New developments in dynamics and computational finite deformation formulations: Methodology, algorithms, and analysis

The present research addresses new developments and framework, methodology, algorithms and the computational formulation aspects of finite deformation dynamic analysis. Of interest here are new theoretical design developments and issues encompassing: (i) a unified theory underlying computational algorithms for time dependent problems, and (ii) an arbitrary reference configuration (ARC) formulation as well as the corresponding constitutive models and stress update formulations for computational finite deformation dynamic analysis.; To provide an understanding of computational algorithms for time dependent problems, a unified theory for linear first-order systems is first formulated with the following objectives: (i) provides a standard methodology to classify algorithms, (ii) enables computational algorithms to be evaluated by standardized comparisons, and (iii) permits new avenues leading to the notion of algorithms by design. In this theory and as a special case, a framework of generalized single step single solve representation of the LMS methods exists, within which optimal controllable numerical dissipative algorithms have been designed in terms of the following aspects: (i) unconditionally stable, (ii) second-order accuracy, (iii) no more than zero-order displacement and velocity overshooting behavior, and (iv) self-starting features. In the approach of the Lie group based geometrical integrators, a class of time integration algorithms is developed and eventually leads to an algorithm with nonlinearly explicit second-order accurate L-stable features.; A fundamental question is raised in this thesis to challenge the application of the two point field strain measure to describe the finite deformation problems. We propose to resolve the problem via an arbitrary reference configuration (ARC) formulation which includes the ARC elasticity theory, the corresponding stress update formulation and ARC Lagrangian formulation. It includes the followings: (i) the ARC elasticity theory is a generalized elasticity bridging hyperelasticity and hypoelasticity, and (ii) the ARC Lagrangian formulation is a generalized finite element formulation bridging the total Lagrangian formulation and the updated Lagrangian formulation. Numerous theoretical proofs, numerical applications and verifications are demonstrated throughout the thesis.