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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.
Download the programme here (PDF, 46 kB).
| 10.00 | Registration and coffee in the Mathematical Institute Common Room |
| 10.40 | Introduction by Endre Suli |
| 10.45 | Andrea Braides (Università di Roma II, Italy) : Variational lattice models of fracture |
| 11.30 | Chris Larsen (Worcester Polytechnic Institute, USA) : Fracture evolution and locality (finish at 12.15) |
| 12.30 | Buffet lunch at St Anne’s College |
| 14.00 | Adriana Garroni (Università di Roma, “La Sapienza”, Italy) : Threshold based quasi-static evolution for damage |
| 14.45 | Robert Rudd (Lawrence Livermore National Laboratory, USA) : Void growth and ductile fracture from the atomistic level |
| 15.30 | Tea in the Mathematical Institute Common Room |
| 16.00 | Matteo Negri (Università di Pavia, Italy) : Quasi-static evolutions of a brittle crack: analytical and numerical aspects in a model case |
| 16.45 | Closing discussion |
| 17.00 | Close |
Download the poster here (PDF, 44 kB).
The workshop will take place on Monday, 10 March 2008 at the Mathematical Institute, University of Oxford , 24–29 St. Giles, Oxford. Registration with coffee will be at 10.00–10.30 am. Lunch will be at St Anne’s College and is free if the box is ticked on the registration page. You can register online or by contacting OxMOS. There is no registration fee. Limited funding is available to cover expenses. Applications will be treated on a case by case basis. Students and researchers at the beginning of their careers are particularly encouraged to apply.
If you need to stay overnight, you will need to arrange your own accommodation unless you are speaking. The Travelodge at Oxford Pear Tree is on an easy Park and Ride bus route which will take you very close to the Maths Institute, and their rooms start at £19 if booked and paid for 21 days in advance. If you would like something more central in Oxford, prices will reflect this. The Tower House Hotel is very central and is quite reasonably priced but the rooms are rather quirky and individual. The Linton Lodge Hotel is well placed for the Maths Institute. Oxford City Council has an accommodation search which may also be useful. Please note that to be close to the Maths Institute or on an easy bus route you will want to search the Central and North areas. Please contact oxmos@maths.ox.ac.uk if you need further advice or help.
We present an overview of models of lattice interactions from which continuous variational fracture models are derived. Starting from Lennard–Jones type interactions energies accounting also for micro-fracture and internal boundary layers are derived. Links with quasicontinuum theories are given and open problems connected to rigidity estimates in spaces of discontinuous functions.
The main goal of recent mathematical work on quasi-static fracture triggered by the Francfort–Marigo approach is to turn Griffith’s criterion for crack growth into a model for predicting crack paths. The main drawback of the Francfort–Marigo model is its reliance on global minimization, which in particular results in violations of Griffith’s criterion. I will discuss some efforts towards creating a variational model based on local minimization that also predicts crack paths, and why a fully satisfactory (variational) approach seems impossible. One possible fix, naturally, is to instead consider dynamics. A computational model, based only on elastic dynamics away from the crack and a localized Griffith’s criterion at the crack tip, will be described, along with a corresponding (speculative) analytical model for fracture dynamics.
We consider a variational model for elastic damage proposed by Francfort and Marigo. This energy based model is nonconvex since only to extreme states (damaged and undameged material) are possible, and in the minimization procedure microstructures can be produced. A relaxed incremental problem that accounts for irreversibility can be defined and, by means of time discretization, a relaxed quasi-static evolution can be obtained. This relaxed quasi-static evolution accounts for a damage process that in principle in completely predictive and does not require any a priori assumption on the damage path.
We give an alternative model for damage based on a threshold criterion. We prove that an ‘energy based’ solution is also a ‘threshold’ solution. As a byproduct we also obtain that local minimizers for the energy based model are actually global minimizers.
Foremost in the design of any structural element is that it be able to withstand the loads applied to it throughout its lifetime. The failure properties of a material depend on its gross mechanical properties such as its strength as well as more microscopic features such as the nature of fracture nucleation sites like second phase particles. In modern applications our understanding of fracture is being driven to develop in new directions for systems that are smaller, faster and under more extreme conditions.
At the microscopic level ductile fracture involves the nucleation, growth and coalescence of voids. The fracture surface results from this linking of voids. Ductile fracture is conventionally modeled at the continuum level, in a variety of models that may or may not model voids explicitly. With the advent of massively parallel supercomputers and multiscale modeling techniques, it has become possible to study these fracture processes starting at the level of atoms. The use of atomistic simulation provides a new insight into the fracture processes. For example, the simulations generate detailed information about the dislocation flow in the plastically deformed material around a growing void and in the interacting plastic zones as voids coalesce. In this lecture we briefly review the conventional theory of ductile fracture, with an emphasis on dynamic fracture, and then discuss the results of atomistic simulations of void growth and coalescence processes. We also describe multiscale models that are being developed for ductile fracture and damage.
This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.
We consider a model problem that mimics the geometry of the Compact Tension specimen, considering a crack that propagates in quasi-static regime, under displacement control. In a wide perspective, our goal is to move towards robust models for the standard experimental tests used to measure material toughness. In the long run this will of course lead to more efficient and reliable tools in the analysis and simulation of real life problems.
A tentative summary of the talk is the following: We give first a detailed description of an evolution derived from Griffith’s criterion and based on the critical points of the energy. Other approaches, proposed in the last few years, are briefly recalled, showing the main differences. Then, we show how this evolution can be derived from a “dynamic” theory of crack propagation, employing rate dependent energies. Finally, we present some results about the finite element discretization of the model.
The content of the talk is essentially contained in:
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