Engineering Elasticity: Elasticity with less Stress and Strain

✍ Scribed by Humphrey Hardy

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 275
• Category: Library

✦ Synopsis

This textbook aimed at upper-level undergraduate and graduate engineering students who need to describe the large deformation of elastic materials like soft plastics, rubber, and biological materials.  The classical approaches to finite deformations of elastic materials describe a dozen or more measures of stress and strain.  These classical approaches require an in-depth knowledge of tensor analysis and provide little instruction as to how to relate the derived equations to the materials to be described.  This text, by contrast, introduces only one strain measure and one stress measure.  No tensor analysis is required.  The theory is applied by showing how to measure material properties and to perform computer simulations for both isotropic and anisotropic materials.  The theory can be covered in one chapter for students familiar with Euler-Lagrange techniques, but is also introduced more slowly in several chapters for students not familiar with these techniques.  The connection to linear elasticity is provided along with a comparison of this approach to classical elasticity.

✦ Table of Contents

Preface for Students
Preface for Instructors
Contents
About the Author
Chapter 1: Getting Ready (Mostly Review)
Linear Algebra
Scalars, Vector, and Matrices
Addition and Subtraction of Vectors
Multiplication of a Scalar Times a Vector
Multiplication of Vectors
Addition and Subtraction of Matrices
A Scalar Times a Matrix
Multiplication of Matrices
Multiplication of a Matrix Times a Vector
Transpose of a Matrix
Inverse of a Matrix
Length of a Vector
A Vector Field and Matrix Field
A Matrix Times a Vector Field
A Matrix Field Times a Vector Field
Calculus-Based Physics 1 Review
Newton´s Laws in Equation Form
Force, Work, and Energy
Miscellaneous
The Difference Between Point, Point Vector, and a General Vector
Material Properties
Isotropic and Anisotropic Materials
Homogeneous and Non-homogeneous Materials
Taylor Expansion and Physical Interpretation
Derivative of a Vector by a Scalar
Problems
Chapter 2: Deformations
A Map
Mathematics of a Mapping
The Application of a Mapping
Problems
Chapter 3: Forces
Problems
Chapter 4: Force-Energy Relationships
Springs
Young´s Modulus
Internal Forces
Energy from Surface Forces
Equation of Motion in Terms of Energy
Problems
Chapter 5: Isotropic Materials
Rotations and Translations
Rotational Invariants
Problems
Chapter 6: Minimizing Energy
Spring Model
Discrete 3D Model
Continuous 3D Model
Problems
Chapter 7: Simulations
Introduction
User Input
Define Nodes and Their Connectivity
Position Boundary Conditions
Force Boundary Conditions
Calculate Initial Volumes
Calculate Deformation Energy per Unit Initial Volume
Define Variable List
Gravity
Go
Plots
Simulating Incompressible Materials
Scaling in Simulations
Problems
Chapter 8: Quasi-static Simulation Examples
Introduction
Single Grid Study
Cylindrical Compression Test
Real Compression
Matching Fig. 2.1
Bending Paper
Extrusion Study
Water Drop
Problems
Chapter 9: The Invariants
Another Set of Invariants
Problems
Chapter 10: Experiments
Experiments
Compression Experiment
Extension Experiments
Extension in One Direction
Extending in Two Directions
Checking Forces
Simulations
Problems
Chapter 11: Time-Dependent Simulations
Example in 1D
Notation Alert
Equivalence of Equations of Motion
Force on Fixed Nodes
Numerical Simulation
Problems
Chapter 12: Anisotropic Materials
Anisotropic Materials
Measuring the Energy of Deformation of an Anisotropic Material
How to Find the Jig Coordinates Within a Sample
Vector Projection
Example 1: Layered Anisotropy
Example 2: Fibrous Sample
Finding (Gram-Schmidt QRD)
Example Deformation of a Simple Spring Model
Numerical Simulation Example
Alternate Set of Anisotropic Invariants
Alternate Experimental Jig
Problems
Chapter 13: Plot Deformation, Displacements, and Forces
Deformation of Local Regions
Internal Forces on Local Regions
Displacement of Nodes
Boundary Forces
Grid Refinement and the Finite Element Method
Problems
Chapter 14: Euler-Lagrange Elasticity
Euler-Lagrange Equations
Define Point Locations Within the Body
Equations of Motion
Force
Quasi-static Deformations
Force Boundary Conditions in Simulations
Discrete Energy Equations Used in Numerical Simulations
Energy per Unit Initial Volume
Problems
Chapter 15: Linear Elasticity
Comparison to Linear Elasticity
Linear Elasticity
Zeroth-Order Term in Equation 15.5
First-Order Term in Equation 15.5
Second-Order Term in Equation 15.5
Comparison to Standard Linear Elasticity Equations
Landau´s Energy Equation Leads to Linear Elastic Force-Displacement Equations
Some Limitations of Linear Elasticity
Advantage of Linear Elasticity
Small Deformations with Rotations
Problems
Chapter 16: Classical Finite Elasticity
Classical Elasticity
Direct Comparisons with Classical Elasticity
Stress
The Invariants
Direct Comparisons with Rubber Elasticity
Problems
Appendices
Appendix A: Deformation in Jig Coordinates
Appendix B: Origins of Anisotropic Invariants
Why ?
Gram-Schmidt QRD
Example of Gram-Schmidt QRD
Subroutine for Calculating Gram-Schmidt QRD
Subroutine for Calculating Equation 12.21 Values
General Numerical Proof That Equation 12.21 Provides the Same Matrix as Gram-Schmidt QRD
Why Cannot We Just Apply Gram-Schmidt QRD to m Instead of jig?
Appendix C: Euler-Lagrange Equations
Euler-Lagrange Equations in 1D
Euler-Lagrange Equations in 2D
Euler-Lagrange Equations with Two Dependent and One Independent Variables
Appendix D: Finite Element Example
Appendix E: Project Ideas
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Index