infinitesimal strains is equal to Comparison with Hooke's law shows that and . For a compressible Rivlin neo-Hookean material the Cauchy stress is given by where is the left Cauchy-Green deformation tensor, and For infinitesimal strains () and the Cauchy stress can be expressed as Comparison with Hooke's law shows that and . {| class="toccolours collapsible collapsed" width="80%" style="text-align:left" !Proof: |- | The Cauchy stress in a compressible hyperelastic material is given by For a compressible Rivlin neo-Hookean material, while, for a compressible Ogden neo-Hookean material, Therefore, the Cauchy stress in a compressible Rivlin neo-Hookean material is given by while that for the corresponding Ogden material is If the isochoric part of the left Cauchy-Green deformation tensor is defined as , then we can write the Rivlin neo-Heooken stress as and the Ogden neo-Hookean stress as The quantities have the form of pressures and are usually treated as such. The Rivlin neo-Hookean stress can then be expressed in the form while the Ogden neo-Hookean stress has the form |} Incompressible neo-Hookean material For an incompressible neo-Hookean material with where 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 {| class="toccolours collapsible collapsed" width="80%" style="text-align:left" !Proof: |- | For a compressible hyperelastic material, the principal components of the Cauchy stress are given by The strain energy density function for a compressible neo Hookean material is Therefore, Since we have Hence, The principal Cauchy stresses are therefore given by |} 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 . Then Equating the two expressions for gives a relation for as a function of , 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, and . Therefore, Assuming no traction on the sides, , so we can write where is the engineering strain. This equation is often written in