arXiv subjects

Raz Kupferman

Publications and source records attributed to Raz Kupferman.

At least 37 records · Page 2Linked to original sources

arXiv2017

On the Bending Energy of Buckled Edge-Dislocations

The study of elastic membranes carrying topological defects has a longstanding history, going back at least to the 1950s. When allowed to buckle in three-dimensional space, membranes with defects can totally relieve their in-plane strain, remaining with a bending energy, whose rigidity modulus is small compared to the stretching modulus. In this paper, we study membranes with a single edge-dislocation. We prove that the minimum bending energy associated with strain-free configurations diverges logarithmically with the size of the system.

cond-mat.soft↗

arXiv2017

Stress theory for classical fields

Classical field theories together with the Lagrangian and Eulerian approaches to continuum mechanics are embraced under a geometric setting of a fiber bundle. The base manifold can be either the body manifold of continuum mechanics, space manifold, or space-time. Differentiable sections of the fiber bundle represent configurations of the system and the configuration space containing them is given the structure of an infinite dimensional manifold. Elements of the cotangent bundle of the configuration space are interpreted as generalized forces and a representation theorem implies that there exist a stress object representing forces, non-uniquely. The properties of stresses are studies as well as the role of constitutive relations in the present general setting.

math-ph↗

arXiv2017

Non-metricity in the continuum limit of randomly-distributed point defects

We present a homogenization theorem for isotropically-distributed point defects, by considering a sequence of manifolds with increasingly dense point defects. The loci of the defects are chosen randomly according to a weighted Poisson point process, making it a continuous version of the first passage percolation model. We show that the sequence of manifolds converges to a smooth Riemannian manifold, while the Levi-Civita connections converge to a non-metric connection on the limit manifold. Thus, we obtain rigorously the emergence of a non-metricity tensor, which was postulated in the literature to represent continuous distribution of point defects.

math.PR↗

arXiv2016

Continuum Dynamics on Manifolds: Application to Elasticity of Residually-Stressed Bodies

This paper is concerned with the dynamics of continua on differentiable manifolds. We present a covariant derivation of equations of motion, viewing motion as a curve in an infinite-dimensional Banach space of embeddings of a body manifold in a space manifold. Our main application is the motion of residually-stressed elastic bodies; residual stress results from a geometric incompatibility between body and space manifolds. We then study a particular example of elastic vibrations of a two- dimensional curved annulus embedded in a sphere.

math-ph↗

arXiv2016

On strain measures and the geodesic distance to $\text{SO}_n$ in the general linear group

We consider various notions of strains; quantitative measures for the deviation of a linear transformation from an isometry. The main approach, which is motivated by physical applications and follows the work of Patrizio Neff and co-workers , is to select a Riemannian metric on $\text{GL}_n$, and use its induced geodesic distance to measure the distance of a linear transformation from the set of isometries. We give a short geometric derivation of the formula for the strain measure for the case where the metric is left-$\text{GL}_n$-invariant and right-$\text{O}_n$-invariant. We proceed to investigate alternative distance functions on $\text{GL}_n$, and the properties of their induced strain measures. We start by analyzing Euclidean distances, both intrinsic and extrinsic. Next, we prove that there are no bi-invariant distances on $\text{GL}_n$. Lastly, we investigate strain measures induced by inverse-invariant distances.

math.DG↗

arXiv2016

Limits of elastic models of converging Riemannian manifolds

In non-linear incompatible elasticity, the configurations are maps from a non-Euclidean body manifold into the ambient Euclidean space, $\mathbb{R}^k$. We prove the $Γ$-convergence of elastic energies for configurations of a converging sequence, $\mathcal{M}_n\to\mathcal{M}$, of body manifolds. This convergence result has several implications: (i) It can be viewed as a general structural stability property of the elastic model. (ii) It applies to certain classes of bodies with defects, and in particular, to the limit of bodies with increasingly dense edge-dislocations. (iii) It applies to approximation of elastic bodies by piecewise-affine manifolds. In the context of continuously-distributed dislocations, it reveals that the torsion field, which has been used traditionally to quantify the density of dislocations, is immaterial in the limiting elastic model.

math.AP↗

arXiv2015

Elastic interactions between 2D geometric defects

In this paper, we introduce a methodology applicable to a wide range of localized two-dimensional sources of stress. This methodology is based on a geometric formulation of elasticity. Localized sources of stress are viewed as singular defects---point charges of the curvature associated with a reference metric. The stress field in the presence of defects can be solved using a scalar stress function that generalizes the classical Airy stress function to the case of materials with nontrivial geometry. This approach allows the calculation of interaction energies between various types of defects. We apply our methodology to two physical systems: shear-induced failure of amorphous materials and the mechanical interaction between contracting cells.

cond-mat.soft↗

arXiv2015

The plane stress state of residually stressed bodies: a stress function approach

The stressed state of flattened thin elastic sheet, as well as that of translationally symmetric 3D solids, are effectively 2D problems. This paper study equilibrium state-of-stress in metrically-incompatible 2D elastic materials. The solution is represented by a scalar stress function, generalizing the Airy stress function, which is determined by geometric compatibility conditions. We develop a perturbative approximation method for solving this stress function, valid for any constitutive relation. We apply the method for the case of a Hookean solid to solve prototypical examples in which the classical Airy approach is either inaccurate or inapplicable. Results are shown to agree well with numerical results obtained in previous works.

cond-mat.soft↗

arXiv2015

Riemannian surfaces with torsion as homogenization limits of locally-Euclidean surfaces with dislocation-type singularities

We reconcile between two classical models of edge-dislocations in solids. The first model, dating from the early 1900s models isolated edge-dislocations as line singularities in locally-Euclidean manifolds. The second model, dating from the 1950s, models continuously-distributed edge-dislocations as smooth manifolds endowed with non-symmetric affine connections (equivalently, endowed with torsion fields). In both models, the solid is modeled as a Weitzenböck manifold. We prove, using a weak notion of convergence [KM15], that the second model can be obtained rigorously as a homogenization limit of the first model, as the density of singular edge-dislocation tends to infinity.

math.DG↗

arXiv2014

A Riemannian approach to the membrane limit in non-Euclidean elasticity

Non-Euclidean, or incompatible elasticity is an elastic theory for pre-stressed materials, which is based on a modeling of the elastic body as a Riemannian manifold. In this paper we derive a dimensionally-reduced model of the so-called membrane limit of a thin incompatible body. By generalizing classical dimension reduction techniques to the Riemannian setting, we are able to prove a general theorem that applies to an elastic body of arbitrary dimension, arbitrary slender dimension, and arbitrary metric. The limiting model implies the minimization of an integral functional defined over immersions of a limiting submanifold in Euclidean space. The limiting energy only depends on the first derivative of the immersion, and for frame-indifferent models, only on the resulting pullback metric induced on the submanifold, i.e., there are no bending contributions.

math.AP↗

arXiv2014

Two-dimensional defects in amorphous materials

We present a new definition of defects which is based on a Riemannian formulation of incompatible elasticity. Defects are viewed as local deviations of the material's reference metric field, $\bar{\mathfrak{g}}$, from a Euclidian metric. This definition allows the description of defects in amorphous materials and the formulation of the elastic problem, using a single field, $\bar{\mathfrak{g}}$. We provide a multipole expansion of reference metrics that represent a large family of two-dimensional (2D) localized defects. The case of a dipole, which corresponds to an edge dislocation is studied analytically, experimentally and numerically. The quadrupole term, which is studied analytically, as well as higher multipoles of curvature carry local deformations. These multipoles are good candidates for fundamental strain carrying entities in plasticity theories of amorphous materials and for a continuous modeling of recently developed meta-materials.

cond-mat.soft↗

arXiv2013

Metric Description of Defects in Amorphous Materials

Classical elasticity is concerned with bodies that can be modeled as smooth manifolds endowed with a reference metric that represents local equilibrium distances between neighboring material elements. The elastic energy associated with a configuration of a body in classical elasticity is the sum of local contributions that arise from a discrepancy between the actual metric and the reference metric. In contrast, the modeling of defects in solids has traditionally involved extra structure on the material manifold, notably torsion to quantify the density of dislocations and non-metricity to represent the density of point defects. We show that all the classical defects can be described within the framework of classical elasticity using tensor fields that only assume a metric structure. Specifically, bodies with singular defects can be viewed as affine manifolds; both disclinations and dislocations are captured by the monodromy that maps curves that surround the loci of the defects into affine transformations. Finally, we show that two dimensional defects with trivial monodromy are purely local in the sense that if we remove from the manifold a compact set that contains the locus of the defect, the punctured manifold can be isometrically embedded in Euclidean space.

cond-mat.mtrl-sci↗

arXiv2013

A Riemannian Approach to Reduced Plate, Shell, and Rod Theories

We derive a dimensionally-reduced limit theory for an $n$-dimensional nonlinear elastic body that is slender along $k$ dimensions. The starting point is to view an elastic body as an $n$-dimensional Riemannian manifold together with a not necessarily isometric $W^{1,2}$-immersion in $n$-dimensional Euclidean space. The equilibrium configuration is the immersion that minimizes the average discrepancy between the induced and intrinsic metrics. The dimensionally reduced limit theory views the elastic body as a $k$-dimensional Riemannian manifold along with an isometric $W^{2,2}$-immersion in $n$-dimensional Euclidean space and linear data in the normal directions. The equilibrium configuration minimizes a functional depending on the average covariant derivatives of the linear data. The dimensionally-reduced limit is obtained using a $Γ$-convergence approach. The limit includes as particular cases plate, shell, and rod theories. It applies equally to "standard" elasticity and to "incompatible" elasticity, thus including as particular cases so-called non-Euclidean plate, shell, and rod theories.

math.DG↗

arXiv2012

Pattern selection and multiscale behavior in metrically discontinuous non-Euclidean plates

We study equilibrium configurations of non-Euclidean plates, in which the reference metric is uniaxially periodic. This work is motivated by recent experiments on thin sheets of composite thermally responsive gels [1]. Such sheets bend perpendicularly to the periodic axis in order to alleviate the metric discrepancy. For abruptly varying metrics, we identify multiple scaling regimes with different power law dependences of the elastic energy $\En$ and the axial curvature $κ$ on the sheet's thickness $h$. In the $h\to0$ limit the equilibrium configuration tends to an isometric embedding of the reference metric, and $\En\sim h^2$. Two intermediate asymptotic regimes emerge in between the buckling threshold and the $h\to0$ limit, in which the energy scales either like $h^{4/5}$ or like $h^{2/3}$. We believe that this system exemplifies a much more general phenomenon, in which the thickness of the sheet induces a cutoff length scale below which finer structures of the metric cannot be observed. When the reference metric consists of several separated length scales, a decrease of the sheet's thickness results in a sequence of conformational changes, as finer properties of the reference metric are revealed.

nlin.PS↗

arXiv2012

Dimension Reduction in Singularly Perturbed Continuous-Time Bayesian Networks

Continuous-time Bayesian networks (CTBNs) are graphical representations of multi-component continuous-time Markov processes as directed graphs. The edges in the network represent direct influences among components. The joint rate matrix of the multi-component process is specified by means of conditional rate matrices for each component separately. This paper addresses the situation where some of the components evolve on a time scale that is much shorter compared to the time scale of the other components. In this paper, we prove that in the limit where the separation of scales is infinite, the Markov process converges (in distribution, or weakly) to a reduced, or effective Markov process that only involves the slow components. We also demonstrate that for reasonable separation of scale (an order of magnitude) the reduced process is a good approximation of the marginal process over the slow components. We provide a simple procedure for building a reduced CTBN for this effective process, with conditional rate matrices that can be directly calculated from the original CTBN, and discuss the implications for approximate reasoning in large systems.

cs.AI↗

arXiv2012

Continuous Time Markov Networks

A central task in many applications is reasoning about processes that change in a continuous time. The mathematical framework of Continuous Time Markov Processes provides the basic foundations for modeling such systems. Recently, Nodelman et al introduced continuous time Bayesian networks (CTBNs), which allow a compact representation of continuous-time processes over a factored state space. In this paper, we introduce continuous time Markov networks (CTMNs), an alternative representation language that represents a different type of continuous-time dynamics. In many real life processes, such as biological and chemical systems, the dynamics of the process can be naturally described as an interplay between two forces - the tendency of each entity to change its state, and the overall fitness or energy function of the entire system. In our model, the first force is described by a continuous-time proposal process that suggests possible local changes to the state of the system at different rates. The second force is represented by a Markov network that encodes the fitness, or desirability, of different states; a proposed local change is then accepted with a probability that is a function of the change in the fitness distribution. We show that the fitness distribution is also the stationary distribution of the Markov process, so that this representation provides a characterization of a temporal process whose stationary distribution has a compact graphical representation. This allows us to naturally capture a different type of structure in complex dynamical processes, such as evolving biological sequences. We describe the semantics of the representation, its basic properties, and how it compares to CTBNs. We also provide algorithms for learning such models from data, and discuss its applicability to biological sequence evolution.

cs.AI↗

arXiv2012

Gibbs Sampling in Factorized Continuous-Time Markov Processes

A central task in many applications is reasoning about processes that change over continuous time. Continuous-Time Bayesian Networks is a general compact representation language for multi-component continuous-time processes. However, exact inference in such processes is exponential in the number of components, and thus infeasible for most models of interest. Here we develop a novel Gibbs sampling procedure for multi-component processes. This procedure iteratively samples a trajectory for one of the components given the remaining ones. We show how to perform exact sampling that adapts to the natural time scale of the sampled process. Moreover, we show that this sampling procedure naturally exploits the structure of the network to reduce the computational cost of each step. This procedure is the first that can provide asymptotically unbiased approximation in such processes.

cs.AI↗

arXiv2012

Mean Field Variational Approximation for Continuous-Time Bayesian Networks

Continuous-time Bayesian networks is a natural structured representation language for multicomponent stochastic processes that evolve continuously over time. Despite the compact representation, inference in such models is intractable even in relatively simple structured networks. Here we introduce a mean field variational approximation in which we use a product of inhomogeneous Markov processes to approximate a distribution over trajectories. This variational approach leads to a globally consistent distribution, which can be efficiently queried. Additionally, it provides a lower bound on the probability of observations, thus making it attractive for learning tasks. We provide the theoretical foundations for the approximation, an efficient implementation that exploits the wide range of highly optimized ordinary differential equations (ODE) solvers, experimentally explore characterizations of processes for which this approximation is suitable, and show applications to a large-scale realworld inference problem.

cs.AI↗