EN224: Linear Elasticity
Division of Engineering
3.8 Axisymmetric Contact
One of the most successful applications of linear elasticity has been to predict the behavior of two solids in contact. The results have provided a basis for designing gears, bearin
EN224: Linear Elasticity
Division of Engineering
2. Theorems of Linear Elasticity
We proceed to prove several useful theorems which follow as a consequence of the structure of the field equations of linear elasticity, and which are useful in interpreting
EN224: Linear Elasticity
Division of Engineering
3. 3D Static Boundary Value Problems
Objective: Find elastostatic states in 3D solids with prescribed boundary conditions This is very difficult to do in general! A few useful techniques: (1) Represent the
EN224: Linear Elasticity
Division of Engineering
3.2 Singular solutions for the infinite solid.
Our first 3D boundary value problems will be the simplest: we will derive certain important solutions for an infinite solid. Although one rarely encounters inf
EN224: Linear Elasticity
Division of Engineering
2. Theorems of Linear Elasticity
We proceed to prove several useful theorems which follow as a consequence of the structure of the field equations of linear elasticity, and which are useful in interpreting
EN224: Linear Elasticity
Division of Engineering
1.2 Kinetics of Deformable Solids
We begin by reviewing general measures of force and balance laws. Internal Forces
Define Note To characterize internal forces in a solid, we define the Cauchy Stress tensor
EN224: Linear Elasticity
Division of Engineering
1. Review of the Field Equations of Linear Elasticity
Objective: derive field equations governing the behavior of linear elastic solids from the perspective of finite deformations of general solids. Review
EN224: Linear Elasticity
Division of Engineering
3.7 Singular Solutions for the Half-Space
Solutions for infinite solids have found many applications in the field of mechanics of materials. Often, however, we cannot neglect the influence of a solids bound
EN224: Linear Elasticity
Division of Engineering
3.6 Eshelby Inclusion Problems
Eshelby found an important application of the results outlined in the preceding section.
Consider an infinite, homogeneous, isotropic, linear elastic solid. Suppose we introdu
EN224: Linear Elasticity
Division of Engineering
1.4 Summary of the Field Equations of Linear Elasticity
Displacement field u(t): Define infinitesimal strain tensor
; assume
Stress measure
Linear Momentum
Angular Momentum: Constitutive Law (or ) with
Spec
EN224: Linear Elasticity
Division of Engineering
1.5 Field Equations Implied by the Fundamental System
We now derive several auxiliary field equations which follow as a consequence of the field equations listed in the preceding section. These field equati
EN224: Linear Elasticity
Division of Engineering
3.5 Eigenstrains
One can also use the Kelvin solution to derive fields due to eigenstrains within an infinite region.
Consider an unbounded, homogeneous linear elastic solid, which is free of stress. Suppos
EN224: Linear Elasticity
Division of Engineering
3.4 Solutions for 3D dislocation loops in an infinite solid.
The Kelvin state, together with the reciprocal theorem, provides a neat way to determine fields associated with dislocation loops.
First, recall
EN224: Linear Elasticity
Division of Engineering
1.3 Constitutive Law for Linear Elastic Solids
Objective: find relationship between , assuming infinitesimal motion
As before, we will begin by reviewing constitutive models for large deformations. We will
EN224: Linear Elasticity
Division of Engineering
3.3 The Boundary Element Method
There is an important application of the singular solutions developed in Section 3.2. It is generally very difficult to find exact solutions to boundary value problems in lin
EN224: Linear Elasticity
Division of Engineering
2. Theorems of Linear Elasticity
We proceed to prove several useful theorems which follow as a consequence of the structure of the field equations of linear elasticity, and which are useful in interpreting