Hyperelastic materials undergo large deformations without any permanent damage and return to their initial state when unloaded. This behavior arises from their highly elastic molecular structure, which allows the material to store and release significant amounts of strain energy.
The unique property for hyperelastic materials comes from their distinct molecular arrangement composed of long, flexible, crosslinked polymer chains. When the material is deformed, the chains uncoil and reorient without permanent damage. Upon unloading, the polymer chains return to their coiled configuration. Because of the large strains in these hyperelastic materials, we have to consider the higher order terms in the Green strain tensor. For more information about strain tensors and their formulations, please refer to our previous blog here. Examples of hyperelastic materials include rubber and certain biological tissues like skin, muscles and blood vessels.
When working with hyperelastic material models, we often encounter the concept of the strain energy density function along with different material models such as Neo-Hookean, Mooney-Rivlin and Ogden models. Different material models for hyperelastic materials arise from different mathematical choices for strain energy density function. In this article, we will first review the basics of the strain energy density function and how stress tensor can be derived from it. We will then apply this framework for computing the stress tensor for a basic incompressible Neo-Hookean material model, which is one of the simplest and most commonly used hyperelastic models.
Strain energy density function (U)
Due to their highly nonlinear mechanical behavior, the relationship between stress and strain in hyperelastic materials is not expressed directly through Hooke’s law as in linear elasticity. The stress-strain behavior of elastic and hyperelastic models are shown in the figure below.
It is defined by using a scalar strain energy density function (U). Physically, this represents the amount of energy stored in the material per unit volume as it deforms. This stored energy comes from molecular level rearrangements such as stretching, uncoiling and reorientation of polymers chains in the hyperelastic materials.
The stress state only depends on the current deformation of the material and not on the deformation history. The strain energy density function has to be differentiated to obtain stresses. The stress tensor (Piola-Kirchhoff), S is computed from the strain energy density as,
The strain energy density is a function of invariants of Cauchy -Green tensor (C).
Using the chain rule for differentiation, derivative of 
can be evaluated as,
The invariants for the Cauchy – Green tensor can be obtained as,
Calculating the derivatives of the invariants,
Substituting these invariant derivatives into the chain rule and multiplying it by 2 gives,
The above equation represents the stress tensor for isotropic hyperelastic materials in terms of invariants.
Incompressible Hyperelastic Materials
For incompressible materials where bulk modulus >> shear modulus, the volume stays constant during deformation.
This is mathematically expressed as,
Substituting this into the above Piola Kirchoff stress equation gives,
where 
is hydrostatic pressure represented by Lagrange multiplier. This acts as an unknown field to enforce the incompressibility constraint and meet the specified boundary conditions. The volumetric stress is calculated using 
for incompressible materials.
Incompressible Neo-Hookean Model
Now, let’s use the above theory to derive the stress tensor for incompressible Neo-Hookean material model.
The strain energy density function for Neo-Hookean model is given by the equation,
where 
is the shear modulus of the material.
Calculating the first order derivatives for strain energy density function gives,
Substituting this in the stress equation gives,
The above equation represents the classic Neo-Hookean Cauchy stress.
Final Thoughts
The strain energy density function provides a powerful framework for understanding and modeling the nonlinear behavior of hyperelastic materials. By relating the stored energy in a material to its deformation, it allows us to derive stress tensors and predict material response under large strains. Using invariants of the Cauchy-Green tensor, this approach can be applied to a wide range of isotropic hyperelastic models like Neo-Hookean, Mooney-Rivlin, and Ogden models. Although commercial FEA tools handle these formulations automatically, understanding the theory helps in interpreting results more effectively.
We’re always here to help, so if you have questions about hyperelastic materials in your models, or just FEA in general, don’t hesitate to reach out!
