Gent (hyperelastic model)

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The Gent hyperelastic material model [1] is a phenomenological model of rubber elasticity that is based on the concept of limiting chain extensibility. In this model, the strain energy density function is designed such that it has a singularity when the first invariant of the left Cauchy-Green deformation tensor reaches a limiting value I_m.

The strain energy density function for the Gent model is [1]


  W = -\cfrac{\mu J_m}{2} \ln\left(1 - \cfrac{I_1-3}{J_m}\right)

where \mu is the shear modulus and J_m = I_m -3.

In the limit where I_m \rightarrow \infty, the Gent model reduces to the Neo-Hookean solid model. This can be seen by expressing the Gent model in the form


   W = \cfrac{\mu}{2x}\ln\left[1 - (I_1-3)x\right] ~;~~ x := \cfrac{1}{J_m}

A Taylor series expansion of \ln\left[1 - (I_1-3)x\right] around x = 0 and taking the limit as x\rightarrow 0 leads to


  W = \cfrac{\mu}{2} (I_1-3)

which is the expression for the strain energy density of a Neo-Hookean solid.

Several compressible versions of the Gent model have been designed. One such model has the form[2]


    W = -\cfrac{\mu J_m}{2} \ln\left(1 - \cfrac{I_1-3}{J_m}\right) + \cfrac{\kappa}{2}\left(\cfrac{J^2-1}{2} - \ln J\right)^4

where J = \det(\boldsymbol{F}), \kappa is the bulk modulus, and \boldsymbol{F} is the deformation gradient.

Consistency condition[edit]

We may alternatively express the Gent model in the form


  W = C_0 \ln\left(1 - \cfrac{I_1-3}{J_m}\right)

For the model to be consistent with linear elasticity, the following condition has to be satisfied:


2\cfrac{\partial W}{\partial I_1}(3)  = \mu

where \mu is the shear modulus of the material. Now, at I_1 = 3 (\lambda_i = \lambda_j = 1),


   \cfrac{\partial W}{\partial I_1} = -\cfrac{C_0}{J_m}

Therefore, the consistency condition for the Gent model is


   -\cfrac{2C_0}{J_m} = \mu\, \qquad \implies \qquad C_0 = -\cfrac{\mu J_m}{2}

The Gent model assumes that J_m \gg 1

Stress-deformation relations[edit]

The Cauchy stress for the incompressible Gent model is given by


   \boldsymbol{\sigma}  = -p~\boldsymbol{\mathit{1}} + 
     2~\cfrac{\partial W}{\partial I_1}~\boldsymbol{B} 
     = -p~\boldsymbol{\mathit{1}} + \cfrac{\mu J_m}{J_m - I_1 + 3}~\boldsymbol{B}

Uniaxial extension[edit]

Stress-strain curves under uniaxial extension for Gent model compared with various hyperelastic material models.

For uniaxial extension in the \mathbf{n}_1-direction, the principal stretches are \lambda_1 = \lambda,~ \lambda_2=\lambda_3. From incompressibility \lambda_1~\lambda_2~\lambda_3=1. Hence \lambda_2^2=\lambda_3^2=1/\lambda. Therefore,


   I_1 = \lambda_1^2+\lambda_2^2+\lambda_3^2 = \lambda^2 + \cfrac{2}{\lambda} ~.

The left Cauchy-Green deformation tensor can then be expressed as


   \boldsymbol{B} = \lambda^2~\mathbf{n}_1\otimes\mathbf{n}_1 + \cfrac{1}{\lambda}~(\mathbf{n}_2\otimes\mathbf{n}_2+\mathbf{n}_3\otimes\mathbf{n}_3) ~.

If the directions of the principal stretches are oriented with the coordinate basis vectors, we have


     \sigma_{11} = -p + \cfrac{\lambda^2\mu J_m}{J_m - I_1 + 3} ~;~~
     \sigma_{22} = -p + \cfrac{\mu J_m}{\lambda(J_m - I_1 + 3)} = \sigma_{33} ~.

If \sigma_{22} = \sigma_{33} = 0, we have


   p =  \cfrac{\mu J_m}{\lambda(J_m - I_1 + 3)}~.

Therefore,


   \sigma_{11} = \left(\lambda^2 - \cfrac{1}{\lambda}\right)\left(\cfrac{\mu J_m}{J_m - I_1 + 3}\right)~.

The engineering strain is \lambda-1\,. The engineering stress is


  T_{11} = \sigma_{11}/\lambda = 
     \left(\lambda - \cfrac{1}{\lambda^2}\right)\left(\cfrac{\mu J_m}{J_m - I_1 + 3}\right)~.

Equibiaxial extension[edit]

For equibiaxial extension in the \mathbf{n}_1 and \mathbf{n}_2 directions, the principal stretches are \lambda_1 = \lambda_2 = \lambda\,. From incompressibility \lambda_1~\lambda_2~\lambda_3=1. Hence \lambda_3=1/\lambda^2\,. Therefore,


   I_1 = \lambda_1^2+\lambda_2^2+\lambda_3^2 = 2~\lambda^2 + \cfrac{1}{\lambda^4} ~.

The left Cauchy-Green deformation tensor can then be expressed as


   \boldsymbol{B} = \lambda^2~\mathbf{n}_1\otimes\mathbf{n}_1 + \lambda^2~\mathbf{n}_2\otimes\mathbf{n}_2+ \cfrac{1}{\lambda^4}~\mathbf{n}_3\otimes\mathbf{n}_3 ~.

If the directions of the principal stretches are oriented with the coordinate basis vectors, we have


   \sigma_{11} = \left(\lambda^2 - \cfrac{1}{\lambda^4}\right)\left(\cfrac{\mu J_m}{J_m - I_1 + 3}\right) = \sigma_{22} ~.

The engineering strain is \lambda-1\,. The engineering stress is


  T_{11} = \cfrac{\sigma_{11}}{\lambda} = 
     \left(\lambda - \cfrac{1}{\lambda^5}\right)\left(\cfrac{\mu J_m}{J_m - I_1 + 3}\right) = T_{22}~.

Planar extension[edit]

Planar extension tests are carried out on thin specimens which are constrained from deforming in one direction. For planar extension in the \mathbf{n}_1 directions with the \mathbf{n}_3 direction constrained, the principal stretches are \lambda_1=\lambda, ~\lambda_3=1. From incompressibility \lambda_1~\lambda_2~\lambda_3=1. Hence \lambda_2=1/\lambda\,. Therefore,


   I_1 = \lambda_1^2+\lambda_2^2+\lambda_3^2 = \lambda^2 + \cfrac{1}{\lambda^2} + 1 ~.

The left Cauchy-Green deformation tensor can then be expressed as


   \boldsymbol{B} = \lambda^2~\mathbf{n}_1\otimes\mathbf{n}_1 + \cfrac{1}{\lambda^2}~\mathbf{n}_2\otimes\mathbf{n}_2+ \mathbf{n}_3\otimes\mathbf{n}_3 ~.

If the directions of the principal stretches are oriented with the coordinate basis vectors, we have


   \sigma_{11} = \left(\lambda^2 - \cfrac{1}{\lambda^2}\right)\left(\cfrac{\mu J_m}{J_m - I_1 + 3}\right) ~;~~ \sigma_{22} = 0 ~;~~ \sigma_{33} = \left(1 - \cfrac{1}{\lambda^2}\right)\left(\cfrac{\mu J_m}{J_m - I_1 + 3}\right)~.

The engineering strain is \lambda-1\,. The engineering stress is


  T_{11} = \cfrac{\sigma_{11}}{\lambda} = 
     \left(\lambda - \cfrac{1}{\lambda^3}\right)\left(\cfrac{\mu J_m}{J_m - I_1 + 3}\right)~.

Simple shear[edit]

The deformation gradient for a simple shear deformation has the form[3]


   \boldsymbol{F} = \boldsymbol{1} + \gamma~\mathbf{e}_1\otimes\mathbf{e}_2

where \mathbf{e}_1,\mathbf{e}_2 are reference orthonormal basis vectors in the plane of deformation and the shear deformation is given by


   \gamma = \lambda - \cfrac{1}{\lambda} ~;~~ \lambda_1 = \lambda ~;~~ \lambda_2 = \cfrac{1}{\lambda} ~;~~ \lambda_3 = 1

In matrix form, the deformation gradient and the left Cauchy-Green deformation tensor may then be expressed as


   \boldsymbol{F} = \begin{bmatrix} 1 & \gamma & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} ~;~~
   \boldsymbol{B} = \boldsymbol{F}\cdot\boldsymbol{F}^T = \begin{bmatrix} 1+\gamma^2 & \gamma & 0 \\ \gamma & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}

Therefore,


   I_1 = \mathrm{tr}(\boldsymbol{B}) = 3 + \gamma^2

and the Cauchy stress is given by


   \boldsymbol{\sigma} = -p~\boldsymbol{\mathit{1}} + \cfrac{\mu J_m}{J_m - \gamma^2}~\boldsymbol{B}

In matrix form,


   \boldsymbol{\sigma} = \begin{bmatrix} -p +\cfrac{\mu J_m (1+\gamma^2)}{J_m - \gamma^2} & \cfrac{\mu J_m \gamma}{J_m - \gamma^2} & 0 \\ \cfrac{\mu J_m \gamma}{J_m - \gamma^2} & -p + \cfrac{\mu J_m}{J_m - \gamma^2} & 0 \\ 0 & 0 & -p + \cfrac{\mu J_m}{J_m - \gamma^2}
 \end{bmatrix}

References[edit]

  1. ^ a b Gent, A.N., 1996, A new constitutive relation for rubber, Rubber Chemistry Tech., 69, pp. 59-61.
  2. ^ Mac Donald, B. J., 2007, Practical stress analysis with finite elements, Glasnevin, Ireland.
  3. ^ Ogden, R. W., 1984, Non-linear elastic deformations, Dover.

See also[edit]