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Analytical calculations involving hyperelastic materials

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I recently became interested in analytical calculations involving hyperelastic materials (just for educational purposes and to verify simple benchmarks in various FEA software). So far I've found only a single formula that could be used for this purpose - stress equation commonly derived from Mooney-Rivlin model's definition. It can be used to calculate the stress in uniaxial tension from known strain (displacement control). But I couldn't find anything like that for other simple loadings such as compression, bending, shear or torsion. Are there any formulas that would let me solve such basic problems (other than tension) analytically, assuming Mooney-Rivlin or other simple hyperelastic model? If yes, where can I find them?

Thanks in advance for your help.