The focus of this thesis is to understand the behavior of composite plates reinforced with

rigid bars that are free to twist and bend with respect to the medium. Such composites are

ubiquitous in nature and in industry, especially with increased interest in modeling biological elements as well as nano-technology. Structured fabrics abound in nature and industry – from cytoskeletons to kevlar sheets.

In the first part of this thesis existing theory on such materials is reviewed. A particular

microstructure - that of a fiber-reinforced medium - is the subject of further study. Such

a medium is treated as a special case of a nematic elastomer with constrained directors.

The salient feature of such a material is the presence of only along-fiber derivatives in the

problem, precluding certain boundary data.

The second part of the thesis focuses on the development of a two-dimensional plate

theory from the three-dimensional fiber reinforced medium studied previously. The resulting two-director model shows behavior similar to a nematic elastomer with directors that are unable to shear. This is then specialized to the case of a laminate with a single family of fibers. To obtain an understanding of the theory it is applied to a simple controllable deformation; that of a fiber-reinforced plate bent into a cylindrical shell. The orientation of the fibers is selected to result in both bending and twist. This controllable deformation helps us understand the possible boundary data and its e↵ect on the solution. The effect of boundary data on existence and uniqueness of solution is highlighted through the example problem.

A nonlinear two-dimensional theory is developed for thin magnetoelastic films capable of large deformations. This is derived directly from three-dimensional theory. Significant simplifications emerge in the descent from three dimensions to two, permitting the self field generated by the body to be computed a posteriori. The model is specialized to isotropic elastomers with two material models. First weak magnetization is investigated leading to a free energy where magnetization and deformation are un-coupled. The second closely couples the magnetization and deformation. Numerical solutions are obtained to equilibrium boundary-value problems in which the membrane is subjected to lateral pressure and an applied magnetic field. An instability is inferred and investigated for the weak magnetization material model.

The mechanical response of solids exhibiting complex material behavior has traditionally been determined by fitting constitutive models of specified functional form to experimentally derived force-displacement (stress-strain) data. However, characterizing the nonlinear mechanical behavior of complex materials requires a method of quantifying material behavior that is not restricted by a specific constitutive relation. To this end, a new method, termed \textit{the reverse updated Lagrangian finite element method} (RULFEM), which is based on the three-dimensional displacement field of the deformed solid and finite element method, is developed for incrementally linear materials. In RULEFM, the body is discretized by finite elements and its material properties are determined element-wise, i.e., the properties are assumed to be uniform at the element level and may vary from one element to another. The validity of RULFEM is demonstrated by three noise-free numerical examples and three numerical examples with various input noise levels. Two methods to assess the global and local errors of the results due to error in the measured input data (noisy data) are also discussed.

The life expectancy of a solid is traditionally predicted by assessing its expected stress cycle and comparing it to the experimentally determined stress state at failure. The accuracy of this procedure is often compromised by unforeseen extremes in the loading cycle or progressive material degradation. Often residually stressed part can either produce longer or shorter lifespans than predicted. Thus determination of the current state of stress (i.e., the residual stress in the absence of external loading) and material properties is particularly important. Typically, the material properties of a solid are determined by fitting the experimental data collected from the measured response to deformations of a stress-free configuration. However, the characterization of the mechanical behavior of a residually stressed body requires, in principle, a method that is not restricted to specific constitutive models. Complementing RULFEM \cite{tartibi2014reverse}, a new method, called \textit{estimating the current-state-of-stress} (ECSS) is developed in this work. Similar to RULFEM, ECSS also uses as input three-dimensional full-field displacement and force data of the body, which is perturbed by small displacements. ECSS complements the first step of the incremental RULFEM method. It generates the nodal current-state-of-stress (or residual stress in the absence of external tractions) as well as the incremental elasticity tensor pertaining to each of the finite elements used to discretize the body. The ECSS method is used to simulate two noise-free examples.

The linear dynamic response of a solid body has been used for material identification and even defect detection in low-damping (negligible viscoelastic behavior) materials. RULFEM and ECSS input full-field data, obtained from the statically perturbed body, are used to determine the initial state of stress and the current material elasticity tensors. A new formulation of ECSS, called \textit{dynamically estimating residual stress} (DERS) is developed in the fifth chapter of this dissertation. This is based on three-dimensional full-field displacement and force data of the dynamically perturbed body. In addition to nodal residual stress tensors and element-wise incremental elasticity tensor, DRES generates element-wise material density of the discretized body.

Abstract

Design, Analysis, and Testing of Large Capacity Kinetic Energy Storage Systems

by

Siyuan Xin

Doctor of Philosophy in Engineering – Mechanical Engineering

University of California, Berkeley

Professor David Steigmann, Chair

Global warming, climate change and pollution caused by the traditional energy generation technologies have become some of the biggest threats in today’s world. And the need for more energy is ever increasing. Development of renewable energy, such as solar and wind, as well as distributed smart grids are needed to replace the old energy production methods. Governments have set goals of increasing the proportion of clean energy generation. One major limiting factor of the development of renewable energy is that these energy sources is not as reliable and as stable as the traditional power sources to meet the energy demand. This problem can only be effectively solved by the use of energy storage to store excess power produced by the renewable energy sources. Therefore, energy storage, especially large scale energy storage systems, will be the key factor for enabling renewable energy to become the primary sources of energy. Current energy storage technologies include batteries, flywheel, supercapacitors, pumped hydro, and so on. But due to the issues of capacity, efficiency, life cycles or costs of these current energy storage technologies, a large scale energy storage system that has low cost, long life cycles and high efficiency is yet to be developed to be integrated with renewable energy generation and smart grids. In this work, the design, analysis and tests of large scale flywheel energy storage systems are carried out. The flywheel energy storage system will feature a large steel rotor. Theoretical analysis of the stress distribution of the rotor is discussed to give a parametric view of the key elements in the design. Material testing such as tensile tests, fracture toughness tests, and fatigue crack propagation tests are conducted to characterize the behavior of the rotor material. Finite element method is used to analyze the stress-strain distribution of the rotor in order to optimize the shape of the rotor and determine the rotational speed. Also frequency analysis using finite element method gives the dynamic response of the rotor. Catastrophic failure of the rotor can cause big damages and need to be prevented from happening. And therefore, fracture mechanics is applied to analyze the safety margin and estimate the lifetime of the flywheel system. Post failure analysis is also discussed in case of rotor failure. Energy storage efficiency is a crucial factor in order for the flywheel energy storage system to be applicable. The rotor is kept in low vacuum pressure to minimize aerodynamic drag and improve energy storage efficiency. An analytical model of aerodynamic drag on the rotor is developed. Experimental measurements of drag power loss on a disc rotor inside a controlled vacuum chamber are compared to the predicted results of the analytical model. Design of sub-systems such as bearings and vacuum system are discussed at the end.

The purpose of this thesis is to examine the mechanical aspects of biological lipid bilayers. Constituting the cell membranes of nearly every organism, lipid bilayers are composed of amphiphilic lipid molecules which self-assemble into bilayers in aqueous solution. They exhibit a fascinating combination of flexural resistance as in shell structures and lateral flow as in two-dimensional fluid sheets, giving rise to a host of complex phenomena. Lipid bilayers represent a promising object of inquiry for medical innovation, as dysfunctional lipid bilayers have been linked to disease formation. This work seeks to mathematically depict previously unexplored lipid bilayer phenomena.A simple model of lipid tilt and distension inspired by parallel research in molecular dynamics is outlined and demonstrated numerically. This simple model assumes reflection symmetry of lipid molecule orientation about the bilayer midsurface. Equilibrium configurations for a membrane of this type are presented for domains containing many closely packed voids representing transmembrane proteins. Lipid tilt and distension patterns arise due to the amphipathic nature of these proteins. Then, a model for a lipid bilayer involving independent tilt fields for top and bottom leaflets of the bilayer is developed. This loosened restriction leads to a complex but more general theory. A model for a lipid bilayer with a conforming elastic cytoskeleton is then proposed and equilibrium equations are established. The spontaneous curvature of the conventional Helfrich theory is shown to arise naturally as a mechanical aspect of this system.

Classical continuum mechanics has been widely used in the failure analysis of materials for decades. However, spatial partial derivatives in governing equations of the conventional theory are not valid along the discontinuities. Alternatively, peridynamic (PD) theory, as a nonlocal continuum theory, eliminates this shortcoming by using integro-differential equations which do not contain spatial derivatives. This feature makes PD theory very attractive for problems including discontinuities such as cracks.

In this study we have extended PD formulation to multiscale problems involving friction, wear, and delamination of thin films. We have formulated a nonlocal ordinary state-based peridynamic formulation for plastic deformation based on the idea of mechanical sublayers which is successfully applied in modeling ductile fracture. In addition, we demonstrated how PD can be used as an efficient and accurate analysis tool in designing real-world applications such as body armor systems using bio-inspired structures with the goal of minimizing the effect of the ballistic impact and bullet penetration depth while being lightweight and comfortable to wear. All our obtained results are validated against experimental observations and an excellent agreement has been achieved. Similar to other mesh-free methods, PD is massively parallelizable. We built parallel PD algorithms leveraging shared and distributed memory systems on CPU as well as CUDA architecture on GPU. We provide extensive experiments showing scalability and bottlenecks associated with each parallelization technique.

In the second part of this dissertation, we introduce a new class of learnable forward and inference models, using graph neural networks (GNN) which develops relational behavior between material points. We demonstrate these models are surprisingly accurate to generalize remarkably well to challenging unseen loading conditions. Our framework offers new opportunities for harnessing and exploiting non-local continuum theory and powerful statistical learning frameworks to take a key step toward building accurate, robust, and efficient patterns of reasoning about materials behavior.

Understanding the mechanical behavior of arterial walls under various physiological loading and boundary conditions is essential for achieving the following: (1) improved therapeutics that are based on mechanical procedures (e.g. arterial segmenting and suturing), (2) study of mechanical factors that may trigger the onset of arterial aneurysms (i.e. focal blood-filled dilatation of the vessel wall caused by disease) and (3) investigations on tissue variations due to health, age, hypertension and atherosclerosis, all of which hold immense clinical relevance. In general, the physiological conditions on an any arterial segment can include axial stretch, torsional twist and transmural (internal, radial) pressure which often provoke large wall-tissue deformations that require theories of continuum hyperelasticity. Further, the presence of collagen fibers throughout the two structural layers (media, adventitia) of the arterial wall require anisotropic strain energy functions for more histological accurate models. Nonlinear computational methods are therefore essential for this class of boundary-value-problems (BVPs) which often do not contain closed-form solutions.

We begin by modeling the arterial vessel wall as a thin sheet in the form of a circular cylinder in the reference configuration. We seek to employ a bio-type strain energy function on this constitutive framework to investigate the onset of non-linear instabilities in a thin-walled, hyperelastic tube under (remote) axial stretch and internal pressure. Viscoelastic effects are also considered in this model. We then build to investigating the effects of various combinations of axial stretch and transmural pressure on the global deformation and through-thickness stress and strain fields of an arterial segment modeled as a two-layer, fiber-reinforced composite and idealized as a thick-walled cylinder in the reference configuration. We further consider (in both models) the presence of local tissue lesions, or portions of the arterial wall having either stiffer (i.e. thrombosis or scar tissue) or softer (i.e. diseased tissue) material characteristics, relative to the surrounding tissue. We account for this by appropriately scaling the elastic constants of the strain energy functions for regions with a lesion and without. For the three-dimensional model, we employ the strain energy function of Holzapfel et al. which has been modified by constraints on the principal invariants by Balzani et al. in order to ensure material polyconvexity. We choose a particular vessel, the human common carotid (HCC) artery, with appropriate geometric and material properties found from various experimentally-based studies (e.g. Fung et al.). We focus on distinct elastic constants for each layer (media, adventitia) that have been obtained through biaxial (i.e. not simply uniaxial data - reasons for this are discussed later) testing of in vitro HCC arteries. The loading conditions are combinations of axial extension and transmural pressure, in the presence and absence of material lesions. The loading is consistent with in vivo conditions on a general segment of the vessel wall.

We find that as a two-dimensional surface, the overall deformation from internal pressure (i.e. the bulge) depends on the magnitude and, more importantly, the rate of axial stretch and transmural pressure, the elastic material parameters of the bio-strain energy function, and of course local inhomogeneities in the material description of the tissue. When modeled as a three-dimensional solid undergoing pure axial stretch, the majority of the stress is in the medial tissue, which displays a significant gradient in the axial direction, whereas the stress in the adventitia is constant throughout the length of the vessel. For supra-physiological pressures (i.e. 20-30 kPa, or about 50% higher than in vivo conditions) the adventitia contributes to the load sharing and the gradient in the medial layer evens out. For narrow (2% of the length), stiff (100x stiffer than surrounding tissue), ring-like lesions under the same pressures and axial stretch, the overall vessel deformation is considerably smaller in the radial direction. The overall segment shape is stabilized by this type of material abnormality. For local spot-like stiff (100x stiffer than surrounding tissue) lesions, the deformation leads to an inward bulge (i.e. a clot) that will likely affect fluid flow characteristics, hence growth and remodeling of the tissue at the wall. For these loading conditions, when the spot-like and ring-like lesions are approximately two-times softer than the surrounding tissue, no significant differences appear in the stress and strain fields.

This work explores the field of continuum plasticity from two fronts. On the theory side, we establish a complete specification of a phenomenological theory of plasticity for single crystals. The model serves as an alternative to the popular crystal plasticity formulation. Such a model has been previously proposed in the literature; the new contribution made here is the constitutive framework and resulting simulations. We calibrate the model to available data and use a simple numerical method to explore resulting predictions in plane strain boundary value problems. Results show promise for further investigation of the plasticity model. Conveniently, this theory comes with a corresponding experimental tool in X-ray diffraction. Recent advances in hardware technology at synchrotron sources have led to an increased use of the technique for studies of plasticity in the bulk of materials. The method has been successful in qualitative observations of material behavior, but its use in quantitative studies seeking to extract material properties is open for investigation. Therefore in the second component of the thesis several contributions are made to synchrotron X-ray diffraction experiments, in terms of method development as well as the quantitative reporting of constitutive parameters. In the area of method development, analytical tools are developed to determine the available precision of this type of experiment - a crucial aspect to determine if the method is to be used for quantitative studies. We also extract kinematic information relating to intragranular inhomogeneity which is not accessible with traditional methods of data analysis. In the area of constitutive parameter identification, we use the method to extract parameters corresponding to the proposed formulation of plasticity for a titanium alloy (HCP) which is continuously sampled by X-ray diffraction during uniaxial extension. These results and the lessons learned from the efforts constitute early reporting of the quantitative profitability of undertaking such a line of experimentation for the study of plastic deformation processes.