The Earthquake Engineering Online Archive

Anisotropic bending-torsion coupling for warping in a non-linear beam

Klinkel, Sven; Govindjee, Sanjay

UCB/SEMM-2002/04, Dept. of Civil and Environmental Engineering, University of California, Berkeley, 2002, 19 pages

MEMS (micro-electro-mechanical system) devices made from single crystal silicon often contain rod-like structures that are operated in bending and/or torsion. The design of these devices usually relies upon simple mechanical theories that ignore the coupling between these two modes of operation. In this paper, the authors develop a theory that is capable of accounting for the material coupling in the bending and twisting of single crystal beams which arises from anisotropic elastic properties and apply it in selected examples to the case of silicon. The generalized Saint-Venant torsion theory, which is valid for isotropic materials, is extended to arbitrary anisotropic linear elastic materials. The anisotropic material behavior couples the bending and torsion behavior. Thus, for the geometrically linear case, two warping functions associated with the bending moments and one warping function that is associated with the torsion moment are found. These warping patterns or functions are then taken as inputs to a geometrically nonlinear formulation. Due to the presence of the additional warping functions, it is found that the existence of non-standard bi-moment and bi-shears play an important role under certain conditions of extreme deformations. The final complexity of the nonlinear formulation dictates the usage of a numerical solution procedure for practical computations. A finite element scheme is employed to solve the governing equations. Example computations elucidate the importance of the coupling effects by examining beams cut from (100) type silicon wafers.

Available online: http://nisee.berkeley.edu/documents/SEMM/SEMM-2002-04.pdf (1 MB)