Advanced elasticity/Mooney-Rivlin material

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A Mooney-Rivlin solid is a generalization of the w:Neo-Hookean solid model, where the strain energy W is a linear combination of two invariants of the w:Finger tensor :

,

where and are the first and the second invariant of w:deviatoric component of the w:Finger tensor:[1]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle I_1 = \lambda_1^2 + \lambda_2 ^2+ \lambda_3 ^2} ,
,
,

where: and are constants.

If (where G is the w:shear modulus) and , we obtain a w:Neo-Hookean solid, a special case of a Mooney-Rivlin solid.

The stress tensor depends upon Finger tensor by the following equation:

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


Uniaxial extension[edit]

Comparison of experimental results (dots) and predictions for w:Hooke's law(1, blue line), w:Neo-Hookean solid(2, red line) and Mooney-Rivlin solid models(3, green line)

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

and w:engineering stress can be calculated as:

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

Rubber[edit]

Elastic response of rubber-like materials are often modelled based on the Mooney-Rivlin model.

Source[edit]

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

Notes and References[edit]

  1. The characteristic polynomial of the linear operator corresponding to the second rank three-dimensional Finger tensor is usually written
    In this article, the trace is written , the next coefficient is written , and the determinant would be written .