Introduction to Continuum Mechanics for Engineers: (39 Mathematical Concepts and Methods in Science and Engineering)

This textbook is intended to introduce engineering graduate students to the essentials of modern continuum mechanics. The objective of an introductory course is to establish certain classical continuum models within a modern framework. Engineering students need a firm understanding of classical models such as linear viscous fluids (Navier-Stokes theory) and infinitesimal elasticity. This understanding should include an appreciation for the status of the classical models as special cases of general nonlinear continuum models. The relationship of the classical models to nonlinear models is essential in light of the increasing reliance, by engineering designers and researchers, on prepackaged computer codes. These codes are based upon models which have a specific and limited range of validity. Given the danger associated with the use of these computer codes in circumstances where the model is not valid, engineers have a need for an in-depth understanding of continuum mechanics and the continuum models which can be formu- lated by use of continuum mechanics techniques. Classical continuum models and others involve a utilization of the balance equations of continuum mechanics, the second law of thermo- dynamics, and the principles of material frame indifference and material symmetry. In addition, they involve linearizations of various types. In this text, an effort is made to explain carefully how the governing principles, linearizations, and other approximations combine to yield classical con- tinuum models. A fundamental understanding of how these models evolve is most helpful when one attempts to study models which account for a wider array of physical phenomena.

Table of Contents:
1. One-Dimensional Continuum Mechanics.- 1.1. Kinematics of Motion and Strain.- 1.2. Balance of Mass.- 1.3. Balance of Linear Momentum.- 1.4. Balance of Energy.- 1.5. General Balance.- 1.6. The Entropy Inequality.- 1.7. Example Constitutive Equations.- 1.8. Thermodynamic Restrictions.- 1.9. Small Departures from Thermodynamic Equilibrium.- 1.10. Small Departures from Static Equilibrium.- 1.11. Some Features of the Linear Model.- 2. Kinematics of Motion.- 2.1. Bodies and Deformations.- 2.2. Velocity, Acceleration, and Deformation Gradients.- 2.3. Transformation of Linear, Surface, and Volume Elements.- 2.4. Strain Kinematics.- 2.5. Infinitesimal Strain Kinematics.- References.- 3. Equations of Balance.- 3.1. Balance of Mass.- 3.2. Balance of Linear Momentum.- 3.3. Balance of Angular Momentum.- 3.4. Balance of Energy.- 3.5. The Entropy Inequality.- 3.6. Jump Equations of Balance-Material Versions.- References.- 4. Models of Material Behavior.- 4.1. Examples.- 4.2. Isothermal Elasticity-Thermodynamic Restrictions.- 4.3. Isothermal Elasticity-Material Frame Indifference.- 4.4. Isothermal Elasticity-Material Symmetry.- 4.5. Incompressible Isothermal Elasticity.- 4.6. Thermoelastic Material with Heat Conduction and Viscous Dissipation-Constitutive Assumptions.- 4.7. Thermoelastic Material with Heat Conduction and Viscous Dissipation-General Thermodynamic Restrictions.- 4.8. Thermoelastic Material with Heat Conduction and Viscous Dissipation-Equilibrium Thermodynamic Restrictions.- 4.9. Thermoelastic Material with Heat Conduction and Viscous Dissipation-Material Frame Indifference.- 4.10. Thermoelastic Material with Heat Conduction and Viscous Dissipation-Material Symmetry.- 4.11. Constitutive Equations for a Compressible, Conducting, Viscous Fluid.- 4.12. Constitutive Equations for an Isotropic Linear Thermoelastic Solid with Heat Conduction.- References.- 5. Materials with Internal State Variables.- 5.1. Constitutive Assumptions and Thermodynamic Results.- 5.2. Maxwell-Cattaneo Heat Conductor.- 5.3. Maxwellian Materials.- 5.4. Closing Remarks-Alternate Forms of the Entropy Inequality.- References.- Appendix A. Mathematical Preliminaries.- A.1. Vector Spaces.- A.2. Linear Transformations.- A.3. Inner Product Spaces.- A.4. Components of Vectors and Linear Transformations.- A.5. Cross Products, Determinants, and the Polar Decomposition Theorem.- A.6. Multilinear Functionals and Tensor Algebra.- A.7. Euclidean Point Spaces, Coordinate Systems.- A.8. Vector Analysis.- Appendix B. Representation Theorems.