Dynamic higher order equations

University dissertation from Chalmers University of Technology

Abstract: The subject of this thesis is to derive and evaluate governing equations and corresponding boundary conditions for solid cylinders and rectangular plates, where the material constituting the cylinder or plate are governed by classical elasticity, micropolar elasticity or a functionally graded case of the previously mentioned models. This is achieved by a systematic power series expansion approach, by either adopting a generalized Hamilton's principle or a direct approach. For the solid cylinders a power series expansion in the radial coordinate for the sought fields are adopted. Equations of motion together with consistent sets of end boundary conditions are derived in a systematic fashion up to arbitrary order using a generalized Hamilton's principle. Governing equations are obtained for longitudinal, torsional, and exural modes. In the case of the rectangular plate, a power series expansion of the sought fields are adopted in the thickness coordinate. Governing equations of motion, for extensional and exural case, together with consistent sets of edge boundary conditions are derived in a systematic fashion up to arbitrary order with use of the direct approach. Both the governing equations for the solid cylinder and the rectangular plate are asymptotically correct to all studied orders. Numerical examples are presented for different sorts of problems, using exact theory, the present series expansion theories of different order, various classical theories and other newly developed approximate theories. These results cover dispersion curves, eigenfrequencies, various curves of cross sectional quantities such as displacements, stresses and micro-rotations, as well as fixed frequency motion due to prescribed end displacement or lateral distributed forces. The results illustrate that the present approach may render benchmark solutions provided higher order truncations are used, and act as engineering equations when using low order truncations.

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