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I think I have a pretty good understanding of the physical interpretation of young's modulus $E$ and poisson ratio $\nu$ in solid mechanics. However, I often find in mathematical papers that the equations are formulated in terms of lame's 1st & 2nd parameters $\mu$ & $\lambda$, respectively. I know there are formulas to relate the two in terms of young's modulus and poisson ratio, but I'm curious how can I best interpret their physical meaning.

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$\mu$ is quite easy - it is the shear modulus. In engineering texts it is often called $G$.

I do not believe that $\lambda$ has a straightforward physical interpretation. However, the bulk modulus $\kappa=\lambda+2\mu/3$, so it is sometimes useful to think about $\lambda$ as something closely related to the bulk modulus. For example, the bulk modulus of a nearly incompressible material can become arbitrarily large; you can see from the formula above that in this case, asymptotically, $\kappa\approx\lambda$.

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The Lame parameter $\lambda$ is equal to the off-diagonal term of the stiffness matrix, i.e. $C_{13}$. Where $\sigma_i=C_{ij} * \epsilon_j$. The stiffness matrix, $S$ is the inverse of the stiffness matrix, which has diagonal elements $1/E$ and off-diagonal elements $-v/E$ (for the upper left quadrant, the lower right is $1/G$ on the diagonal). Thus the top left corner of the $S$ matrix is:

$$S=\begin{bmatrix}1/E & -v/E & -v/E\\-v/E & 1/E & -v/E\\-v/E & -v/E & 1/E\end{bmatrix}$$

inverting gives $\frac{E(1-v)}{(1+v)(1-2v)}$ on the diagonal and $\frac{Ev}{(1+v)(1-2v)}$ (which is $\lambda$) on the off-diagonals. Some the lame parameter relates the transverse stress to uniaxial strain. Meanwhile, the diagonal terms are equivalent to the P-Wave modulus, or which relates axial stress to unaxial strain.

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    $\begingroup$ Welcome to Physics! Note that this site has MathJax enabled, which means you can use Latex-like syntax to add in equations for readability. $\endgroup$ – Kyle Kanos Jul 20 '15 at 17:29

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