Mixed Lagrangian Multiplier Formulation for Size-dependent Couple Stress Elastodynamic Response
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In this thesis, a mixed Lagrangian multiplier formulation is developed for size-dependent couple stress elastodynamic response. This extends previous work on quasistatic couple stress response that also requires the rotation to be one half of the curl of displacement. A Lagrange multiplier, which turns out to be related to the skew-symmetric part of stress, is used to constrain the relation of rotation and displacement, so that rotation becomes an independent variable and C1 continuity is avoided in the weak form. The finite element method is applied first to obtain the matrix form of the Lagrangian formulation in space and then discrete time integration is introduced to produce the discrete action. After that, the variation of discrete action is applied to derive the equations, along with the initial equations developed, for obtaining the solutions at finite time steps. Finally, the problems of uniform deformation of a square plate, transverse deformation of a cantilever, and uniform traction on a square plate with a hole, have been investigated under this formulation, where the results have been compared to existing methods, when possible.