Material Science Cheat Sheet Define Stiffness and Sideways Squish with Bulk and Shear Moduli

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The material provides a detailed solution to the problem of expressing the derived material constants, Young's modulus ($E$) and Poisson's ratio ($\nu$), in terms of the fundamental Bulk Modulus ($K$) and Shear Modulus ($G$). This derivation uses the core constitutive equations relating the stress tensor ($\sigma_{i j}$) in terms of $K$ and $G$ and the strain tensor ($\epsilon_{i j}$) in terms of $E$ and $\nu$ . The process establishes two bridging relationships: first, by taking the trace of the strain equation, the bulk modulus is related to $E$ and $\nu$ via $K=\frac{E}{3(1-2 \nu)}$; second, by considering a purely shear stress state, the relationship $E=2 G(1+\nu)$ is found. By substituting the shear relationship into the rearranged bulk modulus expression, Poisson's ratio is derived as $\nu=\frac{3 K-2 G}{6 K+2 G}$. Finally, substituting this resulting expression for $\nu$ back into $E=2 G(1+\nu)$ yields the final expression for Young's modulus (stiffness): $E=\frac{9 K G}{3 K+G}$.

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