Neo Hookean
325 wordsAnsys Hyperelasticity course where the model is mentioned and some notes on it.
The Neo-Hookean Model
The Neo-Hookean model is a simple yet effective constitutive model used to describe the nonlinear elastic behavior of materials, especially soft materials like rubbers, biological tissues, and inflatables. It is a special case of hyperelastic material models, meaning that it derives from a strain energy function.
1. Strain Energy Function
In the Neo-Hookean model, the strain energy density function ( W ) (also called the strain energy potential) is given by:
\[W = \frac{\mu}{2} (I_1 - 3) + \frac{\kappa}{2} (J - 1)^2\]where:
- $\mu$ is the shear modulus, related to material stiffness.
- $\kappa$ is the bulk modulus, which controls volumetric compressibility.
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$I_1$ is the first invariant of the right Cauchy-Green deformation tensor, given by:
\[I_1 = \lambda_1^2 + \lambda_2^2 + \lambda_3^2\] - where $\lambda_1$, $\lambda_2$, $\lambda_3$ are the principal stretches.
- $J =$ $\lambda_1$ $\lambda_2$ $\lambda_3$ is the Jacobian determinant, representing the volume change.
For an incompressible material, $J = 1$ , so the second term in $W$ is omitted.
2. Stress-Strain Relationship
The Cauchy stress tensor for an incompressible Neo-Hookean material is:
\[\sigma = -pI + \mu B\]where:
- $p$ is a Lagrange multiplier (acting as a pressure term to enforce incompressibility).
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$B$ is the left Cauchy-Green deformation tensor, defined as:
\[B = F F^T\]where $F$ is the deformation gradient.
For a uniaxial extension along the x-direction with stretch $\lambda$, the stress simplifies to:
\[\sigma_x = \mu \left( \lambda^2 - \frac{1}{\lambda} \right)\]For a pure shear deformation, the shear stress is:
\[\sigma_{xy} = \mu \gamma\]where $\gamma$ is the shear strain.
3. Comparison to Hookean Elasticity
The Neo-Hookean model is a nonlinear extension of Hooke’s Law, useful for large deformations.
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Hooke’s Law (linear elasticity) assumes small strains and is expressed as:
\[\sigma = E \epsilon\] -
The Neo-Hookean model accounts for large deformations while maintaining a simple formulation.