This dissertation describes the development of a class of new locally conservative, consistent and convergent mixed discontinuous Galerkin methods for linear and non-linear elastostatics, elastodynamics and viscoelasticity. Unlike classical continuous Galerkin finite element methods and popular interior penalty discontinuous Galerkin methods these schemes are locally conservative and possess higher order accuracy, robustness and simplicity. When coupled with a suitable symplectic time integration scheme, which exhibits controllable dissipation while conserving linear and angular momenta of the system, the new methods provide accurate and stable alternatives to explicit updated Lagrangian dynamics schemes for classes of problems involving large deformation, non-linear materials and long duration events.

The primary motivation of the work is the observed inability of explicit Lagrangian dynamics based codes to simulate long duration events with large deformations and non-linear material properties e.g. analysis of non-impact traumatic brain injury events lasting of the order of 20 seconds. This failure is attributed to under integration for long duration and lack of accuracy and local conservation. For the new mixed discontinuous Galerkin formulations introduced in this dissertation, we prove consistency, optimal convergence and conservation. The difficulties associated with the choices of a compatible and symmetric approximation spaces for classical mixed formulations of elasticity are surmounted for the proposed schemes by first, an appropriate choice of primal and dual variables, namely the displacement and its gradients instead of the conventional choice of displacements and stresses and secondly, suitable polynomial orders for the approximation of these variables. For the applications, both geometric and material non-linearities are considered and the governing equations are discretized in an updated Lagrangian framework. Using suitable error indicators we also establish the suitability of these schemes to adaptive implementations. A series of numerical examples are designed to illustrate salient features of our methodology including accuracy, convergence and long duration simulation.