Session P33: Focus Session: Friction, Fracture and Deformation III

Atomistic Simulation of Size Effects in Bending a Single Crystal

We perform atomistic Monte Carlo simulations of bending a Lennard-Jones single crystal in two dimensions. In examining initial yield, we find an apparent ``reverse'' size effect. However, when strain rate effects are taken into account, we demonstrate that the size effect disappears. Once geometrically necessary dislocations coalesce to form grain boundaries, we observe a size effect of the usual kind, e.g. smaller samples support a higher scaled bending moment than larger samples. We compare simulation results with recent experiments on bending of highly annealed nanowires [B. Wu et al, Nature Matls 4, 525, 2005.] Finally, we observe a topological instability in the evolution of a grain boundary intersecting a free surface under compressive stress. The grain boundary buckles and nucleates a protruding grain, suggesting a novel mechanism for the formation of a hillock on a compressed metal surface.   

Analytical calculation of energy barrier for dislocation nucleation from a crack tip

In a ductile material a crack subjected to a subcritical applied load may respond by emitting dislocations via thermal activation. Computer simulations show the activation energy to be strongly dependent on the applied stress. To understand this result we use conformal mapping techniques to analyze the interaction of a straight screw dislocation with a parallel crack in a strip geometry. The energy barrier for dislocation escape from the crack tip is calculated explicitly and it is found to be a sensitive function of the applied stress, in qualitative agreement with simulation results. Scaling properties of the activation energy are also determined. This analytical result permits us to formulate hypotheses regarding the factors controlling the observed strain rate. To test such hypotheses we finally calculate the strain rate as a function of temperature and applied stress and compare our results with observations.   

XRay Scattering in a Deformed Crystal by a Phase Field Method

We demonstrate the use of a phase field method for dislocated crystals, developed by one of us, for computing the scattering of Xrays. The model addresses deformation on a single slip plane by dislocations of a single burgers vector interacting with a set of point obstacles. The obstacles are introduced in two modes; one randomly on the slip plane, and the second in straight ``walls.'' The obstacles simulate blocking interactions by dislocations on different slip planes, and the ``walls'' represent the intersection of a secondary slip plane with the primary plane being simulated. In the small angle case, the scattering source is the local dilatation induced by the dislocations on the slip plane, and in the Bragg case, the scattering source is the change in local lattice constant. The small angle results show scattering with oscillations attributable to the width of the ``walls.'' In the Bragg case, the Laue spots are broadened by the dislocations, and the results directly confirm the picture of dipolar wall scattering introduced many years ago by H. Mughrabi.   

Atomistic Dislocation Dynamics in Phase Field Crystals: Long Time Scale Properties

The fundamental dislocation processes of glide, climb, and annihilation are studied on diffusive time scales within the framework of a continuum field theory, the Phase Field Crystals (PFC) model. Glide and climb are examined for single edge dislocations subjected to shear and compressive strain, respectively, in a two dimensional hexagonal lattice. It is shown that the natural features of these processes are reproduced without any explicit consideration of elasticity theory or ad hoc construction of microscopic Peierls potentials. Particular attention is paid to the Peierls barrier for dislocation glide/climb and the ensuing dynamic behavior as functions of strain rate, temperature, and dislocation density. It is shown that the dynamics are accurately described by simple viscous motion equations for an overdamped point mass, where the dislocation mobility is the only adjustable parameter. The critical distance for the annihilation of two edge dislocations as a function of separation angle is also presented.   

Studies of the Dislocation Glass

We report the large scale simulations of 2D dislocation systems with overdamped dynamics. 40,000-1,000,000 dislocations were studied with a combination of coarse graining, Fast Fourier Transform and stochastic methods. Both glide and climb processes were considered, as well as the local rotation of crystal axes. Simulations were performed at zero and finite temperatures, with and without dislocation annihilation. When climb processes were included, the system exhibited the formation of dislocation cells/patterns even in equilibrium, without the application of shear. This is in close correspondence with recent experiments on GaAs by P. Rudolph et al. (2005). The distribution function of cell sizes can exhibit a fractal dimension. At long times the system shows glassy dynamics. In particular, aging was observed through the waiting time dependence of the correlations and the effective diffusion. In certain parameter ranges the formation of cells leads to an initial exponential decay of correlations. This is followed by the growth of cells, generating a power law temporal decay in the long time domain. Data for both time domains and for all waiting times can be collapsed onto a single master curve when a t/tw scaling is applied.   

Nonlinear acoustic effects from dislocation-based hysteretic kinking solids under stresses

We argue that proposed recently mechanism explaining inelastic hysteresis in non-linear elastic systems indeed can be explained by means of formation of dislocation-based incipient kink bands (IKB). Using acoustic waves we investigated possible dislocation related mechanisms responsible for nonlinear dynamic response of IKB solids. In this work, for the first time we observed IKB formation and reversibility directly using acoustic coupling technique (ACT) measuring ultrasonic waves attenuation as a function of stress and acoustic emission signatures during compression test of nanolaminated layered ternary carbide (MAX phases) samples. We confirm here that the dynamic behavior of these non-linear elastic systems is due to the interaction of dislocations with the stress waves.   

Simulations of Nano-indentation and Shear Banding in Amorphous Solids

Molecular dynamics simulations of a number of amorphous systems reveal the structural changes that accompany plastic localization. We have simulated both two-dimensional and three-dimensional systems in nanoindentation\footnote{Y. Shi and M.L. Falk, ``Structural transformation and localization during simulated nanoindentation of a non-crystalline metal film,'' Applied Physics Letters, Vol. 86, pp. 011914 (2005).}, uniaxial tension\footnote{Y. Shi and M.L. Falk, ``Strain localization and percolation of stable structure in amorphous solids,'' Physical Review Letters, Vol. 95, pp. 095502 (2005).} and compression in plane strain\footnote{Y. Shi and M.L. Falk, ``Does metallic glass have a backbone? The role of percolating short range order in strength and failure,'' Scripta Materialia, Vol. 54, pp. 381 (2005).}. The degree of strain localization depends sensitively on the quench rate during sample preparation, with localization only arising in more gradually quenched samples. Careful analysis of the strain rate dependence of the localization allows us to extrapolate to the low strain rate limit. This analysis reveals a transition from localized flow to homogeneous flow at a critical value of the potential energy per atom prior to testing. This transition occurs in both two- and three- dimensional systems. The transition appears to be associated with the k-core percolation of short range order (SRO) in the two-dimensional system$^{2}$. We have used a generalization of the Frank-Kasper criterion to identify SRO in the three-dimensional systems. Only in certain systems does this method predict a percolation transition corresponding to the transition in mechanical behavior. We discuss the non-uniqueness of this measure of SRO, and consider whether a more rigorous definition could be derived which applies to systems far from the hard-sphere limit.   

A statistical model of plastic deformation in disordered media

Plastic deformation at the macroscopic scale is assumed to stem from series of successive localized plastic events. A random elastic limit is associated to each site of a discrete mesh. Using a quasi-static driving, one site at a time undergoes plastic shear. The local plastic threshold is then renewed. The localized slip induces long range elastic interactions of quadrupolar symmetry. These additional internal stresses are then used to determine the next weakest site. The model gives rise to a macroscopic plastic flow, corresponding to a genuine depinning transition. We obtain an asymptotic macroscopic yield stress. The transient regime can be associated to a hardening phenomenon of pure statistical origin. Beyond the average plastic behavior we observe stress fluctuations following a universal distribution (only dependent on the system size $L$). Shear deformation presents at all scales spatial and temporal fluctuations of universal character. We observe shear band-like structures which persist only during a finite time $\tau \propto L^z$ and which present a clear anisotropic character with a system size dependent width $w\propto L^\zeta$.   

Scaling laws in fracture of metallic glasses

Brittle metallic glasses themselves can be seen as a model system to study the mechanical properties of metallic based glassy materials. We report a brittle Mg-based bulk metallic glass which approaches the ideal brittle behavior. However, a dimple-like structure is observed at the fracture surface by high resolution scanning electron microscopy, indicating some type of `ductile' fracture mechanism in this very brittle glass. We also show a clear scaling correlation between the fracture toughness and plastic process zone size for various glasses. The results indicate that the fracture in brittle metallic glassy materials might also proceed through the local softening mechanism but at different length scales. The full text of this work has been published under the title \textit{Fracture of Brittle Metallic Glasses: Brittleness or Plasticity} by the authors in Physical Review Letters 94, 125510 (2005).   

Mapping Elasticity at the Nanoscale

In the last few years Atomic Force Acoustic Microscopy has been developed to investigate the elastic response of materials at the nanoscale $^{[1],[2]}$. We have extended this technique to the real-time mapping of nanomechanical properties of material surfaces. This mapping allows us to investigate the local variation of elastic properties with nanometer resolution and to reduce the uncertainties that arise from single measurements. Quantitative measurements are acquired by first performing an accurate calibration of the elastic properties of the Atomic Force Microscope’s probes with respect to single crystal reference materials. A wide variety of surfaces with different mechanical properties have been investigated to illustrate the applicability of this technique. \\ $^{[1]}$ U. Rabe \emph{et al.}, Surf. Interface Anal. $\bf{33} $, 65 (2002)\\ $^{[2]}$ D.C. Hurley \emph{et al.}, J. Appl. Phys. $\bf{94}$, 2347 (2003)