- #1
FEAnalyst
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- TL;DR
- What makes one of the equations for stress based on Mooney-Rivlin potential wrong?
Hi,
as I've mentioned in this thread, I am looking for analytical solutions for simple loading cases involving hyperelastic materials. It turned out that the literature on rubber part design might actually be a good lead. In a rather old (written in 1989) Polish book "Gumowe elementy sprężyste" ("Rubber Elastic Parts") by M. Pękalak and S. Radkowski, I've found a discussion of calculations for several basic load cases. Most of the formulas there are based on a specific derivation of the hyperelastic potential, but for uniaxial tension, there is also an equation derived from the Mooney-Rivlin potential: $$\sigma_{eng}=2 \left( \lambda - \frac{1}{\lambda^{2}} \right) \left( C_{10}+C_{01} \lambda \right)$$ where: ##\lambda## - stretch ratio, ##\lambda=\frac{L}{L_{0}}##, ##L## - final length, ##L_{0}## - initial length, ##C_{10}## and ##C_{01}## - Mooney-Rivlin constants.
Unfortunately, this equation gives incorrect values, but in the article "Hyperelastic Constitutive Modeling of Rubber and Rubber-Like Materials under Finite Strain" by M.N. Hamza and H.M. Alwan, I´ve found another version of this equation, which gives results that fully coincide with those obtained from FEA: $$\sigma=2 \left( \lambda^{2} - \frac{1}{\lambda} \right) \left( C_{10} + \frac{C_{01}}{\lambda} \right)$$ I don't know what's wrong with this first equation - is there a mistake in the book or is it another form that should be used differently? The textbook says that this first equation gives the engineering (nominal) stress, while the article most likely gives the formula for the true stress. However, the relationship between engineering stress and true stress is: $$\sigma_{true}=\sigma_{eng} \lambda$$ Applying this transformation on the first formula doesn't give the second equation. Does anyone know where the error is?
as I've mentioned in this thread, I am looking for analytical solutions for simple loading cases involving hyperelastic materials. It turned out that the literature on rubber part design might actually be a good lead. In a rather old (written in 1989) Polish book "Gumowe elementy sprężyste" ("Rubber Elastic Parts") by M. Pękalak and S. Radkowski, I've found a discussion of calculations for several basic load cases. Most of the formulas there are based on a specific derivation of the hyperelastic potential, but for uniaxial tension, there is also an equation derived from the Mooney-Rivlin potential: $$\sigma_{eng}=2 \left( \lambda - \frac{1}{\lambda^{2}} \right) \left( C_{10}+C_{01} \lambda \right)$$ where: ##\lambda## - stretch ratio, ##\lambda=\frac{L}{L_{0}}##, ##L## - final length, ##L_{0}## - initial length, ##C_{10}## and ##C_{01}## - Mooney-Rivlin constants.
Unfortunately, this equation gives incorrect values, but in the article "Hyperelastic Constitutive Modeling of Rubber and Rubber-Like Materials under Finite Strain" by M.N. Hamza and H.M. Alwan, I´ve found another version of this equation, which gives results that fully coincide with those obtained from FEA: $$\sigma=2 \left( \lambda^{2} - \frac{1}{\lambda} \right) \left( C_{10} + \frac{C_{01}}{\lambda} \right)$$ I don't know what's wrong with this first equation - is there a mistake in the book or is it another form that should be used differently? The textbook says that this first equation gives the engineering (nominal) stress, while the article most likely gives the formula for the true stress. However, the relationship between engineering stress and true stress is: $$\sigma_{true}=\sigma_{eng} \lambda$$ Applying this transformation on the first formula doesn't give the second equation. Does anyone know where the error is?