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Stress, Strain and Hooke’s Law

In very early studies of mechanical deformation under applied stress
Hooke found that the deformation was linearly proportional to the stress for
small deformations. The simplest of these observations was made by Young
for extensions or contractions of a thin rod under axial load. For a model of
a rod with rectangular cross section of area dA and length l the ratio of the
stress, the applied axial force divided by the cross sectional area, to the
strain, the change in length dl divided by the length is a constant. The
constant now known as Young’s modulus, is given by:

F dl
stress dA
Young’s modulus, E = = l
strain dl
l F
dA

The uniform compression of a fluid or solid under hydrostatic
pressure (again force divided by area) produces a strain which is the
fractional change in volume and the constant of proportionality is the bulk
modulus K. For the following cube:

P

P P

P

P

P
Bulk modulus, K =
dV
V

Finally for an element in shear, the force is applied tangentially to one
of the sides of the cube (the other is assumed glued to an immovable plane in
the following cartoon). The stress is now the force divided by the area over
which the shear is applied, and the strain is the lateral deformation dl divided
by height of the element, l. For small deformations the latter ratio is simply
the angular deformation θ.

dl
F
dl/l=θ
F l
2
Shear modulus, µ = l
θ
l

Collectively, these stress strain relationships are known as Hooke’s Law.
One more important physical property is Poisson’s relationship.
When an object is compressed (or extended) it is observed that it expands (or
contracts) perpendicular to the direction of the applied stress. This is
illustrated by the compression of the rectangular block in the following
cartoon. The ratio of the transverse strain to the axial strain is Poisson’s
ratio.

dr
Poisson’s ratio, σ = r l
dl
l
F z
r
dr

dl

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Hookes law

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