Neo-Hookean solid

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A neo-Hookean solid[1][2] 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accurate only for strains less than 20%.[3] The model is also inadequate for biaxial states of stress and has been superseded by the Mooney-Rivlin model.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 is


   W = C_{1} (I_1-3) \,

where C_{1} is a material constant, and I_1 is the first invariant of the left Cauchy-Green deformation tensor, i.e.,


   I_1 = \lambda_1^2 +  \lambda_2^2 + \lambda_3^2~

where \lambda_i are the principal stretches. For three-dimensional problems the compressible neo-Hookean material the strain energy density function is given by


   W = C_{1}~(\bar{I}_1 - 3) + D_1~(J-1)^2 ~;~~ J = \det(\boldsymbol{F}) = \lambda_1\lambda_2\lambda_3

where D_1 is a material constant, \bar{I}_1 = J^{-2/3} I_1 is the first invariant of the deviatoric part of the left Cauchy-Green deformation tensor, and \boldsymbol{F} is the deformation gradient. It can be shown that in 2D, the strain energy density function now becomes


    W = C_{1}~(\bar{I}_1 - 2) + D_1~(J-1)^2 ~;

where \bar{I}_1 =J^{-2/3} I_1.

Several alternative formulations exist for compressible neo-Hookean materials, for example [1]


   W = C_{1}~(I_1 - 3 - 2\ln J) + D_1~(J-1)^2

For consistency with linear elasticity,


   C_{1} = \cfrac{\mu}{2} ~;~~ D_1 = \cfrac{\kappa}{2}

where \mu is the shear modulus and \kappa is the bulk modulus.

Cauchy stress in terms of deformation tensors[edit]

Compressible neo-Hookean material[edit]

For a compressible Rivlin neo-Hookean material the Cauchy stress is given by


  J~\boldsymbol{\sigma} = -p~\boldsymbol{\mathit{1}} + 2C_1~\mathrm{dev}(\bar{\boldsymbol{B}})
    = -p~\boldsymbol{\mathit{1}} + \frac{2C_1}{J^{2/3}}~\mathrm{dev}(\boldsymbol{B})

where \boldsymbol{B} is the left Cauchy-Green deformation tensor, and


   p := -2D_1~J(J-1) ~;~~
   \mathrm{dev}(\bar{\boldsymbol{B}}) = \bar{\boldsymbol{B}} - \tfrac{1}{3}\bar{I}_1\boldsymbol{\mathit{1}} ~;~~
   \bar{\boldsymbol{B}} = J^{-2/3}\boldsymbol{B} ~.

For infinitesimal strains (\boldsymbol{\varepsilon})


  J \approx 1 + \mathrm{tr}(\boldsymbol{\varepsilon}) ~;~~ \boldsymbol{B} \approx \boldsymbol{\mathit{1}} + 2\boldsymbol{\varepsilon}

and the Cauchy stress can be expressed as


   \boldsymbol{\sigma} \approx 4C_1\left(\boldsymbol{\varepsilon} - \tfrac{1}{3}\mathrm{tr}(\boldsymbol{\varepsilon})\boldsymbol{\mathit{1}}\right) + 2D_1\mathrm{tr}(\boldsymbol{\varepsilon})\boldsymbol{\mathit{1}}

Comparison with Hooke's law shows that \mu = 2C_1 and \kappa = 2D_1.

Incompressible neo-Hookean material[edit]

For an incompressible neo-Hookean material with  J = 1


  \boldsymbol{\sigma} = -p~\boldsymbol{\mathit{1}} + 2C_1\boldsymbol{B}

where p is an undetermined pressure.

Cauchy stress in terms of principal stretches[edit]

Compressible neo-Hookean material[edit]

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[edit]

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[edit]

Compressible neo-Hookean material[edit]

The true stress as a function of uniaxial stretch predicted by a compressible neo-Hookean material for various values of C_1,D_1. The material properties are representative of natural rubber.

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[edit]

Comparison of experimental results (dots) and predictions for Hooke's law(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_{11} - \sigma_{33}  = 2C_1\left(\lambda^2 - \cfrac{1}{\lambda}\right)  ~;~~
  \sigma_{22} - \sigma_{33}  = 0

Assuming no traction on the sides, \sigma_{22}=\sigma_{33}=0, so we can write


  \sigma_{11}= 2C_1 \left(\lambda^2 - \cfrac{1}{\lambda}\right)
   = 2C_1\left(\frac {3\varepsilon_{11} + 3\varepsilon_{11}^2 +\varepsilon_{11}^3} {1+\varepsilon_{11}}\right)

where  \varepsilon_{11}=\lambda-1 is the engineering strain. This equation is often written in alternative notation as


  T_{11}= 2C_1 \left(\alpha^2 - \cfrac{1}{\alpha}\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}}= 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[edit]

Compressible neo-Hookean material[edit]

The true stress as a function of biaxial stretch predicted by a compressible neo-Hookean material for various values of C_1,D_1. The material properties are representative of natural rubber.

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[edit]

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[edit]

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. Note also that the magnitude of the stress is quite sensitive to the bulk modulus but not to the shear modulus.

The true stress as a function of equi-triaxial stretch predicted by a compressible neo-Hookean material for various values of C_1,D_1. The material properties are representative of natural rubber.
The true stress as a function of J predicted by a compressible neo-Hookean material for various values of C_1,D_1. The material properties are representative of natural rubber.

Simple shear[edit]

For the case of simple shear the deformation gradient in terms of components with respect to a reference basis is of the form [1]


   \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[edit]

In this case J = \det(\boldsymbol{F}) = 1. Hence,  \boldsymbol{\sigma} = 2C_1\mathrm{dev}(\boldsymbol{B}) . Now,


   \mathrm{dev}(\boldsymbol{B}) = \boldsymbol{B} - \tfrac{1}{3}\mathrm{tr}(\boldsymbol{B})\boldsymbol{\mathit{1}}
     = \boldsymbol{B} - \tfrac{1}{3}(3+\gamma^2)\boldsymbol{\mathit{1}} = 
     \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[edit]

Using the relation for the Cauchy stress for an incompressible neo-Hookean material we get


   \boldsymbol{\sigma} = -p\boldsymbol{\mathit{1}} + 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. Note that 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[edit]

  1. ^ a b c Ogden, R. W. , 1998, Nonlinear Elastic Deformations, Dover.
  2. ^ C. W. Macosko, 1994, Rheology: principles, measurement and applications, VCH Publishers, ISBN 1-56081-579-5.
  3. ^ Gent, A. N., ed., 2001, Engineering with rubber, Carl Hanser Verlag, Munich.

See also[edit]