Workshop on Multiscale Models in Solid Mechanics

3rd June 2009
Mathematical Institute, University of Oxford

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.

You may download a poster for the event here.

Speakers include:

F. Legoll (ENPC, France)
M. Luskin (Minnesota, USA)
M. Ortiz (Caltech, USA)
O. Pierre-Louis (UJF, France)
A. Raoult (Université Paris Descartes, France)
M. Thompson (Oxford, UK)

Registration

The workshop will take place on Wednesday, 3rd June 2009 at the Mathematical Institute, University of Oxford, 24–29 St Giles’, Oxford. Registration with coffee will be at 9.45–10.15 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.

Accommodation

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.

Provisional programme

9.45	Registration and coffee
10.15	Introduction
10.20	M. Ortiz. “Multiscale modelling of energetic materials”
11.10	A. Raoult. “Lattice equivalent energy. A study originating from cardiac modelling”
12.00	Lunch/Steering Committee
14.00	M. Luskin. “Accurate prediction of instability by quasicontinuum methods”
14.50	F. Legoll. “Finite temperature coarse-graining of atomistic models: a possible computational approach”
15.40	Tea
16.10	O. Pierre-Louis. “Dewetting of solid film”
17.00	M. Thompson. “Microstructural modelling for tendon mechanobiology”
17.50	Concluding remarks

Abstracts

Multiscale modelling of energetic materials, M. Ortiz.

To follow.

Lattice equivalent energy. A study originating from cardiac modelling, A. Raoult.

We first describe the discrete homogenization procedure that we introduced some years ago in a joint work with Denis Caillerie and Ayman Mourad. This method can be applied to any network consisting of bars and of nodes, our two preferred examples being the arrangement of myocytes and graphene sheets. It should be noted that in the governing lattice model adjacent bars interact by means of moments. Therefore we deal with interactions between three points. In this general setting our results remain formal. Then, we concentrate on square lattices. This is a simple case because such lattices are Bravais lattices, but our model still incorporates moments. We describe the rigorous convergence results that we recently obtained in a joint work with Nicolas Meunier and Olivier Pantz.

Accurate prediction of instability by quasicontinuum methods, M. Luskin.

Important applications of atomistic-to-continuum coupling methods include the quasistatic study of lattice instabilities such as dislocation formation and movement during nanoindentation, crack tip growth, and the deformation and growth of grain boundaries. In each of these applications, the quasistatic deformation provides an accurate approximation of the crystal deformation until the equilibrium equations become singular, which occurs, for example, when a dislocation forms or moves or when a crack tip advances. Depending on the nature of the singularity, the crystal will then typically undergo a dynamic process when further loaded.

We propose sharp stability analysis as a theoretical criterion for evaluating the predictive capability of atomistic-to-continuum coupling methods. Some error estimates have been obtained that give theoretical justification for the accuracy of a quasicontinuum atomistic-to-continuum coupling method for all loads up to the critical atomistic load for the singularity (the limit load for the atomistic model), but other error estimates that have been presented do not hold near the atomistic limit loads. It is important to understand whether the break-down of these error estimates is an artifact of the analysis, or whether the particular quasicontinuum method actually does incorrectly predict an instability before the applied load has reached the correct limit load of the atomistic model.

Our sharp stability results show that consistent QC methods such as the quasi-nonlocal coupling method reproduce the stability of the atomistic system, whereas the inconsistent energy-based quasicontinuum method incorrectly predicts instability at a significantly reduced applied load.

Our results for the force-based quasicontinuum method (QCF) are less conclusive. The QCF operator is not conservative, and, in fact, we show that the linearization of the QCF operator is not generally positive definite even at applied strains for which all of its eigenvalues are positive. However, we prove that the QCF method as a mapping from w^2,2_ε to ℓ²_ε reproduces the stability of the atomistic system.

Joint work with Matthew Dobson and Christoph Ortner.

Finite temperature coarse-graining of atomistic models: a possible computational approach, F. Legoll.

Molecular dynamics is a classical method to compute constant temperature thermodynamical averages, in the statistical mechanics framework. For instance, the evolution of an atomistic system according to the Langevin equation is simulated, while some relevant observables are averaged along the trajectory.

It is often the case that interesting observables actually do not depend on all the particles, but only on the position of some of them, called repatoms, following the QuasiContinuum terminology. In this case, it is natural to try and design a strategy to compute more efficiently the canonical averages under study. The free energy of the coarse-grained model is another interesting quantity, which leads to the macroscopic constitutive relation of the material, at a given temperature.

In this talk, we first consider one-dimensional chains of atoms, and we present a rigorous and efficient method to compute ensemble averages and free energies, based on a thermodynamic limit procedure. The obtained theoretical results are illustrated with numerical simulations. The extension of our approach to a simple two-dimensional setting will also be presented.

Dewetting of solid film, O. Pierre-Louis.

Liquid films, once spread on a substrate, may break-up into droplets to lower the surface energy. Such a process is called dewetting. As for liquids, thin solid films may break-up into droplets. However two main differences may be pointed out. Firstly, solids exhibit strong surface anisotropy whereas liquids are usually isotropic. Secondly, mass transport mainly occurs via surface diffusion on solids at small scales, while it is mediated by hydrodynamics in liquids.

We shall discuss how the usual macroscopic modelling of solid surface evolution breaks-down at the nanoscale. We also point out that how a precise analysis of the microscopic process involved in dewetting of solid films allows one to build a suitable dynamical model.

Microstructural modelling for tendon mechanobiology, M. Thompson.

Consitutive models of tendon have succeeded well in capturing the tissue-level deformation of this complex, hierarchical, fibre-reinforced composite of collagen and proteoglycan. In order to understand and exploit the mechanosensitivity of this and other tissues in clinical situations however, it is essential to be able to predict the cell-level deformation resulting from tissue-level loading. The aim of this project is to implement and justify a new microstructural model for tendon, and to compare its predictions of the cell mechanical environment with published microscopy-based measurements.