Dislocation patterns beyond optimality

Lead Research Organisation: University of Bath
Department Name: Mathematical Sciences

Abstract

Metals, and steel in particular, play a fundamental role in our everyday life and are used in countless applications, from the construction industry to transport, energy, packaging, and house appliances. Different applications, however, require different material properties of the metals, which need to be designed to meet the requirements. Materials design in its turn requires a deep understanding of the dependence of the mechanical behaviour of the metal on its chemical composition and microstructure. This is not possible without a good understanding of dislocations.

Dislocations are defects in the atomic structure of metals which collectively, at the macroscopic scale, determine how metals deform. For this reason, any macroscopic model that aims for a predictive power has to take into account the presence of these defects. However, since the typical number of dislocations in even a small sample of metal is very high, formulating a model that keeps track of every single dislocation is unfeasible except for very small-scale problems.

Although good models for dislocations are available at the level of the atomic lattice, it is not yet well-understood how to incorporate the effect of their presence and motion in a model at the macroscopic, engineering scale. This challenge has become the focus of intensive research in the last decade - in the engineering community because of the need of good macroscopic models for the development of new metals and alloys, and in the mathematical community for the central role it plays in understanding complex multiscale systems. Over the years, and in different communities, this challenge has been approached in several ways. The great advantage of a rigorous approach is that it is exact; the price to pay, however, is a strong limitation on the configurations that can be analysed. In the majority of the mathematical literature, dislocations are modelled as straight and parallel lines, while they are in fact three-dimensional curves. This assumption reduces the complexity of the theory enormously, although the mathematical challenges in this idealised setting are still countless.

A special configuration that has received great attention in recent years is that of vertically periodic dislocation walls, similar to low-angle grain boundaries. One of the reasons why they are so popular is the general belief that they represent minimum energy arrangements for dislocations.

The proposed research poses a more fundamental question: what are the energetically favourable configurations of dislocations? And, also, how do low-energy dislocation structures vary when a small but non-zero temperature is introduced in the model?

Low-energy dislocation structures (LEDS) like walls, clusters and cells are one of the main features of the microstructure in metals. These high-density configurations increase the resistance of the material against plastic slip, thus leading to a stronger material. Characterising and hence exploiting and optimising LEDS is a key step in materials design: it would allow designers to construct lightweight structures which nevertheless have a high resistance to deformation, resulting, e.g., in safer and more fuel-efficient cars. Therefore, every advance in our research is relevant to mechanical engineering and industry.

The research in this project however goes beyond the specific example of defects in metals. Dislocations are a paradigmatic example of a complex particle system and the analysis developed here would be applicable to a variety of problems dealing with the derivation of the collective behaviour of a large number of individual agents, e.g., crowd and traffic dynamics, swarming, networks.

Planned Impact

Metals, and steel in particular, play a fundamental role in our everyday life and are used in countless applications, from the construction industry to transport, energy, packaging, and house appliances.

Different applications, however, require different material properties of the metals, which need to be designed to meet the requirements. Materials design in its turn requires a deep understanding of the dependence of the mechanical behaviour of the metal on its chemical composition and microstructure. This is not possible without a good understanding of dislocations, which are the main microscopic carriers of the permanent (or plastic) deformation of metals. Therefore every advance in dislocation theory is relevant to mechanical engineering and industry.

The proposed project aims at a threefold impact: (i) to derive new models of the collective dynamics of dislocations and promote them in the engineering community and in industrial research labs; (ii) to develop a rigorous mathematical methodology for upscaling and facilitate knowledge exchange between research in dislocations and dynamic complex systems; (iii) to train PhD students.

(i) Low-energy dislocation structures (LEDS) like clusters, walls and cells are one of the main features of the microstructure. These high-density configurations increase the resistance of the material against plastic slip, thus leading to a stronger material. Exploiting and optimising LEDS would allow designers to construct lightweight structures which nevertheless have a high resistance to deformation, resulting, e.g., in safer and more fuel-efficient cars.

The proposed project addresses exactly this point, its main aim being the characterisation of pattern-forming properties of dislocations at zero and finite temperature, and of how these patterns determine the upscaled evolution.

(ii) The research in this project goes beyond the specific example of defects in metals. Dislocations are a paradigmatic example of a complex particle system and the analysis developed here would be applicable to a variety of problems dealing with the derivation of the collective behaviour of a large number of individual agents, e.g., crowd and traffic dynamics, swarming, networks.

(iii) The training component of my proposal consists of several activities. The two-week visit of Marc Geers proposed in the project, and the course he will give at the University of Bath during his visit, as well as the lectures delivered by the Jose Antonio Carrillo, Maria Giovanna Mora and Mark Peletier will be very valuable for the PhD students. Moreover, all the students of the SAMBa Centre for Doctoral Training (CDT) at Bath will be invited to attend them, and this will contribute to their interdisciplinary training, which is the focus of the CDT. All the students and staff will be moreover invited to take part in Workshop 1 and Workshop 2 that will be held during the time of this project.

Publications

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Cagnetti F (2019) G-convergence of free-discontinuity problems in Annales de l'Institut Henri Poincaré C, Analyse non linéaire

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Cagnetti F (2019) Stochastic Homogenisation of Free-Discontinuity Problems in Archive for Rational Mechanics and Analysis

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Carrillo J (2019) The Ellipse Law: Kirchhoff Meets Dislocations in Communications in Mathematical Physics

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Garroni A (2019) Convergence and Non-convergence of Many-Particle Evolutions with Multiple Signs in Archive for Rational Mechanics and Analysis

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Mora M (2019) The Equilibrium Measure for a Nonlocal Dislocation Energy in Communications on Pure and Applied Mathematics

 
Description The main achievement of this grant is the successful bridging between the dislocation community and the nonlocal community. This has been achieved in a number of ways. First of all, with the paper "The equilibrium measure for a nonlocal dislocation energy", now publishes in Communications on Pure and Applied Mathematics, we pushed the methods developed for nonlocal energies beyond the case of radially symmetric potentials. In doing so, we also discovered surprising connections with random matrices and fluid dynamics. This paper has already led to another article, "The ellipse law: Kirchhoff meets dislocations", in collaboration with a group of harmonic analysts in Barcelona. This paper has just been accepted in Commun. Math. Phys.
The research initiated in these papers has opened up several research avenues that we are currently exploring.

To bring the two communities further together, I have organised a conference at Bath (co-organised with Prof. Mora from Pavia) titled "Nonlocal interactions: Dislocations and beyond" in June (11-14) 2018, which was very successful and attracted big names in both fields. I have also written an article on my research on the London Mathematical Newsletter, to spread the results to a wider audience.
Exploitation Route The results we obtained have already triggered further questions which are being explored by myself, and others. The long-term goal is to share these findings with engineers and steel industry.
Sectors Manufacturing, including Industrial Biotechology,Transport

URL http://cvgmt.sns.it/person/62/
 
Description Collaboration with a research group in Barcelona (Joan Verdera and Joan Mateu). 
Organisation University of Barcelona
Department Faculty of Mathematics and Computer Science
Country Spain 
Sector Academic/University 
PI Contribution We wrote a paper together and we are currently working on another joint project.
Collaborator Contribution We wrote a paper together and we are currently working on another joint project.
Impact A joint paper so far.
Start Year 2017
 
Description Organisation of a conference "Nonlocal interactions: Dislocations and beyond", 11-14 June 2018, University of Bath 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other audiences
Results and Impact It was a conference for experts in two different fields, dislocations and nonlocal interactions, that my research is bridging across.
Year(s) Of Engagement Activity 2018
URL http://www.macs.hw.ac.uk/~ls121/Dislocations-conference.html
 
Description Writing an article on my research for the London Mathematical Society newsletter, "Minimisers for nonlocal energies" 
Form Of Engagement Activity A magazine, newsletter or online publication
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Other audiences
Results and Impact I was approached by one of the editors of the London Mathematical Society Newsletter after one of my talks and I was asked to write an article on my research for the newsletter. At the moment I have submitted my draft, and the article should appear on the issue in March or the coming one.
Year(s) Of Engagement Activity 2018
URL https://www.lms.ac.uk/sites/lms.ac.uk/files/files/NLMS_477_reduced_0.pdf