Continuum mechanics/Nonlinear elasticity

There are two types models of nonlinear elastic behavior that are in common use. These are :

  • Hyperelasticity
  • Hypoelasticity



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 stress-strain behavior for such a material is given by


where   is the Cauchy stress,   is the current mass density,   is the deformation gradient,   is the Lagrangian Green strain tensor, and   is the left Cauchy-Green deformation tensor.

We can use the relationship between the Cauchy stress and the 2nd Piola-Kirchhoff stress to obtain an alternative relation between stress and strain.


where   is the 2nd Piola-Kirchhoff stress and   is the mass density in the reference configuration.

Isotropic hyperelasticity


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 Cayley-Hamilton theorem we can show that


Hence we can also write


The stress-strain relation can then be written as


A similar relation can be obtained for the Cauchy stress which has the form


where   is the right Cauchy-Green deformation tensor.

Cauchy stress in terms of invariants


For w: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 w:strain energy density function is  , then


(See the page on the left Cauchy-Green deformation tensor for the definitions of these symbols).

Saint-Venant–Kirchhoff material


The simplest constitutive relationship that satisfies the requirements of hyperelasticity is the Saint-Venant–Kirchhoff material, which has a response function of the form


where   and   are material constants that have to be determined by experiments. Such a linear relation is physically possible only for small strains.