# Abstracts

*Variational problems for antiferromagnetic systems*

Abstract. Antiferromagnetic spin systems exhibit some features, as frustration or pattern formation, that bring some interesting issues in their continuum approximation. I will present some questions and (few) answers on this subject

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*Stress regularity for a new quasistatic evolution model of perfectly plastic plates*

Abstract. In this talk we will dicuss some properties of solutions to a quasistatic evolution problem for perfectly plastic plates, that has been recently derived from three-dimensional Prandtl-Reuss plasticity. We will prove that the stress tensor has locally square-integrable first derivatives with respect to the space variables. We will also exhibit an example showing that the model under consideration has in general a genuinely three-dimensional nature and cannot be reduced to a two-dimensional setting. The talk is based on a joint work with Prof. Maria Giovanna Mora.

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*A Coupled Framework For Climb-Assisted Glide In Discrete Dislocation Plasticity*

Abstract. It is now well established that the plastic deformation of crystalline solids is size dependent at the micron scale for a range of loading conditions. While there are many underlying reasons for these size effects, attention has been primarily focussed on situations where plastic strain gradients are generated. Models typically tend to over-predict the experimentally observed size effects as they neglect a range of dislocation relaxation mechanisms. These mechanisms include dislocation cross-slip and dislocation climb. Dislocation climb requires the diffusion of vacancies and hence signiﬁcant amounts of dislocation climb only occur at temperatures above a third of the melting temperature - in these cases mass transport reduces the plastic strain gradients and thereby reducing the effect of specimen size. However, even at lower temperatures, dislocations can surmount small obstacles with the aid of small amounts of climb. This prevents the build-up of large dislocation pile-ups which consequently again relaxes stresses. The coupling of vacancy diffusion with dislocation motion is a true "multi-scale" problem as vacancy/dislocation interaction is essentially a dislocation core effect. We present a two-dimensional discrete dislocation plasticity framework coupled with vacancy diffusion wherein dislocation motion occurs by both climb and glide. The effect of dislocation climb is explored for a range of problems including size effects in bending of crystals, metal-matrix composites and passivated films. Dislocation climb typically tends to reduce strength enhancements that occur with decreasing size but in some surprising cases can also result in strength increases.

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*Multi-scale modeling from the top down.*

Abstract. In this talk I would like to discuss methods of modeling microstructure formation on mesoscopic time scales. On mesoscopic length scales, sharp interface descriptions are often used to describe such phenomena. In this approach interfaces or domain walls are replaced with boundary conditions that are then coupled to equations used to describe bulk phenomena, such as diffusion. While such a description is quite useful it can be difficult to implement for a complex set of boundaries. To overcome this problem various methods, such as phase field modeling and the level set method, were developed to deal with the complex microstructures that emerge in everyday processes such as solidification. While these approaches have many advantages it is often difficult to naturally incorporate atomistic features, such as dislocations, grain boundaries and crystalline anisotropy (particularly in polycrystalline materials), that can alter both microstructure formation and material properties. These features can be introduced (at a computational cost) by considering fields that vary on atomic length scales as is done in phase field crystal models. In this talk I would to discuss the connection between these different approaches and introduce the so-called amplitude approach which provides a natural bridge from phase field crystal to phase field models.

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*Stability of screw dislocations in an anti-plane lattice model*

Abstract: "Dislocations are geometric lattice defects which are the carriers of plastic deformation in crystals, and the classical approach to modelling them has been to use continuum linear elasticity theory. However, such an approach introduces singularities with infinite elastic energy, indicating the breakdown of the assumption that the body is well-approximated by a continuum. To better understand the behaviour of dislocations at the microscopic level, this talk studies a lattice model in which these singularities are not present, proving the existence of stable configurations of screw dislocations in a variety of domains with arbitrary net Burgers vector. Along the way, we show that continuum linear elasticity does indeed provide a good approximation of the equilibrium configurations."

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*Recent review on nanoscale deformation mechanisms in coarse-grained and nanocrystalline materials using advanced TEM.*

Abstract. The present work focuses on the investigation of the elementary defect mechanisms controlling the plastic deformation of coarse-grained and nanocrystalline (nc) materials. Special attention was paid to the interaction of grain boundaries, phase boundaries and twin boundaries with the nanoscale fundamental atomistic deformation and cracking mechanisms and the resulting mechanical properties involving the strength, ductility and creep. More precisely, the investigations were performed on bulk coarse-grained Fe-Mn-C, Fe-Mn-Si-Al and Ti-Mo alloys exhibiting TWIP (twinning induced plasticity) and TRIP (transformation induced plasticity) effects as well as nc palladium and aluminium systems in the form of thin films and micro/nanowires with a high density of internal interfaces. Advanced nanocharacterization and testing methods were used to unravel the elementary processes activated at the micro and nanoscale. It mainly relies on transmission electron microscopy (TEM) techniques including aberration corrected TEM and automated crystallographic orientation mapping in TEM coupled with various ex-situ and in-situ micro and nanomechanical testing techniques. The practical outcomes consist in the development of predictive models and of improved processing routes as well as improved characterization methods towards the development of new materials with enhanced performances.

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*Avoiding spurious force artifacts in a fully-nonlocal energy-based quasicontinuum*

Abstract. The quasicontinuum (QC) method was introduced to effciently coarse-grain crystalline atomistic ensembles in order to bridge the scales from individual atoms to the micro- and mesoscales. Various flavors of the QC method have been reported which differ by their local vs. nonlocal thermodynamic formulation, their approximation of the total Hamiltonian or of the interatomic forces, their interpolation schemes, and their model adaption techniques, to name but a few. Here, we present a fully-nonlocal 3D QC formulation which does not conceptually differentiate between atomistic and coarse-grained domains. To this end, we introduce optimal energy-based summation rules based on a set of sampling atoms (different from the representative atoms) to approximate the total Hamiltonian. We show that the new QC scheme produces minimal approximation errors and thus marginal residual and spurious force artifacts. Our sampling scheme, similar in spirit to quadrature rules, allows for automatic model adaption and guarantees no force artifacts in the limits of full atomistic resolution as well as in large elements. Approximation errors and force artifacts in the intermediate regime are small compared to all previous schemes of comparable effciency. We present selected examples of small-scale plasticity to demonstrate the capabilities of the new computational framework.

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*Ghost-force free coupled energies based on bond volumes.*

Abstract. In this talk we present a systematic construction of energy based numerical methods free of ghost forces in lattices arising in crystalline materials. The construction hinges on establishing a connection of the coupled system to conforming finite elements. Key ingredients are: (i) a new representation of discrete derivatives related to long range interactions of atoms as volume integrals and (ii) the construction of an underlying globally continuous function representing the coupled modeling method.

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*Atomistic Long-Term Simulation of Heat and Mass Transport*

Abstract: We formulate a theory of non-equilibrium statistical thermodynamics for ensembles of atoms or molecules. The theory is an application of Jayne's maximum entropy principle, which allows the statistical treatment of systems away from equilibrium. In particular, neither temperature nor atomic fractions are required to be uniform but instead are allowed to take different values from particle to particle. In addition, following the Coleman-Noll method of continuum thermodynamics we derive a dissipation inequality expressed in terms of discrete thermodynamic fluxes and forces. This discrete dissipation inequality effectively sets the structure for discrete kinetic potentials that couple the microscopic field rates to the corresponding driving forces, thus resulting in a closed set of equations governing the evolution of the system. We complement the general theory with a variational meanfield theory that provides a basis for the formulation of computationally tractable approximations. We present several validation cases, concerned with equilibrium properties of alloys, quasistatic-to-dynamic nanovoid cavitation in metals, heat conduction in silicon nanowires and hydrogen desorption from palladium thin films, that demonstrate the range and scope of the method and assess its fidelity and predictiveness. These validation cases are characterized by the need or desirability to account for atomic-level properties while simultaneously entailing time scales much longer than those accessible to direct molecular dynamics. The ability of simple meanfield models and discrete kinetic laws to reproduce equilibrium properties and long-term behavior of complex systems is remarkable.

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*Homogenization of surface and length spin energies*

Abstract. The talk will focus on homogenization and Gamma-convergence of surface energies defined on lattice random spin systems in Z^d through bond interactions. We mostly dwell on nearest neighbors interaction systems, and consider both elliptic statistically homogeneous and dilute percolation cases.

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*Discrete variational models for screw dislocations in crystals*

Abstract. I will discuss some recent variational models for discrete screw dislocations in crystals. I will introduce a discrete framework for anti-plane elasticity in the main crystal structures, taking into account defects. Then, I will describe the statics and the dynamics of dislocations in such crystal structures, focusing on the limit as the lattice space tends to zero. This discussion collects some recent results obtained in collaboration with R. Alicandro, L. De Luca and A. Garroni.

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*Macroscopic Response, Instabilities and Phase Transitions in Elastomer Composites with Nematic Symmetry*

Abstract. We consider elastomers that are reinforced by aligned, but randomly positioned, short, stiff fibers and investigate their macroscopic mechanical behavior. For this purpose, we make use of analytical estimates [1] for (two-dimensional) fiber composites that were obtained by means of the second-order linear comparison homogenization technique. The estimates depend explicitly on the rotation of the fibers, which can act as a softening mechanism, leading to loss of strong ellipticity of the macroscopic response of the composite at a critical stretch—even when the pure elastomer is strongly elliptic. The bifurcation instability takes place when there is sufficient compression along the long axis of the fibers, and depends on the volume fraction and shape of the particles. The post-bifurcation solution is found by performing an appropriate lamination procedure leading to the quasi-convexification of the energy [2]. The change from the pre- to post-bifurcation solution can be interpreted as a symmetry-breaking, second-order “phase transition” from the original “nematic” state to a new “smectic” state consisting of layers of alternating fiber orientations. We show that these instabilities can be associated with “soft” deformation modes in the post-bifurcated regime that are somewhat analogous, but different to those observed in nematic liquid crystal elastomers [3].

References

[1] O. Lopez-Pamies and P. Ponte Castañeda. “On the overall behavior, microstructure evolution and macroscopic stability in reinforced rubbers at large deformations: I — Theory and II — Application to cylindrical fibers.” Journal of the Mechanics and Physics of Solids 54 (2006): 807-863.

[2] G. Geymonat, S. Müller and N. Triantafyllidis. “Homogenization of nonlinearly elastic materials, macroscopic bifurcation and macroscopic loss of rank-one convexity.” Archive for Rational Mechanics and Analysis 122 (1993): 231-290.

[3] A. DeSimone and G. Dolzmann. “Macroscopic response of nematic elastomers via relaxation of a class of SO(3)-invariant energies.” Archive for Rational Mechanics and Analysis 161 (1993): 181-204.

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*Defects in striped phases*

Abstract. We'll give an overview of recent results on defects in striped phases, as illustrated in the prototypical Swift-Hohenberg equation. Using ideas from dynamical systems, we study existence, stability, and wavenumber versus phase selection by defects, both analytically, using center manifolds and normal forms, as well as numerically, using continuation and defect/far-field decompositions. We also show how some of these techniques generalize to two-dimensional point-defects.

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*Two-scale homogenization of dynamics of micro-resonant materials*

Abstract. Physicists have recently introduced a term “meta-materials”, to describe specially designed composite materials whose macroscopic physical properties can be radically different from those of conventional materials. These properties are often due to the effects of the so-called "micro-resonances" and include 'negative’ materials, with implications for various desirable effects. Mathematicians have known this before under different names, e.g. high-contrast homogenization, double porosity type models, etc. We will review some background, as well as some more recent generalizations and applications. One is two-scale analysis of general “partially-degenerating” PDEs, with one application being to band gaps in photonic crystal fibers. Another is two-scale homogenization of dynamic problems with random micro-resonances, which appears to yield macroscopic effects akin to Anderson localization. Some of the work is joined with Ilia Kamotski (UCL) and Shane Cooper (Marseilles).

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*Critical behavior of skeletal muscles.*

Abstract. In contrast to inert matter, distributed biological systems are characterized by hierarchical network architecture with domineering long-range interactions. Even in the absence of metabolic fuel this leads to a highly cooperative passive mechanical behavior. In particular, systems with internal unfolding exhibit coherent macroscopic hopping resisting the destabilizing effect of finite temperatures. A minimal mechanical system exemplifying such behavior is a parallel bundle of bi-stable units linked by two shared backbones and its most well known biological prototype is the power-stroke in skeletal muscles. In this talk we review the mechanics of this model in the continuum limit. Our analysis suggests that the actual skeletal muscles function very close to the critical point exhibited by this system.

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*Optimal fine-scale structures in composite materials*

Abstract. A very classical problem consists in optimizing the structure of a composite material, for instance to achieve high rigidity against a prescribed mechanical loading. In the simplest case, the material is a composite of void and the elastic base material. The problem then reduces to finding the optimal topology and geometry of the structure. One typically aims to minimize a weighted sum of material volume, structure perimeter, and structure compliance (a measure of the inverse structure stiffness). This task is not only of interest for optimal material designs, but also represents a prototype problem to better understand biological structures. The high nonconvexity of the problem makes it impossible to find the globally optimal design; if in addition the weight of the perimeter is chosen small, very fine material structures are optimal that cannot even be resolved numerically. However, one can prove an energy scaling law that describes how the minimum of the objective functional scales with the model parameters. Part of such a proof involves the construction of a near-optimal design, which typically exhibits fine multi-scale structure in the form of branching and which gives an idea of how optimal geometries look like. (Joint with Robert Kohn)

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*Bayesian Uncertainty Quantification in the Evaluation of Alloy Properties with the Cluster Expansion Method*

Abstract. Parametrized surrogate models are used in alloy modeling to quickly obtain otherwise expensive properties such as quantum mechanical energies, and thereafter used to optimize, or simply compute, some alloy quantity of interest, e.g., a phase transition, subject to given constraints. Once learned on a data set, the surrogate can compute alloy properties fast, but with an increased uncertainty compared to the computer code. This uncertainty propagates to the quantity of interest and in this work we seek to quantify it. Furthermore, since the alloy property is expensive to compute, we only have available a limited amount of data from which the surrogate is to be learned. Thus, limited data further increases the uncertainties in the quantity of interest, and we show how to capture this as well. We cannot, and should not, trust the surrogate before we quantify the uncertainties in the application at hand. Therefore, in this work we develop a fully Bayesian framework for quantifying the uncertainties in alloy quantities of interest, originating from replacing the expensive computer code with the fast surrogate, and from limited data. We consider a particular surrogate popular in alloy modeling: the cluster expansion, and aim to quantify how well it captures quantum mechanical energies. Our framework is applicable to other surrogates and alloy properties.

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*Construction of Consistent Atomistic/Continuum Coupling Methods*

Abstract. We discuss the construction of consistent energy based atomistic/continuum (A/C) coupling methods for crystalline solids with defects. For general multi-body interactions on the 2D triangular lattice (and potentially for 3D lattices), we show that ghost force removal (patch test consistent) A/C methods can be constructed for arbitrary interface geometries. We prove that all methods within this class are first-order consistent at the atomistic/continuum interface. The rigorous error analysis of the coupling methods leads to their optimal implementation. For a range of benchmark problems, the resulting consistent coupling methods have the same convergence rates as optimal force-based coupling schemes. This is a joint work with Christoph Ortner (Warwick).

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*Scale-bridging for entropic flows in the presence of energy or noise*

Abstract. One often aims to describe systems out of equilibrium by the governing energy E and entropy S, as well as the corresponding evolution laws for E and S. How can we derive these ingredients of the macroscopic evolution from particle models? In recent years, a dynamic scale-bridging approach has been developed and applied to a number of problems; large deviation theory plays an important role. The talk will present some of these results, in particular discuss the Vlasov-Fokker-Planck equation as a system driven by energy and entropy. Time permitting, an approach of deriving stochastic equations mimicking the fluctuations in underlying mesoscopic models will be sketched as well, connecting fluctuating hydrodynamics and Wasserstein evolution.

This is joint work with Hong Duong and Mark. A. Peletier respectively Rob Jack.