Past OxMOS Workshop/Meeting/Lecture

We consider the problem of cavitation in nonlinear elasticity, or the formation of macroscopic cavities in elastic materials from microscopic defects, when subjected to large tension at the boundary.

The main goal is to determine the optimal locations where the body prefers the cavities to open, the preferred number of cavities, their optimal sizes, and their optimal shapes. To this aim it is necessary to analyze the elastic energy of an incompressible deformation creating multiple cavities, in a way that accounts for the interaction between the cavitation singularities. Based on the quantitative version of the isoperimetric inequality, as well as on new explicit constructions of incompressible deformations creating cavities of different shapes and sizes, we provide energy estimates showing that, for certain loading conditions, there are only the following possibilities:

• only one cavity is created, and if the loading is isotropic then it is created at the centre
• multiple cavities are created, they are spherical, and the singularities are well separated
• there are multiple cavities, but they act as a single spherical cavity, they are considerably distorted, and the distance between the cavitation singularities must be of the same order as the size of the initial defects contained in the domain.

In the latter case, the formation of thin structures between the cavities is observed, reminiscent of the initiation of ductile fracture by void coalesence.

This is joint work with Sylvia Serfaty (LJLL, Univ. Paris VI).

Speakers include:

* David Abrahams (Manchester, UK); * Stuart Antman (Maryland, USA); * Martine Ben Amar (Ecole Normale Supérieure, France); * Mary Boyce (MIT, USA); * John Hutchinson (Harvard, USA); * Nadia Lapusta (Caltech, USA); * John Maddocks (Lausanne, Switzerland); * Stefan Mueller (Bonn, Germany); * Christoph Ortner (Oxford, UK); * Ares Rosakis (Caltech, USA); * Hanus Seiner (Academy of Sciences, Czech Republic); * Eran Sharon (Hebrew University, Israel); * Lev Truskinovsky (Lab de Mécanique des Solids, France); * John Willis (Cambridge, UK).

Speakers include:

* Graeme Ackland (School of Physics and Astronomy, Edinburgh) * Andrea Braides (Rome II) * Thierry Bodineau (École Normale Supérieure, Paris) * Matthew Dobson (Minneapolis) * Laurent Dupuy (CEA, Saclay) * Ryan Elliott (Minneapolis) * Roman Kotecky (Warwick) * Carlos Mora-Corral (BCAM, Bilbao) * Stefano Olla (CEREMADE, Paris-Dauphine) * Bernd Schmidt (TU Munich) * Lev Truskinovsky (École Polytechnique, Palaiseau) * Min Zhou (Georgia Tech, Atlanta)

This short course runs from Monday 29th June to Friday 3rd July. For details of the course and how to register, please visit http://www2.maths.ox.ac.uk/oxmos/meetings/moms/.

Macroscopic properties of solids are inherently connected to their micro- and nano-scale details. For example, the microstructure and defect distribution influence the elastic and plastic properties of a crystal while the details of a defect are determined by its elastic far-field. The goal of multi-scale modelling is to understand such connections between microscopic and macroscopic material behaviour. This workshop brings together researchers working on different aspects of multi-scale modelling of solids: mathematical modelling, analysis, numerical computations, and engineering applications.

The workshop will address current issues related to the stability of solutions in nonlinear elasticity, including local energy minimizers, the stability of growing bodies, global existence for small data, bifurcation and continuation of solutions, and Saint-Venant’s principle.

This talk is devoted to adaptivity in optimal control of PDEs with special emphasis on barrier methods for pointwise state constraints. The talk is divided into to major parts, first we will discuss the case of additional pointwise inequality constraints on the state variable, then we will transfer the results to constraints on the gradient of the state. Each part will start with a discussion of necessary optimality conditions and a brief overview about what is known and what is not known concerning a priori analysis. Then a posteriori error estimates for the discretization error as well as for the error from the barrier method will be presented. Finally we show some simple examples to illustrate the behavior of the estimators. 
The talk will be followed by an informal tea in the Gibson Building seminar room giving an opportunity to chat with Winnifried Wollner and Amit Acharya (our other current OxMOS visitor)

The effective properties of composite media are defined by the constituent phase properties (elastic moduli, thermal conductivities,etc), their volume fractions, and their distribution throughout the medium. In the case of constituents distributed periodically, there exist many homogenization theories which can provide exact solutions for the effective properties. However, the case of the effective properties of random media remains largely an open problem.

In this talk we will begin by discussing the notion of homogenization as an extension to the continuum assumption and regimes in which it breaks down. We then discuss various approaches to dealing with randomness whilst determining the effective properties of acoustic, thermal and elastic media.  In particular we show how the effective properties depend on the randomness of the microstructure

Carlos and Benson will give an update on their research.

Fracture mechanics is a significant scientific field of great practical importance. Recently the subject has been invigorated by a number of important accomplishments. From the viewpoint of fundamental science there have been interesting new developments aimed at understanding fracture at the atomic scale; simultaneously, active research programmes have focussed on mathematical modelling, experimentation and computation at macroscopic scales. The workshop aims to examine various different approaches to the modelling, analysis and computation of fracture. The programme will allow time for discussion.

Invited speakers include:

Andrea Braides (Università di Roma II, Italy)

Adriana Garroni (Università di Roma, “La Sapienza”, Italy)

Christopher Larsen (Worcester Polytechnic Institute, USA)

Matteo Negri (Università di Pavia, Italy)

Robert Rudd (Lawrence Livermore National Laboratory, USA)

We are concerned with the derivation of the $\Gamma$-limit to a three-dimensional geometrically exact

Cosserat model as the relative thickness $h>0$ of a flat domain tends to zero. The Cosserat bulk model involves

already exact rotations as a second independent field and this model is meant to describe defective elastic crystals liable to fracture under shear.

It is shown that the $\Gamma$-limit based on a natural scaling assumption

consists of a membrane like energy contribution and a homogenized transverse shear energy both scaling with $h$,

augmented by an additional curvature stiffness due to the underlying Cosserat bulk formulation, also scaling with $h$.

No specific bending term appears in the dimensional homogenization process. The formulation

exhibits an internal length scale $L_c$ which survives the homogenization process.

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A major technical difficulty, which we encounter in applying the $\Gamma$-convergence arguments,

is to establish equi-coercivity of the sequence of

functionals as the relative thickness $h$ tends to zero. Usually, equi-coercivity follows from a local coerciveness assumption.

While the three-dimensional problem is well-posed for the Cosserat couple modulus $\mu_c\ge 0$, equi-coercivity forces us

to assume a strictly positive Cosserat couple modulus $\mu_c>0$. The $\Gamma$-limit model determines the

midsurface deformation $m\in H^{1,2}(\omega,\R^3)$. For the case of zero Cosserat couple modulus $\mu_c=0$

we obtain an estimate of the $\Gamma-\liminf$ and $\Gamma-\limsup$, without equi-coercivity which is then strenghtened to a $\Gamma$-convergence result for zero Cosserat couple modulus. The classical linear

Reissner-Mindlin model is "almost" the linearization of the $\Gamma$-limit for $\mu_c=0$

apart from a stabilizing shear energy term.

Computational Mechanics (CM) has become

a scientific discipline in itself, being High Perfomance Computational

Mechanics (HPCM) a key sub-discipline. The effort for the most efficient use of

distributed memory machines provides a different perspective to CM scientists

relative to a wide range of topics, from the very physics of the problem to

solve to the numerical method used. Marenostrum supercomputer is the largest

facility in Europe and the 5th in the world (top500.org - Spring 2007). This

talk describes the research lines in the CASE Dpt. of the BSC applied to

Aerospace, Bio-mechanics, Geophysics or Environment, through the development of

Alya, the in-house HPCM code for complex coupled problems capable of running

efficiently in large distributed memory facilities.

Professor Qiang Du will go over some work on modelling interface/microstructures with curvature dependent energies and also the effect of elasticity on critical nuclei morphology.

Global Bifurcation Theory: Alternative of Rabinowitz

Introduction to the topological degree: Existence and Uniqueness of the Brouwer degree, Existence and Uniqueness of the Leray-Schauder degree.

Local Bifurcation Theory (II): Principle of exchange of stability, Lyapunov-Schmidt reduction, Theorem of Ize.

Local Bifurcation Theory (I): Theorem of Crandall and Rabinowitz

OxMOS Team Meetings are held regularly during term and are open to all. Two members of OxMOS will give a short update on their recent research.

There will be three discussion meetings based on aspects of the

programme open to all internal project members. Others interested in

attending should contact Carlos Mora-Corral.

Team meetings, held roughly every four weeks, are open to anyone who is

interested. OxMOS post docs and Dphil students will give updates on the

research.

There will be three discussion meetings based on aspects of the

programme open to all internal project members. Others interested in

attending should contact Carlos Mora-Corral.

Team meetings, held roughly every four weeks, are open to anyone who is

interested. OxMOS post docs and Dphil students will give updates on the

research.

There will be three discussion meetings based on aspects of the

programme open to all internal project members. Others interested in

attending should contact Carlos Mora-Corral.