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: [The characteristic polynomial of the linear operator corresponding to the second rank three-dimensional Finger tensor is usually written:$$p_B (lambda) = lambda^3 - a_1 , lambda^2 + a_2 , lambda - a_3In this article, the trace $$a_1 is written $$I_1, the next coefficient $$a_2 is written $$I_2, and the determinant $$a_3 would be written $$I_3.]

:$$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} ight) left( alpha_1^2 - alpha_1^{-1} ight)

and engineering stress can be calculated as:

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

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.

ource

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

Notes and References