Neo-Hookean solid

Compressible neo-Hookean material For a compressible neo-Hookean material, the Cauchy stress is given by : J~\boldsymbol{\sigma} = -p~\boldsymbol{I} + 2C_1 \operatorname{dev}(\bar{\boldsymbol{B}}) = -p~\boldsymbol{I} + \frac{2C_1}{J^{2/3}} \operatorname{dev}(\boldsymbol{B}) where \boldsymbol{B} is the left Cauchy–Green deformation tensor, and : p := -2D_1~J(J-1) ~;~ \operatorname{dev}(\bar{\boldsymbol{B}}) = \bar{\boldsymbol{B}} - \tfrac{1}{3}\bar{I}_1\boldsymbol{I} ~;~~ \bar{\boldsymbol{B}} = J^{-2/3}\boldsymbol{B} ~. For infinitesimal strains (\boldsymbol{\varepsilon}) : J \approx 1 + \operatorname{tr}(\boldsymbol{\varepsilon}) ~;~~ \boldsymbol{B} \approx \boldsymbol{I} + 2\boldsymbol{\varepsilon} and the Cauchy stress can be expressed as : \boldsymbol{\sigma} \approx 4C_1\left(\boldsymbol{\varepsilon} - \tfrac{1}{3}\operatorname{tr}(\boldsymbol{\varepsilon})\boldsymbol{I}\right) + 2D_1\operatorname{tr}(\boldsymbol{\varepsilon})\boldsymbol{I} Comparison with Hooke's law shows that \mu = 2C_1 and \kappa = 2D_1, which are the shear and bulk moduli, respectively. : Incompressible neo-Hookean material For an incompressible neo-Hookean material with J = 1 : \boldsymbol{\sigma} = -p~\boldsymbol{I} + 2C_1\boldsymbol{B} where p 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 : \sigma_{i} = 2C_1 J^{-5/3} \left[ \lambda_i^2 -\cfrac{I_1}{3} \right] + 2D_1(J-1) ~;~~ i=1,2,3 Therefore, the differences between the principal stresses are : \sigma_{11} - \sigma_{33} = \cfrac{2C_1}{J^{5/3}}(\lambda_1^2-\lambda_3^2) ~;~~ \sigma_{22} - \sigma_{33} = \cfrac{2C_1}{J^{5/3}}(\lambda_2^2-\lambda_3^2) : Incompressible neo-Hookean material In terms of the principal stretches, the Cauchy stress differences for an incompressible hyperelastic material are given by : \sigma_{11} - \sigma_{33} = \lambda_1~\cfrac{\partial{W}}{\partial \lambda_1} - \lambda_3~\cfrac{\partial{W}}{\partial \lambda_3}~;~~ \sigma_{22} - \sigma_{33} = \lambda_2~\cfrac{\partial{W}}{\partial \lambda_2} - \lambda_3~\cfrac{\partial{W}}{\partial \lambda_3} For an incompressible neo-Hookean material, : W = C_1(\lambda_1^2 + \lambda_2 ^2 + \lambda_3 ^2 -3) ~;~~ \lambda_1\lambda_2\lambda_3 = 1 Therefore, : \cfrac{\partial{W}}{\partial \lambda_1} = 2C_1\lambda_1 ~;~~ \cfrac{\partial{W}}{\partial \lambda_2} = 2C_1\lambda_2 ~;~~ \cfrac{\partial{W}}{\partial \lambda_3} = 2C_1\lambda_3 which gives : \sigma_{11} - \sigma_{33} = 2(\lambda_1^2-\lambda_3^2)C_1 ~;~~ \sigma_{22} - \sigma_{33} = 2(\lambda_2^2-\lambda_3^2)C_1 == Uniaxial extension ==

Compressible neo-Hookean material . For a compressible material undergoing uniaxial extension, the principal stretches are : \lambda_1 = \lambda ~;~~ \lambda_2 = \lambda_3 = \sqrt{\tfrac{J}{\lambda}} ~;~~ I_1 = \lambda^2 + \tfrac{2J}{\lambda} Hence, the true (Cauchy) stresses for a compressible neo-Hookean material are given by : \begin{align} \sigma_{11} & = \cfrac{4C_1}{3J^{5/3}}\left(\lambda^2 - \tfrac{J}{\lambda}\right) + 2D_1(J-1) \\ \sigma_{22} & = \sigma_{33} = \cfrac{2C_1}{3J^{5/3}}\left(\tfrac{J}{\lambda} - \lambda^2\right) + 2D_1(J-1) \end{align} The stress differences are given by : \sigma_{11} - \sigma_{33} = \cfrac{2C_1}{J^{5/3}}\left(\lambda^2 - \tfrac{J}{\lambda}\right) ~;~~ \sigma_{22} - \sigma_{33} = 0 If the material is unconstrained we have \sigma_{22} = \sigma_{33} = 0. Then : \sigma_{11} = \cfrac{2C_1}{J^{5/3}}\left(\lambda^2 - \tfrac{J}{\lambda}\right) Equating the two expressions for \sigma_{11} gives a relation for J as a function of \lambda, i.e., : \cfrac{4C_1}{3J^{5/3}}\left(\lambda^2 - \tfrac{J}{\lambda}\right) + 2D_1(J-1) = \cfrac{2C_1}{J^{5/3}}\left(\lambda^2 - \tfrac{J}{\lambda}\right) or : D_1 J^{8/3} - D_1 J^{5/3} + \tfrac{C_1}{3\lambda} J - \tfrac{C_1\lambda^2}{3} = 0 The above equation can be solved numerically using a Newton–Raphson iterative root-finding procedure. Incompressible neo-Hookean material (1), neo-Hookean solid(2) and Mooney-Rivlin solid models(3) Under uniaxial extension, \lambda_1 = \lambda\, and \lambda_2 = \lambda_3 = 1/\sqrt{\lambda}. Therefore, : \sigma_{22} - \sigma_{33} = 0 where \varepsilon_{11}=\lambda-1 is the engineering strain. This equation is written as : \sigma_{11}^{\mathrm{true}}= 2C_1 \left(\lambda^2 - \cfrac{1}{\lambda}\right) The equation above is for the true stress (ratio of the elongation force to deformed cross-section). For the engineering stress the equation is: :\sigma_{11}^{\mathrm{eng}}= \frac{1}{\lambda} \sigma_{11}^{\mathrm{true}}; \sigma_{11}^{\mathrm{eng}}= 2C_1 \left(\lambda - \cfrac{1}{\lambda^2}\right) For small deformations \varepsilon \ll 1 we will have: :\sigma_{11}= 6C_1 \varepsilon = 3\mu\varepsilon Thus, the equivalent Young's modulus of a neo-Hookean solid in uniaxial extension is 3\mu, which is in concordance with linear elasticity (E=2\mu(1+\nu) with \nu=0.5 for incompressibility). == Equibiaxial extension ==

Compressible neo-Hookean material . In the case of equibiaxial extension : \lambda_1 = \lambda_2 = \lambda ~;~~ \lambda_3 = \tfrac{J}{\lambda^2} ~;~~ I_1 = 2\lambda^2 + \tfrac{J^2}{\lambda^4} Therefore, : \begin{align} \sigma_{11} & = 2C_1\left[\cfrac{\lambda^2}{J^{5/3}} - \cfrac{1}{3J}\left(2\lambda^2+\cfrac{J^2}{\lambda^4}\right)\right] + 2D_1(J-1) \\ & = \sigma_{22} \\ \sigma_{33} & = 2C_1\left[\cfrac{J^{1/3}}{\lambda^4} - \cfrac{1}{3J}\left(2\lambda^2+\cfrac{J^2}{\lambda^4}\right)\right] + 2D_1(J-1) \end{align} The stress differences are : \sigma_{11} - \sigma_{22} = 0 ~;~~ \sigma_{11} - \sigma_{33} = \cfrac{2C_1}{J^{5/3}}\left(\lambda^2 - \cfrac{J^2}{\lambda^4}\right) If the material is in a state of plane stress then \sigma_{33} = 0 and we have : \sigma_{11} = \sigma_{22} = \cfrac{2C_1}{J^{5/3}}\left(\lambda^2 - \cfrac{J^2}{\lambda^4}\right) We also have a relation between J and \lambda: : 2C_1\left[\cfrac{\lambda^2}{J^{5/3}} - \cfrac{1}{3J}\left(2\lambda^2+\cfrac{J^2}{\lambda^4}\right)\right] + 2D_1(J-1) = \cfrac{2C_1}{J^{5/3}}\left(\lambda^2 - \cfrac{J^2}{\lambda^4}\right) or, : \left(2D_1 - \cfrac{C_1}{\lambda^4}\right)J^2 + \cfrac{3C_1}{\lambda^4}J^{4/3} - 3D_1J - 2C_1\lambda^2 = 0 This equation can be solved for J using Newton's method. Incompressible neo-Hookean material For an incompressible material J=1 and the differences between the principal Cauchy stresses take the form : \sigma_{11} - \sigma_{22} = 0 ~;~~ \sigma_{11} - \sigma_{33} = 2C_1\left(\lambda^2 - \cfrac{1}{\lambda^4}\right) Under plane stress conditions we have : \sigma_{11} = 2C_1\left(\lambda^2 - \cfrac{1}{\lambda^4}\right) == Pure dilation ==

For the case of pure dilation : \lambda_1 = \lambda_2 = \lambda_3 = \lambda ~:~~ J = \lambda^3 ~;~~ I_1 = 3\lambda^2 Therefore, the principal Cauchy stresses for a compressible neo-Hookean material are given by : \sigma_i = 2C_1\left(\cfrac{1}{\lambda^3} - \cfrac{1}{\lambda}\right) + 2D_1(\lambda^3-1) If the material is incompressible then \lambda^3 = 1 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 : \boldsymbol{F} = \begin{bmatrix} 1 & \gamma & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} where \gamma is the shear deformation. Therefore, the left Cauchy-Green deformation tensor is : \boldsymbol{B} = \boldsymbol{F}\cdot\boldsymbol{F}^T = \begin{bmatrix} 1+\gamma^2 & \gamma & 0 \\ \gamma & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} Compressible neo-Hookean material In this case J = \det(\boldsymbol{F}) = 1. Hence, \boldsymbol{\sigma} = 2C_1\operatorname{dev}(\boldsymbol{B}) . Now, : \operatorname{dev}(\boldsymbol{B}) = \boldsymbol{B} - \tfrac{1}{3}\operatorname{tr}(\boldsymbol{B})\boldsymbol{I} = \boldsymbol{B} - \tfrac{1}{3}(3+\gamma^2)\boldsymbol{I} = \begin{bmatrix} \tfrac{2}{3}\gamma^2 & \gamma & 0 \\ \gamma & -\tfrac{1}{3}\gamma^2 & 0 \\ 0 & 0 & -\tfrac{1}{3}\gamma^2 \end{bmatrix} Hence the Cauchy stress is given by : \boldsymbol{\sigma} = \begin{bmatrix} \tfrac{4C_1}{3}\gamma^2 & 2C_1\gamma & 0 \\ 2C_1\gamma & -\tfrac{2C_1}{3}\gamma^2 & 0\\ 0 & 0 & -\tfrac{2C_1}{3}\gamma^2 \end{bmatrix} Incompressible neo-Hookean material Using the relation for the Cauchy stress for an incompressible neo-Hookean material we get : \boldsymbol{\sigma} = -p~\boldsymbol{I} + 2C_1\boldsymbol{B} = \begin{bmatrix} 2C_1(1+\gamma^2)-p & 2C_1\gamma & 0 \\ 2C_1\gamma & 2C_1 - p & 0 \\ 0 & 0 & 2C_1 -p \end{bmatrix} 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 p. == References ==