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The objective of the course is to combine modelling, analysis and computation in the study of materials. The course includes three core sets of lectures on microstructure, fracture, and soft and active matter. As the underlying equations in these different areas are the same or closely related, the course benefits by looking at the three themes together and shows how mathematical issues and discoveries in one theme illuminate the others.
The course is aimed primarily at research students in applied mathematics but the course will also appeal to research students in related disciplines with an interest in materials. It will be accessible to first-year students, but would also be of interest to some postdocs. To ensure participants are better prepared for the three core lecture courses during the rest of the week, the first day takes the form of a three-hour crash course on elasticity.
The course is structured around three lecture courses on each of the three OxMOS themes. Day 1 will be a 3-hour crash course on elasticity given by John Ball. The courses will be complemented by tutorial/trouble-shooting sessions. These will be hosted by post-doctoral assistants from the OxMOS project. The lecturers will set problems during the lectures which will be discussed and worked through in the tutorial sessions. In addition to the three core courses there will also be lectures by two guest speakers. The programme also includes an afternoon visit to an industrial or experimental centre.
Elasticity (John Ball, Oxford): This course will give an introduction from scratch to nonlinear elasticity theory, covering the balance laws, stress, constitutive equations and problem formulation. It will also briefly review the current state of knowledge concerning the existence of solutions and their singularities.
Microstructure (Kaushik Bhattacharya, Caltech): Virtually every solid material contains microstructure, features at a fine scale that undergo processes at fast scales. For example, even the simplest piece of metal is typically made up of many crystallites (grains), which in turn are made up of many atoms. This complexity is only compounded in sophisticated modern materials. Macroscopic applied loads and fields affect the microscopic structure; conversely, the microscopic structure affects the macroscopic behaviour. This series of lectures will provide a broad overview of microstructure in solids, and describe why it forms, how it evolves, and how it affects macroscopic properties. We begin by showing that the appropriate free-energy is non-convex in a variety of situations, and how its optimization leads to fine-scale microstructure. We then introduce a variety of mathematical tools to address microstructure and compute its consequences on effective properties. We conclude with a discussion of evolution and a few open problems. We will draw from a variety of examples to illustrate the ideas throughout the lectures.
Fracture (Blaise Bourdin, Louisiana State University and Gilles Francfort, Paris-Nord): The course will begin with a review of the mainstream theory of brittle fracture, as it has emerged from the works of Griffith and Irwin. We will propose a re-formulation of that theory within the confines of the calculus of variations, focusing on crack path prediction. We will then illustrate the various possible minimality criteria in a simple 1d-case and discuss in some details the only complete mathematical formulation so far, that is that where global minimality for the total energy holds at each time. We will then stress the drawbacks of global minimality and point to several possible remedies. Next we will focus on the numerical treatment of crack evolution and discuss the pros and cons of the competing methods. We will detail crack regularization and explain why it is indeed a good approximation from the standpoint of crack propagation. This will lead to a discussion of the computation of minimizing states for a non-convex functional. Several actual computations will serve to illustrate the analysis.
Soft and Active Matter (L. Mahadevan, Harvard): Living organisms as materials differ from nonliving matter in that they are almost always soft (i.e. they have fairly low elastic moduli with entropic rather than enthalpic origins), wet (i.e. they are often infiltrated by water), warm (i.e. their properties often derive from thermally induced disorder), and active (i.e. they remodel, grow, shrink, consume energy and generate forces). They are also strongly heterogeneous, i.e. amorphous. In these lectures, we will consider the simplest quantitative aspects of these forms of matter. A rough lecture outline for this theme:
| Time | Sunday 28 June | Monday 29 June | Tuesday 30 June | Wednesday 1 July | Thursday 2 July | Friday 3 July |
|---|---|---|---|---|---|---|
| 9:00–10:00 | Elasticity (Ball) Maths Institute (MI) St Giles Lecture Theatre 1 | Microstructure (Bhattacharya) MI St Giles Lecture Theatre 1 | Fracture (Francfort) Comlab Lecture Theatre A | Industrial excursion to BMW. Group B | Fracture (Bourdin) MI St Giles Lecture Theatre 2 | |
| 10:00–11:00 | Elasticity (Ball) MI St Giles Lecture Theatre 1 | Fracture (Francfort) MI St Giles Lecture Theatre 1 | Fracture (Bourdin) Comlab Lecture Theatre A | Soft and Active Matter (Mahadevan) MI St Giles Lecture Theatre 2 | ||
| 11:00–11:30 | Coffee St Giles Common Room | Coffee St Giles Common Room | Coffee Comlab | Coffee St Giles Common Room | ||
| 11:30–12:30 | Elasticity (Ball) MI St Giles Lecture Theatre 1 | Microstructure (Bhattacharya) MI St Giles Lecture Theatre 1 | Microstructure (Bhattacharya) Comlab Lecture Theatre A | Soft and Active Matter (Mahadevan) MI St Giles Lecture Theatre 2 | ||
| 12:30–13:15 | Lunch St Anne’s College Seminar Room 9 | Lunch St Anne’s College Seminar Room 9 | Lunch St Anne’s College Seminar Room 9 | Lunch St Anne’s College Dining Hall | Tutorial (summary) MI St Giles Lecture Theatre 2 | |
| 13:15–14:00 | Lunch St Anne’s College Dining Hall | |||||
| 14:00–15:30 | Tutorial and trouble-shooting session Dartington House (DH) Seminar Rooms 1 and 3 | Tutorial and trouble-shooting session Dartington House (DH) Seminar Rooms 1 and 3 | 13:45 Soft and Active Matter (Mahadevan) Comlab Lecture Theatre A | Tutorial and trouble-shooting session Dartington House (DH) Seminar Rooms 1 and 3 | Depart | |
| 14:45 Microstructure (Bhattacharya) Comlab Lecture Theatre A | ||||||
| 15:30–16:00 | Tea DH Common Room | Tea DH Common Room | Tea DH Common Room | |||
| 16:00–16:30 | Tutorial (summary) DHSR1/3 | Tutorial (summary) DHSR1/3 | 15:45 Tea Comlab | Tutorial (summary) DHSR1/3 | ||
| 16:30–17:00 | 16:15 Soft and Active Matter (Mahadevan) Comlab Lecture Theatre A | |||||
| 17:00–17:30 | Registration St Anne’s College Lodge | Guest lecture: Dimitry Vvedenski “Multiscale Theory of Nanostructure Assembly” MI St Giles Lecture Theatre 2 | ||||
| 17:30–18:00 | 17:30 Early dinner. St Anne’s College Dining Hall 18:00 for 18:30 Industrial excursion to BMW. Group A | |||||
| 18:00–19:00 | Guest lecture: Peter Hunter “Challenges in Characterising the Material Properties of Soft Tissues” MI St Giles Lecture Theatre 2 | |||||
| 19:00–21:00 | Evening reception, buffet, and drinks St Anne’s College Dining Hall | Dinner St Anne’s College Dining Hall | Dinner St Anne’s Foyer B Ruth Deech Building | Barbeque St Anne’s Marquee |
Main registration will be held at St Anne’s College on Sunday 28 June from 5–7pm in the Lodge at St Anne’s College. We request that all delegates register on Sunday as you will receive the detailed information about the course for the rest of the week. If you are unable to register on Sunday then it will be possible to register at the desk in the foyer of the main Mathematical Institute building on 24–29 St Giles’ on Monday 29 June from 8.30am.
Please note that there is no parking at St Anne’s or the Mathematical Institute without prior arrangement and it is very limited even then.
There will be a reception for all delegates on Sunday 28 June from 7–9pm. This will be held in the Dining Hall at St Anne’s College. A cold buffet and drinks provided will be provided. We hope you will be able to join us.
Lunches for all delegates (whether staying at St Anne’s or not) will be held at St Anne’s College. Dinner is provided for those staying at St Anne’s and for other delegates that have requested dinner. On the evening of Thursday 2 July there will be a special barbeque held in the grounds of St Anne’s following the guest lectures. It has been assumed that all delegates will want to attend (unless we have been informed otherwise).
St Anne’s College,
Woodstock Road,
Oxford, OX2 6HS
UK
Tel: +44 (0)1865 274800
For those that are residential, all accommodation is at St Anne’s College . St Anne’s is located on Woodstock Road and is close to the Mathematical Institute and the Computer Laboratory where the course lectures and tutorials will be held. For transport to St Anne’s please see the information on their website. Please note: Attendees should have received an email containing a list of all those for whom we have booked accommodation. Please check that you appear on the list if you are expecting to stay at St Anne’s. You will be able to check into your room from 12 noon on your day of arrival.
Delegate internet access is available via wired ethernet connection in bedrooms. Wireless connection is not available in delegate bedrooms. On or shortly before arrival an internet registration code will be issued through the Conference Office. This code will also be made available to delegates from the Lodge. Delegates will need this code in order to access the internet. Delegates should provide their own ethernet cable and wireless card. Ethernet cables are available to purchase from the College Lodge should any delegate have forgotten their own. Please contact your own IT staff prior to the conference in order to ensure that your system is configured to connect to a network using DHCP and that your web-browser is not configured to use a local proxy. You should also ensure that your e-mail is correctly configured for use away.
Lectures and tutorials for the course will be spread across several sites which are all central and in close vicinity.
The Mathematical Institute in St Giles’ [Directions]
24–29 St Giles’,
Oxford
OX1 3LB
Oxford University Computing Laboratory [Directions]
Wolfson Building,
Parks Road,
OXFORD
OX1 3QD
The Mathematical Institute Annex at Dartington House [Directions]
Little Clarendon Street,
Oxford
OX1 2HS
More detailed location maps and directions to these sites can also be found by visiting Multimap and entering the postcodes provided.
Monday 29 June: Lectures will be held in the St Giles Building of the Mathematical Institute in Lecture Theatre 1.
Tuesday 30 June: Lectures will be held in the St Giles Building of the Mathematical Institute in Lecture Theatre 1.
Wednesday 1 July: Lectures will be held in the Computing Laboratory on Parks Road in Lecture Theatre A.
Thursday 2 July: The guest lectures will be held in the St Giles Building of the Mathematical Institute.
Friday 3 July: Lectures will be held in the St Giles Building of the Mathematical Institute in Lecture Theatre 2.
Tutorial classes will be held on Monday, Tuesday and Thursday afternoons in the Mathematical Institute Annex at Dartington House on Little Clarendon Street in Seminar Rooms 1 and 3. There will be a tutorial summary on Friday after the lectures in the St Giles Building of the Mathematical Institute in Lecture Theatre 2.
As part of the course there will be an excursion to the BMW factory in Cowley to see some materials modelling taking place in an industrial context. These tours will take place on Wednesday evening at 6.30pm and Thursday morning at 9.00am. We have allocated you into a group for the tour (please see PDF document). Taxis will leave St Anne’s at approximately 30 minutes before these times to take you there.
John Ball is Sedleian Professor of Natural Philosophy at the Mathematical Institute, University of Oxford. His main research areas are the calculus of variations, nonlinear PDE, infinite-dimensional dynamical systems, and their applications to solid mechanics, materials science and liquid crystals. His work in solid mechanics includes the first global existence theorems for nonlinear elastostatics, studies of cavitation in solids, and contributions to the understanding of martensitic microstructure.
Kaushik Bhattacharya is a Professor of Mechanics and Materials Science and the Executive Officer for Mechanical Engineering at the California Institute of Technology. His research concerns the mechanical behaviour of solids, and specifically uses theory to guide the development of new materials.
Blaise Bourdin is Associate Professor at the Department of Mathematics, Louisiana State University. His interest lies in computational sciences and more specifically in the modelling, analysis and numerical implementation of problems arising from solid mechanics. In recent years, his main focus has been in the efficient parallel implementation of variational fracture mechanics.
Gilles Francfort is Professor of Mathematics at the Université Paris-Nord. His research topics of interest lie primarily in the modelling and mathematical formulation of problems in Solid Mechanics. In recent years the main focus has been the development of mathematically consistent models for the propagation of various defects (cracks, damage, …) in an otherwise elastic material.
L. Mahadevan is De Valpine Professor of Applied Mathematics, and Professor of Organismic and Evolutionary Biology at Harvard University and Professor of Systems Biology at Harvard Medical School. He uses a combination of theory and experiment to uncover the patterns that characterize the organization of space and time in living and non-living matter.
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