# Partially Ordered Materials

## Mathematical Perspectives and Challenges

## University of Bath, 21 December 2016

This workshop is organised in conjunction with the LMS South Wales and South Wales Regional Meeting on 20 December.10.00 | Coffee and registration |

10.40 | Roger Moser (Bath) Interaction of Neel walls in micromagnetics |

11.20 | David Bourne (Durham) Energy-driven pattern formation in nonlinear elasticity |

12.00 | Nicolas Dirr (Cardiff) Homogenization of a mean field game system in the small noise limit |

12.40 | Lunch |

14.00 | Xianmin Xu (Chinese Academy of Sciences) Modelling and simulations for cavitation and fracture in nonlinear elasticity |

14.40 | Federica Dragoni (Cardiff) Stochastic homogenisation for degenerate Hamilton-Jacobi equations. |

15.20 | Coffee |

15.40 | Kirill Cherednichenko (Bath) Extreme localisation property for eigenfunctions of one-dimensional high-contrast periodic problems with a defect |

16.20 | Isaac Chenchiah (Bristol) Quasi-static brittle damage evolution with multiple damaged elastic states |

All talks take place in the Wolfson Lecture Theatre, room 1.7 in building 4 West. Refreshments are served in the atrium on level 1 in 4 West.

Organiser: Apala Majumdar

Abstracts

- David Bourne
*Energy-driven pattern formation in nonlinear elasticity*In this talk I will discuss self-similar folding patterns in a compressed elastic film that has partially delaminated from a substrate. Such patterns were observed in recent experiments to fabricate nanoscale structures from semiconductor films via self-assembly. We model the system with a suitable energy functional and prove rigorous upper and lower bounds on the minimum value of the energy. Some of the upper bound constructions match the patterns observed in experiments. This is joint work with Sergio Conti and Stefan Mueller.

- Isaac Chenchiah
*Quasi-static brittle damage evolution with multiple damaged elastic states*We consider quasi-static evolution of brittle damage in the context of geometrically-linear elasticity. By allowing for anisotropic elastic moduli and multiple damaged states we present the issues for the first time in a truly elastic setting. We show the existence of solutions, and that energetic evolutions are also threshold evolutions, thus relating (spatially) global and local criteria for damage.

- Kirill Cherednichenko
*Extreme localisation property for eigenfunctions of one-dimensional high-contrast periodic problems with a defect*Following a number of recent studies of resolvent and spectral convergence of differential operators describing the behaviour of periodic composite media with high contrast, I shall discuss the corresponding one-dimensional version that includes a ``defect'': an inclusion of a fixed size with a given set of material parameters. It is shown that the eigenvalues of the related operator in the space of square integrable functions converge to points of the spectrum of a limit operator whose continuous spectrum has a band-gap structure. In cases when a limit eigenvalue is situated in a gap of the continuous spectrum, this leads to the existence of a wave propagating along the defect and extremely localised in its neighbourhood. This is joint work with M. Cherdantsev (Cardiff) and S. Cooper (Bath).

- Federica Dragoni
*Stochastic homogenisation for degenerate Hamilton-Jacobi equations*In the talk I Investigate the limit behaviour for a family of Cauchy problems for Hamilton-Jacobi equations describing a stochastic microscopic model. The Hamiltonian considered is not coercive in the total gradient. The Hamiltonian depends on a lower dimensional gradient variable which is associated to a Carnot group structure. The rescaling is adapted to the Carnot group structure, therefore it is anisotropic. Under suitable stationary-ergodic assumptions on the Hamiltonian, the solutions of the stochastic microscopic models will converge to a function independent of the random variable: the limit function can be characterised as the unique viscosity solution of a deterministic PDE. The key step will be to introduce suitable lower-dimensional constrained variational problems. In collaboration with Nicolas Dirr, Claudio Marchi and Paola Mannucci.

- Nicolas Dirr
*Homogenization of a mean field game system in the small noise limit*(Joint work with Annalisa Cesaroni and Claudio Marchi)

The theory of mean field games, introduced by P.-L. Lions and J.M. Lasry in 2007, describes the effective behaviour of very many rational agents, providing a rigorous microscopic justification for macroscopic models. The resulting forward-backward system of equations couples a Hamilton-Jacobi equation and a Kolmogorov-Fokker-Planck equation.

This talk concerns the simultaneous effect of homogenization and of the small noise limit for a 2nd order mean field games (MFG) system with local coupling and quadratic Hamiltonian. We show under some additional assumptions that the solutions of our system converge to a solution of an effective 1st order system whose effective operators are defined through a cell problem which is a 2nd order system of ergodic MFG type. We provide several properties of the effective operators and we show that in general the effective system loses the MFG structure.

- Roger Moser
*Interaction of Neel walls in micromagnetics*A Neel wall is a type of transition layer between two magnetisation states in thin films of ferromagnetic materials. We study a model that can be regarded as an Allen-Cahn type model for phase transitions with an additional nonlocal term, but which also has a lot in common with models from the theory of Ginzburg-Landau vortices in superconductivity.

When we study several Neel walls in a row, the nonlocal component of the model gives rise to competing interaction terms and we have attraction between some and repulsion between other pairs of Neel walls. This makes the analysis quite delicate and we observe phenomena that have no counterpart in Allen-Cahn models or the theory of Ginzburg-Landau vortices.

This is joint work with Radu Ignat (Toulouse).

- Xianmin Xu
*Modelling and simulations for cavitation and fracture in nonlinear elasticity*Cavitation is a material failure phenomenon in rubber like nonlinear elastic materials. In the talk, we will introduce some numerical approaches to the cavitation problem. In particular, we illustrate a new variational model, which uses a phase-field function to capture material failure areas. The model is relatively easy to implement numerically and can simulate both cavitation and fracture occurred in these materials. We present several numerical experiments, including void coalescence, void creation under tensile stress, failure in perfect materials and in materials with hard inclusions. The experimental results show the ability of the model and the numerical method to study different failure mechanisms in rubber-like materials. This is a joint work with Duvan Henao and Carlos Mora-Carral.