Nonlinear Continuum Mechanics and Large Inelastic Deformations

ISBN-13: 9789400734135 / Angielski / Miękka / 2013 / 721 str.

Yuriy I. Dimitrienko

Opis

The book provides a rigorous axiomatic approach to continuum mechanics under large deformation. In addition to the classical nonlinear continuum mechanics - kinematics, fundamental laws, the theory of functions having jump discontinuities across singular surfaces, etc. - the book presents the theory of co-rotational derivatives, dynamic deformation compatibility equations, and the principles of material indifference and symmetry, all in systematized form. The focus of the book is a new approach to the formulation of the constitutive equations for elastic and inelastic continua under large deformation. This new approach is based on using energetic and quasi-energetic couples of stress and deformation tensors. This approach leads to a unified treatment of large, anisotropic elastic, viscoelastic, and plastic deformations. The author analyses classical problems, including some involving nonlinear wave propagation, using different models for continua under large deformation, and shows how different models lead to different results. The analysis is accompanied by experimental data and detailed numerical results for rubber, the ground, alloys, etc. The book will be an invaluable text for graduate students and researchers in solid mechanics, mechanical engineering, applied mathematics, physics and crystallography, as also for scientists developing advanced materials.

Recenzje

From the reviews:

"The present book is a textbook for master and PhD students studying the mathematical continuum mechanics. Based on a strong mathematical framework, the author presents the basics of continuum mechanics and materials theory. ... For further studies 60 references are given." (Holm Altenbach, Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 92 (6), 2012)

Spis treści

Preface.- Introduction. Fundamental Axioms of Continuum Mechanics.- 1. Kinematics of Continua .- 1.1. Material and Spatial Descriptions of Continuum Motion .- 1.2. Deformation Tensors and Measures.- 1.3. Polar Decomposition.- 1.4. Rate Characteristics of Continuum .- 1.5. Co-rotational Derivatives.- 2. Balance Laws.- 2.1. The Mass Conservation Law.- 2.2. The Momentum Balance Law and the Stress Tensor.- 2.3. The Angular Momentum Balance Law.- 2.4. The First Thermodynamic Law.- 2.5. The Second Thermodynamic Law .- 2.6. Deformation Compatibility Equations.- 2.7. Dynamic Compatibility Equations .- 2.8. Compatibility Equations for Deformation Rates.- 2.9. The Complete System of Continuum Mechanics.- 3. Constitutive Equations .- 3.1. Basic Principles for Derivation of Constitutive Equations.- 3.2. Energetic and Quasienergetic Couples of Tensors .- 3.3. The Principal Thermodynamic Identity.- 3.4. Principles of Thermodynamically Consistent Determinism, Equipresence and Local Action .- 3.5. Definition of Ideal Continua .- 3.6. The Principle of Material Symmetry.- 3.7. Definition of Fluids and Solids.- 3.8. Corollaries of the Principle of Material Symmetry and Constitutive Equations for Ideal Continua .- 3.9. Incompressible continua.- 3.10. The Principle of Material Indifference.- 3.11. Relationships in a Moving System.- 3.12. The Onsager Principle .- 4. Relations at Singular Surfaces .- 4.1. Relations at a Singular Surface in the Material Description.- 4.2. Relations at a Singular Surface in the Spatial Description .- 4.3. Explicit Form of Relations at a Singular Surface.- 4.4. The Main Types of Singular.- 5. Elastic Continua at Large Deformations .- 5.1. Closed Systems in the Spatial Description .- 5.2. Closed Systems in the Material Description .- 5.3. Statements of Problems for Elastic Continua at Large Deformations .- 5.4. The problem on an Elastic Beam in Tension.- 5.5. Tension of an Incompressible Beam .- 5.6. Simple Shear .- 5.7. The Lamé Problem.- 5.8. The Lamé Problem for an Incompressible Continuum.- 6. Continua of the Differential Type .- 6.1. Models An and Bn of Continua of the Differential Type .- 6.2. Models An and Bn of Fluids of the Differential Type .- 6.3. Models Cn and Dn of Continua of the Differential Type .- 6.4. The Problem on a Beam in Tension .- 7. Viscoelastic Continua at Large Deformations.- 7.1. Viscoelastic Continua of the Integral Type .- 7.2. Principal, Quadratic and Linear Models of Viscoelastic Continua .- 7.3. Models of Incompressible Viscoelastic Solids and Viscoelastic Fluids .- 7.4. Statements of Problems in Viscoelasticity Theory at Large Deformations.- 7.5. The Problem on Uniaxial Deforming of a Viscoelastic Beam .- 7.6. Dissipative Heating of a Viscoelastic Continuum under Cyclic Deforming.- 8. Plastic Continua at Large Deformations .- 8.1. Models An of Plastic Continua at Large Deformations .- 8.2. Models Bn of Plastic Continua .- 8.3. Models Cn and Dn of Plastic Continua .- 8.4. Constitutive Equations of Plasticity Theory 'in Rates' .- 8.5. Statements of Problems in Plasticity Theory .- 8.6. The Problem on All-Round Tension-Compression of a Plastic Continuum.- 8.7. The Problem on Tension of a Plastic Beam .- 8.8. Plane Waves in Plastic Continua .- 8.9. Models of Viscoplastic Continua .- References .- Basic Notation.- Subject Index.