Boundary conditions at the edge of a thin or thick plate bonded to an elastic support

Abstract

At the clamped edge of a thin plate, the interior transverse deflection W(Xl, x2) of the mid-plane x3 = 0 is required to satisfy the boundary conditions w = Ow/On = 0. But suppose that the plate is not held fixed at the edge but is supported by being bonded to another elastic body; what now are the boundary conditions which should be applied to the interior solution in the plate? For the case in which the plate and its support are in two-dimensional plane strain, we show that the correct boundary conditions for w must always have the formwith exponentially small error as L/h ~ oo, where 2h is the plate thickness and L is the length scale of w in the xl-direction. The four coefficients W s, W E, 0 B, 0 F are computable constants which depend upon the geometry of the support and the elastic properties of the support and the plate, but are independent of the length of the plate and the loading applied to it. The leading terms in these boundary conditions as L/h ~ oo (with all elastic moduli remainingfixed) are the same as those for a thinplate with a clamped edge. However by obtaining asymptotic formulae and general inequalities for O s, W F, we prove that these constants take large values when the support is 'soft' and so may still have a strong influence even when h/L is small. The coefficient W is also shown to become large as the size of the support becomes large but this effect is unlikely to be significant except for very thick plates. When h/L is small, the first order corrected boundary conditions are w=O~ dw 4oB h d2w dxl 3(i :u--) ~Xl 2 ~'-O' which correspond to a hinged edge with a restoring couple proportional to the angular deflection of the plate at the edge.

Elastic metasurfaces for splitting SV- and P-waves in elastic solids

“…Though the analytic model shows certain mismatch of amplitudes with the simulation results, the present model is still a useful tool since phase modulation is more crucial in metasurface design. This issue could potentially be addressed by using more complicated boundary conditions, 22 but is beyond the scope of this paper. The explicit expression for the transmitted phase can be easily extracted from Eq.…”