A study of nonlinear deformations and defects in the actuation of soft membranes, rupture dynamics, and mesoscale plasticity

This work involves theoretical and computational modeling of interactions in line defects and deformation, with applications in the fields of earthquake rupture dynamics, designed deformations of soft membranes, and metal plasticity at small length scales. This thesis describes the study of four distinct, specific contemporary problems in these fields. 

First, micropillar compression experiments probing size effects in confined plasticity of metal thin films, including the indirect imposition of ‘canonical’ simple shearing boundary conditions, show dramatically different responses in compression and shear of the film. The theoretical and computational framework of Mesoscale field dislocation mechanics (MFDM) is used to study size effects in micropillar confined thin metal films in different orientations (under nominal compression and shear). The formalism is shown to be capable of reproducing drastically different size effects in compression and shear, as observed experimentally, without any ad-hoc modification to the basic structure of the theory (including boundary conditions), or the use of extra fitting parameters. This is a required theoretical advance in the current state-of-the-art of strain gradient plasticity models. It is also shown that significantly different inhomogeneous fields can display qualitatively similar size effect trends in overall agreement with experimental results. Finite deformation effects in elastic-plastic materials such as Swift and Poynting effects are also demonstrated. 

Second, the same MFDM framework is used to understand the salient aspects of kinkband formation in additively manufactured Cu-Nb nano-metallic laminates (NMLs). A conceptually minimal, plane-strain idealization of the three-dimensional geometry, including crystal orientation, is used to model NMLs. Importantly, the natural jump/interface condition of MFDM imposing continuity of (certain components) of plastic strain rates across interfaces allows theory-driven ‘communication’ of plastic flow across the laminate boundaries in our finite element implementation. Kink bands under layer parallel compression of NMLs in accord with experimental observations arise in our numerical simulations. The possible mechanisms for the formation and orientation of kink bands are discussed, within the scope of our idealized framework. We also report results corresponding to various parametric studies that provide preliminary insights and clear questions for future work on understanding the intricate underlying mechanisms for the formation of kink bands. 

Third, a continuum model of rupture dynamics is developed using the field dislocation mechanics (FDM) theory. The energy density function in our model encodes accepted and simple physical facts related to rocks and granular materials under compression. We work within a 2-dimensional ansatz of FDM where the rupture front is allowed to move only in a horizontal fault layer sandwiched between elastic blocks. Damage via the degradation of elastic modulus is allowed to occur only in the fault layer, characterized by the amount of plastic slip. The theory dictates the evolution equation of the plastic shear strain to be a Hamilton-Jacobi (H-J) equation, resulting in the representation of a propagating rupture front. A Central-Upwind scheme is used to solve the H-J equation. The rupture propagation is fully coupled to elastodynamics in the whole domain, and our simulations recover static friction laws as emergent features of our continuum model, without putting in by hand any such discontinuous criteria in our model. Estimates of material parameters of cohesion and friction angle are deduced. Short-slip and slip-weakening (crack-like) behaviors are also reproduced as a function of the degree of damage behind the rupture front. Moreover, a crack is driven towards the undamaged side with an impact loading, and it is observed in our numerical simulations that an upper bound to the crack speed is the dilatational wave speed of the material, unless the material is put under pre-stressed conditions, when supersonic motion can be obtained. Without pre-stress, intersonic super shear is recovered under appropriate conditions. 

Fourth, a novel dual variational principle is developed for inverse and forward design problems in the actuation of liquid crystal glass sheets based on the PDEs arising from continuum mechanics ideas. A gradient flow and a Newton-Raphson algorithm are developed to obtain the approximations of critical point solutions of the dual functional, with a consistent nonlinear mapping between the primal and dual fields. In the case of complicated design shapes, an elliptic regularization of the dual PDE is developed, and the solutions obtained for the regularized problem are used as the initial guess for the unregularized problem. The solutions obtained for the inverse design problems of shapes such as a hemisphere and a hat shape are computationally demonstrated using this framework. The solutions obtained have less than 2.5% error in the L2 norm of the difference between the prescribed stretches and the computed principal stretches of the Right-Cauchy Green tensor of the deformation mapping the actuated and the unactuated shape. 

Finally, some concluding remarks on all aspects of the work accomplished in this thesis are discussed, along with directions for future work.