User:Berland/Elastic moduli

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The stiffness matrix (9 by 9, or 6 by 6 in Voigt notation) in Hooke's law (in 3D) can be parametrized by only two components for homogeneous and isotropic materials. One may choose whichever pair one prefers among the elastic moduli given below. Some of the possible conversions are listed in the table.

Stiffness matrix for different parametrizations
$$(\lambda,\mu)$$ $$\begin{pmatrix} \lambda + 2\mu & \lambda &\lambda \\ \lambda & \lambda + 2\mu & \lambda \\ \lambda & \lambda & \lambda + 2\mu \\ &&& \mu \\ &&&&\mu \\ &&&&&\mu \\ \end{pmatrix} $$

$$(E, \mu)$$ $$\begin{pmatrix} \mu\frac{\mu-E}{3\mu-E} & \mu \frac{E-2\mu}{3\mu-E} & \mu \frac{E-2\mu}{3\mu-E} \\ \mu \frac{E-2\mu}{3\mu-E} & \mu\frac{\mu-E}{3\mu-E} & \mu \frac{E-2\mu}{3\mu-E} \\ \mu \frac{E-2\mu}{3\mu-E} & \mu \frac{E-2\mu}{3\mu-E} & \mu\frac{\mu-E}{3\mu-E} \\ &&& \mu \\ &&&&\mu \\ &&&&&\mu \\ \end{pmatrix} $$

$$(K, \lambda)$$ $$\begin{pmatrix} 3K-2\lambda & \lambda & \lambda \\ \lambda & 3K-2\lambda & \lambda \\ \lambda & \lambda & 3K-2\lambda \\ &&& \tfrac32 (K-\lambda) \\ &&&&\tfrac32 (K-\lambda) \\ &&&&&\tfrac32 (K-\lambda) \\ \end{pmatrix} $$

$$(K, \mu)$$ $$\begin{pmatrix} K+\tfrac43 \mu & K-\tfrac23\mu & K-\tfrac23\mu \\ K-\tfrac23\mu& K+\tfrac43\mu & K-\tfrac23\mu \\ K-\tfrac23\mu& K-\tfrac23\mu & K+\tfrac43\mu \\ &&& \mu \\ &&&& \mu \\ &&&&& \mu \\ \end{pmatrix} $$