Nicolas Van Goethem

Damage and fracture evolution in brittle materials by shape optimization methods

This paper is devoted to a numerical implementation of the Francfort-Marigo model of damage evolution in brittle materials. This quasi-static model is based, at each time step, on the minimization of a total energy which is the sum of an elastic energy and a Griffith-type dissipated energy. Such a minimization is carried over all geometric mixtures of the two, healthy and damaged, elastic phases, respecting an irreversibility constraint. Numerically, we consider a situation where two well-separated phases coexist, and model their interface by a level set function that is transported according to the shape derivative of the minimized total energy. In the context of interface variations (Hadamard method) and using a steepest descent algorithm, we compute local minimizers of this quasi-static damage model. Initially, the damaged zone is nucleated by using the so-called topological derivative. We show that, when the damaged phase is very weak, our numerical method is able to predict crack propagation, including kinking and branching. Several numerical examples in 2d and 3d are discussed.

A distributional approach to 2D Volterra dislocations at the continuum scale

We develop a theory to represent dislocations and disclinations in single crystals at the continuum (or mesoscopic) scale by directly modelling the defect densities as concentrated effects governed by the distribution theory. The displacement and rotation multi-valuedness is resolved by introducing the intrinsic and single-valued Frank and Burgers tensors from the distributional gradients of the strain field. Our approach provides a new understanding of the theory of line defects as developed by Kroner [10] and other authors [6, 9]. The fundamental identity relating the incompatibility tensor to the Frank and Burgers vectors (and which is a cornerstone of the theory of dislocations in single crystals) is proved in the 2D case under appropriate assumptions on the strain and strain curl growth in the vicinity of the assumed isolated defect lines. In general, our theory provides a rigorous framework for the treatment of crystal line defects at the mesoscopic scale and a basis to strengthen the theory of homogenisation from mesoscopic to macroscopic scale.

A level set method for the numerical simulation of damage evolution

The aim of this article is to present an introduction to dissipation inequalities and to present some well known and some recent results in this area. Mathematics Subject Classification (2000). Primary 93C10; Secondary 93D99.The first part of this article concerns visibility, that is the question of determining the internal properties of a medium by making electromagnetic measurements at the boundary of the medium. We concentrate on the problem of Electrical Impedance Tomography (EIT) which consists in determining the electrical conductivity of a medium by making voltage and current measurements at the boundary. We describe the use of complex geometrical optics solutions in EIT. In the second part of this article we will review recent theoretical and experimental progress on making objects invisible to electromagnetic waves. This is joint work with A. Greenleaf, Y. Kurylev and M. Lassas. Maxwell’s equations have transformation laws that allow for design of electromagnetic parameters that would steer light around a hidden region, returning it to its original path on the far side. Not only would observers be unaware of the contents of the hidden region, they would not even be aware that something was hidden. The object would have no shadow. New advances in metamaterials have given some experimental evidence that this indeed can be made possible at certain frequencies. Mathematics Subject Classification (2000). Primary 35R30, 78A46 ; Secondary 58J05, 78A10 .Applied mathematics is concerned with developing models with predictive capability, and with probing those models to obtain qualitative and quantitative insight into the phenomena being modelled. Statistics is data-driven and is aimed at the development of methodologies to optimize the information derived from data. The increasing complexity of phenomena that scientists and engineers wish to model, together with our increased ability to gather, store and interrogate data, mean that the subjects of applied mathematics and statistics are increasingly required to work in conjunction in order to significantly progress understanding. This article is concerned with a research program at the interface between these two disciplines, aimed at problems in differential equations where profusion of data and the sophisticated model combine to produce the mathematical problem of obtaining information from a probability measure on function space. In this context there is an array of problems with a common mathematical structure, namely that the probability measure in question is a change of measure from a Gaussian. We illustrate the wide-ranging applicability of this structure. For problems whose solution is determined by a probability measure on function space, information about the solution can be obtained by sampling from this probability measure. One way to do this is through the use of Markov chain Monte-Carlo (MCMC) methods. We show how the common mathematical structure of the aforementioned problems can be exploited in the design of effective MCMC methods.

Fields of bounded deformation for mesoscopic dislocations

In this paper we discuss the consequences of the distributional approach to dislocations in terms of the mathematical properties of the auxiliary model fields such as displacement and displacement gradient which are obtained directly from the main model field here considered as the linear strain. We show that these fields cannot be introduced rigourously without the introduction of gauge fields or, equivalently, without cuts in the Riemann foliation associated with the dislocated crystal. In a second step we show that the space of bounded deformations follows from the distributional approach in a natural way and discuss the reasons why it is adequate to model dislocations. The case of dislocation clusters is also addressed, as it represents an important issue in industrial crystal growth while from a mathematical point of view, peculiar phenomena might appear at the set of accumulation points. The elastic–plastic decomposition of the strain within this approach is also given a precise meaning.In this paper we discuss the consequences of the distributional approach to dislocations in terms of the mathematical properties of the auxiliary model fields such as displacement and displacement gradient which are obtained directly from the main model field here considered as the linear strain. We show that these fields cannot be introduced rigourously without the introduction of gauge fields or, equivalently, without cuts in the Riemann foliation associated with the dislocated crystal. In a second step we show that the space of bounded deformations follows from the distributional approach in a natural way and discuss the reasons why it is adequate to model dislocations. The case of dislocation clusters is also addressed, as it represents an important issue in industrial crystal growth while from a mathematical point of view, peculiar phenomena might appear at the set of accumulation points. The elastic–plastic decomposition of the strain within this approach is also given a precise meaning.

ANALYSIS OF THE INCOMPATIBILITY OPERATOR AND APPLICATION IN INTRINSIC ELASTICITY WITH DISLOCATIONS

The incompatibility operator arises in the modeling of elastic materials with disloca- tions and in the intrinsic approach to elasticity, where it is related to the Riemannian curvature of the elastic metric. It consists of applying successively the curl to the rows and the columns of a second- rank tensor, usually chosen symmetric and divergence-free. This paper presents a systematic analysis of boundary value problems associated with the incompatibility operator. It provides answers to such questions as existence and uniqueness of solutions, boundary trace lifting, and transmission condi- tions. Physical interpretations in dislocation models are also discussed, but the application range of these results far exceed any specific physical model.

Topology optimization methods with gradient-free perimeter approximation

1be incorporated within topology optimization algorithms. The required mathematical properties, 2 namely the -convergence and the compactness of sequences of minimizers, are first established. 3 Then we propose several methods for the solution of topology optimization problems with perimeter 4 penalization showing different features. We conclude by some numerical illustrations in the contexts 5 of least square problems and compliance minimization. 6

Cauchy elasticity with dislocations in the small strain assumption

Abstract In this letter it is shown how the singularities created by dislocations in an elastic body must not preclude from a linear approach. Cauchy elasticity is considered and hence no variational approach is needed and the displacement field only appears in a second step, as an interpretation of the one of the two model variables, arisen from a Beltrami decomposition of the strain tensor. The second model variable obeys nonclassical PDEs relying on the incompatibility operator, and whose well posedness is here addressed.

Incompatibility-governed elasto-plasticity for continua with dislocations

In this paper, a novel model for elasto-plastic continua is presented and developed from the ground up. It is based on the interdependence between plasticity, dislocation motion and strain incompatibility. A generalized form of the equilibrium equations is provided, with as additional variables, the strain incompatibility and an internal thermodynamic variable called incompatibility modulus, which drives the plastic behaviour of the continuum. The traditional equations of elasticity are recovered as this modulus tends to infinity, while perfect plasticity corresponds to the vanishing limit. The overall nonlinear scheme is determined by the solution of these equations together with the computation of the topological derivative of the dissipation, in order to comply with the second principle of thermodynamics.

Incompatibility-governed singularities in linear elasticity with dislocations:

The purpose of this paper is to prove the relation inc ε = Curl κ relating the elastic strain ε and the contortion tensor κ , related to the density tensor of mesoscopic dislocations. Here, the dislocations are given by a finite family of closed Lipschitz curves in Ω ⊂ ℝ 3 . Moreover the fields are singular at the dislocations, and, in particular, the strain is non square integrable. Moreover, the displacement fields show a constant jump around each isolated dislocation loop. This relation is called after E. Kröner who first derived the same formula for smooth fields at the macroscale.The purpose of this paper is to prove the relation ince=Curlκ relating the elastic strain e and the contortion tensor κ, related to the density tensor of mesoscopic dislocations. Here, the dislocations are given by a finite family of closed Lipschitz curves in Ω⊂ℝ3. Moreover the fields are singular at the dislocations, and, in particular, the strain is non square integrable. Moreover, the displacement fields show a constant jump around each isolated dislocation loop. This relation is called after E. Kroner who first derived the same formula for smooth fields at the macroscale.

Variational Evolution of Dislocations in Single Crystals

In this paper, we provide an existence result for the energetic evolution of a set of dislocation lines in a three-dimensional single crystal. The variational problem consists of a polyconvex stored elastic energy plus a dislocation energy and some higher-order terms. The dislocations are modeled by means of integral one-currents. Moreover, we discuss a novel dissipation structure for such currents, namely the flat distance, that will serve to drive the evolution of the dislocation clusters.