On the Extreme Values of Young’s Modulus, the Shear Modulus, and Poisson’s Ratio for Cubic Materials

For homogeneous cubic elastic materials with positive definite stored energy it is shown that the maximum and minimum values of Young’s modulus E are related to the maximum and minimum values of the shear modulus G through the simple connection $1/Gmin−1/Gmax=3(1/Emin−1/Emax).$ It is deduced that the ratio of compliances $−s12/s44$ is the maximum value of Poisson’s ratio v in the cubic materials with a positive parameter $χ=2s11−2s12−s44$⁠, and the minimum value of ν in the cubic materials with negative χ.