Gent (hyperelastic model)

Laws

Conservations
Energy Mass Momentum
Inequalities
Clausius–Duhem (entropy)

Solid mechanics

Contact mechanics (frictional)

Fluid mechanics

Fluids
Statics · Dynamics Archimedes' principle · Bernoulli's principle Navier–Stokes equations Poiseuille equation · Pascal's law Viscosity (Newtonian · non-Newtonian) Buoyancy · Mixing · Pressure
Liquids
Surface tension Capillary action
Gases
Atmosphere Boyle's law Charles's law Gay-Lussac's law Combined gas law
Plasma

Rheology

Smart fluids
Viscoelasticity Rheometry Rheometer
Magnetorheological Electrorheological Ferrofluids

Scientists

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

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

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

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

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

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

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

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

Gent (hyperelastic model)

Consistency condition

Stress-deformation relations

Uniaxial extension

Equibiaxial extension

Planar extension

Simple shear

References

See also