The most general linear relationship between two symmetric tensors and which is homogeneous (i.e. independent of direction) is:
Where A and B are constants, and is the Kroneker delta tensor. When is the stress and is the strain, this is an expression of Hooke's law, also known as the constitutive equations of linear elasticity.
Consider a cube of linearly elastic material with each side having length L. A purely compressive force consists of a force directed normal to each face, causing each face to move a distance . The stress and strain may be written as:
Thus:
and the constitutive equations become:
The incompressibility or bulk modulusK is defined as
where . Taking the derivative, it follows that , thus:
and it is clear that the constant A is simply the bulk modulus K.
Figure 1: A cube with sides of length L of an isotropic linearly elastic material subject to tension along the x axis, with a Poisson's ratio of 0.5. The green cube is unstressed, the red is expanded in the x direction by due to tension, and contracted in the y and z directions by .
By the definition of Poisson's ratio, if a positive force is applied only to the x-faces of the cube, they will move a distance of , and the other faces will move a distance of . The stress and strain are written:
If equal and opposite forces are applied to the x faces of the cube, and a force is applied to the other faces such that these other faces do not move, then:
The relative change of volume ΔV/V due to the stretch of the material can be calculated using a simplified formula:
which, for small deformations reduces to:
where
is material volume
is change in material volume
is original length, before stretch
is the change of length along the direction of compression:
Note that for an incompressible material, which implies that . For a material which does not have any transverse expansion or contraction, the volume change will be simply , which implies that .
Derivation
A cube with sides of length , made of an isotropic material, has volume . An axial stress gives new dimensions to the cube: axially and transversely.
For small changes in dimension, Poisson's ratio gives the relationship between these new dimensions:
where is the change in length due to the axial stress. The new volume of the cube is: