Compressible isotropic hyperelastic materials
For isotropic hyperelastic materials, the Cauchy stress can be expressed in terms of the invariants of the left Cauchy–Green deformation tensor (or right Cauchy–Green deformation tensor). If the strain energy density function is
then
(See the page on the left Cauchy–Green deformation tensor for the definitions of these symbols).
Proof 1
The second Piola–Kirchhoff stress tensor for a hyperelastic material is given by
where is the right Cauchy–Green deformation tensor and is the deformation gradient. The Cauchy stress is given by
where . Let be the three principal invariants of . Then
The derivatives of the invariants of the symmetric tensor are
Therefore, we can write
Plugging into the expression for the Cauchy stress gives
Using the left Cauchy–Green deformation tensor and noting that , we can write
For an incompressible material and hence .Then
Therefore, the Cauchy stress is given by
where is an undetermined pressure which acts as a Lagrange multiplier to enforce the incompressibility constraint.
If, in addition, , we have and hence
In that case the Cauchy stress can be expressed as
Proof 2
The isochoric deformation gradient is defined as , resulting in the isochoric deformation gradient having a determinant of 1, in other words it is volume stretch free. Using this one can subsequently define the isochoric left Cauchy–Green deformation tensor .
The invariants of are
The set of invariants which are used to define the distortional behavior are the first two invariants of the isochoric left Cauchy–Green deformation tensor tensor, (which are identical to the ones for the right Cauchy Green stretch tensor), and add into the fray to describe the volumetric behaviour.
To express the Cauchy stress in terms of the invariants recall that
The chain rule of differentiation gives us
Recall that the Cauchy stress is given by
In terms of the invariants we have
Plugging in the expressions for the derivatives of in terms of , we have
or,
In terms of the deviatoric part of , we can write
For an incompressible material and hence .Then
the Cauchy stress is given by
where is an undetermined pressure-like Lagrange multiplier term. In addition, if , we have and hence
the Cauchy stress can be expressed as
Proof 3
To express the Cauchy stress in terms of the stretches recall that
The chain rule gives
The Cauchy stress is given by
Plugging in the expression for the derivative of leads to
Using the spectral decomposition of we have
Also note that
Therefore, the expression for the Cauchy stress can be written as
For an incompressible material and hence . Following Ogden[1] p. 485, we may write
Some care is required at this stage because, when an eigenvalue is repeated, it is in general only Gateaux differentiable, but not Fréchet differentiable.[8][9] A rigorous tensor derivative can only be found by solving another eigenvalue problem.
If we express the stress in terms of differences between components,
If in addition to incompressibility we have then a possible solution to the problem
requires and we can write the stress differences as