The variational approach to brittle fracture in materials with anisotropic surface energy and in thin sheets

Fracture mechanics of brittle materials has focused on bulk materials with isotropic surface energy. In this situation different physical principles for crack path selection are very similar or even equivalent. The situation is radically different when considering crack propagation in brittle materials with anisotropic surface energy. Such materials are important in applications involving single crystals, extruded polymers, or geological and organic materials. When this anisotropy is strong, the phenomenology of crack propagation becomes very rich, with forbidden crack propagation directions or complex sawtooth crack patterns. Thus, this situation interrogates fundamental issues in fracture mechanics, including the principles behind the selection of crack direction. Similarly, tearing of brittle thin elastic sheets, ubiquitous in nature, technology and daily life, challenges our understanding of fracture. Since tearing typically involves large geometric nonlinearity, it is not clear whether the stress intensity factors are meaningful or if and how they determine crack propagation. Geometry, together with the interplay between stretching and bending deformation, leads to complex behaviors, restricting analytical approximate solutions to very simplified settings and specific parameter regimes. In both situations, a rich and nontrivial experimental record has been successfully understood in terms of simple energetic models. However, general modeling approaches to either fracture in the presence of strong surface energy anisotropy or to tearing, capable of exploring new physics, have been lacking. The success of energetic simple models suggests that variational theories of brittle fracture may provide a unifying and general framework capable of dealing with the more general situations considered here. To address fracture in materials with strongly anisotropic surface energy, we propose a variational phase-field model resorting to the extended Cahn-Hilliard framework proposed in the context of crystal growth. Previous phase-field models for anisotropic fracture were formulated in a framework only allowing for weak anisotropy. We implement numerically our higher-order phase-field model with smooth local maximum entropy approximants in a direct Galerkin method. The numerical results exhibit all the features of strongly anisotropic fracture, and reproduce strikingly well recent experimental observations. To explore tearing of thin films, we develop a geometrically exact model and a computational framework coupling elasticity (stretching and bending), fracture, and adhesion to a substrate. We numerically implement the model with subdivision surface finite elements. Our simulations qualitatively and quantitatively reproduced the crack patterns observed in tearing experiments. Finally, we examine how shell geometry affects fracture. As suggested by previous results and our own phase-field simulations, shell shape dramatically affects crack evolution and the effective toughness of the shell structure. To gain insight and eventually develop new concepts for optimizing the design of thin shell structures, we derive the configurational force conjugate to crack extension for Koiter's linear thin shell theory. We identify the conservative contribution to this force through an Eshelby tensor, as well as non-conservative contributions arising from curvature.La mécanica de fractura frágil se ha centrado en materiales tridimensionales con una energía de superficie isotrópica. En esta situación, los diferentes principios para la selección del camino de la fisura son muy similares, o incluso equivalentes. La situación es radicalmente opuesta cuando se considera la propagación de fisuras en medios con energía de superficie anisótropa. Estos materiales son importantes en aplicaciones que involucran materiales cristalinos, polímeros extrudidos, o materiales orgánicos y geológicos. Cuando la anisotropía es fuerte, el fenómeno de la propagación de fisuras es muy rico, con direcciones de propagación prohibidas o complejos patrones de ruptura en dientes de sierra. Por tanto, esta situación plantea cuestiones fundamentales en la mecánica de la fractura, incluyendo los principios de selección de la dirección de propagación de la fractura. Igualmente, el proceso de rasgado de láminas delgadas y frágiles, comunes en la naturaleza, la tecnología y la vida diaria, desafía nuestro entendimiento de la fractura. Dado que el rasgado de estas láminas típicamente involucra grandes no linealidades geométricas, no está claro si los factores de intensidad de esfuerzos son válidos o si, y en tal caso cómo determinan la propagación de fisuras. La interacción entre la geometría, las deformaciones y la curvatura da lugar a comportamientos complejos, lo que restringe las soluciones analíticas aproximadas a ejemplos muy simplificados y a regímenes de parámetros limitados. En ambas situaciones, se han podido interpretar experimentos no triviales con modelos energéticos simples. Sin embargo, no se ha profundizado en modelos generales de fractura en presencia de energía de superficie fuertemente anisótropa o en láminas delgadas, ambas interesantes por su capacidad para explorar nueva física. El mencionado éxito de los modelos energéticos simplificados sugiere que las teorías variacionales de fractura en medios frágiles pueden proveer un marco unificador para considerar situaciones más generales, como las que se consideran en este trabajo. Para caracterizar la fractura en materiales con energía de superficie fuertemente anisótropa, proponemos un modelo variacional de campo de fase basado en el modelo extendido de Cahn-Hilliard. Los modelos de campo de fase existentes para la fractura anisótropa fueron formulados en un contexto que sólo admite anisotropía débil. En este trabajo, implementamos numéricamente nuestro modelo de campo de fase de alto orden con aproximantes locales de máxima entropía en un método directo de Garlerkin. Los resultados numéricos muestran todas las características de fractura con anisotropía fuerte, y reproducen llamativamente bien las últimas observaciones experimentales. Para explorar el rasgado de láminas delgadas, desarrollamos un modelo geométricamente exacto y un esquema computacional que acopla elasticidad (estiramiento y flexión), fractura, y la adhesión a un substrato. Implementamos numéricamente el modelo con elementos finitos basados en superficies de subdivisión. Nuestras simulaciones reproducen los patrones de ruptura, tanto cualitativamente como cuantitativamente, observados en los experimentos de rasgado. Finalmente, examinamos cómo la geometría de la lámina afecta la fractura. Como ha sido sugerido en resultados previos y en nuestras propias simulaciones de campo de fase, la forma de la lámina afecta dramáticamente la evolución de fisuras y la resistencia efectiva del material. Para comprender mejor estos fenómenos y con el objetivo de desarrollar nuevos conceptos para la optimización del diseño de estructuras de láminas delgadas, derivamos la fuerza configuracional conjugada a la extensión de la fractura para la teoría lineal de láminas delgadas de Koiter. Identificamos las contribuciones conservativas a esta fuerza a través del tensor de Eshelby, así como las contriuciones no conservativas que aparecen por el efecto de la curvatura

Point-set manifold processing for computational mechanics: thin shells, reduced order modeling, cell motility and molecular conformations

In many applications, one would like to perform calculations on smooth manifolds of dimension d embedded in a high-dimensional space of dimension D. Often, a continuous description of such manifold is not known, and instead it is sampled by a set of scattered points in high dimensions. This poses a serious challenge. In this thesis, we approximate the point-set manifold as an overlapping set of smooth parametric descriptions, whose geometric structure is revealed by statistical learning methods, and then parametrized by meshfree methods. This approach avoids any global parameterization, and hence is applicable to manifolds of any genus and complex geometry. It combines four ingredients: (1) partitioning of the point set into subregions of trivial topology, (2) the automatic detection of the local geometric structure of the manifold by nonlinear dimensionality reduction techniques, (3) the local parameterization of the manifold using smooth meshfree (here local maximum-entropy) approximants, and (4) patching together the local representations by means of a partition of unity. In this thesis we show the generality, flexibility, and accuracy of the method in four different problems. First, we exercise it in the context of Kirchhoff-Love thin shells, (d=2, D=3). We test our methodology against classical linear and non linear benchmarks in thin-shell analysis, and highlight its ability to handle point-set surfaces of complex topology and geometry. We then tackle problems of much higher dimensionality. We perform reduced order modeling in the context of finite deformation elastodynamics, considering a nonlinear reduced configuration space, in contrast with classical linear approaches based on Principal Component Analysis (d=2, D=10000's). We further quantitatively unveil the geometric structure of the motility strategy of a family of micro-organisms called Euglenids from experimental videos (d=1, D~30000's). Finally, in the context of enhanced sampling in molecular dynamics, we automatically construct collective variables for the molecular conformational dynamics (d=1...6, D~30,1000's)

High-order adaptive methods for computing invariant manifolds of maps

The author presents efficient and accurate numerical methods for computing invariant manifolds of maps which arise in the study of dynamical systems. In order to decrease the number of points needed to compute a given curve/surface, he proposes using higher-order interpolation/approximation techniques from geometric modeling. He uses B´ezier curves/triangles, fundamental objects in curve/surface design, to create adaptive methods. The methods are based on tolerance conditions derived from properties of B´ezier curves/triangles. The author develops and tests the methods for an ordinary parametric curve; then he adapts these methods to invariant manifolds of planar maps. Next, he develops and tests the method for parametric surfaces and then he adapts this method to invariant manifolds of three-dimensional maps

New geometric techniques for linear programming and graph partitioning

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.Includes bibliographical references (leaves 79-82).In this thesis, we advance a collection of new geometric techniques for the analysis of combinatorial algorithms. Using these techniques, we resolve several longstanding questions in the theory of linear programming, polytope theory, spectral graph theory, and graph partitioning. The thesis consists of two main parts. In the first part, which is joint work with Daniel Spielman, we present the first randomized polynomial-time simplex algorithm for linear programming, answering a question that has been open for over fifty years. Like the other known polynomial-time algorithms for linear programming, its running time depends polynomially on the number of bits used to represent its input. To do this, we begin by reducing the input linear program to a special form in which we merely need to certify boundedness of the linear program. As boundedness does not depend upon the right-hand-side vector, we run a modified version of the shadow-vertex simplex method in which we start with a random right-hand-side vector and then modify this vector during the course of the algorithm. This allows us to avoid bounding the diameter of the original polytope.(cont.) Our analysis rests on a geometric statement of independent interest: given a polytope ... in isotropic position, if one makes a polynomially small perturbation to b then the number of edges of the projection of the perturbed polytope onto a random 2-dimensional subspace is expected to be polynomial. In the second part of the thesis, we address two long-open questions about finding good separators in graphs of bounded genus and degree: 1. It is a classical result of Gilbert, Hutchinson, and Tarjan [25] that one can find asymptotically optimal separators on these graphs if he is given both the graph and an embedding of it onto a low genus surface. Does there exist a simple, efficient algorithm to find these separators given only the graph and not the embedding? 2. In practice, spectral partitioning heuristics work extremely well on these graphs. Is there a theoretical reason why this should be the case? We resolve these two questions by showing that a simple spectral algorithm finds separators of cut ratio O( g/n) and vertex bisectors of size O(Vng) in these graphs, both of which are optimal. As our main technical lemma, we prove an O(g/n) bound on the second smallest eigenvalue of the Laplacian of such graphs and show that this is tight, thereby resolving a conjecture of Spielman and Teng.(cont.) While this lemma is essentially combinatorial in nature, its proof comes from continuous mathematics, drawing on the theory of circle packings and the geometry of compact Riemann surfaces. While the questions addressed in the two parts of the thesis are quite different, we show that a common methodology runs through their solutions. We believe that this methodology provides a powerful approach to the analysis of algorithms that will prove useful in a variety of broader contexts.by Jonathan A. Kelner.Ph.D

Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere

We present two exact implementations of efficient output-sensitive algorithms that compute Minkowski sums of two convex polyhedra in 3D. We do not assume general position. Namely, we handle degenerate input, and produce exact results. We provide a tight bound on the exact maximum complexity of Minkowski sums of polytopes in 3D in terms of the number of facets of the summand polytopes. The algorithms employ variants of a data structure that represents arrangements embedded on two-dimensional parametric surfaces in 3D, and they make use of many operations applied to arrangements in these representations. We have developed software components that support the arrangement data-structure variants and the operations applied to them. These software components are generic, as they can be instantiated with any number type. However, our algorithms require only (exact) rational arithmetic. These software components together with exact rational-arithmetic enable a robust, efficient, and elegant implementation of the Minkowski-sum constructions and the related applications. These software components are provided through a package of the Computational Geometry Algorithm Library (CGAL) called Arrangement_on_surface_2. We also present exact implementations of other applications that exploit arrangements of arcs of great circles embedded on the sphere. We use them as basic blocks in an exact implementation of an efficient algorithm that partitions an assembly of polyhedra in 3D with two hands using infinite translations. This application distinctly shows the importance of exact computation, as imprecise computation might result with dismissal of valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages long. The advisor was Prof. Dan Halperi