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Stresses interpretation ABAQUS

Why do some strain energy density (SED) function have 1 invariant (like neo-Hookean) while some others have both 1st and 2nd invariant like mooney-rivlin? is there a specific reason behind using 1st invariant in SED vs all invariants in the SED?


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wvmars's picture

For well-written, concise overviews, see

MC Boyce and EM Arruda, Rubber Chemistry and Technology, 73, 504, 2000. 

RS Rivlin Rubber Chemistry and Technology, 65 51 1992. 

Long story short:

I1 measures average length change of material fibers in all possible directions surrounding a material point

I2 measures average area change of material planes of all possible orientations passing through a material point

At small strains, it turns out that I1 and I2 are linearly dependent on each other (see Criscione, J. C., Humphrey, J. D., Douglas, A. S., & Hunter, W. C. (2000). An invariant basis for natural strain which yields orthogonal stress response terms in isotropic hyperelasticity. Journal of the Mechanics and Physics of Solids, 48(12), 2445-2465. )

The originators of finite strain theories of elasticity defined W as a function of I1 and I2 (and I3 (volume ratio)) for the sake of generality in their formulation. However, in practice, it has often been found convenient, and sometimes accurate to neglect dependence on I2.   The argument for neglecting I2 was perhaps stated best here:

Yeoh, O. H. (1990). Characterization of elastic properties of carbon-black-filled rubber vulcanizates. Rubber chemistry and technology, 63(5), 792-805.

Yeoh, O. H. (1993). Some forms of the strain energy function for rubber. Rubber Chemistry and technology, 66(5), 754-771.

best wishes!


Thanks for the references!

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