Thomsen, Erik Vilain1; Reck, Kasper1; Skands, Gustav Erik1; Bertelsen, Christian Vinther6; Hansen, Ole4
1 Department of Micro- and Nanotechnology, Technical University of Denmark2 MEMS-AppliedSensors, Department of Micro- and Nanotechnology, Technical University of Denmark3 Nano Bio Integrated Systems, Department of Micro- and Nanotechnology, Technical University of Denmark4 Experimental Surface and Nanomaterials Physics, Department of Physics, Technical University of Denmark5 Silicon Microtechnology, Department of Micro- and Nanotechnology, Technical University of Denmark6 DTU Danchip, Technical University of Denmark
Deflection of thin crystalline plates
While silicon is an anisotropic material it is often in literature treated as an isotropic material when it comes to plate calculations. This leads to considerable errors in the calculated deflection. To overcome this problem, we present an in-depth analysis of the bending behavior of thin crystalline plates. An analysis of the compliance tensor for the 32 different crystal classes shows, that for thin plates, only 5 different types of plates exist. An anisotropic plate equation valid for crystalline thin plates is derived and solved for circular, elliptic, rectangular and square plates using both exact analytical expressions and approximate expressions calculated by the Galerkin method. The results are applied to plates made on silicon (0 0 1), (0 1 1) and (1 1 1) substrates, respectively, and analytical equations for the deflection, strain energy and resonance frequency of such plates are presented. These expressions are in excellent agreement with anisotropic finite element calculations. The calculated deflection differs less than 0.1%, for both circular and rectangular plates, compared to finite element calculations. The results are presented as ready-to-use facilitating accurate analytical models involving crystalline plates, such as those often found in the field of micro electro mechanical systems. The effect of elastic boundary conditions is taken into account by using an effective radius of the plate.
Sensors and Actuators A: Physical, 2014, Vol 220, p. 347-364