Neo-Hookean solid
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 The model is also inadequate for biaxial states of stress and has been superseded by the Mooney-Rivlin model.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 formula_2 is a material constant, and formula_3 is the first invariant (trace), of the right Cauchy-Green deformation tensor, i.e., where formula_5 are the principal stretches. For a compressible neo-Hookean material the strain energy density function is given by where  formula_7 is a material constant and formula_8 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 formula_11 is the first invariant of the isochoric part formula_12 of the right Cauchy–Green deformation tensor. For consistency with linear elasticity, where formula_14 is the first Piola-Kirchhoff stress. By simplifying the right hand side we arrive at which for infinitesimal strains is equal to Comparison with Hooke's law shows that formula_17 and formula_18. For a compressible Rivlin neo-Hookean material the  Cauchy stress is given by where formula_20 is the left Cauchy-Green deformation tensor, and For infinitesimal strains (formula_22) and the Cauchy stress can be expressed as Comparison with Hooke's law shows that formula_25 and formula_26. Incompressible neo-Hookean material. For an incompressible neo-Hookean material with formula_27 where formula_29 is an undetermined pressure. Cauchy stress in terms of principal stretches. Compressible neo-Hookean material. For a compressible neo-Hookean hyperelastic material, the principal components of the Cauchy stress are given by Therefore, the differences between the principal stresses are Incompressible neo-Hookean material. In terms of the principal  stretches, the Cauchy stress differences for an incompressible hyperelastic material are given by For an incompressible neo-Hookean material, Therefore, which gives Uniaxial extension. Compressible neo-Hookean material. For a compressible material undergoing uniaxial extension, the principal stretches are Hence, the true (Cauchy) stresses for a compressible neo-Hookean material are given by The stress differences are given by If the material is unconstrained we have formula_39. Then Equating the two expressions for formula_41 gives a relation for formula_42 as a function of formula_43, i.e., or The above equation can be solved numerically using a Newton–Raphson iterative root-finding  procedure. Incompressible neo-Hookean material. Under uniaxial extension, formula_46 and formula_47. Therefore, Assuming no traction on the sides, formula_49, so we can write where formula_51 is the engineering strain. This equation is often written in alternative notation as The equation above is for the true stress (ratio of the elongation force to deformed cross-section). For the engineering stress the equation is: For small deformations formula_54 we will have: Thus, the equivalent Young's modulus of a neo-Hookean solid in uniaxial extension is formula_56, which is in concordance with linear elasticity (formula_57 with formula_58 for incompressibility). Equibiaxial extension. Compressible  neo-Hookean material. In the case of equibiaxial extension Therefore, The stress differences are If the material is in a state of plane stress then formula_62 and we have We also have a relation between formula_42 and formula_43: or, This equation can be solved for formula_42 using Newton's method. Incompressible neo-Hookean material. For an incompressible material formula_69 and the differences between the principal Cauchy stresses take the form Under plane stress conditions we have Pure dilation. For the case of pure dilation Therefore, the principal Cauchy stresses for a compressible neo-Hookean material are given by If the  material is incompressible then formula_74 and the principal stresses can be arbitrary. The figures below show that extremely high stresses are needed to achieve large triaxial extensions or compressions. Equivalently, relatively small triaxial stretch states can cause very high stresses to develop in a rubber-like material. The magnitude of the stress is quite sensitive to the bulk modulus but not to the shear modulus. Simple shear. For the case of simple shear the deformation gradient in terms of components with respect to a reference basis is of the form where formula_76 is the shear deformation. Therefore the left Cauchy -Green deformation tensor is Compressible neo-Hookean material. In this case formula_78. Hence, formula_79. Now, Hence the Cauchy stress is given by Incompressible neo-Hookean material. Using the relation for the Cauchy stress for an incompressible neo-Hookean material we get Thus neo-Hookean solid shows linear dependence of shear stresses upon shear deformation and quadratic dependence of the normal stress difference on the shear deformation. The expressions for the Cauchy stress for a compressible and an incompressible neo-Hookean material in simple shear represent the same quantity and provide a means of determining the unknown pressure formula_29. Neo-Hookean