# Applied Mechanics of Solids

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## Book Description

Modern computer simulations make stress analysis easy. As they continue to replace classical mathematical methods of analysis, these software programs require users to have a solid understanding of the fundamental principles on which they are based.

__Develop Intuitive Ability to Identify and Avoid Physically Meaningless Predictions__

**Applied Mechanics of Solids** is a powerful tool for understanding how to take advantage of these revolutionary computer advances in the field of solid mechanics. Beginning with a description of the physical and mathematical laws that govern deformation in solids, the text presents modern constitutive equations, as well as analytical and computational methods of stress analysis and fracture mechanics. It also addresses the nonlinear theory of deformable rods, membranes, plates, and shells, and solutions to important boundary and initial value problems in solid mechanics.

The author uses the step-by-step manner of a blackboard lecture to explain problem solving methods, often providing the solution to a problem before its derivation is presented. This format will be useful for practicing engineers and scientists who need a quick review of some aspect of solid mechanics, as well as for instructors and students.

__Select and Combine Topics Using Self-Contained Modules and Subsections__

Borrowing from the classical literature on linear elasticity, plasticity, and structural mechanics, this book:

- Introduces concepts, analytical techniques, and numerical methods used to analyze deformation, stress, and failure in materials or components
- Discusses the use of finite element software for stress analysis
- Assesses simple analytical solutions to explain how to set up properly posed boundary and initial-value problems
- Provides an understanding of algorithms implemented in software code

Complemented by the author’s website, which features problem sets and sample code for self study, this book offers a crucial overview of problem solving for solid mechanics. It will help readers make optimal use of commercial finite element programs to achieve the most accurate prediction results possible.

## Table of Contents

**1 Overview of Solid Mechanics**

DEFINING A PROBLEM IN SOLID MECHANICS

**2 Governing Equations**

MATHEMATICAL DESCRIPTION OF SHAPE CHANGES IN SOLIDS

MATHEMATICAL DESCRIPTION OF INTERNAL FORCES IN SOLIDS

EQUATIONS OF MOTION AND EQUILIBRIUM FOR DEFORMABLE

SOLIDS

WORK DONE BY STRESSES: PRINCIPLE OF VIRTUAL WORK

**3 Constitutive Models: Relations between Stress and Strain**

GENERAL REQUIREMENTS FOR CONSTITUTIVE EQUATIONS LINEAR ELASTIC MATERIAL BEHAVIORSY

HYPOELASTICITY: ELASTIC MATERIALS WITH A NONLINEAR STRESS-STRAIN RELATION UNDER SMALL DEFORMATION

GENERALIZED HOOKE’S LAW: ELASTIC MATERIALS SUBJECTED TO SMALL STRETCHES BUT LARGE ROTATIONS

HYPERELASTICITY: TIME-INDEPENDENT BEHAVIOR OF RUBBERS AND FOAMS SUBJECTED TO LARGE STRAINS

LINEAR VISCOELASTIC MATERIALS: TIME-DEPENDENT BEHAVIOR OF POLYMERS AT SMALL STRAINS

SMALL STRAIN, RATE-INDEPENDENT PLASTICITY: METALS LOADED BEYOND YIELD

SMALL-STRAIN VISCOPLASTICITY: CREEP AND HIGH STRAIN RATE DEFORMATION OF CRYSTALLINE SOLIDS

LARGE STRAIN, RATE-DEPENDENT PLASTICITY

LARGE STRAIN VISCOELASTICITY

CRITICAL STATE MODELS FOR SOILS

CONSTITUTIVE MODELS FOR METAL SINGLE CRYSTALS

CONSTITUTIVE MODELS FOR CONTACTING SURFACES AND INTERFACES IN SOLIDS

**4 Solutions to Simple Boundary and Initial Value Problems**

AXIALLY AND SPHERICALLY SYMMETRIC SOLUTIONS TO QUASI-STATIC LINEAR ELASTIC PROBLEMS

AXIALLY AND SPHERICALLY SYMMETRIC SOLUTIONS TO QUASI-STATIC ELASTIC-PLASTIC PROBLEMS

SPHERICALLY SYMMETRIC SOLUTION TO QUASI-STATIC LARGE

STRAIN ELASTICITY PROBLEMS

SIMPLE DYNAMIC SOLUTIONS FOR LINEAR ELASTIC MATERIALS

**5 Solutions for Linear Elastic Solids **

GENERAL PRINCIPLES

AIRY FUNCTION SOLUTION TO PLANE STRESS AND STRAIN STATIC LINEAR ELASTIC PROBLEMS

COMPLEX VARIABLE SOLUTION TO PLANE STRAIN STATIC LINEAR ELASTIC PROBLEMS

SOLUTIONS TO 3D STATIC PROBLEMS IN LINEAR ELASTICITY

SOLUTIONS TO GENERALIZED PLANE PROBLEMS FOR ANISOTROPIC LINEAR ELASTIC SOLIDS

SOLUTIONS TO DYNAMIC PROBLEMS FOR ISOTROPIC LINEAR ELASTIC SOLIDS

ENERGY METHODS FOR SOLVING STATIC LINEAR ELASTICITY PROBLEMS

THE RECIPROCAL THEOREM AND APPLICATIONS

ENERGETICS OF DISLOCATIONS IN ELASTIC SOLIDS

RAYLEIGH-RITZ METHOD FOR ESTIMATING NATURAL FREQUENCY OF AN ELASTIC SOLID

**6 Solutions for Plastic Solids**

SLIP-LINE FIELD THEORY

BOUNDING THEOREMS IN PLASTICITY AND THEIR

APPLICATIONS

**7 Finite Element Analysis: An Introduction**

A GUIDE TO USING FINITE ELEMENT SOFTWARE

A SIMPLE FINITE ELEMENT PROGRAM

**8 Finite Element Analysis: Theory and Implementation**

GENERALIZED FEM FOR STATIC LINEAR ELASTICITY

THE FEM FOR DYNAMIC LINEAR ELASTICITY

FEM FOR NONLINEAR (HYPOELASTIC) MATERIALS

FEM FOR LARGE DEFORMATIONS: HYPERELASTIC MATERIALS

THE FEM FOR VISCOPLASTICITY

ADVANCED ELEMENT FORMULATIONS: INCOMPATIBLE MODES, REDUCED INTEGRATION, AND HYBRID ELEMENTS

LIST OF EXAMPLE FEA PROGRAMS AND INPUT FILES

**9 Modeling Material Failure**

SUMMARY OF MECHANISMS OF FRACTURE AND FATIGUE UNDER STATIC AND CYCLIC LOADING

STRESS- AND STRAIN-BASED FRACTURE AND FATIGUE CRITERIA

MODELING FAILURE BY CRACK GROWTH: LINEAR ELASTIC FRACTURE MECHANICS

ENERGY METHODS IN FRACTURE MECHANICS

PLASTIC FRACTURE MECHANICS

LINEAR ELASTIC FRACTURE MECHANICS OF INTERFACES

**10 Solutions for Rods, Beams, Membranes, Plates, and Shells**

PRELIMINARIES: DYADIC NOTATION FOR VECTORS AND TENSORS

MOTION AND DEFORMATION OF SLENDER RODS

SIMPLIFIED VERSIONS OF THE GENERAL THEORY OF DEFORMABLE ROD

EXACT SOLUTIONS TO SIMPLE PROBLEMS INVOLVING ELASTIC RODS

MOTION AND DEFORMATION OF THIN SHELLS: GENERAL THEORY

SIMPLIFIED VERSIONS OF GENERAL SHELL THEORY: FLAT PLATES AND MEMBRANES

SOLUTIONS TO SIMPLE PROBLEMS INVOLVING MEMBRANES, PLATES, AND SHELLS

**Appendix A: Review of Vectors and Matrices**

A.1. VECTORS

A.2. VECTOR FIELDS AND VECTOR CALCULUS

A.3. MATRICES

**Appendix B: Introduction to Tensors and Their Properties**

B.1. BASIC PROPERTIES OF TENSORS

B.2. OPERATIONS ON SECOND-ORDER TENSORS

B.3. SPECIAL TENSORS

**Appendix C: Index Notation for Vector and Tensor Operations**

C.1. VECTOR AND TENSOR COMPONENTS

C.2. CONVENTIONS AND SPECIAL SYMBOLS FOR INDEX

NOTATION

C.3. RULES OF INDEX NOTATION

C.4. VECTOR OPERATIONS EXPRESSED USING INDEX NOTATION

C.5. TENSOR OPERATIONS EXPRESSED USING INDEX NOTATION

C.6. CALCULUS USING INDEX NOTATION

C.7. EXAMPLES OF ALGEBRAIC MANIPULATIONS USING

INDEX NOTATION

**Appendix D: Vectors and Tensor Operations in Polar Coordinates**

D.1. SPHERICAL-POLAR COORDINATES

D.2. CYLINDRICAL-POLAR COORDINATES

**Appendix E: Miscellaneous Derivations**

E.1. RELATION BETWEEN THE AREAS OF THE FACES OF A

TETRAHEDRON

E.2. RELATION BETWEEN AREA ELEMENTS BEFORE AND AFTER DEFORMATION

E.3. TIME DERIVATIVES OF INTEGRALS OVER VOLUMES WITHIN A DEFORMING SOLID

E.4. TIME DERIVATIVES OF THE CURVATURE VECTOR FOR A

DEFORMING ROD

**References**

## Reviews

"It’s an impressive piece of work. It covers an immense amount of material and is well written."

—James R. Barber, University of Michigan, Ann Arbor, USA"A valuable treatise in mechanics of solids."

—David L. McDowell, Georgia Institute of Technology, Atlanta, USA"This book is great for materials science and engineering students who are interested in both classic and state-of-the-art materials research. With both in-depth discussion and authoritative summary, it will be widely used in teaching and research in theoretical and computational mechanics of materials."

—Yanfei Gao, University of Tennessee, Knoxville, USA