Hyperelastic

Hyperelasticity is a constitutive framework in continuum mechanics used to model the nonlinear elastic response of solids undergoing large deformations. In a hyperelastic material, the stress state is derived from a strain energy density function W that depends on the deformation gradient F, or equivalently on the right Cauchy-Green tensor C = F^T F. The work done by stresses during deformation is recoverable, and the material is considered elastic and path-independent under isothermal, quasi-static conditions.

The second Piola-Kirchhoff stress S is given by S = 2 ∂W/∂C, and the Cauchy stress sigma is obtained from sigma = (1/J) F S F^T, with J = det F. For incompressible hyperelastic materials, the constraint J = 1 introduces a Lagrange multiplier p, appearing as an isotropic pressure term in the Cauchy stress.

Common models of hyperelasticity include Neo-Hookean, which is the simplest and based on W proportional to I1 − 3; Mooney-Rivlin, which depends on the invariants I1 and I2; Ogden, which uses principal stretches with power-law forms; as well as Gent and Arruda-Boyce, which reflect network behavior. These models provide closed-form expressions for W and permit straightforward derivation of stresses under finite deformations.

Applications of hyperelastic models are widespread in engineering and biomechanics, particularly for rubber-like polymers, elastomers, and soft tissues such as arteries and skin. They are routinely used in finite element analysis to predict large-strain behavior and to simulate material response under complex loading.

Limitations include the assumption of rate-independent, purely elastic behavior, which excludes time-dependent phenomena such as viscoelasticity, hysteresis, and damage. In practice, hyperelastic models are often complemented by viscoelastic or damage terms, and require careful calibration to experimental data.