There are two types models of nonlinear elastic behavior that are in common use. These are :
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Hyperelastic materials are truly elastic in the sense that if a load is applied to such a material and then removed, the material returns to its original shape without any dissipation of energy in the process. In other word, a hyperelastic material stores energy during loading and releases exactly the same amount of energy during unloading. There is no path dependence.
If is the Helmholtz free energy, then the stressstrain behavior for such a material is given by
where is the Cauchy stress, ρ is the current mass density, is the deformation gradient, is the Lagrangian Green strain tensor, and is the left CauchyGreen deformation tensor.
We can use the relationship between the Cauchy stress and the 2nd PiolaKirchhoff stress to obtain an alternative relation between stress and strain.
where is the 2nd PiolaKirchhoff stress and ρ_{0} is the mass density in the reference configuration.
For isotropic materials, the free energy must be an isotropic function of . This also mean that the free energy must depend only on the principal invariants of which are
In other words,
Therefore, from the chain rule,
From the CayleyHamilton theorem we can show that
Hence we can also write
The stressstrain relation can then be written as
A similar relation can be obtained for the Cauchy stress which has the form
where is the right CauchyGreen deformation tensor.
For w:isotropic hyperelastic materials, the Cauchy stress can be expressed in terms of the invariants of the left CauchyGreen deformation tensor (or right CauchyGreen deformation tensor). If the w:strain energy density function is , then
(See the page on the left CauchyGreen deformation tensor for the definitions of these symbols).
Proof 1: 

The second PiolaKirchhoff stress tensor
for a hyperelastic material is given by
where is the right CauchyGreen deformation tensor and is the deformation gradient. The Cauchy stress is given by where . Let I_{1},I_{2},I_{ 3} 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 CauchyGreen deformation tensor and noting that I_{3} = J^{2}, we can write 
Proof 2: 

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 W in terms of , we have or, 
Proof 3: 

To express the Cauchy stress in terms of the stretches λ_{1},λ_{2},λ_{3}
recall that
The chain rule gives The Cauchy stress is given by Plugging in the expression for the derivative of W leads to Using the spectral decomposition of we have Also note that Therefore the expression for the Cauchy stress can be written as 
The simplest constitutive relationship that satisfies the requirements of hyperelasticity is the SaintVenant–Kirchhoff material, which has a response function of the form
where λ and μ are material constants that have to be determined by experiments. Such a linear relation is physically possible only for small strains.
