Mixed action principles theory and applications to a continuum
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Hamilton's Principle is a variational principle, which is different from Newton's equation of motion in that it looks at the trajectory of the system as a whole, whereas the equation of motion looks only at the local trajectory. It is more efficient and more general than Newton's equation of motion in the sense that it has been widely applied throughout physics, including for electro-magnetic fields, the motion of waves, and special relativity. However, Hamilton's Principle is not compatible with the given initial conditions in the strong form. Furthermore, applying Hamilton's principle to non-conservative systems requires not only a Lagrangian, but also another scalar function, the Rayleigh's dissipation function, which is not completely consistent with a variational scheme. The critical weakness in Hamilton's Principle, associated with non-compatible initial conditions within the weak form, can be resolved by a novel formulation named Extended Hamilton's Principle, as presented in this dissertation. Even though this still requires Rayleigh's dissipation function to account for non-conservative contributions, Extended Hamilton's Principle provides a unified way to account for both initial and boundary conditions in a continuum dynamics formulation through a newly defined action variation. The action variation in Extended Hamilton's Principle is suitable for application within a space-time finite element scheme. This is numerically implemented through Galerkin's method and validated for various 1D and continuum cases. The numerical approach based on Extended Hamilton's Principle shows some advantages compared to the currently used numerical methods, especially in accounting for material nonlinearity. It also provides a good basis to develop more efficient space-time finite element numerical algorithms in the future. Apart from the creation of Extended Hamilton's Principle, this dissertation also presents another new variational principle, named the Mixed Convolved Action Principle. This latter principle is a complete variational principle for dynamics of the system, which solves long pending difficulties in Hamilton's Principle. In particular, Rayleigh's dissipation function is no longer required and the specified initial conditions are compatible with the strong form.