1. STRESS-STRAIN RELATIONSHIP
Applied Load shape change (strain) stress
There must be a relation between stress and strain
Linear Elasticity: Simplest and most commonly used
Uni-axial Stress:
Axial force F will generate stress
In the elastic range, the relation between stress and strain is
Reduction of cross-section
E: Young’s modulus, : Poisson’s ratio

2. UNI-AXIAL TENSION TEST
Terms
Explanations
Proportional limit
The greatest stress for which stress is still proportional to strain
Elastic limit
The greatest stress without resulting in any permanent strain
Yield stress
The stress required to produce 0.2% plastic strain
Strain hardening
A region where more stress is required to deform the material
Ultimate stress
The maximum stress the material can resist
Necking
Cross section of the specimen reduces during deformation
Fracture
Material failure

3. LINEAR ELASTICITY (HOOKE’S LAW)
When the material is in the Proportional Limit (or Elastic Limit)
In General 3-D Relationship
Stress-Strain Matrix
For homogeneous, isotropic material 36 constants can be reduced to 2 independent constants.

4. LINEAR ELASTICITY (HOOKE’S LAW) cont.
Isotropic Material:
Stress in terms of strain:
Strain in terms of stress
What are the limits on Poisson’s ratio and what happens at these limits?
Shear Modulus

5. DEGENERATION TO 2D Plane Strain Problem
3D engineering problems are often simplified to 2D problems
Deformation in z-dir is constrained
Strains in z-dir are zero y y
Normal stress in z-dir x x z

6. DEGENERATION TO 2D cont. Plane Stress Problem:
Plate-like structure under in-plane loads
No constraints in thickness dir
Stresses in z-dir are zero
Stress-strain relation

7. EQUILIBRIUM EQUATIONS
Equilibrium Relation (2D)
Equilibrium Relation (3D)

8. BOUNDARY-VALUE PROBLEM
When boundary conditions are given, how can we calculate the displacement, stress, and strain of the structure?
Solve for displacement
Equilibrium equation
Constitutive equation (Stress-strain relation)
Strain definition
Get differential equation for displacements
Load and boundary conditions