Page 218 - Intro to Tensor Calculus

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Figure 2.3-1. Section of a rubber band

When the force F is applied to our rubber band example there arises the strains
∆` ∆w ∆h
, , .
` w h
The second concept introduced by our simple example is stress. Stress is defined as a force per unit area. In
particular,
Force force
stress = , with dimension of .
Area over which force acts unit area
We will be interested in studying stress and strain in homogeneous, isotropic materials which are in equilib-
rium with respect to the force system acting on the material.

Hooke’s Law

For linear elastic materials, where the forces are all one dimensional, the stress and strains are related
by Hooke’s law which has two parts. The Hooke’s law, part one, states that stress is proportional to strain
in the stretch direction, where the Young’s modulus E is the proportionality constant. This is written

F ∆`
Hooke’s law part 1 = E . (2.3.1)
A `
A graph of stress vs strain is a straight line with slope E in the linear elastic range of the material.
The Hooke’s law, part two, involves the fact that there is a strain contraction perpendicular to the
stretch direction. The strain contraction is the same for both the width and height and is proportional to
the strain in the stretch direction. The proportionality constant being the Poisson’s ratio ν.
∆w ∆h ∆` 1
Hooke’s law part 2 = = −ν , 0 <ν < . (2.3.2)
w h ` 2
The proportionality constants E and ν depend upon the material being considered. The constant ν is called
the Poisson’s ratio and it is always a positive number which is less than one half. Some representative values
for E and ν are as follows.
6
6
Various types of steel 28 (10) psi ≤ E ≤ 30 (10) psi 0.26 ≤ ν ≤ 0.31
6
6
Various types of aluminium 9.0 (10) psi ≤ E ≤ 11.0 (10) psi 0.3 ≤ ν ≤ 0.35