Materials with Memory: Initial-Boundary Value Problems for Constitutive Equations with Internal Variables

ISBN-13: 9783540640660 / Angielski / Miękka / 1998 / 170 str.

Hans-Dieter Alber; A. Dold; F. Takens

Opis

This book contributes to the mathematical theory of systems of differential equations consisting of the partial differential equations resulting from conservation of mass and momentum, and of constitutive equations with internal variables. The investigations are guided by the objective of proving existence and uniqueness, and are based on the idea of transforming the internal variables and the constitutive equations. A larger number of constitutive equations from the engineering sciences are presented. The book is therefore suitable not only for specialists, but also for mathematicians seeking for an introduction in the field, and for engineers with a sound mathematical background.

Spis treści

>1 Introduction 1 > >2 Initial-Boundary Value Problems for the Inelastic Behavior of > Metals 7 >2.1 Formulation of the Initial-Boundary Value Problems................7 >2.2 Examples of Constitutive Equations...............................11 > >3 Constitutive Equations of Monotone Type and Generalized > Standard Materials 23 >3.1 Energy Estimate and Classes of Constitutive Equations............23 >3.2 Uniqueness and Existence for Dynamic and Quasi-Static Problems: > Basic Ideas of the Proofs and Results in the Literature..........27 >3.3 Examples of Constitutive Equations Revisited.....................31 >3.4 A Criterion for Monotone Type....................................41 > >4 Existence of Solutions for Constitutive Equations of Monotone > Type 45 >4.1 Formulation of the Problem as a First Order Evolution Equation...45 >4.2 Maximality of the Evolution Operator.............................48 >4.3 Existence for the Dynamic Problem................................56 > >5 Transformation of Interior Variables 57 >5.1 Transformation Fields............................................57 >5.2 Properties of the Transformation: Restrictions Imposed by > the e-Independence of the Transformation Field > and Invariance under Linear Transformations......................60 >5.3 The Class of Constitutive Equations Transformable to Monotone > Type........................................................ >5.4 The Class of Constitutive Equations Transformable to Gradient > Type........................................................ >5.5 The Class of Constitutive Equations Transformable to Monotone- > Gradient Type....................................................69 > >6 Classification Conditions 75 >6.1 Transformations which Leave the Class of Pre-Monotone > Equations Invariant..............................................76 >6.2 Transformation of Pre-Monotone Equations to Gradient Type........87 > >7 Transformation of Rate Independent Constitutive Equations 99 >7.1 Transformationof Constitutive Equations Containing Set-Valued > Operators................................................... >7.2 Transformation to Monotone-Gradient Type........................102 >7.3 Example 1: One Variable of Isotropic Hardening..................106 >7.4 Example 2: Several Variables of Isotropic Hardening.............111 > >8 Application of the Theory to Engineering Models 117 >8.1 Pre-Monotone Type of the Model..................................117 >8.2 Conditions for Monotone Type of the Model.......................119 >8.3 Pre-Monotone Type Preserving Transformations of the Model.......125 >8.4 Transformation to Gradient Type.................................126 >8.5 Transformation to Monotone Type and to Monotone-Gradient > Type........................................................ > >9 Open Problems and Related Results 137 >9.1 Transformation Fields Depending on e and z......................137 >9.2 History Functionals.............................................138 >9.3 Hysteresis Operators. Existence Theory in L...................139 >9.4 Constitutive Equations Defining Continuous Operators in > Banach Spaces...................................................14 > >A The Second Law of Thermodynamics and the Dissipation > Inequality 143 >A.1 Consequences of the Second Law for the Constitutive Equations...143 >A.2 The Free Energy.................................................149 > >Bibliography 153 > >Index 165 >