Mechanical and Civil Engineering Seminar
Virtual PhD Thesis Defense
The development of new materials is an important component of many cutting-edge technologies such as space technology, electronics and medical devices. The properties of advanced materials involve phenomenon across multiple scales. The material may be heterogeneous on a scale that is small compared to that of applications, or may spontaneously develop fine-scale structure. Numerical simulation of such phenomena can be an effective tool in understanding the complex physics underlying these materials, thereby assisting the development and refinement of such materials, but can be challenging.
This thesis develops a new method to exploit the use of graphical processing units and other accelerators for the computational study of complex phenomena in heterogeneous materials. The governing equations are nonlinear partial differential equations, typically second order in space and first order in time. We propose an operator-splitting scheme to solve these equations by observing that these equations come about by a composition of linear differential constraints like kinematic compatibility and balance laws, and nonlinear but local constitutive equations. We formulate the governing equation as an incremental variational principle. We treat both the deformation and the deformation gradient as independent variables, but enforce kinematic compatibility between them as a constraint using an augmented Lagrangian. The resulting local-global problem is solved using the alternating direction method of multipliers. This enables efficient implementation on massively parallel graphical processing units and other accelerators. We use the study elastic composites in finite elasticity to verify the method, and to demonstrate its numerical performance. We also compare the performance of the proposed method with that of other emerging approaches.
We apply the method to understand the mechanisms responsible for a remarkable in-plane liquid-like property of liquid crystal elastomers (LCEs). LCEs are rubber-like solids where rod-like nematic molecules are incorporated into the main or a side polymer chain. They undergo isotropic to nematic phase transition accompanied by spontaneous deformation which can be exploited for actuation. Further, they display a soft behavior at low temperatures due to the reorientation of the nematic directors. Recent experiments show that LCEs exhibits an in-plane liquid-like behavior under multiaxial loading, where there is shear strain with no shear stress. Our numerical studies of LCEs provides insights into the director distribution and reorientation in polydomain specimens, and how these lead to the observed liquid-lie behavior. The results show good agreement with experimental observations. In addition to providing insight, this demonstrates the ability of our computational approach to study multiple coupled fields.
The core ideas behind the method developed in this thesis are then applied elsewhere. First, we use it to study multi-stable deployable engineering structures motivated by origami. The approach uses two descriptions of origami kinematics, angle/face based approach and vertex/truss based approach independently, and enforces the relationship between them as a constraint. This is analogous to the treatment of kinematic compatibility above where both the deformation and deformation gradient are used as independent variables. The constraint is treated using a penalty. Stable and rigid-foldable configurations are identified by minimizing the penalty using alternate directions, and pathways between stable states are found using the nudged elastic band method. The approach is demonstrated using various examples.
Second, we use a balance law or equilibrium to the problem of determining the stress field from high resolution x-ray diffraction. This experimental approach determines the stress field locally, and errors lead to non-equilibriated fields. It is hypothesized that imposing equilibrium leads to a more accurate stress reconstruction. We use Hodge decomposition to project a non-equilibriated stress field onto the divergence-free (equilibriated) subspace. This projection is numerically implemented using fast Fourier transforms. This method is first verified using synthetic data, and then applied to experimental data obtained from a beta-Ti alloy. It results in large corrections near grain boundaries.
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