mathematics and, more generally, into the field of Advanced Applications.