Question

In some materials the relationship between stress and strain during plastic deformation is given by σ = σ_y+ K(ε − ε_y) ⁿ  where K and n are constants. What is the energy absorbed by the specimen during plastic deformation in a tensile test if the material yields at a strain of ε_y and fails at a strain of ε_f?

Answer 1

Plastic deformation means when the stress is removed, the material does not return to its previous dimension but there is a permanent (irreversible) deformation.

In tensile tests, if the deformation is elastic, the stress-strain relationship is called Hooke's law:

s = E e

E is the slope of the stress-strain curve. E is Young's modulus or modulus of elasticity. In some cases, the relationship is not linear so that E can be defined alternatively as the local slope:

E = ds/de

Shear stresses produce strains according to:

t = G g

where G is the shear modulus.

Elastic moduli measure the stiffness of the material. They are related to the second derivative of the interatomic potential, or the first derivative of the force vs. internuclear distance. Due to thermal vibrations the elastic modulus decreases with temperature. E is large for ceramics (stronger ionic bond) and small for polymers (weak covalent bond). Since the interatomic distances depend on direction in the crystal, E depends on direction (i.e., it is anisotropic) for single crystals. For randomly oriented policrystals, E is isotropic.