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Hyperelastic models – question about distortional and deviatoric parts of Cauchy-Green tensor

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  • Hyperelastic models – question about distortional and deviatoric parts of Cauchy-Green tensor

    Dear Mr. Bergstrom,

    I am currently pursuing PhD which topic is connected to modelling of rubber springs. At the beginning I would like to thank you for your Bergstrom-Boyce material model and your publications which proved to be very helpful in my work.

    I have question about general stress equations for several hyperelastic models which are described in your book “Mechanics of solid polymers”. In equations (5.92 – Yeoh model) and (5.108 – EC model) there is used a deviator of distortional part of Cauchy-Green tensor dev[b*]. I understand why use the b* (as it is defined in 5.37 for C tensor) - it is because we are interested only in isochoric part of tensor. The problem I have is with deviator which comes afterward. I have no idea what is the physical interpretation for deviator of distortional part of Cauchy-Green tensor.

    As I remember from small strains theory, deviatoric strains represent shape change at constant volume. But on the other side, deviator has such physical meaning only for small strains, which generally is not a case for hyperelastic materials. So, besides having b* as an input which should have already described what is needed, what is the purpose of using a deviator for large strains/stretches?

    For the Mooney-Rivlin model (eq. 5.87) there is no deviator of b* tensor. I can’t see a reason for this difference. One thing that has come to my mind is the appearance of second invariant I2 in MR model in contrast to EC and Yeoh models. Maybe it is a false trail, but if it is not, I still have no idea how second invariant presence or absence can affect usage of deviator of b*.

    I would be grateful for any advice for this matter. I can’t say I am proficient in solid mechanics, so if my problem is trivial, I beg your pardon.

    Wojciech Sikora
    Last edited by wsikora; 2017-07-20, 14:54.

  • #2
    I have solved my problem. When second invariant is absent, the remaining expressions in stress equation resemble deviator. It seems that in this case deviator doesn't have any physical meaning, it is just a mathematical formula used for convenience.