An Introduction to Mathematical Modeling

A Course in Mechanics

J. Tinsley Oden

• Publisher: Wiley
• Year: 2012
• ISBN:
  9781118105740
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

A modern approach to mathematical modeling, featuring unique applications from the field of mechanics

An Introduction to Mathematical Modeling: A Course in Mechanics is designed to survey the mathematical models that form the foundations of modern science and incorporates examples that illustrate how the most successful models arise from basic principles in modern and classical mathematical physics. Written by a world authority on mathematical theory and computational mechanics, the book presents an account of continuum mechanics, electromagnetic field theory, quantum mechanics, and statistical mechanics for readers with varied backgrounds in engineering, computer science, mathematics, and physics.

The author streamlines a comprehensive understanding of the topic in three clearly organized sections:

• Nonlinear Continuum Mechanics introduces kinematics as well as force and stress in deformable bodies; mass and momentum; balance of linear and angular momentum; conservation of energy; and constitutive equations
  
  

• Electromagnetic Field Theory and Quantum Mechanics contains a brief account of electromagnetic wave theory and Maxwell's equations as well as an introductory account of quantum mechanics with related topics including ab initio methods and Spin and Pauli's principles
  
  

• Statistical Mechanics presents an introduction to statistical mechanics of systems in thermodynamic equilibrium as well as continuum mechanics, quantum mechanics, and molecular dynamics
  
  

Each part of the book concludes with exercise sets that allow readers to test their understanding of the presented material. Key theorems and fundamental equations are highlighted throughout, and an extensive bibliography outlines resources for further study.

Extensively class-tested to ensure an accessible presentation, An Introduction to Mathematical Modeling is an excellent book for courses on introductory mathematical modeling and statistical mechanics at the upper-undergraduate and graduate levels. The book also serves as a valuable reference for professionals working in the areas of modeling and simulation, physics, and computational engineering.

Part I

Nonlinear Continuum Mechanics

CHAPTER 1

KINEMATICS OF DEFORMABLE BODIES

Continuum mechanics models the physical universe as a collection of “deformable bodies,” a concept that is easily accepted from our everyday experiences with observable phenomena. Deformable bodies occupy regions in three-dimensional Euclidean space

, and a given body will occupy different regions at different times. The subsets of

occupied by a body

are called its configurations. It is always convenient to identify one configuration in which the geometry and physical state of the body are known and to use that as the reference configuration; then other configurations of the body can be characterized by comparing them with the reference configuration (in ways we will make precise later).

For a given body, we will assume that the reference configuration is an open, bounded, connected subset Ω0 of

3 with a smooth boundary

. The body is made up of physical points called material points. To identify these points, we assign each a vector X and we identify the components of X as the coordinates of the place occupied by the material point when the body is in its reference configuration relative to a fixed Cartesian coordinate system.

It is thus important to understand that the body

is a non-denumerable set of material points X. This is the fundamental hypothesis of continuum mechanics: Matter is not discrete; it is continuously distributed in one-to-one correspondence with points in some subset of

3. Bodies are thus “conti...

Table of contents

• Cover
• Half Title page
• Title page
• Copyright page
• Dedication
• Preface
• Part I: Nonlinear Continuum Mechanics
• Part II: Electromagnetic Field Theory and Quantum Mechanics
• Part III: Statistical Mechanics
• Exercises
• Bibliography
• Index