(PHD, 2020)

Abstract:

The method of *oncotripsy* (from Greek, *onco-* meaning “tumor” and *–tripsy* “to break”) exploits aberrations in the material properties and morphology of cancerous cells to target them selectively using tuned low-intensity pulsed ultrasound. Compared to other noninvasive ultrasound treatments that ablate malignant tissue, oncotripsy has the capability of targeting unhealthy tissue with minimal damage to healthy cells in the ablation process.

We propose a model of oncotripsy that follows as an application of cell dynamics, statistical mechanical theory of network elasticity and ‘birth-death’ kinetics to describe processes of damage and repair of the cytoskeleton. We also develop a reduced dynamical model that approximates the three-dimensional dynamics of the cell and facilitates parameter studies, including sensitivity analysis and process optimization.

The dynamical system encompasses the relative motion of the nucleus to the cell membrane and a state variable measuring the extent of damage to the cytoskeleton. The dynamical system evolves in time as a result of structural dynamics and kinetics of cytoskeletal damage and repair. The resulting dynamics are complex and exhibits behavior on multiple time scales, including the period of vibration and attenuation, the characteristic time of cytoskeletal healing, the pulsing period and the time of exposure to the ultrasound. Damage on the cells develops in the order of millions of ultrasound cycles, and the failure mechanism is explained as a fatigue process. We also account for cell variability and estimate the attendant variance of the time-to-death of a cell population. We show that the dynamical model predicts — and provides a conceptual basis for understanding — the oncotripsy effect and other trends observed in experiments.

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(PHD, 2020)

Abstract:

We develop a critical-state model of fused silica plasticity on the basis of data mined from molecular dynamics (MD) calculations. The MD data is suggestive of an irreversible densification transition in volumetric compression resulting in permanent, or plastic, densification upon unloading. Moreover, this data exhibits dependence on temperature and the rate of deformation. We show that these characteristic behaviors are well-captured by a critical state model of plasticity, where the densification law for glass takes the place of the classical consolidation law of granular media and the locus of constant volume states denotes the critical-state line. A salient feature of the critical-state line of fused silica, as identified from the MD data, that renders its yield behavior anomalous is that it is strongly non-convex, owing to the existence of two well-differentiated phases at low and high pressures. We argue that this strong non-convexity of yield explains the patterning that is observed in molecular dynamics calculations of amorphous solids deforming in shear. We employ an explicit and exact rank-2 envelope construction to upscale the microscopic critical-state model to the macroscale. Remarkably, owing to the equilibrium constraint the resulting effective macroscopic behavior is still characterized by a non-convex critical-state line. Despite this lack of convexity, the effective macroscopic model is stable against microstructure formation and defines well-posed boundary-value problems. We present examples of ballistic impact of silica glass rods by way of the optimal transport meshfree method. We extend the study of the inelastic behavior of silica glass to include the effect of many different temperatures, pressures, and strain rates using MD and maximum entropy atomistics (MXE) calculations. Owing to the temperature dependence of the model, the macroscopic model becomes unstable against adiabatic shear localization. Thus, the material adopts small inter-facial regions where the shear strain is extremely high. We characterize the shear band size, thereby predicting a yield knockdown factor at the macroscale, and compare the results to behavior reported in flyer plate impact experiments.

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(PHD, 2019)

Abstract:

Despite the emergence of architected materials for various applications, metals still play a key role in engineering in general and aeronautics in particular. Turbine blades in jets engines for instance are made from single-crystal Nickel superalloys. As a result, studying the failure mechanism of these crystalline materials would help understand the limits of their applications. At the core of this mechanism are line defects called *dislocations*. Indeed, the plastic deformation of metals is governed by the motion of dislocation ensembles inside the crystal. In this thesis, we propose a novel approach to dislocation dynamics through the *method of monopoles*. In this approach, we discretize the dislocation line as a collection of points (or *monopoles*), each of which carries a Burgers “charge” and an element of line. The fundamental difference between our method and current methods for dislocation dynamics lies in the fact that the latter discretize the dislocation as a collection of line segments from which spans a need to keep track of the connectivity of the nodes. In our approach, we propose a “line-free” discretization where a linear connectivity or sequence between monopoles need not be defined. This attribute of the formulation offers significant computational advantages in terms of simplicity and efficiency. Through verification examples, we show that our method is consistent with existing results for simple configurations. We then build on this success to investigate increasingly complex examples, this with the ultimate goal of simulating the plastic deformation of a BCC grain in an elastic matrix.

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(PHD, 2018)

Abstract:

Data Driven Computing is a new field of computational analysis which uses provided data to directly produce predictive outcomes. This thesis first establishes definitions of Data-Driven solvers and working examples of static mechanics problems to demonstrate efficacy. Significant extensions are then explored to both accommodate noisy data sets and apply the deveoloped methods to dynamic problems within mechanics. Possible method improvements discuss incorporation of data quality metrics and adaptive data sampling, while new applications focus on multi-scale analysis and the need for public databases to support constitutive data collaboration.

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(PHD, 2018)

Abstract:

In the search for new alloys with a great strength-to-weight ratio, magnesium has emerged at the forefront. With a strength rivaling that of steel and aluminum alloys — materials which are deployed widely in real world applications today — but only a fraction of the density, magnesium holds great promise in a variety of next-generation applications. Unfortunately, the widespread adoption of magnesium is hindered by the fact that it fails in a brittle fashion, which is undesirable when it comes to plastic deformation mechanisms. Consequently, one must design magnesium alloys to navigate around this shortcoming and fail in a more ductile fashion.

However, such designs are not possible without a thorough understanding of the underlying mechanisms of deformation in magnesium, which is somewhat contested at the moment. In addition to slip, which is one of the dominant mechanisms in metallic alloys, a mechanism known as twinning is also present, especially in hexagonal close-packed (HCP) materials such as magnesium. Twinning involves the reorientation of the material lattice about a planar discontinuity and has been shown as one of the preferred mechanisms by which magnesium accommodates out-of-plane deformation. Unfortunately, twinning is not particularly well-understood in magnesium, and needs to be addressed before progress can be made in materials design. In particular, though two specific modes of twinning have been acknowledged, various works in the literature have identified a host of additional modes, many of which have been cast aside as “anomalous” observations.

To this end, we introduce a new framework for predicting the modes by which a material can twin, for any given material. Focusing on magnesium, we begin our investigation by introducing a kinematic framework that predicts novel twin configurations, cataloging these twins modes by their planar normal and twinning shear. We then subject the predicted twin modes to a series of atomistic simulations, primarily in molecular statics but with supplementary calculations using density functional theory, giving us insight on both the energy of the twin interface and barriers to formation. We then perform a stress analysis and identify the twin modes which are most likely to be activated, thus finding the ones most likely to affect the yield surface of magnesium.

Over the course of our investigation, we show that many different modes actually participate on the yield surface of magnesium; the two classical modes which are accepted by the community are confirmed, but many additional modes — some of which are close to modes which have been previously regarded as anomalies — are also observed. We also perform some extensional work, showing the flexibility of our framework in predicting twins in other materials and in other environments and highlighting the complicated nature of twinning, especially in HCP materials.

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(PHD, 2016)

Abstract:

It has been well-established that interfaces in crystalline materials are key players in the mechanics of a variety of mesoscopic processes such as solidification, recrystallization, grain boundary migration, and severe plastic deformation. In particular, interfaces with complex morphologies have been observed to play a crucial role in many micromechanical phenomena such as grain boundary migration, stability, and twinning. Interfaces are a unique type of material defect in that they demonstrate a breadth of behavior and characteristics eluding simplified descriptions. Indeed, modeling the complex and diverse behavior of interfaces is still an active area of research, and to the author’s knowledge there are as yet no predictive models for the energy and morphology of interfaces with arbitrary character. The aim of this thesis is to develop a novel model for interface energy and morphology that i) provides accurate results (especially regarding “energy cusp” locations) for interfaces with arbitrary character, ii) depends on a small set of material parameters, and iii) is fast enough to incorporate into large scale simulations.

In the first half of the work, a model for planar, immiscible grain boundary is formulated. By building on the assumption that anisotropic grain boundary energetics are dominated by geometry and crystallography, a construction on lattice density functions (referred to as “covariance”) is introduced that provides a geometric measure of the order of an interface. Covariance forms the basis for a fully general model of the energy of a planar interface, and it is demonstrated by comparison with a wide selection of molecular dynamics energy data for FCC and BCC tilt and twist boundaries that the model accurately reproduces the energy landscape using only three material parameters. It is observed that the planar constraint on the model is, in some cases, over-restrictive; this motivates an extension of the model.

In the second half of the work, the theory of faceting in interfaces is developed and applied to the planar interface model for grain boundaries. Building on previous work in mathematics and materials science, an algorithm is formulated that returns the minimal possible energy attainable by relaxation and the corresponding relaxed morphology for a given planar energy model. It is shown that the relaxation significantly improves the energy results of the planar covariance model for FCC and BCC tilt and twist boundaries. The ability of the model to accurately predict faceting patterns is demonstrated by comparison to molecular dynamics energy data and experimental morphological observation for asymmetric tilt grain boundaries. It is also demonstrated that by varying the temperature in the planar covariance model, it is possible to reproduce a priori the experimentally observed effects of temperature on facet formation.

Finally, the range and scope of the covariance and relaxation models, having been demonstrated by means of extensive MD and experimental comparison, future applications and implementations of the model are explored.

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(PHD, 2016)

Abstract: This work presents the development and investigation of a new type of concrete for the attenuation of waves induced by dynamic excitation. Recent progress in the field of metamaterials science has led to a range of novel composites which display unusual properties when interacting with electromagnetic, acoustic, and elastic waves. A new structural metamaterial with enhanced properties for dynamic loading applications is presented, which is named *metaconcrete*. In this new composite material the standard stone and gravel aggregates of regular concrete are replaced with spherical engineered inclusions. Each metaconcrete aggregate has a layered structure, consisting of a heavy core and a thin compliant outer coating. This structure allows for resonance at or near the eigenfrequencies of the inclusions, and the aggregates can be tuned so that resonant oscillations will be activated by particular frequencies of an applied dynamic loading. The activation of resonance within the aggregates causes the overall system to exhibit negative effective mass, which leads to attenuation of the applied wave motion. To investigate the behavior of metaconcrete slabs under a variety of different loading conditions a finite element slab model containing a periodic array of aggregates is utilized. The frequency dependent nature of metaconcrete is investigated by considering the transmission of wave energy through a slab, which indicates the presence of large attenuation bands near the resonant frequencies of the aggregates. Applying a blast wave loading to both an elastic slab and a slab model that incorporates the fracture characteristics of the mortar matrix reveals that a significant portion of the supplied energy can be absorbed by aggregates which are activated by the chosen blast wave profile. The transfer of energy from the mortar matrix to the metaconcrete aggregates leads to a significant reduction in the maximum longitudinal stress, greatly improving the ability of the material to resist damage induced by a propagating shock wave. The various analyses presented in this work provide the theoretical and numerical background necessary for the informed design and development of metaconcrete aggregates for dynamic loading applications, such as blast shielding, impact protection, and seismic mitigation.

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(PHD, 2016)

Abstract: In this thesis we study the growth of a Li electrode-electrolyte interface in the presence of an elastic prestress. In particular, we focus our interest on Li-air batteries with a solid electrolyte, LIPON, which is a new type of secondary or rechargeable battery. Theoretical studies and experimental evidence show that during the process of charging the battery the replated lithium adds unevenly to the electrode surface. This phenomenon eventually leads to dendrite formation as the battery is charged and discharged numerous times. In order to suppress or alleviate this deleterious effect of dendrite growth, we put forth a study based on a linear stability analysis. Taking into account all the mechanisms of mass transport and interfacial kinetics, we model the evolution of the interface. We find that, in the absence of stress, the stability of a planar interface depends on interfacial diffusion properties and interfacial energy. Specifically, if Herring-Mullins capillarity-driven interfacial diffusion is accounted for, interfaces are unstable against all perturbations of wavenumber larger than a critical value. We find that the effect of an elastic prestress is always to stabilize planar interfacial growth by increasing the critical wavenumber for instability. A parametric study results in quantifying the extent of the prestress stabilization in a manner that can potentially be used in the design of Li-air batteries. Moreover, employing the theory of finite differences we numerically solve the equation that describes the evolution of the surface profile and present visualization results of the surface evolution by time. Lastly, numerical simulations performed in a commercial finite element software validate the theoretical formulation of the interfacial elastic energy change with respect to the planar interface.

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(PHD, 2016)

Abstract: This thesis presents a topology optimization methodology for the systematic design of optimal multifunctional silicon anode structures in lithium-ion batteries. In order to develop next generation high performance lithium-ion batteries, key design challenges relating to the silicon anode structure must be addressed, namely the lithiation-induced mechanical degradation and the low intrinsic electrical conductivity of silicon. As such, this work considers two design objectives of minimum compliance under design dependent volume expansion, and maximum electrical conduction through the structure, both of which are subject to a constraint on material volume. Density-based topology optimization methods are employed in conjunction with regularization techniques, a continuation scheme, and mathematical programming methods. The objectives are first considered individually, during which the iteration history, mesh independence, and influence of prescribed volume fraction and minimum length scale are investigated. The methodology is subsequently extended to a bi-objective formulation to simultaneously address both the compliance and conduction design criteria. A weighting method is used to derive the Pareto fronts, which demonstrate a clear trade-off between the competing design objectives. Furthermore, a systematic parameter study is undertaken to determine the influence of the prescribed volume fraction and minimum length scale on the optimal combined topologies. The developments presented in this work provide a foundation for the informed design and development of silicon anode structures for high performance lithium-ion batteries.

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(PHD, 2015)

Abstract: Kohn-Sham density functional theory (KSDFT) is currently the main work-horse of quantum
mechanical calculations in physics, chemistry, and materials science. From a mechanical
engineering perspective, we are interested in studying the role of defects in the
mechanical properties in materials. In real materials, defects are typically found at
very small concentrations e.g., vacancies occur at parts per million,
dislocation density in metals ranges from 10^{10}*m*^{ − 2} to 10^{15}*m*^{ − 2},
and grain sizes vary from nanometers to micrometers in polycrystalline materials, etc. In order to model materials at
realistic defect concentrations using DFT, we would need
to work with system sizes beyond millions of atoms. Due to the cubic-scaling
computational cost with respect to the number of atoms in conventional DFT implementations, such system sizes are
unreachable. Since the early 1990s, there has been a huge interest in developing DFT
implementations that have linear-scaling computational cost. A promising
approach to achieving linear-scaling cost is to approximate the density matrix in
KSDFT. The focus of this
thesis is to provide a firm mathematical framework to study the convergence of
these approximations. We reformulate the Kohn-Sham density
functional theory as a nested variational problem in the density matrix,
the electrostatic potential, and a field dual to the electron density. The
corresponding functional is linear in the density matrix and thus amenable to
spectral representation. Based on this reformulation, we introduce a new
approximation scheme, called spectral binning, which does not require smoothing
of the occupancy function and thus applies at arbitrarily low temperatures. We
proof convergence of the approximate solutions with respect to spectral binning
and with respect to an additional spatial discretization of the domain. For a
standard one-dimensional benchmark problem, we present numerical experiments for
which spectral binning exhibits excellent convergence characteristics and
outperforms other linear-scaling methods.

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(PHD, 2015)

Abstract:

This thesis aims at a simple one-parameter macroscopic model of distributed damage and fracture of polymers that is amenable to a straightforward and efficient numerical implementation. The failure model is motivated by post-mortem fractographic observations of void nucleation, growth and coalescence in polyurea stretched to failure, and accounts for the specific fracture energy per unit area attendant to rupture of the material.

Furthermore, it is shown that the macroscopic model can be rigorously derived, in the sense of optimal scaling, from a micromechanical model of chain elasticity and failure regularized by means of fractional strain-gradient elasticity. Optimal scaling laws that supply a link between the single parameter of the macroscopic model, namely the critical energy-release rate of the material, and micromechanical parameters pertaining to the elasticity and strength of the polymer chains, and to the strain-gradient elasticity regularization, are derived. Based on optimal scaling laws, it is shown how the critical energy-release rate of specific materials can be determined from test data. In addition, the scope and fidelity of the model is demonstrated by means of an example of application, namely Taylor-impact experiments of polyurea rods. Hereby, optimal transportation meshfree approximation schemes using maximum-entropy interpolation functions are employed.

Finally, a different crazing model using full derivatives of the deformation gradient and a core cut-off is presented, along with a numerical non-local regularization model. The numerical model takes into account higher-order deformation gradients in a finite element framework. It is shown how the introduction of non-locality into the model stabilizes the effect of strain localization to small volumes in materials undergoing softening. From an investigation of craze formation in the limit of large deformations, convergence studies verifying scaling properties of both local- and non-local energy contributions are presented.

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(PHD, 2014)

Abstract: This work is concerned with the derivation of optimal scaling laws, in the sense of matching lower and upper bounds on the energy, for a solid undergoing ductile fracture. The specific problem considered concerns a material sample in the form of an infinite slab of finite thickness subjected to prescribed opening displacements on its two surfaces. The solid is assumed to obey deformation-theory of plasticity and, in order to further simplify the analysis, we assume isotropic rigid-plastic deformations with zero plastic spin. When hardening exponents are given values consistent with observation, the energy is found to exhibit sublinear growth. We regularize the energy through the addition of nonlocal energy terms of the strain-gradient plasticity type. This nonlocal regularization has the effect of introducing an intrinsic length scale into the energy. We also put forth a physical argument that identifies the intrinsic length and suggests a linear growth of the nonlocal energy. Under these assumptions, ductile fracture emerges as the net result of two competing effects: whereas the sublinear growth of the local energy promotes localization of deformation to failure planes, the nonlocal regularization stabilizes this process, thus resulting in an orderly progression towards failure and a well-defined specific fracture energy. The optimal scaling laws derived here show that ductile fracture results from localization of deformations to void sheets, and that it requires a well-defined energy per unit fracture area. In particular, fractal modes of fracture are ruled out under the assumptions of the analysis. The optimal scaling laws additionally show that ductile fracture is cohesive in nature, i.e., it obeys a well-defined relation between tractions and opening displacements. Finally, the scaling laws supply a link between micromechanical properties and macroscopic fracture properties. In particular, they reveal the relative roles that surface energy and microplasticity play as contributors to the specific fracture energy of the material. Next, we present an experimental assessment of the optimal scaling laws. We show that when the specific fracture energy is renormalized in a manner suggested by the optimal scaling laws, the data falls within the bounds predicted by the analysis and, moreover, they ostensibly collapse—with allowances made for experimental scatter—on a master curve dependent on the hardening exponent, but otherwise material independent.

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(PHD, 2013)

Abstract: Motivated by issues inherent in modeling and designing self-assembling systems (e.g. multiple collisions, collisions between non-smooth bodies, clumping and jamming behaviors, etc.), the goal of this thesis is to develop robust numerical tools that enable ecient and accurate direct simulation of self assembling systems and the application of optimal control methods to this type of system. The systems will be alternately modeled using linear nite elements, rigid bodies, or chains of rigid bodies. To this end, this work begins with development of a linear programming based collision detection algorithm for general convex polyhedral bodies. The resulting linear program has several features which render it extremely useful in determining the force system at the time of contact in numerical collision integrators. With robust collision detection in hand, three related numerical integration methods for dynamics with collisions are treated; a direct potential-based approach, and exact collision integrator in a discrete variational setting, and a decomposition-based algorithm, again in the discrete variational setting. Finally, several control problems are treated in the Discrete Mechanics and Optimal Control{Constrained (DMOCC) framework in which collisions between non-smooth bodies either need to be avoided or explicitly included in the optimal control problem. A globally stable feedback controller and a family of trajectories for spacecraft docking are also developed and tested with an accurate representation of an optimized CubeSat docking system.

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(PHD, 2011)

Abstract:

Defects, though present in relatively minute concentrations, play a significant role in determining macroscopic properties. Even vacancies, the simplest and most common type of defect, are fundamental to phenomena like creep, spall and radiation ageing. This necessitates an accurate characterization of defects at physically relevant concentrations, which is typically in parts per million. This represents a unique challenge since both the electronic structure of the defect core as well as the long range elastic field need to be resolved simultaneously. Unfortunately, accurate ab-initio electronic structure calculations are limited to a few hundred atoms, which is orders of magnitude smaller than that necessary for a complete description. Thus, defects represent a truly challenging multiscale problem.

Density functional theory developed by Hohenberg, Kohn and Sham (DFT) is a widely accepted, reliable ab-initio method for computing a wide range of material properties. We present a real-space, non-periodic, finite-element and max-ent formulation for DFT. We transform the original variational problem into a local saddle-point problem, and show its well-posedness by proving the existence of minimizers. Further, we prove the convergence of finite-element approximations including numerical quadratures. Based on domain decomposition, we develop parallel finite-element and max-ent implementations of this formulation capable of performing both all-electron and pseudopotential calculations. We assess the accuracy of the formulation through selected test cases and demonstrate good agreement with the literature.

Traditional implementations of DFT solve for the wavefunctions, a procedure which has cubic-scaling with respect to the number of atoms. This places serious limitations on the size of the system which can be studied. Further, they are not amenable to coarse-graining since the wavefunctions need to be orthonormal, a global constraint. To overcome this, we develop a linear-scaling method for DFT where the key idea is to directly evaluate the electron density without solving for the individual wavefunctions. Based on this linear-scaling method, we develop a numerical scheme to coarse-grain DFT derived solely based on approximation theory, without the introduction of any new equations and resultant spurious physics. This allows us to study defects at a fraction of the original computational cost, without any significant loss of accuracy. We demonstrate the efficiency and efficacy of the proposed methods through examples. This work enables the study of defects like vacancies, dislocations, interfaces and crack tips using DFT to be computationally viable.

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(PHD, 2011)

Abstract:

The overarching goal of this thesis is to develop new numerical time integration schemes for Lagrangian mechanics that better cope with the challenges of understanding the dynamic behavior of materials. We specifically address the formulation of convergent time integration schemes that exhibit good long-term behavior—such as conferred by symplecticity and exact conservation properties—and that have the ability to automatically and asynchronously modulate the time step in different regions of the domain. We achieve these properties in a progression of three developments: (i) energy-stepping, (ii) force-stepping, and (iii) asynchronous energy-stepping integrators. These developments are based on a new method of approximation for Lagrangian mechanics, proposed in this thesis, that consists of replacing the Lagrangian of the system by a sequence of approximate Lagrangians that can be solved exactly. Then, energy-stepping integrators result from replacing the potential energy by a piecewise constant approximation, force-stepping integrators result from replacing the potential energy by a piecewise affine approximation, and asynchronous energy-stepping integrators result from replacing localized potential energies by piecewise constant approximations. Throughout the dissertation, the properties of these time integrators are theoretically predicted and born out by a number of selected examples of application. Furthermore, we address the challenges of understanding the propagation of solitary waves in granular crystals at low impact velocity conditions by investigating the role of energy-trapping effects with the numerical time integration schemes developed in this work.

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(PHD, 2011)

Abstract:

Fiber composite material panels and sandwich panels possess both a high resistance to weight ratio and a high stiffness to weight ratio. Due to these features, fiber composite panels are used widely in aeronautic and marine structures, where the improvement of the structural performance while keeping a low weight is crucial. Sandwich structures, consisting of a foam core enclosed by two external layers of fiber reinforced material, seem to be promising in minimizing the total weight, maintaining structural rigidity and improving the resistance under exceptional loads, such as those due to explosions. Full scale experiments to test the performance of real fiber composite sandwich structures subjected to underwater explosions would be very complex and expensive. Therefore, the capability to numerically simulate the response of sandwich structures undergoing explosive loading will provide a powerful and unique tool to analyze and optimize their design by investigating the influence of different parameters. Obviously, small scale laboratory tests will still be essential to validate and calibrate the computational model before its use.

The present research focuses on the development of a computational scheme to model the behavior of large sandwich panels subjected to underwater explosions. The description of the sandwich requires the definition of the material behavior of the components, i.e., the foam core and the external sheets, of the structural behavior of the thin shell structure, and of the interaction with the surrounding fluid. Several finite kinematics material models taken from the recent literature have been used, and a new simple model for fiber reinforced composites has been developed and validated. The thin shell structure is modeled with an existing in-house built non-local shell finite element code (SFC), equipped with fracturing capabilities. The coupling between the behavior of the shells and the action of the fluid as a consequence of an underwater explosion is modeled here with the aid of an existing fluid-solid interaction (FSI) code. In this study, the FSI code has been expanded in order to include the possibility of simulating fiber composite materials. New algorithms and new control indicators, such as global measures of energy dissipation, have also been developed. The new capabilities of the fluid-solid coupled solver have been verified and validated before applying the solver to realistic problems. In the applications part of the present research, two different methods for applying the pressure load due to an underwater explosion are compared. The first method is simpler, and consists in applying a prescribed pressure profile without considering FSI. In the second method, the explosive charge is modeled as a spherical energy deposition and the full FSI is considered. The simpler method is used to assess the role of different design parameters of the face sheets on the overall response of sandwich panels when subjected to impulsive loads. Subsequently, the best sandwich design obtained from these initial simulations is used for the evaluation of the mechanical performance of the hull section of an existing Argentinean navy vessel. The final application of the proposed computational scheme is a parametric analysis of the hull section, considering different weights of the explosive charge and different distances of the explosion location from the hull wall.

Finally, with awareness of the limits of the adopted approach, several alternative schemes to improve the dynamical analysis of sandwich panels impulsively loaded are presented and discussed. In particular, two different kinds of shell finite elements are introduced. The proposed shell elements are based on alternative approximation schemes, which may model in a more realistic way the behavior of sandwich structures under extreme loads.

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(PHD, 2011)

Abstract:

Voids are observed to be generated under sufficient loading in many materials, ranging from polymers and metals to biological tissues. The presence of these voids can have drastic implications at the macroscopic level including strong material softening and more incipient fracture. Developing tools to appropriately account for these effects is therefore very desirable.

This thesis is concerned with both, the appearance of voids (nucleation process) and the modeling and simulation of materials in the presence of voids. A particular nucleation mechanism based on vacancy aggregation in high purity metallic single crystals is analyzed. A multiscale model is developed in order to obtain an approximate value of the time required for vacancies to form sufficiently large clusters for further growth by plastic deformation. It is based on quantum mechanical results, kinetic Monte Carlo methods and continuum mechanics estimates calibrated with quasi-continuum results. The ultimate goal of these simulations is to determine the feasibility of this nucleation mechanism under shock loading conditions, where the temperature and tensions are high and vacancy diffusion is promoted.

On the other hand, the effective behavior of materials with pre-existent voids is analyzed within the general framework of continuum mechanics and is therefore applicable to any material. The overall properties of the heterogeneous material are obtained through a two-level characterization: a representative volume element consisting of a hollow sphere is used to describe the “microscopic” fields, and an equivalent homogeneous material is used for the “macroscopic” behavior. A variational formulation of this two-scale model is presented. It provides a consistent definition of the macro-variables under general loading conditions, extending the well-known static averaging results so as to include microdynamic effects under finite deformations. This variational framework also provides a suitable starting point for time discretization and consistent definitions within discrete time. The spatial boundary value problem resulting from this multiscale model is solved with a particular spherical shell element specially developed for this problem. The approximation space is based on spherical harmonics, which respects the symmetries of the porous material and allows the representation of the fields on the sphere with very few degrees of freedom. Numerical tools, such as the exact representation of the boundary conditions and an exact quadrature rule, are also provided. The resulting numerical model is verified extensively, demonstrating good convergence results, and its applicability is shown through several material point calculations and a full two-scale finite element implementation.

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(PHD, 2011)

Abstract:

Materials with micrometer dimensions and their distinct mechanical properties have generated a great interest in the material science community over the last couple of decades. There is strong experimental evidence showing that microcrystalline materials are capable of achieving much higher yield and fracture strength values than bulk mesoscopic samples as they decrease in size. Several theories have been proposed to explain the size effect found in micromaterials, but a predictive physics-based model suitable for numerical simulations remains an open avenue of research. Since the successful design of micro-electro-mechanical systems (MEMS) and novel engineered materials hinges upon the mechanical properties at the micrometer scale, there is a compelling need for a quantitative and accurate characterization of the size effects exhibited by metallic micromaterials.

This work is concerned with the multiscale material modeling and simulation of strength in crystalline materials with micrometer dimensions. The elasto-viscoplastic response is modeled using a continuum crystal plasticity formulation suitable for large-deformation problems. Crystallographic dislocation motion is accounted for by stating the crystal kinematics within the framework of continuously distributed dislocation theory. The consideration of the dislocation self-energy and the step formation energy in the thermodynamic formulation of the constitutive relations renders the model non-local and introduces a length scale. Exploiting the concept of total variation we are able to recover an equivalent model that is local under a staggered approach, and therefore amenable to time integration using variational constitutive updates. Numerical simulations of compression tests in nickel micropillars using the proposed multiscale framework quantitatively capture the size dependence found in experimental results, showcasing the predictive capabilities of the model.

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(PHD, 2011)

Abstract:

In a number of areas of application, the behavior of systems depends sensitively on properties that pertain to the atomistic scale, i. e., the angstrom and femtosecond scales. However, generally the behaviors of interest are macroscopic and are characterized by slow evolution on the scale of meters and years. This broad disparity of length and time scales places extraordinary challenges in computational material science.

The overarching objective of this dissertation is to address the problem of multiple space and time scales in atomistic systems undergoing slow macroscopic evolution while retaining full atomistic detail. Our approach may be summarized as follows:

- The issue of accounting for finite temperature in coarse grained systems has not been solved entirely. For finite temperature systems at equilibrium, constructing an effective free energy in terms of a reduced set of atomic degrees of freedom is still an open area of research. In particular, the thermal vibrations of the missing degrees of freedom need to be accounted for. This is specially important if the aim of the simulation is to determine the dynamic properties of a system, or to allow the transmission of dynamic information between regions of different spatial discretization. To this end, we introduce a framework to simulate (spatially) coarse dynamic systems using the Quasicontinuum method (QC). The equations of motion are strictly derived from dissipative Lagrangian mechanics, which provides a classical Langevin implementation where the characteristic time is governed by the vibrations of the finest length scale in the computational cell. In order to assess the framework’s ability to transmit information across scales, we study the phonon impoverish spectra in coarse regions and the resulting underestimation of thermal equilibrium properties.
- Atomistic simulations have been employed for the past thirty years to determine structural and thermodynamic (equilibrium) properties of solids and their defects over a wide range of temperatures and pressures. The traditional Monte Carlo (MC) and Molecular Dynamics (MD) methods, while ideally suited to these calculations, require appreciable computational resources in order to calculate the long-time averages from which properties are obtained. In order to permit a reasonably quick, but accurate determination of the equilibrium properties of interest, we present an extension of the “maximum entropy” method to build effective alloy potentials while avoiding the treatment of all the system’s atomic degrees of freedom. We assess the validity of the model by testing its ability to reproduce experimental measurements.
- Based upon these effective potentials, we present a numerical framework capable of following the time evolution of atomistic systems over time windows currently beyond the scope of traditional atomistic methods such as Molecular Dynamics (MD) or Monte Carlo (MC). This is accomplished while retaining the underlying atomistic description of the material. We formulate a discrete variational setting in which the simulation of time-dependent phenomena is reduced to a sequence of incremental problems, each characterized by a variational principle. In this fashion we are able to study the interplay between deformation and diffusion using time steps or strain rates that are orders of magnitude larger or smaller than their MD|MC counterparts.
- We formulate a new class of “Replica Time Integrators” (RTIs) that allows for the two-way transmission of thermal phonons across mesh interfaces. This two-way transmission is accomplished by representing the state of the coarse region by a collection of identical copies or “replicas” of itself. Each replica runs at its own slow time step and is out-of-phase with respect to the others by one fast time step. Then, each replica is capable of absorbing from the fine region the elementary signal that is in phase with the replica. Conversely, each replica is capable of supporting –and transmitting to the fine region– an elementary signal of a certain phase. Since fine and coarse regions evolve asynchronously in time, RTIs permit both spatial and temporal coarse graining of the system of interest. Using a combination of phase-error analysis and numerical testing we find that RTIs are convergent, and allow step waves and thermal phonons to cross mesh interfaces in both directions losslessly.

- Based upon these effective potentials, we present a numerical framework capable of following the time evolution of atomistic systems over time windows currently beyond the scope of traditional atomistic methods such as Molecular Dynamics (MD) or Monte Carlo (MC). This is accomplished while retaining the underlying atomistic description of the material. We formulate a discrete variational setting in which the simulation of time-dependent phenomena is reduced to a sequence of incremental problems, each characterized by a variational principle. In this fashion we are able to study the interplay between deformation and diffusion using time steps or strain rates that are orders of magnitude larger or smaller than their MD|MC counterparts.

- Atomistic simulations have been employed for the past thirty years to determine structural and thermodynamic (equilibrium) properties of solids and their defects over a wide range of temperatures and pressures. The traditional Monte Carlo (MC) and Molecular Dynamics (MD) methods, while ideally suited to these calculations, require appreciable computational resources in order to calculate the long-time averages from which properties are obtained. In order to permit a reasonably quick, but accurate determination of the equilibrium properties of interest, we present an extension of the “maximum entropy” method to build effective alloy potentials while avoiding the treatment of all the system’s atomic degrees of freedom. We assess the validity of the model by testing its ability to reproduce experimental measurements.

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(PHD, 2009)

Abstract:

Stress corrosion cracking (SCC) is a very common failure mechanism characterized by a slow, environmentally induced crack propagation in structural components. Time-to-failure tests and crack-growth-rate tests are widespread practices for studying the response of various materials undergoing SCC. However, due to the large amount of factors affecting the phenomenon and the scattered data, they do not provide enough information for quantifying the effects of main SCC mechanisms. This thesis is concerned with the development of a novel 3-dimensional, multiphysics model for understanding the intergranular SCC of polycrystalline materials under the effect of impurity-enhanced decohesion. This new model is based upon: (i) a robust algorithm capable of generating the geometry of polycrystals for objects of arbitrary shape; (ii) a continuum finite element model of the crystals including crystal plasticity; (iii) a grain boundary diffusion model informed with first-principles computations of diffusion coefficients; and (iv) an intergranular cohesive model described by concentration-dependent constitutive relations also derived from first-principles. Results are validated and compared against crack-growth-rate and initiation time tests.

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(PHD, 2009)

Abstract:

A single crystal plasticity theory for insertion into finite element simulation is formulated using sequential laminates to model subgrain dislocation structures. It is known that local models do not adequately account for latent hardening, as latent hardening is not only a material property, but a nonlocal property (e.g., grain size and shape). The addition of the nonlocal energy from the formation of subgrain structure dislocation walls and the boundary layer misfits provide both latent and self hardening of crystal slip. Latent hardening occurs as the formation of new dislocation walls limit motion of new mobile dislocations, thus hardening future slip systems. Self hardening is accomplished by evolution of the subgrain structure length scale. No multiple slip hardening terms are included.

The substructure length scale is computed by minimizing the nonlocal energy. The minimization of the nonlocal energy is a competition between the dislocation wall and boundary layer energy. The nonlocal terms are also directly minimized within the subgrain model as they impact deformation response. The geometrical relationship between the dislocation walls and slip planes affecting dislocation mean free path is accounted for giving a first-order approximation to shape effects. A coplanar slip model is developed due to requirements when modeling the subgrain structure. This subgrain structure plasticity model is noteworthy as all material parameters are experimentally determined rather than fit. The model also has an inherit path dependency due to the formation of the subgrain structures. Validation is accomplished by comparison to single crystal tension test results.

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(PHD, 2009)

Abstract:

This thesis develops a novel meshfree numerical method for simulating general fluid flows. Drawing from concepts in optimal mass transport theory and in combination with the notion of material point sampling and meshfree interpolation, the optimal transport meshfree (OTM) method provides a rigorous mathematical framework for numerically simulating three-dimensional general fluid flows with general, and possibly moving boundaries (as in fluid-structure interaction simulations). Specifically, the proposed OTM method generalizes the Benamou-Brenier differential formulation of optimal mass transportation problems which leads to a multi-field variational characterization of general fluid flows including viscosity, equations of state and general geometries and boundary conditions. With the use of material point sampling in conjunction with local max-entropy shape functions, the OTM method leads to a meshfree formulation bearing a number of salient features. Compared with other meshfree methods that face significant challenges to enforce essential boundary conditions as well as couple to other methods, such as the finite element method, the OTM method provides a new paradigm in meshfree methods. The OTM method is numerically validated by simulating the classical Riemann benchmark example for Euler flow. Furthermore, in order to highlight the ability of the OTM to simulate Navier-Stokes flows within general, moving three-dimensional domains, and naturally couple with finite elements, an illustrative strongly coupled FSI example is simulated. This illustrative FSI example, consisting of a gas-inflated sphere impacting the ground, is simulated as a toy model of the final phase of NASA’s landing scheme devised for Mars missions, where a network of airbags are deployed to dissipate the energy of impact.

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(PHD, 2009)

Abstract:

This dissertation is concerned with the development of a robust and efficient meshless method, the Optimal Transportation Method (OTM), for general solid flows involving extremely large deformation, fast, transient loading and hydrodynamic phenomena. This method is a Lagrangian particle method through an integration of optimal transportation theory with meshless interpolation and material point integrations. The theoretical framework developed in this thesis generalized the Benamou-Brenier differential formulation of optimal transportation problems and leads to a multi-field variational characterization of solid flows, including elasticity, inelasticity, equation of state, and general geometries and boundary conditions. To this end, the accuracy, robustness and versatility of OTM is assessed and demonstrated with convergence and stability test, Taylor anvil test and a series of full three-dimensional simulations of high/hyper-velocity impact examples with the aid of a novel meshless dynamic contact algorithm presented in this thesis.

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(PHD, 2009)

Abstract: This work introduces a rigorous uncertainty quantification framework that exploits concentration–of–measure inequalities to bound failure probabilities using a well-defined certification campaign regarding the performance of engineering systems. The framework is constructed to be used as a tool for deciding whether a system is likely to perform safely and reliably within design specifications. Concentration-of-measure inequalities rigorously bound probabilities-of-failure and thus supply conservative certification criteria, in addition to supplying unambiguous quantitative definitions of terms such as margins, epistemic and aleatoric uncertainties, verification and validation measures, and confidence factors. This methodology unveils clear procedures for computing the latter quantities by means of concerted simulation and experimental campaigns. Extensions to the theory include hierarchical uncertainty quantification, and validation with experimentally uncontrollable random variables.

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(PHD, 2008)

Abstract:

Soft materials such as polymers and biological tissues have several engineering and biomechanical applications. These materials exhibit complex mechanical behavior, characterized by large strains, hysteresis, rate sensitivity, stress softening (Mullins effect), and deviatoric and volumetric plasticity. The need to accurately predict the behavior of such materials has been a tremendous challenge for scientists and engineers.

This thesis presents a seamless, fully variational constitutive model capable of capturing all of the above complex characteristics. Also, this work describes a fitting procedure based on the use of Genetic Algorithms, which proves to be necessary for the multi-modal, non-convex optimization required to identify fitting material parameters.

The capabilities of the presented model are demonstrated via several fits of experimental tests on a wide range of materials. These tests involve monotonic and cyclic loading of polyurea, high-density polyethylene, and brain tissue, and also involve cyclic hysteresis, softening, rate effects, shear, and cavitation plasticity.

Application to ballistic impact on a polyurea retrofitted DH36 steel plate is simulated and validated, utilizing the soft material model presented in this thesis for the polymer and a porous plasticity model for the metal. Localization elements are also included in this application to capture adiabatic shear bands. Moreover, computational capability for assessing the blast performance of metal/elastomer composite shells utilizing the soft material model for the elastomer is also presented.

Another implemented application is in the area of traumatic brain injuries under impact/acceleration loading. Clinically observed brain damage is reproduced utilizing the model presented in this work and a predictive capability of the distribution, intensity, and reversibility/irreversibility of brain tissue damage is demonstrated.

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(PHD, 2007)

Abstract:

This thesis presents a computational method for seamlessly bridging the atomistic and the continuum realms at finite temperature. The theoretical formulation is based on the static theory of the quasicontinuum and extends it to model non-equilibrium finite temperature material response.

At non-zero temperature, the problem of coarse-graining is compounded by the presence of multiple time scales in addition to multiple spatial scales. We address this problem by first averaging over the thermal motion of atoms to obtain an effective temperature-dependent energy on the macroscopic scale. Two methods are proposed to this end. The first method is developed as a variational mean field approximation which yields local thermodynamic potentials such as the internal energy, the free energy, and the entropy as phase averages of appropriate phase functions. The chief advantage of this theory is that it accounts for the anharmonicity of the interaction potentials, albeit numerically, unlike many methods based on statistical mechanics which require the quasi-harmonic approximation for computational feasibility. Furthermore, the theory reduces to the classical canonical ensemble approach of Gibbs under the quasi-harmonic approximation for perfect, isotropic, infinite crystals subjected to uniform temperature. In the second method, based on perturbation analysis, the internal energy is derived as an effective Hamiltonian of the atomistic system by treating the thermal fluctuations as perturbations about an equilibrium configuration.

These energy functionals are then introduced into the quasicontinuum theory, which facilitates spatial coarse-graining of the atomistic description. Finally, a variational formulation for simulating rate problems, such as heat conduction, using the quasicontinuum method is developed. This is achieved by constructing a joint incremental energy functional whose Euler-Lagrange equations yield the equilibrium equations as well as the time-discretized heat equation.

We conclude by presenting the results for numerical validation tests for the thermal expansion coefficient and the specific heat for some materials and compare them with classical theory, molecular dynamics results, and experimental data. Some illustrative examples of thermo-mechanical coupled problems such as heat conduction in a deformable solid, adiabatic tension test, and finite temperature nanoindentation are also presented which show qualitative agreement with expected behavior and demonstrate the applicability of the method.

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(PHD, 2007)

Abstract:

Electronic structure calculations, especially those using density-functional theory have provided many insights into various materials properties in the recent decade. However, the computational complexity associated with electronic structure calculations has restricted these investigations to periodic geometries with small cell-sizes (computational domains) consisting of few atoms (about 200 atoms). But material properties are influenced by defects—vacancies, dopants, dislocations, cracks, free surfaces—in small concentrations (parts per million). A complete description of such defects must include both the electronic structure of the core at the fine (sub-nanometer) scale and also elastic and electrostatic interactions at the coarse (micrometer and beyond) scale. This in turn requires electronic structure calculations at macroscopic scales, involving millions of atoms, well beyond the current capability. This thesis presents the development of a seamless multi-scale scheme, Quasi-Continuum Orbital-Free Density-Functional Theory (QC-OFDFT) to address this significant issue. This multi-scale scheme has enabled for the first time a calculation of the electronic structure of multi-million atom systems using orbital-free density-functional theory, thus, paving the way to an accurate electronic structure study of defects in materials.

The key ideas in the development of QC-OFDFT are (i) a real-space variational formulation of orbital-free density-functional theory, (ii) a nested finite-element discretization of the formulation, and (iii) a systematic means of adaptive coarse-graining retaining full resolution where necessary, and coarsening elsewhere with no patches, assumptions, or structure. The real-space formulation and the finite-element discretization gives freedom from periodicity, which is important in the study of defects in materials. More importantly, the real-space formulation and its finite-element discretization support unstructured coarse-graining of the basis functions, which is exploited to advantage in developing the QC-OFDFT method. This method has enabled for the first time a calculation of the electronic structure of samples with millions of atoms subjected to arbitrary boundary conditions. Importantly, the method is completely seamless, does not require any ad hoc assumptions, uses orbital-free density-functional theory as its only input, and enables convergence studies of its accuracy. From the viewpoint of mathematical analysis, the convergence of the finite-element approximation is established rigorously using Gamma-convergence, thus adding strength and validity to the formulation.

The accuracy of the proposed multi-scale method under modest computational cost, and the physical insights it offers into properties of materials with defects, have been demonstrated by the study of vacancies in aluminum. One of the important results of this study is the strong cell-size effect observed on the formation energies of vacancies, where cells as large as tens of thousands of atoms were required to obtain convergence. This indicates the prevalence of long-range physics in materials with defects, and the need to calculate the electronic structure of materials at macroscopic scales, thus underscoring the importance of QC-OFDFT.

Finally, QC-OFDFT was used to study a problem of great practical importance: the embrittlement of metals subjected to radiation. The brittle nature of metals exposed to radiation is associated with the formation of prismatic dislocation loops—dislocation loops whose Burgers vector has a component normal to their plane. QC-OFDFT provides an insight into the mechanism of prismatic dislocation loop nucleation, which has remained unclear to date. This study, for the first time using electronic structure calculations, establishes vacancy clustering as an energetically favorable process. Also, from direct numerical simulations, it is demonstrated that vacancy clusters collapse to form stable prismatic dislocation loops. This establishes vacancy clustering and collapse of these clusters as a possible mechanism for prismatic dislocation loop nucleation. The study also suggests that prismatic loops as small as those formed from a 7-vacancy cluster are stable, thus shedding new light on the nucleation size of these defects which was hitherto unknown.

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(PHD, 2006)

Abstract: This thesis is concerned with the exploration and development of a variational finite element mesh adaption framework for non-linear solid dynamics and its conceptual links with the theory of dynamic configurational forces. The distinctive attribute of this methodology is that the underlying variational principle of the problem under study is used to supply both the discretized fields and the mesh on which the discretization is supported. To this end a mixed-multifield version of Hamilton’s principle of stationary action and Lagrange-d’Alembert principle is proposed, a fresh perspective on the theory of dynamic configurational forces is presented, and a unifying variational formulation that generalizes the framework to systems with general dissipative behavior is developed. A mixed finite element formulation with independent spatial interpolations for deformations and velocities and a mixed variational integrator with independent time interpolations for the resulting nodal parameters is constructed. This discretization is supported on a continuously deforming mesh that is not prescribed at the outset but computed as part of the solution. The resulting space-time discretization satisfies exact discrete configurational force balance and exhibits excellent long term global energy stability behavior. The robustness of the mesh adaption framework is assessed and demonstrated with a set of examples and convergence tests.

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(PHD, 2005)

Abstract: The goal of this thesis is to develop a general framework for lattice statics analysis of defects in ferroelectric Perovskites. The techniques presented here are general and can be easily applied to other systems as well. We present all the calculations and numerical examples for two technologically important ferroelectric materials, namely, PbTiO3 and BaTiO3. We use shell potentials, that are derived using quantum mechanics calculations, and analyze three types of defects: (i) 180° and 90° domain walls, (ii) free surfaces and (iii) steps in 180° domain walls. Our formulation assumes that an interatomic potential is given. In other words, there is no need to have the force constants or restrict the number of nearest neighbor interactions a priori. Depending on the defect and symmetry, the discrete governing equations are reduced to those for representatives of some equivalence classes. The idea of symmetry reduction in lattice statics calculations is one of the contributions of this thesis. We call our formulation of lattice statics ‘inhomogeneous lattice statics’ as we consider the fact that close to defects force constants (stiffness matrices) change. For defects with one-dimensional symmetry reduction we solve the discrete governing equations directly using a novel method in the setting of the theory of difference equations. This will be compared with the solutions obtained using discrete Fourier transform. For defects with two-dimensional symmetry reduction we solve the discrete governing equations using discrete Fourier transform. We calculate the fully nonlinear solutions using modified Newton-Raphson iterations and call the method ‘inhomogeneous anharmonic lattice statics’. This work is aimed to fill the gap between quantum mechanics ab initio calculations and continuum models (based on Landau-Ginzberg-Devonshire theory) of ferroelectric domain walls.

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(PHD, 2005)

Abstract:

Complex lattice Quasicontinuum theory is developed and applied to the description of ferroelectric phenomena. Quasicontinuum theory is a multiscale theory that provides a unified description of materials by combining atomistic and continuum approaches. It provides a seamless transition between atomistics and continuum, but the description of the material is derived directly from the underlying atomic structure, using the computationally expensive atomistics only where needed, at the location of phenomena of atomistic origin.

Complex Lattice Quasicontinuum theory can be applied to complex lattice crystals consisting of many kinds of atoms. One highlight of it is treatment of each component lattice as separately and independently as possible. The component Quasicontinua are coupled through the microscopic forces within nodal clusters, making the complex atomistics of the heterogeneous lattice the basis of the description.

Ferroelectrics are especially suited to the application of Quasicontinuum theory. The nature of defects in ferroelectric materials is atomistic, but their influence over the material is long ranged due to induced elastic fields. Many different ferroelectric phenomena involving the perovskite ferroelectrics Barium Titanate and Lead Titanate are investigated and simulated. For Barium Titanate: the 180 degree domain wall structure and quasistatic crack under load. For Lead Titanate: the 180 degree domain wall structure and a domain wall step.

The results for the domain walls show that the domain wall thickness is atomistically small, of the order of few lattice constants, which is in agreement with recent ab initio molecular dynamics simulations, but we also observe long range effects resulting from the presence of the wall. During crack loading in the sample of Barium Titanate we observe polarization changes around the crack tip which are consistent with experimental observations of an increase of fracture toughness. The quasicontinuum study of a domain wall step gives an atomistical view into the equilibrium structure of the step.

Quasicontinuum is able to model these phenomena with atomistic precision around the defects and non-homogeneities, and also capture the influence of long-ranging effects in the samples. These studies could also give valuable modeling input for larger scale continuum approaches.

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(PHD, 2004)

Abstract:

This thesis is concerned with the formulation of a continuum theory of packaging of DNA in bacterial viruses based on a director-field representation of the encapsidated DNA. The point values of the director field give the local direction and density of the DNA. The continuity of the DNA strand requires that the director field be divergence-free and tangent to the capsid wall. The energy of the DNA is defined as a functional of the director field which accounts for bending, torsion, and for electrostatic interactions through a density-dependent interaction energy. The operative principle which determines the encapsidated DNA conformation is assumed to be energy minimization.

The director-field theory is used for the direct formulation and study of two low-energy DNA conformations: the inverse spool and torsionless toroidal solenoids. Analysis of the inverse spool configuration yields predictions of the interaxial spacing and the dependence of the packing force on the packed genome fraction which are found to be in agreement with experiments. Further analysis shows that torsionless toroidal solenoids can achieve lower energy than the inverse spool configuration.

Also, the theory is adapted to a framework of numerical optimization, wherein all fields are discretized on a computational lattice, and energy minimizing configurations are sought via simulated annealing and the nonlinear conjugate gradient method. It is shown that the inverse spool conformation is stable in all regions of the virus capsid except in a central core, where the DNA tends to buckle out of the spooling plane.

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(PHD, 2004)

Abstract:

A practical algorithm has been developed to construct, through sequential lamination, the partial relaxation of multiwell energy densities such as those characteristic of shape memory alloys. The resulting microstructures are in static and configurational equilibrium, and admit arbitrary deformations. The laminate topology evolves during deformation through branching and pruning operations, while a continuity constraint provides a simple model of metastability and hysteresis. In cases with strict separation of length scales, the method may be integrated into a finite element calculation at the subgrid level. This capability is demonstrated with a calculation of the indentation of a Cu-Al-Ni shape memory alloy by a spherical indenter.

In verification tests the algorithm attained the analytic solution in the computation of three benchmark problems. In the fourth case, the four-well problem (of, e.g., Tartar), results indicate that the method for microstructural evolution imposes an energy barrier for branching, hindering microstructural development in some cases. Although this effect is undesirable for purely mathematical problems, it is reflective of the activation energies and metastabilities present in applications involving natural processes.

The method was further used to model Shield’s tension test experiment, with initial calculations generating reasonable transformation strains and microstructures that compared well with the sequential laminates obtained experimentally.

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(PHD, 2004)

Abstract:

This thesis is concerned with variational principles for general coupled thermomechanical problems in dissipative materials including finite elastic and plastic deformation, non-Newtonian viscosity, rate sensitivity, arbitrary flow and hardening rule, as well as heat conduction. It is shown that there exists a potential function such that both the conservation of energy and balance of linear momentum are the Euler-Lagrange equations of its first variation. Inspired from the time-discretized version of the variational formulation, we present a procedure for variational thermomechanical update, which generalizes the isothermal approach under a variational thermodynamic framework. This variational formulation then serves as a basis for temperature change as well as constitutive updates.

An important application of the variational formulation is to optimize the shear band thickness in strain localization processes. We show that this optimization takes the form of a configurational-force equilibrium and results in a well-defined band thickness. We further implement displacement discontinuities into a class of strain-localization finite elements. These elements consist of two surfaces, attached to the abutting volume elements, which can separate and slip relative to each other, and thus enable the accurate and efficient simulation of the dynamical formation of stain localization.

The variational formulation also leads to a finite-deformation continuum modeling of bulk metallic glasses. It is shown that the strain softening of bulk metallic glasses is due to the increase of free volume (and thus the decrease of viscosity), while temperature rise accelerates the localization of the deformation. The model reproduces the constitutive behavior of Vitreloy 1 bulk metallic glass at various strain rates and temperatures.

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(PHD, 2003)

Abstract:

A phase-field theory of dislocations, strain hardening and hysteresis in ductile single crystals is developed. The theory accounts for an arbitrary number and arrangement of dislocation lines over a slip plane; the long-range elastic interactions between dislocation lines; the core structure of the dislocations; the interaction between the dislocations and an applied resolved shear stress field; and the irreversible interactions with short-range obstacles, resulting in hardening, path dependency and hysteresis.

We introduce a variational formulation for the statistical mechanics of dissipative systems. The influence of finite temperature as well as the mechanics in the phase-field theory are modeled with a Metropolis Monte Carlo algorithm and a mean field approximation.

A chief advantage of the present theory is that at zero temperature it is analytically tractable, in the sense that the complexity of the calculations may be reduced, with the aid of closed form analytical solutions, to the determination of the value of the phase field at point-obstacle sites. The theory predicts a range of behaviors which are in qualitative agreement with observation, including hardening and dislocation multiplication in single slip under monotonic loading; the Bauschinger effect under reverse loading; the fading memory effect; the evolution of the dislocation density under cycling loading; temperature softening; strain rate dependence; and others.

The model also reproduces the formation of dislocation networks observed in grain boundaries for different crystal structures and orientations. Simultaneously with the stable configurations the theory naturally predicts the equilibrium dislocation density independently of initial values or sources.

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(PHD, 2003)

Abstract: This thesis is concerned with the development of Variational Arbitrary Lagrangian-Eulerian method (VALE) method. VALE is essentially finite element method generalized to account for horizontal variations, in particular, variations in nodal coordinates. The distinguishing characteristic of the method is that the variational principle simultaneously supplies the solution, the optimal mesh and, in case problems of shape optimization, optimal shape. This is accomplished by rendering the functional associated with the variational principle stationary with respect to nodal field values as well as with respect to the nodal positions of triangulation of the domain of analysis. Stationarity with respect to the nodal positions has the effect of the equilibriating the energetic or configurational forces acting in the nodes. Further, configurational force equilibrium provides precise criterion for mesh optimality. The solution so obtained corresponds to minimum of energy functional (minimum principle) in static case and to the stationarity of action sum (discrete Hamilton’s stationarity principle) in dynamic case, with respect to both nodal variables and nodal positions. Further, the resulting mesh adaption scheme is devoid of error estimates and mesh-to-mesh transfer interpolation errors. We illustrate the versatility and convergence characteristics of the method by way of selected numerical tests and applications, including the problem of semi-infinite crack, the shape optimization of elastic inclusions and free vibration of 1-d rod.

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(PHD, 2003)

Abstract:

This thesis develops the theory and implementation of variational integrators for computational solid mechanics problems, and to some extent, for fluid mechanics problems as well. Variational integrators for finite dimensional mechanical systems are succinctly reviewed, and used as the foundations for the extension to continuum systems. The latter is accomplished by way of a space-time formulation for Lagrangian continuum mechanics that unifies the derivation of the balance of linear momentum, energy and configurational forces, all of them as Euler-Lagrange equations of an extended Hamilton’s principle. In this formulation, energy conservation and the path independence of the J- and L-integrals are conserved quantities emanating from Noether’s theorem. Variational integrators for continuum mechanics are constructed by mimicking this variational structure, and a discrete Noether’s theorem for rather general space-time discretizations is presented. Additionally, the algorithms are automatically (multi)symplectic, and the (multi)symplectic form is uniquely defined by the theory. For instance, in nonlinear elastodynamics the algorithms exactly preserve linear and angular momenta, whenever the continuous system does.

A class of variational algorithms is constructed, termed asynchronous variational integrators (AVI), which permit the selection of independent time steps in each element of a finite element mesh, and the local time steps need not bear an integral relation to each other. The conservation properties of both synchronous and asynchronous variational integrators are discussed in detail. In particular, AVI are found to nearly conserve energy both locally and globally, a distinguishing feature of variational integrators. The possibility of adapting the elemental time step to exactly satisfy the local energy balance equation, obtained from the extended variational principle, is analyzed. The AVI are also extended to include dissipative systems. The excellent accuracy, conservation and convergence characteristics of AVI are demonstrated via selected numerical examples, both for conservative and dissipative systems. In these tests AVI are found to result in substantial speedups, at equal accuracy, relative to explicit Newmark.

In elastostatics, the variational structure leads to the formulation of discrete path-independent integrals and a characterization of the configurational forces acting in discrete systems. A notable example is a discrete, path-independent J-integral at the tip of a crack in a finite element mesh.

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(PHD, 2002)

Abstract:

The aim of this dissertation was to develop models of fatigue crack growth and stress-corrosion cracking by investigating cohesive theories of fracture. These models were integrated in a finite-element framework embedding a contact algorithm and techniques of remeshing and adaptive meshing. For the fatigue model, we developed a phenomenological cohesive law which exhibits unloading-reloading hysteresis. This model qualitatively predicts fatigue crack growth rates in metals under constant amplitude regime for short and long cracks, as well as growth retardation due to overload. Quantitative predictions were obtained in the case of long cracks. We developed a chemistry-dependent cohesive law which serves as a basis for the stress-corrosion cracking model. In order to determine this cohesive law, two approaches, based on energy relaxation and the renormalization group, were used for coarse-graining interplanar potentials. We analyzed the cohesive behavior of a large–but finite–number of interatomic planes and found that the macroscopic cohesive law adopts a universal asymptotic form. The resulting stress-corrosion crack growth rates agreed well with those observed experimentally in ‘static’ fatigue tests given in the literature.

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(PHD, 2001)

Abstract:

This dissertation is concerned with the development of a robust three-dimensional finite-element framework for the simulation of complex problems in mechanics and physics of solids. This approach is intended to shine light on impact and erosion mechanisms among other multiscale, multiphysics problems. The components of the computational framework are a contact algorithm including friction, wear, finite deformation plasticity, heat generation, heat transfer, and adaptive meshing coupled with error estimation. The adaptive meshing is a key development that enhances the efficiency and robustness of the method. We demonstrate the ability of the methodology to simulate diverse problems such as shear banding, impact, and wear.

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(PHD, 2001)

Abstract:

The objective of this work is to establish the applicability of cohesive theories of fracture in situations involving material interface, material heterogeneity (e.g., layered composites), material anisotropy(e.g., fiber-reinforced composites), shear cracks, intersonic dynamic crack growth and dynamic crack branching. The widely used cohesive model is extended to orthotropic range. The so-developed computational tool, completed by a self-adaptive fracture procedure and a frictional contact algorithm, is capable of following the evolution of three-dimensional damage processes, modeling the progressive decohesion of interfaces and anisotropic materials. The material parameters required by cohesive laws are directly obtained from static experiments. The ability of the methodology to simulate diverse problems such as delamination between fibers of graphite/epoxy composites, as well as sandwich structures and branching within brittle bulk materials has been demonstrated.

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(Engineer, 2000)

Abstract: A micromechanically based constitutive model for the dynamic inelastic behavior of brittle materials, specifically “Dionysus-Pentelicon marble” with distributed microcracking is presented. Dionysus-Pentelicon marble was used in the construction of the Parthenon, in Athens, Greece. The constitutive model is a key component in the ability to simulate this historic explosion and the preceding bombardment form cannon fire that occurred at the Parthenon in 1678. Experiments were performed by Rosakis (1999) that characterized the static and dynamic response of this unique material. A micromechanical constitutive model that was previously successfully used to model the dynamic response of granular brittle materials is presented. The constitutive model was fitted to the experimental data for marble and reproduced the experimentally observed basic uniaxial dynamic behavior quite well. This micromechanical constitutive model was then implemented into the three dimensional nonlinear lagrangain finite element code Dyna3d(1998). Implementing this methodology into the three dimensional nonlinear dynamic finite element code allowed the model to be exercised on several preliminary impact experiments. During future simulations, the model is to be used in conjunction with other numerical techniques to simulate projectile impact and blast loading on the Dionysus-Pentelicon marble and on the structure of the Parthenon.

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(PHD, 1998)

Abstract:

This dissertation is concerned with the development of a general computational framework for mesh adaption such as is required in the three-dimensional lagrangian finite element simulation of strongly nonlinear, possibly dynamic, problems. It is shown that, for a very general constitutive framework, the solutions of the incremental boundary value problem obey a minimum principle, provided that the constitutive updates are formulated appropriately. This minimum principle is taken as a basis for asymptotic error estimation. In particular, we chose to monitor the error of a lower-order projection of the finite element solution. The optimal mesh size distribution then follows from a posteriori error indicators which are purely local, i. e., can be computed element-by-element.

A sine qua non condition for the successful accomplishment of the kind of analysis envisioned in this work is the possibility to mesh the deforming domains of analysis. In the first section of this thesis a method is presented for mesh generation in complex geometries and general–possibly non-manifold–topologies.

The robustness and versatility of the computational framework is demonstrated with the aid of convergence studies and selected examples of application and the results contrasted with previous approaches
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(PHD, 1998)

Abstract: A micromechanically based model for fatigue crack nucleation in ductile F.C.C. metals is developed. The theory includes a model of F.C.C. crystal-plasticity in finite deformations that takes into account the Bauschinger effect, dipole annihilation in the persistent slip bands (PSBs), with vacancy generation and PSB elongation as a byproduct, as well as coupled vacancy diffusion and the attendant surface motion due to the flux of vacancies out of the body. Finite element simulations are performed in order to establish the predictive capability of the theory. Detailed modelling of the intersection of the PSB with a free surface, enhanced by the use of remeshing and surface evolution techniques, enable the prediction of nucleation sites, life expectancy, surface profile, alternate slip between the sides of the PSB and strain localization at the grooves. In an attempt to resolve the dislocation structures experimentally observed during cyclic loading, a theory based on the non-convexity of a pseudo-energy density is developed. Non-homogeneous minimizers are found containing variants oriented in coincidence with the dislocation walls observed experimentally. Due to the latent hardening and geometrical softening, the minimizing structures are found to consist of regions of single slip which is in accordance with the observed “patchy slip.”

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