# Theory and modeling of materials

**Statistical models for fracture and plasticity**We have worked in the characterization of fracture and plasticity in disordered media through the use of statistical lattice models, which allow for relatively simple descriptions of disorder and elasticity. The cornerstone in this respect has been for the last twenty years the Random Fuse Model (RFM) a lattice model for the fracture of solid materials in which as a further key simplification vectorial elasticity has been substituted with a scalar field. Recently, we have used the RFM to study the crossover from deterministic to statistical size effects when the sample has a large notch (1). This is a central problem in materials science and engineering, since the dependence on the sample size of the fracture strength has important implication for the safety and reliability of components and structures. In our study, we were able to show that statistical size effects always prevail at sufficiently large sizes even if a large flaw is present in the sample. We have also characterized the elusive fracture process zone ahead of the crack tip, where most of the damage processes take place (Figure 1). While brittle materials show dramatic size effects, with the strength going to zero as the sample size increases, ductile materials show much smaller size effects. A plastic version of the RFM has been used to study ductile yielding of disordered media and the relation to optimization (2). It has been argued in the past that yielding is an optimization process and that the yield surface, where the plastic deformation is concentrated, corresponds to a minimum energy surface. We have shown that this claim does not apply and the two surfaces differ. As a result, the global yield stress is lower than expected from naive optimization and the difference persists as the sample size increases.

**Dislocation dynamics**Our activity in dislocation dynamics involves, 2D simulations of interacting dislocation points, 2D models of flexible dislocation lines gliding in a plane with solute atoms, and 3D dislocation dynamics simulations.

**: We have considered the behavior of parallel straight edge dislocations moving in a single slip system under the action of constant stress. Dislocations interact with each other through the long-range elastic stress field they produce in the host material. The long-range and anisotropic character of this interaction is responsible for the remarkable features of plastic flow. In addition, we have included in the model the multiplication of dislocations and their annihilation. The model displays an intermittent steady-state characterized by dislocation avalanches with a power law distribution of energies that is in close agreement with the experiments (3). Due to their complicated mutual interactions, most of the time dislocations are jammed into metastable configurations, like walls and dipoles (Figure 2). The progressive jamming of dislocations into metastable structure also explains the power law creep relaxation observed in a vast variety of materials. Our analysis allows to reduce the scaling found in Andrade creep to a second-order non-equilibrium critical phenomenon: a jamming transition of the dislocations (4).**

*2D simulations***: We investigated the depinning transition occurring in dislocation assemblies. In particular, we considered the cases of regularly spaced pileups and low angle grain boundaries interacting with a disordered stress landscape provided by solute atoms, or by other immobile dislocations present in non-active slip systems. Using linear elasticity, we computed the stress originated by small deformations of these assemblies and the corresponding energy cost in two and three dimensions. Contrary to the case of isolated dislocation lines, which are usually approximated as elastic strings with an effective line tension, the deformations of a dislocation assembly cannot be described by local elastic interactions with a constant tension or stiffness. A nonlocal elastic kernel results as a consequence of long range interactions between dislocations (5). We have analyzed the effect of this in creep (5) and slip line grown in metallic alloys (6).**

*Flexible dislocation lines***: By combining three-dimensional simulations of the dynamics of interacting dislocations with a statistical analysis of the corresponding deformation behavior, we determined the distribution of strain changes during dislocation avalanches and established its dependence on microcrystal size (7). Our results suggest that for sample dimensions on the micrometer and submicrometer scale, large strain fluctuations may make it difficult to control the resulting shape in a plastic-forming process (Figure 3).**

*3D simulations***Friction at the atomic scale**Understanding the microscopic mechanisms that govern friction represents a fundamental scientific problem with important practical applications. According to the macroscopic description dating back to Amontons and Coulomb, two bodies in contact under a normal force start to slide when subject to a lateral force exceeding the static friction force, while sliding motion can be sustained under a dynamic friction force. The transition from static to dynamic friction is not completely well defined, because even when the lateral force is below the nominal static friction, a body can slowly creep forward due to thermal activation. We have analyzed the onset of slip of a xenon (Xe) monolayer sliding on a copper (Cu) substrate using molecular dynamics simulations (8). We considered thermal-activated creep under a small external lateral force, and observe that slip proceeds by the nucleation and growth of domains in the commensurate interface between the film and the substrate. We measured the activation energy for the nucleation process considering its dependence on the external force, the substrate corrugation, and particle interactions in the film. Our results are relevant to understand experiments on the sliding of adsorbed monolayers.

Mechanical vibrations are known to affect frictional sliding and the associated stick-slip patterns causing sometimes a drastic reduction of the friction force. This issue is relevant for applications in nanotribology and to understand earthquake triggering by small dynamic perturbations. We studied the dynamics of repulsive particles confined between a horizontally driven top plate and a vertically oscillating bottom plate (9). Our numerical results show a suppression of the high dissipative stick-slip regime in a well-defined range of frequencies that depends on the vibrating amplitude, the normal applied load, the system inertia and the damping constant. We proposed a theoretical explanation of the numerical results and derive a phase diagram indicating the region of parameter space where friction is suppressed. Our results allow to define better strategies for the mechanical control of friction.

**Domain walls in ferromagnetic materials**We study the dynamics of domain walls in disordered ferromagnetic material by means of numerical simulation and analytical means. We have investigated the scaling properties of the Barkhausen effect in several soft ferromagnetic materials: polycrystals with different grain sizes and amorphous alloys. In the limit of vanishing external field rate, we can group the samples in two distinct classes, characterized by universal exponents, for the avalanche size distributions. We interpreted these results in terms of the depinning transition of domain walls and obtained an expression relating the cutoff of the distributions to the demagnetizing factor which is in quantitative agreement with experiments (10). Recent work is focusing on the Barkhausen noise in thin films analyzed by magneto-optical methods (Figure 4). We are developing algorithms to correctly estimate the avalanche distribution from images and models to interpret their scaling behavior (11). Current work is also exploring domain wall dynamics in thin strips and wires.

**Selection of published papers**

- M. J. Alava, P. K. V. V. Nukala and S. Zapperi, “Role of disorder in the size scaling of materials strength”, Phys. Rev. Lett. 100, 0555502 (2008)
- C. B. Picallo, J. M. Lopez, S. Zapperi and M. J. Alava, “Optimization and Plasticity in Disordered Media” Phys. Rev. Lett. 103, 225502 (2009)
- M. C. Miguel, A. Vespignani, S. Zapperi, J. Weiss and J. R. Grasso, “Intermittent dislocation flow in viscoplastic deformation”, Nature 410, 667 (2001)
- M. C. Miguel, A. Vespignani, M. Zaiser and S. Zapperi, “Dislocation jamming and Andrade creep” Phys. Rev. Lett. 89, 165501 (2002)
- P. Moretti, M.-C. Miguel, M. Zaiser, and S. Zapperi, “Depinning transition of dislocation assemblies: Pileups and low-angle grain boundaries” Phys. Rev. B 69, 214103 (2004)
- F. Leoni and S. Zapperi, “Slip line growth as a critical phenomenon”, Phys. Rev. Lett. 102, 115502 (2009)
- F. Csikor, C. Motz, D. Weygand, M. Zaiser and S. Zapperi, Dislocation Avalanches, Strain Bursts, and the Problem of Plastic Forming at the Micrometer Scale, Science 318, 251 (2007)
- M. Reguzzoni, M. Ferrario, S. Zapperi and M. C. Righi, “Onset of frictional slip in an adsorbed monolayer”, PNAS 107, 1311 (2010)
- R. Capozza, A. Vanossi, A. Vezzani and S. Zapperi, “Suppression of friction by mechanical vibration”, Phys. Rev. Lett. 103, 085502 (2009)
- G. Durin and S. Zapperi, “The Barkhausen effect” in The Science of Hysteresis”, edited by G. Bertotti and I. Mayergoyz, vol. II pp 181-267 (Academic Press, Amsterdam, 2006)
- A. Magni, G. Durin, J. P. Sethna and S. Zapperi, “Visualization of avalanches in magnetic thin films: temporal processing“, J. Stat. Mech. (2009) P01020

**Contracts**

- FP6 NEST-PATHFINDER “Triggering instabilities in materials and geosystems - TRIGS” www.trigs.eu (2006-2009)
- PRIN 2008 “Tribologia dei nanoclustes: misure AFM e simulazioni numeriche” (2010-2011)
- COMPLEXITY-NET pilot project: “Localizing signatures of catastophic failure” – LOCAT locat.weebly.com (2012-2013)

**Collaborations**

- M. J. Alava (Aalto University, Helsinki)
- G. Durin (INRIM, Torino)
- L. Laurson (ISI, Torino)
- A. Magni (INRIM, Torino)
- A. Mughal (Aberystwyth University)
- P. K. K. V. Nukala (Oak Ridge)
- S. Papanikolaou (Cornell University)
- J. P. Sethna (Cornell University)
- R. Sommer (CBPF, Rio de Janeiro)
- A. Vanossi (Sissa, Trieste)
- A. Vezzani (CNR, Parma)
- M. Zaiser (University of Edinburgh)

**Contacts**

- Dr. Stefano Zapperi, phone +39 02 66173 385
- Dr. Alessandro Taloni, phone +39 02 66173 362
- Dr. Giulio Costantini, phone +39 02 66173 362
- Dr. Daniele Vilone, phone +39 02 66173 404
- Dr. Alessandro Sellerio, phone +39 02 66173 329
- Dr. Carlotta Negri, phone +39 02 66173 325