Mooney-Rivlin solid

Mooney-Rivlin solid

In continuum mechanics, a Mooney-Rivlin solid is a generalization of the Neo-Hookean solid model, where the strain energy W is a linear combination of two invariants of Finger tensor mathbf{B}:

W = C_{10} (overline{I}_1-3) + C_{01} (overline{I}_2-3)+ frac{1}{d}(J_{el}-1)^2,

where overline{I}_1 and overline{I}_2 are the first and the second invariant of deviatoric component of the Finger tensor:

I_1 = lambda_1^2 + lambda_2 ^2+ lambda_3 ^2,

I_2 = lambda_1^2 lambda_2^2 + lambda_2^2 lambda_3^2 + lambda_3^2 lambda_1^2,

I_3 = lambda_1^2 lambda_2^2 lambda_3^2,

where: overline{I_p} = J^{-2/3}I_p.

and C_{10}, C_{01}, and d are constants.

If C_1= frac {1} {2} G (where G is the shear modulus) and C_2=0, we obtain a Neo-Hookean solid, a special case of a Mooney-Rivlin solid.

The stress tensor mathbf{T} depends upon Finger tensor mathbf{B} by the following equation:

mathbf{T} = -pmathbf{I} +2C_1 mathbf{B} +2C_2 mathbf{B}^{-1}

The model was proposed by Melvin Mooney and Ronald Rivlin in two independent papers in 1952.

Uniaxial extension

For the case of uniaxial elongation, true stress can be calculated as:

T_{11} = left(2C_1 - frac {2C_2} {alpha_1} right) left(alpha_1^2 - alpha_1^{-1} right)

and engineering stress can be calculated as:

T_{11eng} = left(2C_1 - frac {2C_2} {alpha_1} right) left(alpha_1 - alpha_1^{-2} right)

The Mooney-Rivlin solid model usually fits experimental data better than Neo-Hookean solid does, but requires an additional empirical constant.

Brain tissues

Elastic response of soft tissues like that in the brain is often modelled based on the Mooney--Rivlin model.

Source

  • C. W. Macosko Rheology: principles, measurement and applications, VCH Publishers, 1994, ISBN 1-56081-579-5

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