Elasticity

Elasticity vs Plasticity

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Material specimen loaded up to point $A$ then unloaded.
If stress-strain curve is traced in reverse, the material is elastic.
Point $E$ represents the transition point (Elastic Limit) beyond which the curve is no longer retraced as the specimen is unloaded.
Unloading curve $B C$ is parallel to the initial portion of the loading curve (parallel to tangent to the loading curve at Point $O$ )
Characteristic by which material undergoes inelastic strains beyond the elastic limit is known as Plasticity.

Linear Elasticity

Design structures and machines to function in the linear elastic region in order to avoid permanent deformations due to yielding.

Hooke's Law

Created by Robert Hooke
Fundamental linear elastic relation between uniaxial Stress and Strain
$σ = E ϵ$
Where:

$E$ is the "modulus of elasticity" also known as "Elastic Modulus" or "Young's Modulus"
$σ$ is Stress
$ϵ$ is Strain

Hooke's Law in Shear

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$τ = G γ$
Where:

$τ$ is the Shear Stress
$G$ is the Shear Modulus
$γ$ is the Shear Strain

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Stress-Strain curve showing typical yield behavior for nonferrous alloys. Stress ( $σ$ ) is shown as a function of strain $ϵ$ .

Elastic region: Hooke's Law applies here only
Proportionality Limit
Elastic Limit
Offset Yield Strength