Gent hyperelastic model explained

The Gent hyperelastic material model 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} → inf ty , the Gent model reduces to the Neo-Hookean solid model. This can be seen by expressing the Gent model in the formW =- \cfrac{\mu}{ 2 x} 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 → 0 leads toW = \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] (the below strain energy function yields a non zero hydrostatic stress at no deformation, refer^[3] for compressible Gent models).

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

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 ( λ _{i} = λ _{j} = 1 ) , \cfrac{\partial W}{\partial I _{1}} = - \cfrac{C _{0}{J} _{m}} Therefore, the consistency condition for the Gent model is- \cfrac{ 2 C _{0}{J} _{m}} = \mu \implies C _{0} = - \cfrac{\mu J _{m}{2}} The Gent model assumes that J _{m} \gg 1

Stress-deformation relations

The Cauchy stress for the incompressible Gent model is given by

\boldsymbol{\sigma} = - p~\boldsymbol{ I } + 2~\cfrac~\boldsymbol = -p~\boldsymbol + \cfrac~\boldsymbol

Uniaxial extension

For uniaxial extension in the

n _{1} -direction, the principal stretches are λ _{1} = λ ,~ λ _{2=λ} _{3} . From incompressibility λ _{1~λ} _{2~λ} _{3=1} . Hence λ ^{2=1 / λ} _{3} . Therefore,I _{1} = λ ^{2} _{3} = λ ^{2} + \cfrac{ 2 }{{ λ }} ~. The left Cauchy-Green deformation tensor can then be expressed as\boldsymbol{B} = λ ^{2 ~ n} _{1 ⊗ n} _{1} + \cfrac{ 1 }{ λ }~( n _{2 ⊗ n} _{2+n} _{3 ⊗ n} _{3)} ~. If the directions of the principal stretches are oriented with the coordinate basis vectors, we have\sigma _{11} = - p + \cfrac{ λ ^{2\mu} J _{m}{J} _{m} - I _{1} + 3 } ~;~~ \sigma _{22} = - p + \cfrac{\mu J _{m}{λ(J} _{m} - I _{1} + 3 )} = \sigma _{33} ~. If \sigma _{22} = \sigma _{33} = 0 , we havep = \cfrac{\mu J _{m}{λ(J} _{m} - I _{1} + 3 )}~. Therefore,\sigma _{11} = \left( λ ^{2} - \cfrac{ 1 }{ λ }\right)\left(\cfrac{\mu J _{m}{J} _{m} - I _{1} + 3 }\right)~. The engineering strain is λ-1 . The engineering stress is T _{11} = \sigma _{11} / λ = \left( λ - \cfrac{ 1 }{ λ ^{2}\right)\left(\cfrac{\mu} J _{m}{J} _{m} - I _{1} + 3 }\right)~.

Equibiaxial extension

For equibiaxial extension in the

n _{1} and n _{2} directions, the principal stretches are λ _{1} = λ _{2} = λ . From incompressibility λ _{1~λ} _{2~λ} _{3=1} . Hence λ ^{2} _{3=1 / λ} . Therefore,I _{1} = λ ^{2} _{3} = 2 ~ λ ^{2} + \cfrac{ 1 }{ λ ^{4}} ~. The left Cauchy-Green deformation tensor can then be expressed as\boldsymbol{B} = λ ^{2 ~ n} _{1 ⊗ n} _{1} + λ ^{2 ~ n} _{2 ⊗ n} _{2+} \cfrac{ 1 }{ λ ^{4 }~ n} _{3 ⊗ n} _{3} ~. If the directions of the principal stretches are oriented with the coordinate basis vectors, we have\sigma _{11} = \left( λ ^{2} - \cfrac{ 1 }{ λ ^{4}\right)\left(\cfrac{\mu} J _{m}{J} _{m} - I _{1} + 3 }\right) = \sigma _{22} ~. The engineering strain is λ-1 . The engineering stress is T _{11} = \cfrac{\sigma _{11} } = \left(\lambda - \cfrac\right)\left(\cfrac\right) = T_~.

Planar extension

Planar extension tests are carried out on thin specimens which are constrained from deforming in one direction. For planar extension in the

n _{1} directions with the n _{3} direction constrained, the principal stretches are λ _{1=λ,} ~ λ _{3=1} . From incompressibility λ _{1~λ} _{2~λ} _{3=1} . Hence λ _{2=1/λ} . Therefore,I _{1} = λ ^{2} _{3} = λ ^{2} + \cfrac{ 1 }{ λ ^{2}} + 1 ~. The left Cauchy-Green deformation tensor can then be expressed as\boldsymbol{B} = λ ^{2 ~ n} _{1 ⊗ n} _{1} + \cfrac{ 1 }{ λ ^{2 }~ n} _{2 ⊗ n} _{2+} n _{3 ⊗ n} _{3} ~. If the directions of the principal stretches are oriented with the coordinate basis vectors, we have\sigma _{11} = \left( λ ^{2} - \cfrac{ 1 }{ λ ^{2}\right)\left(\cfrac{\mu} J _{m}{J} _{m} - I _{1} + 3 }\right) ~;~~ \sigma _{22} = 0 ~;~~ \sigma _{33} = \left( 1 - \cfrac{ 1 }{ λ ^{2}\right)\left(\cfrac{\mu} J _{m}{J} _{m} - I _{1} + 3 }\right)~. The engineering strain is λ-1 . The engineering stress is T _{11} = \cfrac{\sigma _{11} } = \left(\lambda - \cfrac\right)\left(\cfrac\right)~.

Simple shear

The deformation gradient for a simple shear deformation has the form^[4]

\boldsymbol{F} = \boldsymbol{ 1 } + \gamma~ e _{1 ⊗ e} _{2} where e _{1,e} _{2} are reference orthonormal basis vectors in the plane of deformation and the shear deformation is given by\gamma = λ - \cfrac{ 1 }{ λ } ~;~~ λ _{1} = λ ~;~~ λ _{2} = \cfrac{ 1 }{ λ } ~;~~ λ _{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} ⋅ \boldsymbol{F} ^{T} = \begin{bmatrix} 1+ \gamma ^{2} & \gamma & 0 \ \gamma & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} Therefore, I _{1} = tr (\boldsymbol{B}) = 3 + \gamma ^{2} and the Cauchy stress is given by\boldsymbol{\sigma} = - p~\boldsymbol{ 1 } + \cfrac~\boldsymbol 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

1. 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. Horgan . Cornelius O. . Saccomandi . Giuseppe . 2004-11-01 . Constitutive Models for Compressible Nonlinearly Elastic Materials with Limiting Chain Extensibility . Journal of Elasticity . en . 77 . 2 . 123–138 . 10.1007/s10659-005-4408-x . 1573-2681.
4. Ogden, R. W., 1984, Non-linear elastic deformations, Dover.

See also

• Hyperelastic material
• Strain energy density function
• Mooney-Rivlin solid
• Finite strain theory
• Stress measures