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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 [math]\displaystyle{ I_m }[/math]. The strain energy density function for the Gent model is where [math]\displaystyle{ \mu }[/math] is the shear modulus and [math]\displaystyle{ J_m = I_m -3 }[/math]. In the limit where [math]\displaystyle{ I_m \rightarrow \infty }[/math], the Gent model reduces to the Neo-Hookean solid model. This can be seen by expressing the Gent model in the form A Taylor series expansion of [math]\displaystyle{ \ln\left[1 - (I_1-3)x\right] }[/math] around [math]\displaystyle{ x = 0 }[/math] and taking the limit as [math]\displaystyle{ x\rightarrow 0 }[/math] leads to 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 (the below strain energy function yields a non zero hydrostatic stress at no deformation, refer https://link.springer.com/article/10.1007/s10659-005-4408-x for compressible Gent models). where [math]\displaystyle{ J = \det(\boldsymbol{F}) }[/math], [math]\displaystyle{ \kappa }[/math] is the bulk modulus, and [math]\displaystyle{ \boldsymbol{F} }[/math] is the deformation gradient.
We may alternatively express the Gent model in the form
For the model to be consistent with linear elasticity, the following condition has to be satisfied:
where [math]\displaystyle{ \mu }[/math] is the shear modulus of the material. Now, at [math]\displaystyle{ I_1 = 3 (\lambda_i = \lambda_j = 1) }[/math],
Therefore, the consistency condition for the Gent model is
The Gent model assumes that [math]\displaystyle{ J_m \gg 1 }[/math]
The Cauchy stress for the incompressible Gent model is given by
For uniaxial extension in the [math]\displaystyle{ \mathbf{n}_1 }[/math]-direction, the principal stretches are [math]\displaystyle{ \lambda_1 = \lambda,~ \lambda_2=\lambda_3 }[/math]. From incompressibility [math]\displaystyle{ \lambda_1~\lambda_2~\lambda_3=1 }[/math]. Hence [math]\displaystyle{ \lambda_2^2=\lambda_3^2=1/\lambda }[/math]. Therefore,
The left Cauchy-Green deformation tensor can then be expressed as
If the directions of the principal stretches are oriented with the coordinate basis vectors, we have
If [math]\displaystyle{ \sigma_{22} = \sigma_{33} = 0 }[/math], we have
Therefore,
The engineering strain is [math]\displaystyle{ \lambda-1\, }[/math]. The engineering stress is
For equibiaxial extension in the [math]\displaystyle{ \mathbf{n}_1 }[/math] and [math]\displaystyle{ \mathbf{n}_2 }[/math] directions, the principal stretches are [math]\displaystyle{ \lambda_1 = \lambda_2 = \lambda\, }[/math]. From incompressibility [math]\displaystyle{ \lambda_1~\lambda_2~\lambda_3=1 }[/math]. Hence [math]\displaystyle{ \lambda_3=1/\lambda^2\, }[/math]. Therefore,
The left Cauchy-Green deformation tensor can then be expressed as
If the directions of the principal stretches are oriented with the coordinate basis vectors, we have
The engineering strain is [math]\displaystyle{ \lambda-1\, }[/math]. The engineering stress is
Planar extension tests are carried out on thin specimens which are constrained from deforming in one direction. For planar extension in the [math]\displaystyle{ \mathbf{n}_1 }[/math] directions with the [math]\displaystyle{ \mathbf{n}_3 }[/math] direction constrained, the principal stretches are [math]\displaystyle{ \lambda_1=\lambda, ~\lambda_3=1 }[/math]. From incompressibility [math]\displaystyle{ \lambda_1~\lambda_2~\lambda_3=1 }[/math]. Hence [math]\displaystyle{ \lambda_2=1/\lambda\, }[/math]. Therefore,
The left Cauchy-Green deformation tensor can then be expressed as
If the directions of the principal stretches are oriented with the coordinate basis vectors, we have
The engineering strain is [math]\displaystyle{ \lambda-1\, }[/math]. The engineering stress is
The deformation gradient for a simple shear deformation has the form[1]
where [math]\displaystyle{ \mathbf{e}_1,\mathbf{e}_2 }[/math] are reference orthonormal basis vectors in the plane of deformation and the shear deformation is given by
In matrix form, the deformation gradient and the left Cauchy-Green deformation tensor may then be expressed as
Therefore,
and the Cauchy stress is given by
In matrix form,