Hooke’s Law / Poisson’s Ratio, ν

 Key Concepts:  Hooke's Law expresses the relationship between axial stress and axial strain for a given material.  If this relationship is linear then axial stress is directly proportional to axial strain.  The proportionality constant is the modulus of elasticity, E.  As loading increases axial stress may no longer be linear to axial strain.  This critical point is the proportional limit.

In a Nut Shell:  Properties of engineering materials vary widely.  Common examples

of engineering materials with differing stress-strain relations include concrete, glass,

steel, wood, aluminum, composites, and others.  A basic mechanical property is the

relationship between stress and strain for a given material under axial loading.  Also,

as you stretch a rubber band  in one direction it contracts in the other two directions.

This response gives rise to the Poisson Ratio effect.

Hooke’s Law for an Axial Member undergoing Tensile Loading

For an axial member under load with stress. σ, and (extensional) strain, ε, Hooke’s

Law (in the linear range) gives

σ  =  E ε

where  E   is  Young’s modulus (modulus of elasticity) or proportionality constant.

Since strain is dimensionless, the common units for Young’s modulus are psi (English

units) or GPa (metric units).

Stress-strain Relation for Brittle and Ductile Materials

The stress-strain diagram for a brittle and a ductile material may be idealized to

simplify the stress-strain model.  In both cases the slope of the stress-strain curve in

the linear range is Young’s Modulus.  Click here to view typical stress-strain diagrams

Note for a test involving a brittle material, the stress increases in a linear manner to its

proportional limit  and then continues in a nonlinear fashion to failure at its ultimate

strength.

For a ductile material, the stress increases in a linear manner to its yield point.  For a

linear-elastic model the strain continues to increase with no change in the stress beyond the

yield stress in this idealized model of the stress-strain relationship.

Values of Modulus of Elasticity, E, and Poisson’s Ratio, ν

 Material E in ksi E in GPa ν Aluminum 10 70 0.33 Mild Steel 30 200 0.3 Concrete 3.6 25 0.1-0.2 Glass 1.75 12 0.2-0.27