The Wikipedia article has a completely separate table for Lamé parameter conversions in the 2D case. Philosophically, why is there even a different table for 2D if these parameters are meant to express intrinsic material qualities? Practically, how does one derive and use the 2D values?
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$\begingroup$ Must be better references for the Lamé equations than Wikipedia… $\endgroup$– Solar MikeCommented Feb 23, 2023 at 20:12
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1$\begingroup$ Perhaps, in some cases, the 2D model is considered either under plane strain conditions or under plane stress conditions. In this case, one of the deformed coordinates either does not change (in the ideal case), or does it change negligibly? Maybe this is the point of using a two-dimensional table of Lame parameters? For example here, the 2D case of deformations is investigated: comet-fenics.readthedocs.io/en/latest/demo/elasticity/… $\endgroup$– ayrCommented Feb 24, 2023 at 6:21
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Let me answer here with my own findings, since I am currently working on elasticity of membranes (non-linear membrane model, publications are to appear):
For any isotropic membrane (2D) material, there are two membrane Lamé parameters $(\lambda,\mu)$. We call $\lambda$: first membrane Lamé parameter and $\mu$: second membrane Lamé parameter of that material. They are defined analogously to the classical three dimensional Lamé parameters, by expansion of the stresses close to the natural state/reference configuration:
For any membrane motion (family of time-dependend membrane deformations) starting in the reference configuration, there are two parameters $\lambda$ and $\mu$, such that the derivative of the time-dependend membrane second Piola Kirchhoff stress tensor $S(t)$ with respect to time at $t=0$ can be computed from the derivative of the time-dependend membrane Cauchy-Green deformation tensor $C(t)$ at $t=0$ as follows:
$$ S'(0)= \frac{\lambda}{2}\mathrm{tr}(C'(0))\mathrm{Id}+\mu C'(0). $$
They obtain a physical interpretation by choosing specific membrane motions, like pure shear ($\mathrm{tr}(C'(0))=0$) and pure dilation ($C'(0)=a\cdot\mathrm{Id}$). Then for membranes it turns out that $\mu$ is the membrane shear modulus $g$ and $\lambda+\mu$ is the membrane bulk modulus $k$. (Note that a membrane pure dilation would correspond to an equibiaxial extension in three dimensions.)
Concerning the relation inbetween the 3D and 2D values I think:
Let $\Lambda$ and $M$ denote here the classical 3D Lamé parameters of a given isotropic material (Note the difference in the units: $M,\Lambda$ are in $\tfrac{N}{m^2}$, whereas $\lambda,\mu$ are in $\tfrac{N}{m}$). Then the membrane Lamé parameters for an isotropic elastic film with thickness $d$ in terms of the 3D Lamé parameters of that material are given by $$ \mu =d\cdot M $$ and $$ \lambda =d\cdot\frac{2M\Lambda}{\Lambda+2M} $$
(Any elastic modulus of an isotropic material has its own membrane version, and these can be computed as in the table in Wikipedia. I would also very much like to know the source for that table. Thanks for your question.)
