OxMOS / MULTIMAT

Workshop on Microstructure

Workshop on Microstructure

2–5 September 2007, University of Oxford

The programme will combine some longer talks by invited speakers (see below) and some shorter invited talks by participants.

14:00–19.00 | Registration in Mary Ogilvie Foyer |

19:00–21:00 | Reception and cold fork buffet dinner in Dining Hall |

9:30–10:30 | Charlie Elliott (Sussex) — Computational approach to dealloying by surface dissolution |

10:30–11:00 | Coffee in Mary Ogilvie Foyer |

11:00–12:00 | Mike Finnis (Imperial) — Grain boundaries at the atomic scale, theory, mathematics and simulation |

12:00–12:30 | Ralf Drautz (Materials, Oxford) — Valence-dependent analytic bond-order potential for transition metals |

12:30–14:00 | Lunch in the Dining Hall |

14:00–14:30 | Christoph Ortner (Oxford) — On local minimizers in rate-independent fracture |

14:30–15:00 | Chiara Zanini (Leipzig) — Quasistatic crack growth and viscous approximations |

15:00–15:30 | Steven Kenny (Loughborough) — Multiscale modelling of nanoindentation |

15:30–16:00 | Tea in Mary Ogilvie Foyer |

16:00–17:00 | Zhiping Li (Beijing) — Numerical methods for crystalline microstructure |

17:00–19:00 | Tour of the Museum of the History of Science (17:30 and 18:30) |

(17:00–19:30) | OxMOS Steering Committee Drinks and Meeting, Seminar Room 5 |

20:00–21:00 | Dinner in the Dining Hall |

(20:00–21:00) | OxMOS Steering Committee Dinner at Queen’s College |

From 9.30–12.30 there will be two parallel activities:

- a MULTIMAT Board meeting on Framework 7 in Seminar Room 5 (with break for coffee)
- a session of introductory lectures designed primarily for research students and postdocs in Mary Ogilvie Lecture Theatre. The programme for these lectures is as follows.

9:30–10:00 | Nicolas Van Goethem (École Polytechnique) — A brief review of the geometry of dislocations in single crystals |

10:00–10:30 | Erell Bonnot (Barcelona) — Acoustic emission during martensitic phase transformations |

10:30–11:00 | Francisco-Jose Perez-Reche (Padova) — Martensitic transformations: Models for hysteresis, cycling and criticality |

11:00–11:30 | Coffee in Mary Ogilvie Foyer |

11:30–12:00 | Nils Wiese (Glasgow) — Investigating magnetic materials in the TEM |

12:00–12:30 | Antonio Capella-Kort (Bonn) — Introduction to the Micromagnetic Model |

12:30–14:00 | Lunch in the Dining Hall |

14:00–14:30 | Andrea Braides (Rome) — The use of Gamma-convergence in the analysis of multiscale problems |

14:30–15:00 | Rémi Delville (Antwerp) — TEM investigation of twinning in NiTi ternary alloys with special lattice parameters |

15:00–15:30 | Carlos Mora Corral (Oxford) — A variational model for sharp and smooth interfaces in solids |

15:30–16:00 | Tea in Mary Ogilvie Foyer |

16:00–17:00 | Wim Vanroose (University of Antwerp) — How to find macroscopic states based on the microscopic evolution law? |

17:00–20:00 | Free / |

MULTIMAT Board meeting in Upper Common Room | |

20:00–21:30 | BBQ dinner on the Claire Palley Lawn |

9:30–10:00 | Benedetta Cerruti (University of Barcelona) — Random field Potts model. Hysteresis and microstructure |

10:00–10:30 | Duvan Henao (Oxford) — Convergence of regularized minimizers for cavitation problems in nonlinear elasticity |

10:30–11:00 | Michal Landa (Prague) — Material characterisation and elastic constants investigated by ultrasound |

11:00–11:30 | Coffee in Mary Ogilvie Foyer |

11:30–12:00 | Zaoyang Guo (Glasgow) — Constitutive modelling of fibre-reinforced hyperelastic composites in finite elasticity with application to soft tissue |

12:00–12:30 | Anja Schlömerkemper (Leipzig) — On a discrete-to-continuum limit involving multiple scales and its application to magnetic forces |

12:30–13:00 | Martin Kružík (Prague) — A continuum model of misoriented dislocation cell structure formation |

13:00–14:00 | Lunch in the Dining Hall |

14:00 | Close |

*Ralf Drautz and David Pettifor, Department of Materials, University of Oxford, Oxford, UK*

The derivation of robust interatomic potentials is a key step for bridging from the electronic to the atomistic modelling hierarchy in materials science. We present an analytic interatomic bond-order potential (BOP) that depends explicitly on the valence of the transition metal element [1]. This analytic potential predicts the structural trend from hcp to bcc to hcp to fcc that is observed across the non-magnetic 4d and 5d transition metal series. The potential also describes the different ferromagnetic moments of the alpha (bcc), gamma (fcc) and epsilon (hcp) phase of the 3d transition metal iron, the difference between the ferromagnetic and anti-ferromagnetic states as well as non-collinear spin-configurations. In addition, this new potential includes a correct description of alloy bonding within its remit.

In this talk we will show how the potential is derived from the tight-binding electronic structure and demonstrate that it may be regarded as a systematic extension of the second-moment Finnis-Sinclair potential to include higher moments. We will briefly discuss the application of the new bond-order potential to the modelling of wall materials for fusion reactors and the stability and kinetics of topologically close-packed phases in Ni-based superalloys.

[1] R. Drautz and D.G. Pettifor, Phys. Rev. B 74, 174117 (2006).

*Christoph Ortner (Oxford)*

I will review the theory of Francfort/Marigo (1998) as well as several “classical” theories for rate-independent fracture mechanics and use the insights gained there to define a natural notion of local minimizer (and the corresponding notion of stable solutions) for the Griffith functional. I then use this definition to analyse a standard numerical approximation scheme in a time-discrete setting.

*Chiara Zanini (Leipzig)*

We consider the propagation of a crack along a prescribed crack path and propose a notion of irreversible and quasistatic evolution of brittle fractures inspired by Griffith’s theory and based on a local stability criterion (instead of the global stability one introduced by Francfort and Marigo). Existence is obtained by means of a vanishing viscosity approach.

This talk collects a joint work with Rodica Toader (University of Udine) and more recent developments with Dorothee Knees and Alexander Mielke (WIAS, Berlin).

*E. McGee, S.D. Kenny and R. Smith
Department of Mathematical Sciences
Loughborough University
Loughborough
LE11 3TU*

Many processes in materials science occur on multiple length scales. We will present a multiscale model that links finite element methods to classical molecular dynamics. The methodology is novel in that it allows dynamical simulations to be performed in 3D and with arbitrary potential functions. We will illustate the methodology by demonstrating its use in simulating the nanoindentation process.

Nanoindentation is an experimental technique used to measure the hardness of materials on the nanoscale. Traditionally either molecular dynamics or finite element methods have been used to model this process. The difficulty in applying finite element models to these problems comes from producing a suitable model of the plastic behaviour of the material. Whilst in molecular dynamics models the problem is one of the influence of the boundary conditions on the results.

We will present a 3D multiscale model of nanoindentation that couples a molecular dynamics (MD) model with a finite element (FE) model. The MD model is used to describe the tip and the material around the tip at the atomic scale, allowing for an accurate description of plastic deformation. The FE model is used to describe the long range elastic fields in the material. We will show how this model gives contact pressures, which are in better agreement with experimental results than traditional atomistic only models, illustrating that a correct description of the long range elastic field is essential.

Nanoindentation of the Au (100) surface using a second nearest neighbour embedded atom potential will be used to illustrate the methodology.

*Zhiping Li (Beijing)*

In this talk, I will report the numerical methods for crystalline microstructure I have been working on in recent years. There are basically three classes of methods. The first is the mesh transformation method developed to reduce the strong mesh dependency of the finite element solutions for microstructure problems. The second is the finite order rank-one convex envelope method, which is developed to calculate nested laminated microstructures. The third is the multiscale method, which is developed to compute nonhomogeneous microstructures. The ideas of the methods, as well as numerical and analytical results will be presented.

*Nicolas Van Goethem (École Polytechnique)*

In this talk, I will briefly review the static theory of dislocations and disclinations in single crystals at the mesoscopic scale. It will be emphasised that a rigorous description should not involve the notion of reference configuration. As a main result, a relation between the defect invariants and the strain incompatibility will be given in the 2D and 3D cases. Moreover, the homogenisation from the meso- to the macroscale will be discussed, together with the non-Riemannian description of the macroscopic dislocated crystal.

*E. Bonnot
Departament d’Estructura i Constituents de la Matèria, Universitat de Barcelona
Diagonal 647, Facultat de Física, 08028 Barcelona, Catalonia*

I will present the key aspects of the theory of the acoustic emission and will discuss why it is an interesting technique when dealing with martensitic transformations. This will be illustrated with results recently obtained corresponding to mechanical and thermal experiments in different martensitic materials. I will show that these results enable to study kinetic features of martensite propagation fronts.

*Francisco-Jose Perez-Reche (Università di Padova)*

Reversible martensitic transformations involve a coordinated distortion of the crystal lattice and belong to the class of ferroelastic first-order phase changes with athermal character. In such systems the macroscopic strain discontinuity typically splits into a set of bursts (avalanches) corresponding to transitions between neighboring metastable states. The hysteresis observed in martensites is related to the evolution of the system following a metastable path. The kinetic features of such systems evolve during the process known as training that consists in cycling through the martensitic transformation. In particular, the hysteresis profile smoothens and the size distribution of the avalanches evolves towards a power law (critical behavior) under thermal cycling. This lecture will be an introduction to the modeling of hysteresis, cycling, and criticality in first-order phase transitions with particular emphasis in martensites.

*N. Wiese, Dept. of Physics and Astronomy, University of Glasgow, U.K.*

The talk is intended to give a broad overview about the experimental work on magnetic materials in Glasgow. An introduction of several sample preparation techniques, the high-resolution Lorentz microscopy modes of transmission electron microscopy (TEM) and examples of domain wall investigations on various material systems will be given.

After the introduction, several examples of our studies on magnetic domain walls in sub-micrometre scale NiFe elements and wires will be highlighted, including recent results on the cross-tie domain state, and creation, propagation, and transformation of domain walls in magnetic nanowires.

*Antonio Capella-Kort (Bonn)*

The Micromagnetic Model is a model problem for applied calculus of variations. This is a well accepted model and there is a enormous well documented experimental data to test the model. The model is based in a variational principle for a non convex, non local energy functional for the magnetization vector. These characteristics make it an interesting mathematical problem, with many local minima that account for the different magnetization patterns found in the experiments. The aim of the talk is to present the model explaining its different elements, and over an specific setting, go over its analysis and outcomes.

The talk is intended to be an introduction/overview to the subject.

*Andrea Braides (Rome)*

Gamma-convergence is a very useful tool for the description of complex variational problems involving small parameters (homogenization, phase transitions, atomistic systems, etc.). The underlying idea of the Gamma-convergence approach is to substitute an energy with a small parameter by another one where the dependence on this parameter has been averaged out (Gamma-limit), or simplified (Gamma-development). The resulting energy ‘justified by Gamma- convergence’ may however fail to represent the relevant behaviour of the original energy in its sensitiveness on relevant external parameters (boundary conditions, forcing terms, etc.) or be of a type different from a desired form commonly used by practitioners. I will present some proposals on how to use Gamma-convergence to overcome (some of) those drawbacks. On the one hand Gamma-convergence may be used as an equivalence relation (so that a wide range of energies may be equivalent to the same Gamma-limit, or Gamma-development, even if defined on different function spaces). On the other hand we may focus on the ‘singular’ external parameters where the description given by the Gamma-convergence approach is not ‘uniformly close’ to the original one. Those parameters may be singled out by simple necessary conditions and in many situations constitute a simple set of isolated points. Close to those values a finer analysis is necessary by the computation of a family of Gamma-limits. Once this analysis is performed, the issue of the construction of ‘uniformly-equivalent theories’ can be addressed.

Joint work with Lev Truskinovsky.

*Rémi Delville — EMAT, University of Antwerp
Dominique Schryvers — EMAT, University of Antwerp
Jerry Zhang — University of Minnesota
Richard D. James — University of Minnesota*

Alloys with special lattice parameters with a middle eigenvalue of their transformation matrix satisfying λ_{2}=1 are supposed to present an austenite phase compatible with a single variant of martensite. Work conducted by the team of Minneapolis suggests that this could be a relevant parameter to characterize hysteresis. Theoretical considerations predict also that compound twins should be observed as λ_{2}<1 and type I/II as λ_{2}>1. TEM investigation of TiNiPd, TiNiAu and TiNiPt over a range of composition leading to a λ_{2} crossing over 1 is used to characterize twinning modes in these alloys. First observations indicate that in real systems with λ_{2} close to 1, large twinless areas indeed exist but also single twins or sequences of twin stacking can occur. Whether this is due to local composition or stress distribution remains to be investigated.

*Carlos Mora Corral (Oxford)*

We propose a variational model to study interfaces in one-dimensional elastic solids. The model consists of a perturbation of a double-well potential, and allows both smooth interfaces. We introduce three parameters in the model: two of them are constants of the body and measure the resistance of the material to the creation of, respectively, smooth and sharp interfaces; the third one controls the boundary conditions. We analyze the regime of the parameters for which there are smooth interfaces, sharp interfaces, or no interfaces at all.

*Wim Vanroose (Dept. of Mathematics, University of Antwerp)*

We discuss how traditional methods such as Newton–Krylov to find fixed points of partial differential equations can be used to find the fixed points of problems with multiple scales. These Newton–Krylov methods only require a matrix-vector product to construct a basis for the Krylov subspace and calculate the Newton corrections. We show that it is possible to find macroscopic states as steady states, traveling waves or periodic solutions of a system that is described by microscopic evolution law only and no macroscopic description is available. We use a special matrix-vector product that lifts the macroscopic state vector to a microscopic state that is subsequently evolved over a short time with the help of the microscopic evolution law. The result of this evolution is then projected back on the macroscopic variables and is interpreted as the result of the matrix-vector product. This method was first proposed by Kevrekidis and co-workers and is known as the equation-free method. It has been applied to study the motion of material defects in the presence of defects, to study macroscopic behaviour of water molecules in nanotubes, bi-stability in gene regularory networks and several other physical systems.

*Benedetta Cerruti (University of Barcelona)*

A model for the study of microstructures and hysteresis in a first-order phase transitions from a single varian phase to a multivariant phase is presented. It is based on a modification of the Random Field Ising model with metastable dynamics. We study the hysteresis loops shape varying the hamiltonian parameters, focusing on the loop area and asymmetry. Depending on the parameters values we are able to reproduce three kind of microstructures. Linear domains, parallel to the spin direction, planar domains or disordered phase. Moreover we study the avalanche statistics for the two loop branches. In the model we include a ‘dipolar’ term truncated to nearest neighbor, thus at this level it is not suitable to simulate the behavior of real microstructures. Nevertheless, the results are quite promising and the inclusion of higher order interaction is in progress.

*Duvan Henao (Oxford)*

In a recent paper, J. Sivaloganathan, S. Spector and V. Tilakraj considered energy minimizers for nonlinear elasticity in which cavitation is allowed only at a single prescribed point in the reference configuration. They showed that such energy minimizers can be seen as the deformation of a body containing an infinitesimal pre-existing void. We generalize their result, which constitutes a first step towards numerical simulation of cavitation, to the case when the possibility of more than one point discontinuity is considered.

*Michal Landa, Prague*

Resonant ultrasound spectroscopy (RUS) is a well-known technique for investigation of elastic properties of solids based on the inversion of natural frequencies of free elastic vibrations of a small simple shaped specimen. However, low symmetry phases appearing in ferroelastic materials (shape memory alloys) commonly exhibit specific behaviors, particularly (i) strong elastic anisotropy, (ii) unusual temperature dependence of some elastic constants and (iii) natural tendency of ferroelastics to form twinned microstructures.

We summarize recent experimental and theoretical improvements in particular: (i) the dedicated sample preparation methods, (ii) the reliable vibration mode identification exploiting scanning laser interferometry measurement, (iii) well-posed resonance inversion based on analytical expression of the gradient and the Hessian of the minimized error function and (iv) estimation of accuracy of evaluated elastic constants or their combinations.

It is not always possible to obtain an interface free single crystal of some SMAs. Then the measurement must be done on microtwinned crystals. For such case, a homogenization algorithm based on the macroscopic deformation response of the layered structure is applied.

Selected results, recently obtained by applying the RUS method to investigate phases in shape memory alloys, will be presented and discussed.

*Zaoyang Guo
Departments of Mechanical and Civil Engineering, University of Glasgow, UK*

Recently we developed a homogenization procedure to model the mechanical behaviour of fibre-reinforced hyperelastic composites in finite elasticity and applied it to soft tissue. In this constitutive modelling framework, for incompressible composites, the deformation gradient is decomposed multiplicatively into two parts: a uniaxial deformation along the fibre direction and a subsequent shear deformation. This permits the fibreâ€“matrix interaction caused by inhomogeneous deformation to be estimated by using effective properties from conventional composites theory based on small strain linear elasticity and suitably generalized to the present large deformation case. To show the efficiency of this framework, a transversely isotropic hyperelastic model is proposed to describe the mechanical behaviour of fibre-reinforced soft tissue. The multiplicative decomposition of deformation can be extended for compressible composites by adding an equi-biaxial deformation on the transverse plane to allow volume change. The mechanical behaviour of a neo-Hookean solid with aligned cylindrical voids is modelled based on this extended decomposition of deformation.

*Anja Schlömerkemper (Leipzig)*

I will present recent results about magnetic force formulae for two rigid magnetic bodies. Using different scales to describe the distance between the bodies, we discuss continuum limits of atomistic dipole-dipole interactions. This allows to link a classical force formula and a limiting force formula obtained earlier as a discrete-to-continuum limit for bodies being in contact on the atomistic scale.

This is joint work with B. Schmidt.

*Martin Kružík
Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic;
Faculty of Civil Engineering, Czech Technical University*

Within the framework of continuum mechanics, the formation of misoriented dislocation cells can be explained as a result of a trend to reduce the energetically costly hardening in multi slip by decreasing locally the number of active slip systems. The finite size of real cells is controlled by short-range dislocation interactions. Within the proposed framework the cell size is a result of a compromise: bulk strain and dissipative energy tend to decrease the size, while the short-range interactions restrict that tendency. We analyze effects of the short-range interactions using a model of a crystal deformed by symmetric double slip, where plastic strain is carried by straight, parallel, edge dislocations. This is a joint work with J. Kratochvíl (Prague) and R. Sedláček (Munich).