Table of Contents: Introduction to Nonlinear Thermomechanics of Solids

Introduction to Nonlinear Thermomechanics of Solids

The first part of this textbook presents the mathematical background needed to precisely describe the basic problem of continuum thermomechanics. The book then concentrates on developing governing equations for the problem dealing in turn with the kinematics of material continuum, description of the...

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Main Authors:	Kleiber, Michał. (Author, http://id.loc.gov/vocabulary/relators/aut), Kowalczyk, Piotr. (http://id.loc.gov/vocabulary/relators/aut)
Corporate Author:	SpringerLink (Online service)
Language:	English
Published:	Cham : Springer International Publishing : Imprint: Springer, 2016.
Edition:	1st ed. 2016.
Series:	Lecture Notes on Numerical Methods in Engineering and Sciences,
Subjects:	Mechanics. Mechanics, Applied. Mathematical physics. Mechanical engineering. Computer-aided engineering. Computer mathematics. Solid Mechanics. Classical Mechanics. Theoretical, Mathematical and Computational Physics. Mechanical Engineering. Computer-Aided Engineering (CAD, CAE) and Design. Computational Science and Engineering.
Online Access:	https://doi.org/10.1007/978-3-319-33455-4

Table of Contents:

1. Introduction
1.1. General remarks on the book content
1.2. Nonlinear continuum thermomechanics as a field of research and its industrial applications
2. Fundamental concepts of mechanics
2.1. Statics of a bar
2.2. Trusses
2.3. Two-dimensional continuum generalization
3. Fundamentals of tensor algebra and analysis
3.1. Introduction
3.1.1. Euclidean space and coordinate systems
3.1.2. Scalars and vectors
3.1.3. Basis of vector space
3.2. Tensors
3.2.1. Definitions
3.2.2. Operations on tensors
3.2.3. Isotropic tensors
3.3. Second order tensors
3.3.1. Definitions and properties
3.3.2. Tensor eigenproblem
3.3.3. Spectral decomposition of symmetric tensor
3.3.4. Polar decomposition of tensor
3.4. Tensor functions and fields
3.4.1. Integration and differentiation of tensor fields
3.4.2. Gauss–Ostrogradski theorem
3.5. Curvilinear coordinate systems
3.6. Notations used in tensor description
4. Motion, deformation and strain in material continuum
4.1. Motion of bodies
4.2. Strain
4.2.1. Definitions
4.2.2. Physical meaning of strain in one dimension
4.2.3. Physical meaning of strain components
4.2.4. Some other strain tensor properties
4.3. Area and volumetric deformation
4.4. Strain rate and strain increments
4.4.1. Time derivative of a tensor field. Lagrangian and Eulerian description of motion
4.4.2. Increments and rates of strain tensor measures
4.4.3. Strain increments and rates in one dimension
4.5. Strain compatibility equations
5. Description of stress state
5.1. Introduction
5.1.1. Forces, stress vectors and stress tensor in continuum
5.1.2. Principal stress directions. Extreme stress values
5.2. Description of stress in deformable body
5.2.1. Cauchy and Piola–Kirchhoff stress tensors
5.2.2. Objectivity and invariance of stress measures
5.3. Increments and rates of stress tensors
5.4. Work of internal forces. Conjugate stress–strain pairs
6. Conservation laws in continuum mechanics
6.1. Mass conservation law
6.2. Momentum conservation law
6.3. Angular momentum conservation law
6.4. Mechanical energy conservation law
7. Constitutive equations
7.1. Introductory remarks
7.2. Elastic materials
7.2.1. Linear elasticity
7.2.2. Nonlinear elasticity
7.3. Viscoelastic materials
7.3.1. One-dimensional models
7.3.2. Continuum formulation
7.3.3. Energy dissipation in viscoelastic materials
7.4. Elastoplastic materials
7.4.1. One-dimensional models
7.4.2. Three-dimensional formulation in plastic flow theory
8. Fundamental system of solid mechanics equations
8.1. Field equations and initial-boundary conditions
8.2. Incremental form of equations
8.3. Some special cases
8.4. Example of analytical solution
9. Fundamentals of thermomechanics and heat conduction problem
9.1. Laws of thermodynamics
9.1.1. The first law of thermodynamics
9.1.2. The second law of thermodynamics
9.2. Heat conduction problem
9.3. Fundamental system of solid thermomechanics equations. Thermomechanical Couplings
9.4. Thermal expansion in constitutive equations of linear elasticity
10. Variational formulations in solid thermomechanics
10.1. Variational principles — introduction
10.2. Variational formulations for linear mechanics problems
10.2.1. Virtual work principle and potential energy
10.2.2. Extended variational formulations
10.3. Variational formulations for nonlinear mechanics problems
10.3.1. Elasticity at large deformations
10.3.2. Incremental problem of nonlinear mechanics
10.4. Variational formulations for heat conduction problems
11. Discrete formulations in thermomechanics
11.1. Discrete formulations in heat conduction problems
11.1.1. Linear problem of stationary heat conduction
11.1.2. General form of the heat conduction problem
11.2. Discrete formulations in solid mechanics problems
11.2.1. Linear problem of statics
11.2.2. Linear problem of dynamics
11.2.3. Nonlinear elastic problem with large deformations
11.2.4. Incremental form of nonlinear mechanics problem
11.3. Weighted residual method
12. Fundamentals of finite element method
12.1. Introduction
12.1.1. FEM formulation for linear heat conduction problem
12.1.2. FEM formulation for linear static elasticity problem
12.2. FEM approximation at the element level
12.2.1. Simple one-dimensional elements
12.2.2. Constant strain elements
12.2.3. Isoparametric elements
13. Solution of FEM equation systems
13.1. Introduction
13.2. Solution methods for linear algebraic equation systems
13.2.1. Elimination methods
13.2.2. Iterative methods
13.3. Multigrid methods
13.4. Solution methods for nonlinear algebraic equation systems
13.5. Solution methods for linear and nonlinear systems of first order ordinary differential equations
13.6. Solution methods for linear and nonlinear systems of second order ordinary differential equations
Bibliography
Index. .

Introduction to Nonlinear Thermomechanics of Solids

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