Stress and Strain
Stress
Basically the force exerted

F is in N, A0 is in m^2 (or in^2) 
Stress is in N/m^2 (= 1 Pa)
Strain
Basically the change in length, as a result of stress
Types of stress/strain
Can be tension, compression, shear, rotational.
For compression
Same as tensile test, except you get negative strain (length gets smaller)
Hooke's Law
For most materials stressed in tension and at relatively low levels: 

E is the modulus of Elasticity (in GPa or psi)
- metals and ceramics have similar E's, with ceramics having a slightly larger E generally
- polymers have a very low E
- when stress and strain are proportional, elastic deformation occurs (no permanent deformation)
Relationship with temperature
As temperature increases, E decreases (for most materials, except for some rubbers).
Similarly, for shear stress/strain:

where G is modulus of elasticity

What happens on the atomic scale?
Elastic strain is manifested as stretching of interatomic bonds. The stronger the bonds are, the more force (so more strain) required to pull the bonds apart: 

E is the slope of the above graph.

Poisson's Ratio

Basically the ratio of the lateral and axial strains.

Resilience
Capacity of a material to absorb energy when it is deformed elastically, and then upon unloading, to have this energy recovered. 
Given by % elongation

Plastic deformation
Permanent, nonrecoverable deformation.  On the atomic scale, bonds break with atom neighbors, and reform with new neighbors - called "slip"
Yielding
Point at which plastic deformation begins (aka proportional limit). Point P on graph:

Ductility

Measure of the degree of plastic deformation that has been sustained at fracture.

Tensile Strength
Maximum stress the material can ever take. Point M (with fracture at point F):

Toughness
Several definitions:
- resistance to fracture when a crack (defect) is present 
- ability of material to absorb energy and plastically deform before fracturing 
Formula (assuming linear elastic region)

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