1. Consistency Condition

We may alternatively express the Gent model in the form

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

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where ${\displaystyle \mu }$ is the shear modulus of the material. Now, at ${\displaystyle I_1=3({\lambda }_i={\lambda }_j=1)}$,

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Therefore, the consistency condition for the Gent model is

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The Gent model assumes that ${\displaystyle J_m\gg 1}$

2. Stress-Deformation Relations

The Cauchy stress for the incompressible Gent model is given by

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2.1. Uniaxial Extension

Stress-strain curves under uniaxial extension for Gent model compared with various hyperelastic material models. https://handwiki.org/wiki/index.php?curid=2087400

For uniaxial extension in the ${\displaystyle {\mathbf{n}}_1}$-direction, the principal stretches are ${\displaystyle {\lambda }_1=\lambda ,{\lambda }_2={\lambda }_3}$. From incompressibility ${\displaystyle {\lambda }_1{\lambda }_2{\lambda }_3=1}$. Hence ${\displaystyle {\lambda }_2^2={\lambda }_3^2=1/\lambda }$. Therefore,

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

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

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If ${\displaystyle {\sigma }_{22}={\sigma }_{33}=0}$, we have

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Therefore,

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The engineering strain is ${\displaystyle \lambda -1}$. The engineering stress is

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2.2. Equibiaxial Extension

For equibiaxial extension in the ${\displaystyle {\mathbf{n}}_1}$ and ${\displaystyle {\mathbf{n}}_2}$ directions, the principal stretches are ${\displaystyle {\lambda }_1={\lambda }_2=\lambda }$. From incompressibility ${\displaystyle {\lambda }_1{\lambda }_2{\lambda }_3=1}$. Hence ${\displaystyle {\lambda }_3=1/{\lambda }^2}$. Therefore,

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

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

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The engineering strain is ${\displaystyle \lambda -1}$. The engineering stress is

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2.3. Planar Extension

Planar extension tests are carried out on thin specimens which are constrained from deforming in one direction. For planar extension in the ${\displaystyle {\mathbf{n}}_1}$ directions with the ${\displaystyle {\mathbf{n}}_3}$ direction constrained, the principal stretches are ${\displaystyle {\lambda }_1=\lambda ,{\lambda }_3=1}$. From incompressibility ${\displaystyle {\lambda }_1{\lambda }_2{\lambda }_3=1}$. Hence ${\displaystyle {\lambda }_2=1/\lambda }$. Therefore,

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

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

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The engineering strain is ${\displaystyle \lambda -1}$. The engineering stress is

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2.4. Simple Shear

The deformation gradient for a simple shear deformation has the form^[1]

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

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

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Therefore,

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and the Cauchy stress is given by

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In matrix form,

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