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 we will have: Thus, the equivalent Young's modulus of a neo-Hookean solid in uniaxial extension is , which is in concordance with linear elasticity ( with 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 and we have We also have a relation between and : or, This equation can be solved for using Newton's method. Incompressible neo-Hookean material For an incompressible material 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 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 is the shear deformation. Therefore the left Cauchy-Green deformation tensor is Compressible neo-Hookean material In this case . Hence, . 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 . References See also Hyperelastic material Strain energy density function Mooney-Rivlin solid Finite strain theory Stress measures Continuum mechanics Elasticity (physics) Non-Newtonian fluids Rubber properties Solid mechanics