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
Thomas Young, , England, polymath and physician with contribution to Rosetta stone, astigmatism, electric theory, ‘The last person who knew everything’/
Simeon Denis Poisson , France, student of Lagrange and Laplace, most known for the Poisson equation for potential fields.
Robert Hooke English natural philosopher, architect and polymath. Came close to experimental proof of the inverse distance square in the law of gravity, and an early proponent of evolution.

2. UNI-AXIAL TENSION TEST
Definitions in table are material for quiz
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
Energy considerations dictate that C is symmetric (21 constants)
For homogeneous, isotropic material 21 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? See notes page.
Shear Modulus
From the equations it is obvious that we get singularities when Poisson’s ratio is either 0.5 or -1. From the equation for G we see that at -1 we will get infinite shear rigidity, and at 0.5 we will need infinite stresses to generate any normal strains. When Poisson’s ratio approaches 0,5 the material becomes incompressible, and this is true of many plastics. Negative Poisson’s ratios are more exotic and do not normally happen in isotropic materials, but one can generate foams with negative Poisson’s ratios.

5. Quiz-like questions What is the proportional limit?
What are the limits on Poisson’s ratio?
What happens when Poisson’s ratio is 0.5?
How many independent elastic constants can a general material have?
For an isotropic material with E=3GPa, and G=1.2GPa, if the only non-zero strain component is exx=0.005, what are the non-zero stress components?
Answers in the notes page.
Proportional limit: The greatest stress for which stress is still proportional to strain
-1<n<0.5
n=0.5 corresponds to incompressible material where you need infinite stresses to change volume, so we must have exx+eyy+ezz=0
General elastic material can have up to 21 independent elastic constants
To calculate the stresses we ues Hooke’s law. Using Matlab we get
exx=0.005; E=3;G=1.2;nu=E/(2*G)-1
nu =
>> sig=E/((1+nu)*(1-2*nu))*[(1-nu), nu, nu]'*exx
sig =
0.0180
0.0060 So

6. 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

7. 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

8. Quiz-like Questions
Airplane wings and fuselages are analyzed using plane stress, but almost all their loading is pressure, which is in the z direction and generates szz. How come?
A point in the skin of the wing has exx=eyy=0.001, gxy=0. What is ezz if n=0.25?
Answers in the notes page.
The szz is a fraction of atmospheric pressure, that is a few psi. On the other hand the in-plane stresses due to bending of the wing or fuselage are thousands or ten of thousands psi, so the z stress can be neglected.
From Hooke’s law for plane stress

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

10. 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