I know that for isotropic materials, a rule for the maximum element size L_max is:
L_max < lambda_min / 10
where lambda_min is the min. wavelength, defined as:
lambda_min = c_T / f
where f is the excitation frequency and c_T the transverse wave speed:
c_T = sqrt(G/rho)
Now I wonder:
- Can I take the same approach for an orthotropic material?
- If so, how can I calculate or obtain the transverse wave speed?
Dominik Itner can answer this. We just came across the same problem recently.
In general it is possible to formulate the Christoffel equation for any given elasticity tensor assuming plane wave motion. This allows you to assemble the Christoffel matrix Gamma with
Gammaij = Cijkl * dk * dl
for a particular (unit) direction vector d = {d1 d2 d3} (in which the assumed plane wave is moving) and with elasticity tensor C.
This in turn allows you to compute the quasi long wave speed and quasi shear wave speeds in any given direction as the eigenvalues of Gamma with
ci = sqrt(lambdai / rho) with i in {1,2,3}.
Consequently, you can find the minimum wave speed in all directions, which, in theory, should be allowed to be plucked in as a "substitute" to cT.
Indeed, the minimum wave speed for an isotropic elasticity tensor converges exactly to sqrt(G/rho). In case of weak transverse isotropy, it converges to sqrt(Gmin/rho). I have observed relevant effects on the minimum wave speed especially in cases of strong orthotropy or anisotropy.
(edit: Christoffel matrix assembly and wave speed computation)
The element size depends on the error you are allowed to have. In an analysis we come to the conclusion that 15-20 elments per wavelength gives an error of 1% in group velocity.
see Fig. 10
Article Comparison of different higher order finite element schemes ...
In anisotropic composites (transverse isotropy), the wave speed depends on the propagation direction. We have quasilongitudinal (L), fast quasishear (Sfast), and slow shear (Sslow) waves. Their speeds are obtained through solution of Christoffel's equation. The lowest possible speed is that of Sslow for wave propagation normal to the fibers, namely
c_Sslow_perpendicular = sqrt(C44/rho).
I would use this speed for the calculation of L_max for composites.
You can use the Dispersion Calculator (Bulk waves tab) to plot the wave speeds in 2-D and 3-D (see attached plot). In the material editor, you get the wave speeds of all three bulk wave types for propagation parallel and normal to the fibers once you have entered the material parameters (engineering constants or stiffness matrix components).
The Dispersion Calculator can be downloaded here:
https://www.dlr.de/zlp/en/desktopdefault.aspx/tabid-14332/24874_read-61142/#/gallery/33485
In addition to my above answer:
The wave speeds of L, Sfast, and Sslow for propagation parallel and normal to the fibers are:
c_L_parallel = sqrt(C11/rho)
c_Sfast_parallel = sqrt(C66/rho)
c_Sslow_parallel = sqrt(C55/rho)
c_L_perpendicular = sqrt(C22/rho)
c_Sfast_perpendicular = sqrt(C66/rho)
c_Sslow_perpendicular = sqrt(C44/rho)
All of these are found in the attached plot. Therein, the fibers are oriented horizontally. c_Sslow_perpendicular, as marked by the arrow, is the global minimum.
Velocity= frequency* wavelength
minimum wavelength= minimum velocity/frequency
element size= (minimum wavelength/15) to (minimum wavelength/8)
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