Obstacles and sources in dislocation dynamics: Strengthening and statistics of abrupt plastic events in nanopillar compression

Abstract

Mechanical deformation of nanopillars displays features that are distinctly different from the bulk behavior of single crystals: Yield strength increases with decreasing size and plastic deformation comes together with strain bursts or/and stress drops (depending on loading conditions) with a very strong sensitivity of the stochasticity character on material preparation and conditions. The character of the phenomenon is standing as a paradox: While these bursts resemble the universal, widely independent of material conditions, noise heard in bulk crystals using acoustic emission techniques, they emerge primarily with decreasing size and increasing strength in nanopillars. In this paper, we present a realistic but minimal discrete dislocation plasticity model for the elasto-plastic deformation of nanopillars that is consistent with the main experimental observations of nano pillar compression experiments and provides a clear insight to this paradox. With increasing sample size, the model naturally transitions between the typical progressive behavior of nanopillars to a behavior that resembles evidence for bulk mesoscale plasticity. The combination of consistent strengthening, large flow stress fluctuations and critical avalanches is only observed in the depinning regime where obstacles are much stronger than dislocation sources; in contrast, when dislocation source strength becomes comparable to obstacle barriers, then yield strength size effects are absent but plasticity avalanche dynamics is strongly universal, across sample width and aspect-ratio scales. Finally, we elucidate the mechanism that leads to the connection between depinning and size effects in our model dislocation dynamics. In this way, our model builds a way towards unifying statistical aspects of mechanical deformation across scales.

Introduction

The dynamical character of crystal plasticity at the nanoscale has been under scrutiny for more than a decade  (Dimiduk, Woodward, LeSar, Uchic, 2006, Greer, De Hosson, 2011, Kraft, Gruber, Mönig, Weygand, 2010, Uchic, Dimiduk, 2005, Uchic, Dimiduk, Florando, Nix, 2003, Uchic, Dimiduk, Florando, Nix, 2004, Uchic, Shade, Dimiduk, 2009). This interest is driven by the identification of unconventional plasticity size effects in the uniaxial deformation of samples made by the focused ion beam technique. Experiments of nanocrystalline pillar tension and compression have convincingly shown apparent strengthening with decreasing pillar width w, with the yield strength varying as ${\sigma }_Y\sim w^{-n}$ with n ∈ (0.4, 0.8) (Greer, De Hosson, 2011, Uchic, Shade, Dimiduk, 2009), and a mild decrease with slenderness $\alpha =h/w$ (Kiener, Grosinger, Dehm, Pippan, 2008, Senger, Weygand, Motz, Gumbsch, Kraft, 2011, Volkert, Lilleodden, 2006). The mechanism of strengthening in nanopillars has been attributed to the exhaustion of typical dislocation mechanisms and to a transition from typical Frank-Read sources in the bulk to the predominance of atypical sources such as surface sources (Diao, Gall, Dunn, Zimmerman, 2006, Gall, Diao, Dunn, 2004, Park, Gall, Zimmerman, 2006) and single-arm sources (Lu, Huang, Wang, Sun, Lou, 2010, Oh, Legros, Kiener, Dehm, 2009, Ryu, Cai, Nix, Gao, 2015, Weinberger, Cai, 2008, Zheng, Cao, Weinberger, Huang, 2010).

Nano-strengthening is accompanied by large, abrupt strain jumps when the experiment is performed under load control) or by stress drops when under displacement control (Dimiduk, Uchic, Parthasarathy, 2005, Dimiduk, Uchic, Rao, Woodward, Parthasarathy, 2007, Dimiduk, Woodward, LeSar, Uchic, 2006, Greer, Nix, 2006, Greer, Oliver, Nix, 2005, Ng, Ngan, 2007, Ng, Ngan, 2008a, Ng, Ngan, 2008b, Shan, Mishra, Asif, Warren, Minor, 2008, Uchic, Dimiduk, 2005, Uchic, Dimiduk, Florando, Nix, 2004). The stochastic abrupt events resemble noise/avalanches in disordered magnets or earthquakes (Dimiduk, Woodward, LeSar, Uchic, 2006, Papanikolaou, Bohn, Durin, Sommer, Zapperi, Sethna, 2011, Papanikolaou, Dimiduk, Choi, Sethna, Uchic, Woodward, Zapperi, 2012, Uchic, Dimiduk, 2005, Weiss, Lahaie, Grasso, 2000). Analysis of the statistics of abrupt plastic events has revealed that nanopillar events, statistically, appear to follow power-law-tailed distributions for strain steps with a large event cutoff that depends on specimen width (Miguel, Vespignani, Zapperi, Weiss, Grasso, 2001, Miguel, Vespignani, Zapperi, Weiss, Grasso, 2001, Weiss, Lahaie, Grasso, 2000, Weiss, Marsan, 2003). The actual nature of these events, however has remained somewhat elusive, partially due to the complexity of loading paths, intertwining slip and stress events. In a complementary approach, acoustic emission (AE) measurements in a multitude of materials have revealed the presence of ubiquitous power-law plastic events that are independent of loading paths as well as of sample dimensions (Miguel, Vespignani, Zapperi, Weiss, Grasso, 2001, Weiss, Rhouma, Richeton, Dechanel, Louchet, Truskinovsky, 2015). The energy release during such bulk events statistically displays a power law distribution with exponent τ ∈ (1.4, 1.9) and no apparent cutoff dependence on sample parameters (Weiss et al., 2015); similarly, the exponent range in nanopillars is τ ∈ (1.3, 2.1) for strain jumps or stress drops (depending on the selected loading path), but it shows strong fluctuations with dislocation density and sample dimensions (Greer and De Hosson, 2011).

While it may appear natural, it is not clear whether there is a causal relation between the nano-strengthening effects and the plastic avalanche statistics. Based on this premise, theoretical studies of the stochastic/abrupt plastic flow have mainly adopted continuum methods (Alava, Laurson, Zapperi, 2014, Koslowski, LeSar, Thomson, 2004, Miguel, Vespignani, Zapperi, Weiss, Grasso, 2001, Papanikolaou, Dimiduk, Choi, Sethna, Uchic, Woodward, Zapperi, 2012, Zaiser, 2006, Zaiser, Nikitas, 2007) thus addressing bulk properties, or two-dimensional discrete dislocation dynamics in the special environment of randomly placed edge dislocations in periodic systems with no obstacles or dislocation sources (Alava, Laurson, Zapperi, 2014, Ispánovity, Groma, Györgyi, Csikor, Weygand, 2010, Ispánovity, Hegyi, Groma, Györgyi, Ratter, Weygand, 2013, Miguel, Vespignani, Zapperi, Weiss, Grasso, 2001). Such approaches have provided useful insights for predicting unconventional power-law statistics of abrupt plastic events in crystals (Weiss and Marsan, 2003), but they lack consideration of the mechanisms that give rise to nano-strengthening (starvation, source/obstacle exhaustion, single-arm source proliferation etc.).

Dislocation sources can be either bulk –typically of the Frank-Read type– or unconventional, such as surface and single-arm sources; it is believed that bulk sources typically require much smaller average stress to be activated than surface ones (Greer and De Hosson, 2011). Dislocation obstacles capture the effect caused by precipitates, as well as forest dislocations that cross the glide slip planes. As molecular dynamics (MD) and three-dimensional discrete dislocation dynamics (3D-DDD) simulations (Madec et al., 2002) have shown, the strength of such obstacles varies strongly with dislocation configurations. Actual statistical properties of dislocation obstacle distributions currently remain unknown and thus, it is currently not feasible to investigate, using MD or 3D-DDD, the effect of obstacles on dislocation avalanche statistics. However, two-dimensional discrete edge dislocation dynamics (2D-DDD) simulations are ideal for the investigation of various obstacle-related statistical properties, since obstacles are minimally and randomly introduced with a pre-defined statistical distribution. Although the resulting statistical properties have not been explored in much detail, the interplay between dislocation sources and obstacles has been demonstrated in various 2D-DDD studies  (e.g., Cleveringa, Van der Giessen, Needleman, 1999, Balint, Deshpande, Needleman, Van der Giessen, 2007, Deshpande, Needleman, Van der Giessen, 2005, Chakravarty, Curtin, 2010). In fact, Chakravarty and Curtin (2010) have developed a direct connection between the yield strength of a system of edge dislocations in single slip and obstacle spacing, obstacle strength, source nucleation strength and average source spacing.

In this paper, we propose a minimal 2D discrete dislocation plasticity model for elastoplastic deformation of nanopillars and show that its predictions are realistic in that they are consistent with the main experimental observations of strengthening as well as stochastic plastic flow. The explanation of the apparent paradox mentioned above naturally emerges through the stochastic competition among dislocation sources and obstacles.

The remainder of this paper is organized as follows: Section 2 describes the methodology of our 2D-DDD model that is designed to phenomenologically but minimally capture the details of nanopillar compression experiments by incorporating relatively strong surface and weak bulk sources. Also we present details of our statistical analysis of dislocation dynamics, similarly to common experimental protocols. Section 3 is focused on simulation results that are experimentally relevant, such as the behavior of the yield stress as a function of sample width and aspect ratio. In addition, we present the statistical aspect of pillar compression beyond yielding and we show qualitative agreement with uniaxial nanopillar compression experiments. Furthermore, we investigate the role of surface sources in our strengthening and statistics results, by performing extensive simulations in a simplified model where surface dislocation sources are absent, but the bulk properties remain unchanged. The relative role of source strength to obstacle strength in our simplified model is investigated in more detail in Section 4, showing that there are two qualitatively different regimes in terms of strengthening and statistics that we fully characterize. In Section 5, we summarize our results. In the Appendix, we show that our main conclusions do not alter when a second slip system is added.