A neo-Hookean solid is a hyperelastic material model, similar to Hooke's law, that can be used for predicting the nonlinear stress-strain behavior of materials undergoing large deformations. The model was proposed by Ronald Rivlin in 1948. In contrast to linear elastic materials, the stress-strain curve of a neo-Hookean material is not linear. Instead, the relationship between applied stress and strain is initially linear, but at a certain point the stress-strain curve will plateau. The neo-Hookean model does not account for the dissipative release of energy as heat while straining the material and perfect elasticity is assumed at all stages of deformation. The neo-Hookean model is based on the statistical thermodynamics of cross-linked polymer chains and is usable for plastics and rubber-like substances. Cross-linked polymers will act in a neo-Hookean manner because initially the polymer chains can move relative to each other when a stress is applied. However, at a certain point the polymer chains will be stretched to the maximum point that the covalent cross links will allow, and this will cause a dramatic increase in the elastic modulus of the material. The neo-Hookean material model does not predict that increase in modulus at large strains and is typically accurate only for strains less than 20%. The model is also inadequate for biaxial states of stress and has been superseded by the Mooney-Rivlin model. The strain energy density function for an incompressible neo-Hookean material in a three-dimensional description is where is a material constant, and is the first invariant (trace), of the right Cauchy-Green deformation tensor, i.e., where are the principal stretches. For a compressible neo-Hookean material the strain energy density function is given by where is a material constant and is the deformation gradient. It can be shown that in 2D, the strain energy density function is Several alternative formulations exist for compressible neo-Hookean materials, for example where is the first invariant of the isochoric part of the right Cauchy–Green deformation tensor. For consistency with linear elasticity, where is the first Lamé parameter and is the shear modulus or the second Lamé parameter. Alternative definitions of and are sometimes used, notably in commercial finite element analysis software such as Abaqus. Cauchy stress in terms of deformation tensors Compressible neo-Hookean material For a compressible Ogden neo-Hookean material the Cauchy stress is given by where is the first Piola-Kirchhoff stress. By simplifying the right hand side we arrive at which for