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Elastic Strain

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Nomenclature

σ	normal stress
E	modulus of elasticity
ε	normal strain
τ	shear stress
G	modulus of rigidity
γ	shear strain

Details

Normal strain ε is defined in the lesson Normal Strain for the tensile specimen and is given by Eq1 as ε = δ/L, where δ is the total elongation of the bar within the length L. Hooke's law for the tensile specimen is given by Eq2 of the lesson Hooke's Law as:

σ = Eε

where the constant E is called Young's modulus or the modulus of elasticity.

When a material is placed in tension, there exists not only an axial strain, but also negative strain (contraction) perpendicular to the axial strain. Assuming a linear, homogeneous, isotropic material, this lateral strain is proportional to the axial strain. If the axial direction is x, then the lateral strains are ε_y = ε_z = −νε_x. The constant of proportionality ν is called Poisson's ratio, which is about 0.3 for most structural metals.

If the axial stress is in the x direction, then from Hooke's law for normal stress and strain:

(Eq1)

εx = σx E

and:

(Eq2)

εy = εz = −ν σx E

For a stress element undergoing σ_x, σ_y, and σ_z simultaneously, the normal strains are given by:

(Eq3)

εx = 1 E [σx − ν(σy + σz)]

(Eq4)

εx = 1 E [σx − ν(σy + σz)]

and:

(Eq5)

εx = 1 E [σx − ν(σy + σz)]

Shear strain γ is the change in a right angle of a stress element when subjected to pure shear stress, and Hooke's law for shear is given by:

τ = Gγ

It can be shown for a linear, isotropic, homogeneous material, the three elastic constants are related to each other by:

(Eq6)

E = 2G(1 + ν)