Symposium CM05 : Strain Localization, Avalanches and Intermittent Deformation Mechanisms

In the traditional view of dislocation-mediated plasticity, the mechanical properties of crystalline solids are predominantly controlled by the internal microstructure, which can be artificially tailored, from hardening or the introduction of solutes or precipitates, to improve the performance in service. The external size L, with a length scale much greater than microstructural scales, was usually regarded as unimportant. Meanwhile, at bulk scales, plastic deformation generally occurs in a smooth and continuous (“mild”) way without detectable fluctuations or instabilities. Over recent years, as a consequence of progressive miniaturization of systems and devices, the mechanical properties of metallic materials at micro- to nano-scales became a major concern. The compression of micro/nano-sized single crystals revealed an unexpected external size effect on strength. This “smaller is stronger” phenomenon is unfortunately accompanied by “wild” fluctuations manifesting directly on the loading curves through intermittent, catastrophic strain bursts with a broad range of sizes following a power-law distribution, even in FCC metals deforming smoothly at macro scales. Consequently, despite the high strength achieved at small scales, the plastic process is uncontrolled due to the stochastic nature of strain bursts, and the dislocation avalanches, possibly spanning the system size, may poison forming processes and the load-carrying capacity.
This calls for new metallurgical strategies to mitigate these plastic instabilities at small scales. From compression tests on micro-pillars of pure Al and Al-alloys single crystals strengthened by different types of solutes or precipitates, we showed that:
- Diminishing the external length scale L (miniaturization) intensifies fluctuations and contributes to criticality.
- Introducing quenched disorder shifts the transition from wild to mild plasticity towards smaller external length scales. This “dirtier is milder” effect opens the possibility to mitigate plastic instabilities at small length scales.
- The mild-to-wild transition is closely related with a transition from forest- to exhaustion-hardening, as dislocation-starved states characterizing small samples prevent dislocation entanglements that frustrate avalanches at bulk scales.
- The inter-relation of size and disorder effects reveals itself through a material-independent mapping between the power law exponent of avalanche size distribution and the degree of wildness quantifying the proportion of plastic strain occurring through dislocation avalanches. Translating the pinning strength of obstacles into an internal length scale l, we showed that a single parameter R=L/l controls both the transition from exhaustion- to forest-hardening, and the transition from wild to mild fluctuations. This gives clues towards a possible tailoring of the materials through the introduction the solutes or precipitates in order to tame undesired plastic avalanches at small scales.

Based upon extensive micropillar compression experiments, molecular dynamics and DDD simulations, here we show that the size and frequency distributions of dislocation avalanches triggered under progressive straining are characterized by transitional cut-off slip s_c from incipient to large avalanche domains, each satisfying universal power-law functions with distinctly different exponents. As compared to conventional load-controlled straining, where large bursts result in plastic collapse of the material sample, presently applied progressive straining prevents collapse as the applied stress decreases and the dislocation network evolves during large avalanche propagations. Our results reveal that the critical slip s_c is the distinctive length scale that challenges the prevailing conception on the scale invariance of dislocation avalanches irrespectively of underlying dislocation glide mechanisms. Therefore, s_c is ruled by crystalline structure, sample size and temperature vis-à-vis onset of specific dislocation glide, dislocation interaction and dislocation annihilation mechanisms.

In FCCs, a reduction in s_c is indicative of severe interactions between the mobile dislocations and a heavily entangled dislocation network, as well as of the enhancement in the dislocation pinning capacity in crystals where dislocation cross-slip is inhibited. Increasing s_c marks onset of surface dislocation annihilation hindering dislocation network development in micrometer-sized samples, whereas dislocation source starvation reduces s_c in submicrometer sizes. Parameter s_c is smaller in BCCs than in FCCs. In the latter, s_c is governed by the mobility of the screw dislocations present in various arrangements as a function of sample size and temperature. High-temperature mutual annihilation of mobile dislocations is found to effectively increase s_c.

Stress binning of the avalanche distributions emerges as a powerful element in the assessment of sample size effects in dislocation-mediated plasticity. In the spirit of self-organized criticality (SOC), large FCC and BCC micropillars show constancy in the slip distribution irrespectively of the stress level. On the other hand, the likelihood for large avalanche emissions in confining FCC sample sizes is associated with the sudden destabilization of the dislocation network at large stresses, leading to substantial stress drops and to stress-tuned (binned) avalanche distributions. The enhancement in the mobility of the screw dislocation population in BCCs at large applied stresses and elevated temperatures leads to large avalanche emissions, promoting onset of stress-binned distributions.

Recent research efforts have focused on intermittency in crystal plasticity. For examples, mechanical testing on microcrystals detects sudden strain bursts, which demonstrates criticality among relatively large and rapid dislocation events or avalanches. However, the origin of the dislocation events is not clear. Here we report on the dislocation avalanches in the slowly-compressed nanopillars of high entropy alloy AlxCoCrFeNi (x=0.1) as visualized by simultaneous electron dislocation imaging and nanoscopic mechanical measurements. The observation of dislocations is correlated with the load and displacement. By detecting larger nanopillar slips and observing dislocation events that precedes before and after, we show that the dislocation avalanches arise from the build-up of high density dislocation waves on different slip planes, which act as the source of intermittent emittance of dislocation arrays and avalanches. By correlating the dislocation wave dynamics, we gain further insights into the avalanche triggering mechanism [1].

[1] The work is performed in collaborations with Li Shu, Peter Liaw, and Karin A Dahmen and supported by NSF DMR and DOE BES.

It has been well established that plastic deformation of microcrystals proceeds intermittently in time, with this behavior generally attributed to sudden, collective dislocation rearrangements (dislocation avalanches). However, previously it was not possible to access the temporal sub-structure of these intermittent slip events. In this talk, we discuss use of the recently validated high-data-acquisition-rate nanoindentation technique to investigate the evolution of slip velocity during intermittent slip events in metallic microcrystals from two different crystal systems, namely FCC (represented by Au) and BCC (represented by Nb). We will discuss the (previously demonstrated) similarity of the event-size distributions between the two systems in order to contrast this with the notable differences observed in event velocities and mean profile shape. In particular, we will focus on the dramatically extended velocity relaxation for slip events in the Nb samples relative to the Au samples.

The transition from elastic to plastic deformation in crystalline metals shares both history dependence and scale-invariant avalanche behaviors with other non-equilibrium systems under external loading. Many of these other systems, however, typically exhibit purely elastic behavior only after training through repeated cyclic loading; recent studies in these other systems show power laws and scaling of the hysteresis magnitude and training time as the peak load approaches a reversible—irreversible transition (RIT). We discover here that deformation of small crystals shares these key features. Yielding and hysteresis in uniaxial compression experiments of single-crystal Cu nano- and micropillars decay under repeated cyclic loading; the amplitude and decay time diverge as the peak stress approaches the failure stress, with power laws and scaling as seen in RITs in other nonequilibrium systems. We observe that these effects become smaller as the pillars become larger, perhaps explaining why scale-invariant training effects have not been observed in macroscopic samples.

About 15 years ago, plasticity strain bursts were first observed during microcrystal compression deformation experiments. Soon, nano-scale crystals we also explored and these also exhibited strain bursts. Subsequently, the nano- micro-compression technique was applied to a wide variety of both crystalline and glassy materials. New insights into deformation and dislocation ensemble behavior emerged, suggesting that aspects of the kinematics of deformation common to representations of crystalline flow in simulations, are invalid when “size effects” begin to dominate physically small samples. These occur when certain mean-field conditions no longer hold. For example, the forest-hardening model for slip resistance fails as the dislocation ensemble deviates from mean-field multiplication and storage rates. Likewise, the deformation velocity gradient cannot be described as a homogeneous sum of slips since dislocation sources are not uniformly available at small scales and heterogeneous slip localization dominates. For these conditions, the smooth, power-law like strain-rate dependence on stress, gives way to discrete dislocation avalanches. However, aspects of the experiments and inconsistencies between experimental results, simulations, and theory leave lingering open questions. One may even argue that these inconsistencies are leading to a somewhat confusing body of literature that attempts too many generalizations from far too simplified simulations, or from theory that is only reconciled within a narrow set of experiments while neglecting other results that fall outside of the range of study. This presentation focuses on dislocation behavior in crystalline materials tested using the microcrystal compression method. After a brief synopsis of prior results, numerous aspects of the state of the art are considered including strain burst behavior over the loading stage and Stages I-III of FCC metal deformation; maximum avalanche sizes; effects of loading rate; effects of sample size; aspects of materials types; experimental conditions and settings; and the space-time coupling and other aspects of plasticity avalanche phenomena. The work emphasizes well-supported observations versus those aspects needing further investigation.

Reducing the sample size has become increasingly popular for quantifications of various mechanical size-effects, one of which is that virtually all stress-strain curves from metallic single crystals become intermittent, underlining that plastic flow is not a smooth process. Previous investigations on the intermittent stress-strain behavior of small-scale crystals have focused on the scale-freeness of displacement jump-size distributions, reporting pure or truncated power-law behavior that, via similar scaling exponents, suggest a universal aspect of plasticity, which is shared by vastly different microstructures and materials. However, investigating the time-resolved spatiotemporal avalanche dynamics of the same intermittent flow, revealed that there are indications for a materials-specific response, as expressed by different velocity relaxation behavior and average avalanche shape-functions in fcc and bcc crystals [1].
One fundamental aspect of our recent efforts in tracing nano-scale slip dynamics in single crystals is that there is a materials-dependent and test-system dependent range of finite slip velocities [2, 3]. Whilst the scatter of the recorded velocities is large and exhibits some agreement with recently predicted scaling laws, each so far investigated material has a recorded maximum slip velocity. The fact that finite values are seen, suggests that macroscopically smooth flow may be a manifestation of a time-scale mismatch between the internal processes, that is nano-scale slip (dislocation avalanches), and the applied rate. Here we investigate this proposition experimentally, by tracing the time-resolved nano-scale slip dynamics of Nb microcrystals across more than five orders of magnitude in displacement rate.

[1] G. Sparks, J. Sickle, K.A. Dahmen, R. Maass. Shapes and velocity relaxation of dislocation avalanches in fcc and bcc crystals, http://arxiv.org/abs/1705.06636 (2017).
[2] R. Maass, P.M. Derlet, J.R. Greer. Independence of Slip Velocities on Applied Stress in Small Crystals, Small 11 (2015) 341-351.
[3] G. Sparks, P.S. Phani, U. Hangen, R. Maass. Spatiotemporal slip dynamics during deformation of gold micro-crystals, Acta Materialia 122 (2017) 109-119.

Deformation of micron-sized single crystals of a dual-phase high entropy alloy, Al_0.7CoCrFeNi i.e., a face-centered-cubic (FCC) and body-centered-cubic (BCC) structure, has been previously studied and revealed different deformation mechanisms during uniaxial compression. These mechanisms are explained in terms of common nanoplasticity mechanisms: collective dislocation glide and nucleation-governed plasticity in the FCC phase and dislocation crossslip in the BCC phase. In this study, vertically aligned ~ 2 um –diameter pillars made from the same alloy, which contain a single-phase boundary in each sample, were subjected to uniaxial compression. Results reveal that for a high symmetry orientation of FCC [101]/BCC [001] samples, the high symmetry [101] orientation of the FCC phase drives the elastic deformation with a calculated yield stress of 0.5 GPa but the continuous deformation in the plastic regime is a signature that is attributed to the cross slip in the BCC phase. We observed slip transmission through the boundary from FCC to the BCC half-pillar, which suggests dislocation nucleation in the BCC phase at the onset of yield in the FCC phase as well as the dislocation-boundary interaction, which was not observed during the deformation of single crystalline BCC samples of the same orientation [001]. Micro-pillars that contain different crystal orientations (low and high symmetry) were studied; we discuss their deformation mechanisms in the context of different contributions of each phase and orientation to the elastic and plastic deformation of the overall sample. Yield strengths of Nanopillars with high symmetry orientation of BCC half-pillars have a higher yield strength of a factor of 2.5 greater than pillars with high symmetry orientation FCC half-pillars. We show that in addition to strength, the strain-hardening rate also varies with the different orientations and combinations studied.

High-Energy Diffraction Microscopy (HEDM) is a technique for use of synchrotron radiation in study of polycrystalline materials. Advances in detector technology and supporting software play a significant role in the development of this technique. Applied to the in situ study of deformation, HEDM enables the study of transient response in structural alloys. The identification of slip system activity, kinetics of different slip modes, intragranular slip gradients and the intermittency of plastic flow will be shown. Of particular note, the capability to resolve the (tensorial) state of stress at the scale of individual grains provides for drawing association between transient response and particular modes of crystallographic slip. The presentation will survey several examples that show intermittent and transient response. Attention will be given to the development of constitutive models accounting for the kinetics of crystallographic slip.

Alloys with real microstructures involving second phase precipitates of duralumin and alternant lamellas from eutectic Sn/Pb system are investigated on mechanical properties during micron-scaled deformation. A series of unusual phenomena about size effect is reported as far from conventional notion, i.e., “smaller is stronger”: (i) precipitated duralumin much stronger than pure Al exhibits a prominent indentation creep with a shear viscosity approaching to pure Al within submicron depth but shear viscosity turns 4~5 times higher than pure Al in larger depth; (ii) precipitate duralumin micropillar exhibits a weakest strength at a critical size ~7µm meanwhile strain-hardening is also slowest and creep fastest at the same size. Moreover, the reduction of strength at the critical size is more significant in Peak-aged specimens than naturally aged ones; (iii) eutectic Sn/Pb micropillar presents an inverse size-dependent strength when the thickness of alternant lamellas approaches unit micron. Nevertheless, the scenario of inverse size effect will be suppressed with thickening the lamellar structure of Sn/Pb alloys. Theoretical modeling based on dislocation dynamics points out that those real microstructures are introducing an intrinsic length scale which will coact with external length scale coming from specimen size to control mechanical properties of micro-metals. The results indicate that the commonly known ‘smaller is stronger’ notion is not always right especially when intrinsic length scale is introduced by real microstructures to affect strength.

It is generally claimed that physical processes which display scale-invariant power law distributions are subjected to a dynamic criticality that prohibits a well-defined kinetic law. In this paper, we demonstrate the coexistence of these two apparently contradicting behaviors during the same physical process – the motion of type II twin boundaries in martensite Ni-Mn-Ga. The process is investigated by combined measurements of the temporal twin boundary velocity and the acoustic emitted energy. Velocity values are extracted from high-resolution force measurements taken during displacement driven mechanical tests, as well as from force driven magnetic tests, and cover an overall range of six orders of magnitudes. Acoustic emission (AE) is measured during mechanical tests.
Velocity values follow a normal distribution whose characteristic value is determined by the material’s kinetic relation, and its width is scaled with the average velocity. In addition, it is observed that velocity distributions are characterized by a heavy tail at the right (i.e., faster) end that exhibits a power law over more than one and a half orders of magnitude. At the same time, the AE signals follow a scale-invariant power law distribution, which is not sensitive to the average twin boundary velocity. The coexistence of these two different statistical behaviors reflects the complex nature of twin boundary motion and suggests the possibility that the transformation proceeds through physical sub-processes that are close to criticality alongside other processes that are not.

Because of the disordered nature of amorphous systems, their fracture involves local structural rearrangements as well as micro-fracturing processes, and is not fully understood. Several experiments performed on silicate glasses have shown that, indeed, a crack progresses in these materials by creating first a series of nano-cracks around it tip. This "damaged" region was shown to extend over ~10 nanometers. Because of the smallness of this size, direct observation is impossible, and the scope of our work is to build, and fracture in a controlled way amorphous structures made from basic bricks much larger than the silica tetrahedron (the size of which is ~5A). We have worked on an agar gel, for which the basic entity is the junction between chains (~15nm), on a so-called Casimir gel where microgels play the role of "atoms" (diameter~1µm) and on a gel made from an emulsion with droplet size ~50µm.

These materials being very soft gels (Young's modulus E<100kPa), we had to imagine new microfluidic devices in order to ensure their controlled fracture. Cracks were observed using conventional or confocal microscopy. By studying both the shape of the crack and the displacement field in the vicinity of the crack tip, we could evaluate the distance to linear elasticity, and disclose the relevant time and length scales.

In the case of the agar gel, it is still impossible to observe directly dissipative processes, but their large scale consequences can be fully characterized. In the case of the Casimir gel, micro-cracking ahead of the propagating crack tip is observed directly, and the process zone size is measured as a function of temperature, which rules the interaction potential between microgels. Finally, in the emulsion-based gel, local rearrangements which do not involve cracks are also observed.

Levy Fight in Mayonnaise

Glassy systems with long-range interactions often present avalanche type-response under slow driving, whose statistics is similar to that of earthquakes.
They also present a vanishing density of excitation at low energy or “pseudo gap”. I will explain why these facts must come together, and discuss in particular the plasticity of amorphous solids (for example, how does a mayonnaise flow when one slowly pushes it with a spoon). I will argue that the mean-field description of plasticity maps into the problem of Levy Flights near an absorbing boundary, and draw consequences of this analogy.

Fault zone dynamics hold the key to resolving many outstanding geophysical problems including the heat flow paradox, discrepancy between fault static and dynamic strength, and energy partitioning. Most fault zones that generate tectonic events are gouge filled and fluid saturated posing the need for formulating gouge-specific constitutive models that capture spatially heterogeneous compaction and dilation, non-monotonic rate dependence, and transition between localized and distributed deformation.
In this presentation, we focus primarily on elucidating microscopic underpinnings for shear banding and stick-slip instabilities in sheared saturated granular materials and explore their implications for earthquake dynamics. We use a non-equilibrium thermodynamics model, the Shear Transformation Zone theory, to investigate the dynamics of strain localization and its connection to stability of sliding in the presence and absence of pore fluids. We also consider the possible influence of self-induced mechanical vibrations as well as the role of external acoustic vibrations as analogue for triggering by a distant event. For the dry case, our results suggest that at low and intermediate strain rates, persistent shear bands develop only in the absence of vibrations. Vibrations tend to fluidize the granular network and de-localize slip at these rates. Stick-slip is only observed for rough grains and it is confined to the shear band. At high strain rates, stick-slip disappears and the different systems exhibit similar stress-slip response. Changing the vibration intensity, duration or time of application alters the system response and may cause long-lasting rheological changes. The presence of pore fluids modifies the stick slip pattern and may lead to both loss and development of slip instability depending on the value of the confining pressure, imposed strain rate and hydraulic parameters.
We analyze these observations in terms of possible transitions between rate strengthening and rate weakening response facilitated by a competition between shear induced dilation and acoustic compaction. We discuss the implications of our results on dynamic triggering, quiescence and strength evolution in gouge filled fault zones.

The shear transformation zone (STZ) theory provides a rigorous framework for continuum modeling of plastic deformation in amorphous solids. The STZ theory relies upon the notion of an effective disorder temperature defined by T_eff=∂U_c/∂S_c, where U_c is the configurational energy and S_c the configurational entropy – making it a “structural” analog to the classical vibrational thermodynamic temperature. Over sufficiently large length-scales it is hypothesized that the density of STZs scales with the effective temperature (i.e. higher effective temperature implies higher density of STZs). The STZ theory further defines a flow rule such that plastic deformation at a given stress state scales with the STZ density. Hence, regions of the material with higher effective temperature are more susceptible to plastic rearrangement than regions with low effective temperature.

In this work, we explore the inherently stochastic nature of effective temperature fluctuations of metallic glasses. We start by quantifying random variations in potential energies at the atomistic scale and utilize an affine transformation [1] to coarse-grain to a continuum level random field description of the effective temperature [2]. Careful statistical investigation of the atomic potential energies and their coarse-grained fields may provide evidence for medium-to-long range correlations in the potential energies and a minimum coarse-graining length-scale when defining effective temperature initial conditions from atomistic simulations. The continuum STZ model is calibrated using a surrogate-based efficient global optimization (EGO) [3] routine and the sensitivity of the STZ model to coarse-graining parameters is investigated [4]. This serves to motivate the need for, at a minimum, stochastic modeling of continuum STZ equations based on random field effective temperature initial conditions. More generally though, this study coupled with some further observations suggest a roadmap toward a fully stochastic model of plastic deformation in amorphous metals that builds from the structural disorder at the atomistic level and enables statistically consistent continuum modeling at higher length-scales.

[1] Y. Shi, M. B. Katz, H. Li, and M. L. Falk, “Evaluation of the Disorder Temperature and Free-Volume Formalisms via Simulations of Shear Banding in Amorphous Solids,” Phys. Rev. Lett., vol. 98, no. 18, p. 185505, May 2007.
[2] A. R. Hinkle, C. H. Rycroft, M. D. Shields, and M. L. Falk, “Coarse graining atomistic simulations of plastically deforming amorphous solids,” Phys. Rev. E, vol. 95, no. 5, 2017.
[3] D. R. Jones, M. Schonlau, and W. J. Welch, “Efficient Global Optimization of Expensive Black-Box Functions,” J. Glob. Optim., vol. 13, no. 4, pp. 455–492, 1998.
[4] D. Giovanis and M. D. Shields, (In Review) “Uncertainty quantification for complex systems with very high dimensional response using Grassmann manifold variations,” J. Comput. Phys.

Glasses, and the more general category of materials known as amorphous solids, lack crystal structure and find wide application from consumer goods to photovoltaics. Yet, issues quantifying disorder have stymied the construction of physically grounded mechanical constitutive laws for these materials suitable for failure prediction. Atomistic simulation methods can provide some insight regarding the mechanisms of plastic deformation, strain localization, cavitation and fracture. I will briefly discuss some applications of atomistic simulation methods to investigate each of these phenomena and the physical insights that have been gained. Recent investigations have aimed at quantifying the defects that control plastic flow. These have confirmed some of the assumptions built into the shear transformation zone theory of amorphous plasticity. I will further discuss methods for quantitatively predicting strain localization, a limiting failure process in high-strength metallic glasses and other amorphous materials by parameterizing the effective-temperature shear transformation zone theory from molecular dynamics simulations. We have directly cross-compared molecular dynamics simulations and continuum representations of these same materials in order to test and validate our constitutive theories. The role of coarse graining in the linkage of continuum and atomistic methods is crucial, and convergence only arises above a critical length scale on the order of tens of angstroms. The investigation makes clear the need to separate out the relevant fluctuations in material structure from the shorter wavelength fluctuations that serve to obscure them. It is, in the end, the interactions between these larger-scale relevant fluctuations via the material’s mechanical response that controls the failure process during strain localization.

Plastic deformation in metallic glasses and other amorphous solids is believed to be mediated by point defects referred to as shear transformation zones (STZs). STZs are defined as clusters of atoms rearranging cooperatively when strained plastically. However, locating STZs in the glass microstructure is a significant challenge. We deploy a local yield stress (LYS) method to evaluate the local stress at which instabilities are triggered in the microstructure of a simulated Cu₆₄Zr₃₆ metallic glass. We then use this data to locate STZs by finding the minima in this scalar field. The predictions of the LYS method are cross-correlated with molecular statics and molecular dynamics simulations of shear to verify the predictive nature of this nano-scale measure of material structure. We analyze the persistence of the LYS measure as well as the statistics of the defects identified.

Using broadband nanoindentation creep (BNC), we measure room temperature hardness-strain rate properties of Zr₅₀Cu₄₅Al₅ metallic glass ribbons and ingots. We estimate the shear stress and strain rate inside the shear bands. We observe (and propose) the following:

1. Aging the glass near Tg increases flow stress but has a secondary effect on the slope of the stress-strain rate curve. This is indication that a back stress exists, which contributes to the measured hardness. We propose that the magnitude of this flow stress is related to the stress required to initiate new shear bands. The result also suggests that deformation kinetics inside the shear band are independent of prior thermal history (consistent with loss of memory of the aged structure because of the large strains encountered inside shear bands).
2. Individual stress-strain rate curves generated from BNC depend on deformation history, particularly initial rate of loading (the same thing is observed in amorphous polymers). We propose that structure inside the shear band varies depending on strain rate.
3. The high strain rate portions of shear stress-strain rates data are dominated by transients, which (we propose) are caused by partial relaxation of the internal stresses related to shear band initiation.
4. At strain strain rates below about 10^-3/s there is a downward inflection in flow stress. This feature suggests the onset of a new mechanism at low strain rates.
5. We observe a number of subtle size effects including one in the amount of creep during hold at constant hold. Otherwise, there is not a measureable size effect in either the hardness or the modulus for loads between between 0.1 and 10 mN.

Lastly, the stress-strain rate data generated from BNC, while broadly consistent with data obtained from other experimental methods, seem inconsistent with molecular dynamics simulations extrapolated to laboratory strain rates. We speculate on the origins of the discrepancy.

This work was supported by NSF Grant NSF CMMI-1232731. M.J. Kramer, I. Kalay, and Y.E. Kalay of Ames Laboratory kindly provided the materials.

The structural description of even the most basic amorphous materials are under considerable debate. In this work, an intuitive computational technique has been developed to construct 3D statistical density maps to directly visualize and identify local atomic structures from simple monatomic amorphous germanium (a-Ge) to complex multi-atom systems such as copper zirconium metallic glass. This approach clearly reveals structural differences between molecular dynamics and reverse monte carlo models of a-Ge as well as changes due to annealing. We show motifs in copper zirconium that are unresolvable through traditional tools such as Voronoi indexing. This self-sorted local atomic motif (SLAM) method builds upon the Kabsch algorithm incorporating techniques in computer vision to produce least-squares optimized 3D density maps. Simultaneously, the SLAM method incorporates self-contained categorization to define local motifs based upon atomic structures physically present in a model rather than imposing biased criterion inherent in Voronoi indexing methods.

We present the methodology of the SLAM method and also present resulting motifs through stereographic projections. Subtle distortions of the tetrahedral structure of a-Ge is shown to change upon annealing and comparisons to continuous random network models show different dihedral disorder with respect to reverse monte carlo models. A first examination of structures in metallic glass is shown and a discussion of its potential application to shear-band structure and composition.

Slowly-compressed nano-crystals, bulk metallic glasses, rocks, granular materials, and the earth all deform via intermittent slips or “quakes”. We find that although these systems span 12 decades in length scale, they all show the same scaling behavior for their slip size distributions and other statistical properties. Remarkably, they also appear to show similar slip dynamics.

A simple mean field model for avalanches of slipping weak spots explains the agreement across scales. It predicts the observed slip-size distributions, and the temporal slip profiles. The analysis draws on tools from statistical physics and the renormalization group. The results enable extrapolations from one scale to another, and from one force to another, across different materials and structures, from nanocrystals to earthquakes. Connections to neuron avalanches in the brain and recent observations on stars will also be discussed, extending the range of scales to 16 decades in length.

References:

[1] Jonathan T. Uhl, Shivesh Pathak, Danijel Schorlemmer, Xin Liu, Ryan Swindeman, Braden A.W. Brinkman,, Michael LeBlanc, Georgios Tsekenis, Nir Friedman, Robert Behringer, Dmitry Denisov, Peter Schall, Xiaojun Gu, Wendelin J. Wright, Todd Hufnagel, Andrew Jennings, Julia R. Greer, P.K. Liaw, Thorsten Becker, Georg Dresen, and Karin A. Dahmen, Universal Quake Statistics: From Compressed Nanocrystals to Earthquakes, Sci. Rep. 5, 16493 (2015).

[2] N. Friedman, A. T. Jennings, G. Tsekenis, J.-Y. Kim, J. T. Uhl, J. R. Greer, and K. A. Dahmen. Statistics of dislocation slip-avalanches in nano-sized single crystals show tuned critical behavior predicted by a simple mean field model, Phys. Rev. Lett. 109, 095507 (2012).

[3] James Antonaglia, Wendelin J. Wright, Xiaojun Gu, Rachel R. Byer, Todd C. Hufnagel, Michael LeBlanc, Jonathan T. Uhl, and Karin A. Dahmen, Bulk Metallic Glasses Deform via Slip Avalanches, Phys. Rev. Lett. 112, 155501 (2014).

[4] N. Friedman, S. Ito, B.A.W. Brinkman, L. DeVille, K. Dahmen, J. Beggs, and T. Butler. Universal critical dynamics in high resolution neuronal avalanche data. Phys. Rev. Lett. 108, 208102 (2012).

[5] Mohammed A. Sheikh, Richard L. Weaver, and Karin A. Dahmen, Avalanche Statistics Identify Intrinsic Stellar Processes near Criticality in KIC 8462852, Phys. Rev. Lett. 117, 261101 (2016).

[6] Dmitry V. Denisov, Kinga A. Loerincz, Wendelin J. Wright, Todd C. Hufnagel, Aya Nawano, Xiaojun Gu, Jonathan T. Uhl, Karin A. Dahmen & Peter Schall, Universal slip dynamics in metallic glasses and granular matter: linking frictional weakening with inertial effects, Sci. Rep. 7, 43376 (2017).

The fracture of disordered or amorphous materials is not fully understood as it involves structural rearrangements as well as micro-fracturing. In atomic solids, the damage region which precedes the fracture tip is often very small, 10s nanometers, and the crack tip propagates rapidly, meters per second. Therefore, it is exceedingly challenging to image the real crack tip and to follow it dynamically. We have developed a disordered cohesive colloidal material where each colloidal particle of micron diameter plays the role of the atoms. Due to this dramatically increased length scale, the material is very soft with a Young’s modulus of 5Pa and each individual colloid is distinguishable with a confocal microscope. We load a high volume fraction liquid dispersion of fluorescent microgel colloids into a microfluidic device with a flow geometry that permits Mode 1 fracture. A cohesive attraction is then externally triggered using a critical Casimir interaction by gently heating the dispersion within the device. We then hydraulically fracture the colloidal solid by flowing fluid devoid of particles into the device and use confocal microscopy to follow the propagating fracture. After locating the instantaneous centers of each colloidal particle, we calculate the local shear strain matrix; a clear damage zone extends well in front of the crack tip. We utilize this damage zone to predict the motion of crack tip; the magnitude of the accumulation of plastic deformation in front of the crack tip correlates well with the directionality. Lastly, we directly observe that micro-cracking ahead of the tip causes intermittency in the crack tip velocity and position.

Nanocrystalline (nc) materials provide superior mechanical properties due to their unique microstructures. Fine grain size with significant grain boundary area results in enhanced grain boundary-mediated deformation but also suppressed dislocation activities, which leads to low ductility and limits their applications. Extensive theoretical models, simulations, experiments have been proposed and demonstrated that plastic deformation of nc materials can be via various mechanisms such as grain boundary sliding, grain rotation, twinning, etc. Previous experimental literatures mostly focused on nc metals. There is still a lack of experimental data on nc ceramics. Understanding plastic deformation mechanism of nc ceramics is crucial to improve ductility and achieve better toughness. Transition metal nitrides are widely used in industry due to their excellent mechanical properties, high chemical stability, and low electrical resistivity. Furthermore, they are capable of plastic deformation based on von Mises criteria. However, it is usually challenging to study ceramic plasticity owing to their vulnerability to intrinsic flaws. Micro/nanocompression techniques have emerged as a prominent way to study ceramic plasticity since small volume and compressive stress are beneficial to avoid premature fracture. Microstructural response of plastic deformation could be resolved in real time with the power of transmission electron microscopes.
In this study, in situ indentation in TEM was carried out to investigate plastic deformation mechanism of zirconium nitride nanopillar. Zirconium nitride nanopillars were fabricated by focus ion beam from the cross-section of nanocrystalline zirconium nitride hard coatings deposited by unbalance magnetron sputtering. The average grain size is around 20 nm. Scanning electron nanobeam diffraction was utilized to map grain orientation of the pillars. Indentation videos were recorded in bright field and diffraction mode. Frame by frame analysis combining with load-displacement curve was implemented to correlate microstructural activity and mechanical behavior.
In the preliminary results, we observed that zirconium nitride nanopillars showed ductile behavior. Grain rotation is suggested to be the main mechanism to accommodate applied strain. Discrete plastic events occurred in different grains and finally formed shear plane leading to fracture. Future work continues to study the mechanism behind grain rotation. Moreover, the nanopillars of different preferred orientation will also be involved to study the effect of texture on mechanical properties of nc ceramics.

Strain analysis of planarized bulk materials and thick films is often performed using optical techniques such as Moiré interferometry or digital image correlation (DIC). These methods are contactless, capable of measuring strain fields in-situ, with a sample under test strained via mechanical or thermal loading. While both techniques are effective at mapping micro-scale strains, such resolution may be insufficient for observing strain near inference regions of two bonded materials with different coefficients of thermal expansion. Attempts to detect nano-scale strain in interface regions using DIC and Moiré techniques have seen some success, but suffer from limitations not typically encountered at larger length scales.

Practical digital image correlation techniques can effectively observe strain with a resolution of 0.2 pixels, making features 0.7-2 microns in size ideal for nano-strain measurements using white light illumination. However, resolving such feature sizes requires the use of high magnification optics or an SEM. High magnification optics significantly restrict the field of view which can be imaged, while use of an SEM contributes considerable complexity to each measurement. Moiré interferometry techniques have a much wider field of view, but a maximum resolution equal to λ/2 where λ is the wavelength of light used to generate the projected reference grating. In addition to a small field of view and modest strain resolution, both DIC and Moiré interferometry rely on resolving a single image plane, limiting the detection of out-of-plane interface deformations.

An alternative technique to DIC and Moiré interferometry based on laser diffraction will be presented. This technique has the potential to detect out-of-plane sample deformations and nano-scale strain simultaneously over a large field of view. A thin, high-efficiency diffraction grating made of evaporated gold was fabricated on the planarized cross-section of a microelectronics package. Such packages contain microstructures with many combinations of metal, semiconductor, and polymer interfaces arranged in various geometries. Focused light from a He-Ne laser was then directed at the grating, generating a far-field diffraction pattern. The m=1 and m=-1 diffraction angles were determined using two wide area camera sensors and resulting diffraction angle data used to map the surface normal of the interface by stepping the incoming laser beam across the grating area. Strain data was obtained by measuring the difference in observed grating pitch at two sample temperatures. Sample preparation, optical measurement configuration, determination of diffraction angle, and accuracy/error of the surface normal and strain measurements will be presented in detail.

X-ray Bragg coherent diffraction imaging has evolved into a powerful tool for local structure and strain characterization at the nanoscale. Unencumbered by high resolution x-ray optics, or stringent sample preparation requirements, the technique has enabled imaging of strain in nano-scale materials at operando conditions. By measuring the coherent scattering in the vicinity of a Bragg peak of a crystal lattice and computationally inverting to an image, the technique has demonstrated quantitative strain resolution at the 10^-4 level with sub 10nm spatial resolution.
This talk will introduce the method and describe details of the coherent diffraction measurements and image retrieval process. I will also present a handful of recent studies performed at the Advanced Photon Source at Argonne National Laboratory strain imaging was used to elucidate the mechanisms underlying catalytic activity in gold nanocrystals and the mechanisms of damage under focused ion beam milling.

In concentrated disordered solid solutions, the disorder permeates the whole solid and hence cannot be viewed akin to a defect in a perfect crystal. To comprehend the effect of atomic disorder in concentrated alloys on the Bragg peaks of powder XRD patterns, model binary and multicomponent alloys have been used for both computational and experimental studies. The effect of various a priori distortion parameters like 'δ', △Rmax and u on the Bragg peaks is studied. Intensity and FWHM of the Bragg peaks are monitored across alloys to isolate the effects of atomic disorder. Using the results, discrimination is made between true bond distortion and other measures of lattice distortion. This is further used to understand the non-linear variation in the intensity of the Bragg peaks with lattice distortion. The shortcomings of modeling atomic disorder as a thermal effect are pointed out in light of the investigations.

Colloidal particles at low volume fractions can gelate into disordered solids by their aggregation into a space-filling network. The mechanical properties of the resulting colloidal gel depend on the connectivity of the final network as well as on the architecture of its constituent particle strands. Rheological and mechanical tests over colloidal gels have shown interesting results such as yield localization and delayed yielding. However, direct observation the flow and fracture behavior of colloidal gels at mesoscopic and microscopic scales is hindered by difficulties in gripping and imposing a controlled load on a soft material and by the combination of large deformations and viscoelastic processes.

In this work, we use two experimental setups fabricated with microfluidic technologies to analyze the mechanical response of aqueous colloidal systems made with silica nanoparticles and with the synthetic hectorite clay Laponite RD (both with diameters around 25 nm). The first one is a Hele-Shaw-like cell with a built-in notch. By injecting water over the notch, a flow rate controlled crack is initiated. The second one consists of a chamber where a rectangular gel sample is surrounded by an immiscible oil. The aspiration of the oil at a controlled flow rate imposes a proportional rate of displacement to the oil-gel interfaces, resulting in the nucleation of a single crack at a notch.

For given particle volume fraction and gel time, a wide range of behaviors was observed as a function of ionic strength for both materials, from a liquid flowing under load to an elastic solid which breaks. The measurement of the displacement fields in the vicinity of the crack tip by Digital Image Correlation and of the Crack Tip Opening Displacement enable the determination of the stress intensity factor and the energy release rate during fracture. The comparison of both methods allows us to estimate the size of the non-linear fracture zone, which tends to increase for decreasing ionic strengths. We also estimate the characteristic time by considering the influence of the crack speeds.

Twinning, in-addition to dislocation slip, is a prevalent deformation mechanism in low-symmetry (hexagonal close packed - hcp) metals. Deformation twins are shear transformed domains which evolve in three sequential stages: twin nucleation, twin propagation and twin growth. Most of the work carried out on hcp twins has been limited to static and two-dimensional studies. On the contrary, the short- and medium- range interactions among numerous transformed domains in the crystal and their connection to the kinetics of growth of individual domains can only be understood via three-dimensional analysis. In the present work, the newly developed DDD-FFT technique is used to model the three-dimensional nature of twins and study the influence of interfacial defects on twin growth in magnesium single crystal. The stress state in the vicinity of the twin is extracted for an experimentally observed (01-12) [0-111] tensile twin with a shear eigenstrain ~13%. Plastic relaxation around the twin was observed to change the stress state in the vicinity of the twin. Studying the role of surface energy and local stress fields for different transformed domains provides insight into the dynamics of interfacial defects and their connection to twin growth. Dislocation-twin interactions were observed to not only influence local stresses but also twin growth and, in-turn, the overall mechanical response of the magnesium crystal.

We have developed a novel algorithm to generate 3D virtual representative twinned microstructures that are statistically equivalent to experimentally observed microstructures from electron back-scattered diffraction (EBSD) scans of a nickel-based superalloy sample. The distributions and correlations of various morphological and crystallographic information are extracted from EBSD data and then used to create virtual parent grain microstructures. To incorporate twins inside the parent grain microstructures, the distributions of twins with respect to parent grains, including the joint probability of number of twins and parent grain size, along with the conditional probability distributions of twin thickness and twin distance from parent centroids are employed. Then, subsequent to determination of morphological and geometrical information of twins through a Monte-Carlo scheme, the microstructure-based statistically equivalent representative volume elements (M-SERVEs) are generated. The statistics of M-SERVEs are compared and validated with respect to EBSD microstructural statistics, and the convergence of the results is studied with the aid of the Kolmogorov-Smirnov test. To study the mechanical behavior of the material, a crystal plasticity finite element model is presented, and the material parameters are calibrated and then validated using the experimental tests. Through the convergence study, the minimum size of M-SERVE required to capture the morphological and crystallographic statistics of EBSD scans as well as the minimum size of the property-based statistically equivalent representative volume elements (P-SERVE) necessary to reproduce the macroscopic material response in mechanical tests are reported. The framework is implemented to study the performance of polycrystalline microstructures under monotonic and fatigue loading. Finally, the weak regions of the polycrystal susceptible to crack nucleation are strengthened by locally varying the material parameters, representing the crystallography and morphology of subgrain structures, which in turn result in designing stronger superalloys across length scales.

To understand the formation and evolution of dislocation patterns developing during plastic deformation has long been a central aim of materials research. Several continuum theories have been proposed that are able to model different types of dislocation patterns like PSB or fractal structures. Apart from these, discrete dislocation dynamics has also been employed and demonstrated the tendency of dislocation ensembles to form patterns. Embryonic forms of some well-known structures have been directly observed with this method.

In order to identify the necessary ingredients of a continuum theory and their role in patterning in the talk, contrary to previous approaches, we aim at a multi-scale description. We, therefore, parallelly study the patterning phenomenon in a 2D discrete dislocation system and its continuum counterpart. To this end, we develop a two-dimensional stochastic continuum dislocation dynamics theory that is derived by coarse graining the equations of motion of discrete dislocations. The main ingredients of the continuum theory is the evolution equations of statistically stored and geometrically necessary dislocation densities, which are driven by the long-range internal stress, a stochastic flow stress term and, finally, local strain gradient terms, commonly interpreted as dislocation back-stress.

The agreement between the two models is shown primarily in terms of the patterning characteristics that include the formation of dipolar dense dislocation walls that are perpendicular to the slip plane. The length of these walls gradually increase with increasing strain and reach the simulation box size at yielding. Not only the different parameters of the patterns, but also the stress-strain curves exhibit quantitative agreement between the too models.

The multi-scale approach enables us to identify the back-stress term as the main source of the pattering in the continuum description. We will thoroughly discuss the relation of this finding to local stresses, internal correlations and stochastic and intermittent processes.
Connections of our results to theories of kinematic hardening and strain-gradient plasticity as well as to the Bauschinger effect will also be described followed by prospects for generalization to 3D.

When far from equilibrium, many-body systems display behavior that strongly depends on the initial conditions. A characteristic such example is the phenomenon of plasticity of crystalline and amorphous materials that strongly depends on the material history. In plasticity modeling, the history is captured by a quenched, local and disordered flow stress distribution. While it is this disorder that causes avalanches that are commonly observed during nanoscale plastic deformation, the functional form and scaling properties have remained elusive. In this presentation, a generic formalism is developed for deriving local disorder distributions from field-response (\eg stress/strain) timeseries in models of crackling noise. We demonstrate the efficiency of the method in the hysteretic random-field Ising model and also, models of elastic interface depinning that have been used to model crystalline and amorphous plasticity. We show that the capacity to resolve the quenched disorder distribution improves with the temporal resolution and number of samples.

Designing more reliable irradiation-resistant material is an essential aspect of nuclear engineering. Dislocation channel formation is one of the most important catastrophic failure origins. Since dislocation channels are typically 50~200nm wide, an intriguing question is that if the sample size decreases to the submicron scale, and becomes of the same order of the characteristic length, then how does irradiation influence plastic flow localization? We report the first systematic three dimensional Discrete Dislocation Dynamics (3D-DDD) observations about the size effect on dislocation channel formation. Because of the high density of radiation-induced defects, massive simulations of discrete interactions between dislocations and all radiation defects is computationally prohibitive. To overcome this computational difficulty, we developed a hybrid continuum-discrete model for the collective dynamics of dislocations in dense irradiation defect field. To quantify plastic flow localization, we condense the complex 3D deformation information into some easy-to-handle parameters to describe the flow localization. 3D-DDD simulations reveal that with the reduction of external size, the flow localization mechanism transitions from irradiation-controlled to an intrinsic dislocation source-controlled, and the spatial correlation of plastic deformation decreases due to weaker dislocation interactions and less frequent cross slip, which manifests itself through thinner dislocation channel. A discrete dislocation source activation model coupled with a cross slip tuned channel widening model is developed to reproduce and explain this transition. This finding has implications to the design of radiation-resistant materials.

The statistics and origin of intermittent micro-plasticity in a one and two dimensional dislocation dynamics simulation are studied via a linear stability analysis of the evolving dislocation configuration. When the spatial dimension is unity, a fold catastrophe is found to directly relate the dislocation configuration prior to loading to the stress at which the first plastic event occurs. More generally, it is found that the resulting plasticity of the first event is related to the eigenvector structure of the dynamical matrix evaluated at the emerging instability. Within this context the degree of micro-plastic localization and the system's closeness to criticality will be discussed.

Uniaxial compression/tension at small scales can be carried out using either load control or displacement control, both are commonly used in experimental and theoretical (simulation) studies.
While it is typically suggested that loading protocols should not influence the mechanical properties, this assumption may be violated at the nanoscale due to the emergence of size effects and abrupt strain bursts.
Here, we report a theoretical study carried out through two dimensional discrete dislocation dynamics simulations, showing that two loading protocols indeed have different effects on flow stress and plastic events statistics during uniaxial compression of thin films or pillars. The combination of nanoscale size effects and dislocation drag strain-rate effects provide a complex connection between load control at low rates with displacement control at high rates. Through scaling analysis, we demonstrate how to build a correspondence between load and displacement control at various rates and sizes. Our main focus is on the flow stress and plastic flow stress noise statistics, however our method is generalizable to other measurable quantities.

It is long established that multiply-acting dislocations sources are not expected nor are they ever observed in stage II deformation of bulk materials [1]. By analyzing large scale discrete dislocation dynamics simulations [2] of relaxed fcc microstructures [3] loaded under multislip conditions, dislocation multiplication is quantified in terms of source length and therefore source strength as well as nucleation rates. It is observed that individual nucleated dislocations (by cross-slip or glissile junctions [4]) only have a finite and local contribution to plastic deformation and no source-like object acts multiply.
On average the dislocation source length is correlated with the mean dislocation density, but the distribution of the identified sources varies considerably with the inhomogeneous local stress state. Unexpected small sources are identified and the reason for their activity is described. Nucleated dislocations do not glide very far before they themselves trigger a new nucleation event. This also raises the question when dislocation density transport is more relevant than a physical model for dislocation nucleation occurring in cascades.
The presented measures are useful for the development of a source model in dislocation density based continuum theories, since the three dimensional information as well as the statistics are studied in detail. Especially debunking the idea of source models with Frank-Read sources, which is used e.g. as a nucleation model in discrete dislocations plasticity modeling in 2D, which does not seem to capture wide ranges of material behavior or seem overly complicated.

[1] D. Kuhlmann-Wilsdorf, Chapter 59 The LES theory of solid plasticity, Dislocations ins Solids Vol. 11, Eds. F.R.N. Nabarro and M.S. Duesbery, Elsevier (2002) 211
[2] D. Weygand, L.H. Friedman, E. Van der Gieesen, A. Needleman, Model. Simul. Mater. Sci. Eng 10 (2002) 437
[3] C. Motz, D. Weygand, J. Senger, P. Gumbsch, Acta Mater. 57 (2009) 1744
[4] M. Stricker, D. Weygand, Acta Mater. 99 (2015) 130

A dislocation-density dynamics framework for investigating the “intensive” structure with the resolution lower than the dislocation core is established, based on which the dislocation interaction inside the core can be explicitly taken into consideration and the desired core structure can be maintained. This framework is then implemented numerically by using the divergence preserving finite volume method in 2D. The highly flexible nature of the present framework is further verified by the simulations of various dislocation mechanisms, including the Frank-Read sources, the Orowan bowing, and the loop shrinkage and expansion. The results exhibit the detailed core structures during the dislocation evolutions and the continuity of dislocation densities is well reserved.

A novel Fast Fourier Transform based discrete dislocation dynamics tool is introduced to study plasticity in polycrystalline media and is used to assess the predictive capability of advanced constitutive models to account for the effects of localized strain gradients, resulting from the interactions between dislocations and grain boundaries, on plastic deformation. To this end, discrete dislocation dynamics simulations of plastic deformation in aluminum polycrystals are performed and the ability of dislocation to cross grain boundaries is modulated such as to vary the level of strain localization within the system. In parallel a continuum level Fast Fourier Transform based elasto –viscoplastic strain-gradient crystal plasticity model is used to replicate the simulations performed with discrete dislocation dynamics. The two approaches are then directly compared. In particular, the advantages and shortcomings of hierarchical and concurrent multi-scale modeling frameworks are discussed.

Hart [1] proposed the experimental requirements necessary to establish existance of a mechanical equation of state (MES) with a single internal variable. Hart and Solomon [2] demonstrated that high purity aluminum satisfies this test, at least for limited range of prior deformation and thermal history and under simple (uniaxial) loading. The same observations were later repeated for other crystalline solids including metals, covalently bonded solids, and ionic solids. Key to Hart's demonstration of an MES is the existance of a scaling relationship in which stress-strain rate curves generated from different levels of work hardening, and corresponding to different internal structures, overlap each other when translated on log(stress)-log(strain rate) graphs. Often the translation is a line of slope 1/5, although our own experiments on halite suggest 1/3 is a better fit for high temperatures, in the regime where power-law creep dominates. Furthermore, in the constructed master curves the flow stress, σ, appears to approach a high strain rate limit, σ*, following an exponential form as a function of strain rate that was later called the "Lambda Law."

Here, we discuss how the lambda law can be explained in terms of the statistics of dislocation avalanches, which grow toward a percolation threshold as the stress approaches σ*. Meanwhile, dislocation bursts are coupled with dislocation climb and recovery elsewhere inside the greater dislocation network. Analysis of the coupled process suggests that (strain rate) = k(σ*/σ-1)^-(1/λ) where λ≈0.15, and in which the constant of proportionality, k, scales in proportion to (σ*)⁵ at intermediate homologous temperature and (σ*)³ at high homologous temperature. We provide examples of other materials that behave this way, and we show in experiments on halite how the constant structure load relaxation curves are related to steady state creep.

This research was supported by the National Science Foundation Grant EAR-1620474
[1] Hart, E. W. (1970). A phenomenological theory for plastic deformation of polycrystalline metals. Acta Metall., 18(6), 599–610.
[2] Hart, E. W., & Solomon, H. D. (1973). Load relaxation studies of polycrystalline high purity aluminium. Acta Metallurgica, 21(3), 295–307.

Experimental results are presented which analyze the global nonlinear response of a metallic glass in terms of avalanche statistics.
Samples of Pd₄₀Ni₄₀P₂₀ are measured using a dynamic mechanical analyzer in single cantilever mode at temperatures slightly below the glass transition temperature. The metallic glass is excited by a sinusoidal stress at 1 Hz frequency and fixed field amplitude. Initially, the field amplitude is in the linear regime of the stress strain relation, and is switched to a higher amplitude value sufficiently long to allow for time-resolved analysis.
By a period-by-period Fourier-analysis, the evolution of nonlinear contributions to the first harmonic and the emergence of additional odd harmonics with time are revealed, similar as in preceding dielectric experiments on amorphous media by Richert [1]. The timescale of the evolution of nonlinearity is connected to the structural relaxation time of the material, suggesting long-range interactions between relaxation modes.
A new incremental analysis approach on the highly resolved strain data yields a very different sight on elastic and plastic response regimes [2]. Independent of low or high field amplitude, power-law behavior in strain response distributions identifies the full investigated range of temperatures and stresses to be connected to a superposition of elastic and plastic response.
These findings question the differentiation between linear and nonlinear response regimes. The analysis of the data in terms of intermittency results in the observation of power-law behavior, which is typical for systems with underlying avalanche dynamics as observed for metallic glasses [3, 4]. The data thus suggest a microscopic view on elastic and plastic sample response to mechanical fields, which is based on the existence of interacting shear transformation zones with a coupling via their mechanical Eshelby stress fields.

References
1. Richert, R., Weinstein, S., Nonlinear Dielectric Response and Thermodynamic Heterogeneity in Liquids, Physical Review Letters. 97, 095703 (2006).
2. Riechers, B. and Samwer, K. Nonlinear Response and Avalanche Behavior in Metallic Glasses, The European Physical Journal Special Topics, 226, 14 (2017) 2997.
3. Krisponeit, J.-O., Pitikaris, S., Avila, K. E., Küchemann, S., Krüger, A., Samwer, K., Crossover from random three-dimensional avalanches to correlated nano shear bands in metallic glasses, Nature Communications 5, 2013.
4. Herrero-Gómez, C., Samwer, K., Stress and temperature dependence of the avalanche dynamics during creep deformation of metallic glasses, Scientific Reports 6 (2016).

We develop a new numerical method for simulating three-dimensional elasto-plastic materials in the slow, quasi-static limit. Large-scale simulations are performed in parallel, and make use of a custom multigrid solver to enforce the quasi-staticity constraint. We test the method using an elasto-plastic model of a bulk metallic glass based on the shear transformation zone (STZ) theory. By applying simple shear to heterogeneous samples, we explore and statistically characterize the dynamics of shear band growth.

Molecular dynamics simulations are used to investigate the underlying thermally activated processes contributing to stress relaxation in a model binary amorphous system (J. Mater. Res. 32, 2668 (2017); Acta Mater. 143, 205 (2018)). Depending on the degree to which the amorphous solid is relaxed, significant atomic-scale scale activity is seen well below the glass transition regime, and that this is largely independent of the external load. The observed stress relaxation is mediated by spatially localized and thermally activated structural excitations, which can collectively lead to a global gauge shearing, the location of which correlates with regions of low icosahedral content with lower coordination and higher density.

Strain localization into shear bands in metallic glasses (MGs) has long sought to be a phenomenon confined to the nano-scale, but recent work highlights that shear bands induce a position-dependent long-range signature (Acta Materialia 98 (2015) 94). Here we use x-ray photon correlation spectroscopy to unravel the structural dynamics around a shear band. We show that damage accumulation into a shear band leads to significantly accelerated relaxation dynamics at length scales three orders of magnitude larger than the core of the shear band. The ten-times-faster dynamics in the shear-band vicinity compared to the as-cast matrix occurs in a characteristic three-stage aging regime. These insights highlight how an ubiquitous nano-scale strain-localization mechanism in MGs leads to fundamental changes of the material at the mesoscale.

Plastic deformation of metallic glasses is mostly localized in plate-like mesoscopic defects, so-called shear bands. Their propagation often initiates catastrophic failure. Yet, small compositional variations (“minor alloying”) as well as structural heterogeneities can affect their propagation revealing different interaction mechanisms that are summarized in a qualitative mechanism map that categorizes the various processes that may occur in the course of the interaction between shear bands and structural heterogeneities.
Although the occurrence of shear bands is well known and often determines the mechanical performance of the material, their actual physical properties remain fairly unknown. Here, experimental data on the rate of atomic diffusion within shear bands have been obtained using the radiotracer method on post-deformed specimens. Additionally, novel TEM-based methods served to experimentally determine the local specific volume as well as the local degree of medium-range order and the local chemical composition quantitatively. Moreover, local strain fields at shear bands have also been analyzed and the impact of shear deformation and relaxation on the low-temperature heat capacity anomaly known as the “Boson peak” has been addressed, revealing complex dependencies on time, temperature and strain. Relaxation experiments showed an unexpected temporal evolution of the shear bands, including so-called cross-over behavior. The experimental results are discussed with respect to the underlying mechanism during the early stages of shear band activation and their temporal evolution as well as on the properties characterizing these “defects” in deformed metallic glasses.

Shear band formation is critical in understanding plasticity in non-crystalline solids such as metallic glasses. The shear bands in metallic glasses initiate from generally multiple soft spots, which interact and compete for growth. Eventually one shear band becomes dominant, growing rapidly driven by the elastic energy release of the rest of the sample. In this talk, however, we will focus on isolated, steady-state shear band close to zero-width, in a tip wear system (a flat amorphous tip sliding against a rigid slider). Here, the shear band locates at the interface (highest plastic strain rate is from the moving debris; the strain rate quickly decreases to zero away from the interface). Such steady-state shear bands enable position dependent effective temperature measurement of the shear band. We show that the effective temperature, characterizing the mechanical agitation from the slider, governs both the debris formation and the shear band formation. Furthermore, the plastic activities of the shear band can be characterized by the debris clusters. We show that large fractal clusters emerge as the contact stress approaches a critical value corresponding to global plasticity.

We have studied the statistics of shear banding events during mechanical deformation of metallic glasses using low noise and high temporal resolution data. These statistics are often measured at macroscopic length scales, yet they provide a tool with which we can probe the microscopic deformation mechanisms. A simple mean field model of slipping weak spots predicts that the statistics of these intermittent plastic events should show tuned critical behavior across length scales. The tuning parameters include the stiffness of the mechanical test system, the applied strain rate, temperature, specimen size, the applied stress level, and the extent of weakening in the specific material. In this presentation, we will demonstrate the effects of some of these tuning parameters using experimental results from compression testing and nanoindentation and for several different Zr-based alloys with the key finding being that the tuning parameters only affect the cutoff of the scaling behavior; the power law exponents are unchanged.

We report simulations of a two-dimensional binary Lennard-Jones system, a Cu Zr alloy and amorphous silicon subjected to shear deformation in excess of 1000%. In all cases shear bands are observed to form and to broaden as shear is applied to the system. The broadening of the shear bands can be modeled by assuming the rate of broadening is related to the strain rate in the band. However, initial investigations indicate that this simple assumption is not routinely applicable. The implications for theories of plasticity will be considered to elucidate any additional constraints these observations place on the underlying theory of the constitutive response of these systems.

It is currently beleived that the macroscopic plastic deformation of amorphous materials is the result of the collective behavior of localized particle rearrangements called shear transformations. The triggering of shear transformations is governed by the local structural disorder, whereas their interaction by the elastic bulk in which they are embedded. Consequently, the collective complex dynamics is a result of the competition between elasticity and disorder.

Around yielding, one observes serrated flow curves, intermittency and widely distributed avalanches of rearrangements, all of them signatures of a dynamic phase transition. Therefore, under the claim of universality, it has been argued that the generic properties of such critical systems are independent of the form of the elastic interaction or the underlying disorder. Only recently has become clearer that both the elastic interaction and the form of the disorder may have a strong impact on the critical properties.
Here we compare results of a mesoscopic depinning-like model with several quadrupolar interactions (elastic kernels) and two different flavors of disorder. We work in the quasistatic limit and we find that the particular form of the disorder has little impact on the critical properties. We confirm, however, that the elastic kernel plays an important role when it comes to the scaling of the distribution of avalanche rates and we find good agreement with previous molecular dynamics studies.

The most dramatic impact of the kernel however arises when looking at fluctuations, in particular, fluctuations of strains and displacements. We show that the presence of soft modes in the Eshelby-like elastic interaction kernels allows for a continuous, diffusive increase of these fluctuations. This constant increase of fluctuations and the associated shear banding is known to lead to material failure. Such fluctuations are generally downplayed in most homogenizing procedures, we find, however, that they feature finite-size scaling, and, as such, they may not be simply disregarded. We study the finite size scaling of the associated diffusion coefficient "D" and we find that up to a strain necessary to form a single slip line, D scales linearly with system size (D~L) independently of the kernel, consistent with molecular dynamics simulations. After a crossover regime, however, a new scaling is found (D~L^1.6) for the kernel featuring soft modes, which suggests a more prominent strain localization than the one observed in MD. This later regime is not consistent with MD, it has, however, been observed in previous lattice models. We show that shear bands (slip lines) are the soft deformation modes of the elastic interaction, and as such, they can develop at no energy cost whatsoever. For kernels that lack any soft deformation modes, fluctuations saturate after a finite time and the system becomes subdiffusive.

The physics of random amorphous solids has proven more formidable to unravel and richer in content as compared to crystals. The distinct differences stem from the marginal stability of glassy and jammed systems and reveal themselves in structural aging, the density of vibrational states, the force and gap distributions, yielding response etc. Recent advances in theory show that jamming and glassiness relate through the Gardner transition into a marginally stable glassy phase.

We study infinitesimally polydisperse crystalline packings of hard spheres with the intend to understand how amorphous physics connects with crystalline physics in the high-pressure, low-temperature Gardner phase of marginal glassy stability. Our near-crystalline configurations mostly maintain the translational and bond-orientational symmetries of the crystal yet also exhibit structural and dynamical features characteristic of jammed and marginal glass systems. Our findings suggest that experimental particulate systems that are commercially available can form structures that may appear crystalline but behave as amorphous systems. Our results indicate that a Gardner phase exists for disordered crystals expanding the range of validity of disordered physics continuously up to the vicinity of the perfect crystal.

We present results on the rate dependent flow stress in a coarse grained model of an amorphous material. In particular we study two variants of the model. Both are based on a non-convex local strain-energy function which contains quenched disorder. One variant consists of a piece-wise quadratic strain-energy function, the other variant consists of a smooth strain-energy function. Other groups have recently introduced rate-dependence into models with similar strain-energy functions via an explicit, but ad-hoc, time dependence in the local yielding rate. Here, instead, we simply suppose that the background material surrounding a yielding site acts as a visco-elastic material. We make no other assumptions about the local yielding rate in the shear zone, and it simply follows as a consequence of our model of the viscous background. With this prescription for the dynamics, we show that the smooth-potential version of the model — but not the cuspy-potential version — gives precise agreement with finer-scale particle based simulations for the rate dependence of: i) the flow stress, ii) the plastic correlation length, and iii) the diffusion coefficient. We conclude that correct modeling of both i) the smoothness of the strain-energy function and ii) the background viscosity and time-dependent yielding is necessary to get agreement with finer-scale particle-based models.

The mechanical properties of amorphous systems are expected to be tightly related with the identification and the characterization of the elementary events at the origin of plasticity. Interestingly, these localized events involving rearrangements at the atomic scale can be associated, from a continuous perspective, to mechanical heterogeneities also known as Eshelby inclusions. The representation of plasticity in terms of Eshelby inclusions should help to bridge between length scales, to link the atomistic description with the mesoscopic models of plasticity and to access macroscopic properties. Eshelby inclusions have already been used to understand the structure of shear bands[1], to explore the potential energy landscape of a model sheared glass[2] and also to study the dynamical aspects of the elementary plastic events[3]. More importantly, the signature of Eshelby inclusions have been identified from experiments in colloidal systems[4].
In this presentation I will briefly review our technique to extract Eshelby inclusions from molecular dynamics simulations. Then, I will show how this approach can be used to inform mesoscopic models, for instance by calculating the energy barriers associated to the elementary plastic events. In the last part I will discuss the effect of external parameters and especially the effect of the shear rate which is currently the subject of active research within our group.

[1] R. Dasgupta, H.G.E. Hentschel, I. Procaccia Phys. Rev. Lett. 109:255502 (2012)
[2] F. Boioli, T. Albaret, D. Rodney Phys. Rev. E 95:033005 (2017)
[3] F. Puosi, J. Rottler, J-L Barrat Phys. Rev. E 89:042302 (2014)
[4] P. Schall, D.A. Weitz, F. Spaepen Science 318, 1895 (2007)

The mechanics of failure in disordered materials remain inadequately understood. As these materials are deployed in applications including microelectromechanical systems, electrochemical devices, and nanoprinting, an improved understanding of failure, including factors controlling localization vs homogeneous plastic flow, is necessary. In particular, the mechanisms of plasticity may change under the many environmental conditions in which these devices must be able to operate. Furthermore, the sites where plastic rearrangements occur, and the nature of the ensuing flow, may vary depending on the material. Plastic deformation in disordered materials such as bulk metallic glasses is proposed to originate through rearrangements occurring at soft spots [1]. Evidence suggests that an analog to soft spots can be found in other disordered materials, including packings of colloids [2] and nanoparticles [3]. In this work, we examine the mechanical behavior of disordered nanoparticle packings formed by layer-by-layer assembly. We use atomic force microscopy-based nanoindentation, which provides information about the physics of the nanoparticle packings, with indentations applied to individual nanoparticles. These measurements reveal multiple load drops similar to the pop-ins observed previously in metallic glass [1], which we propose correspond to the activation and propagation of local rearrangements. This hypothesis is supported by topographic imaging of indents with individual particle resolution. The magnitudes of these load drops consistently follow an exponential distribution, which stands in contrast to the power-law relationship often observed in granular materials exhibiting slip avalanches. We propose that the exponential distributions observed here are the result of the small volume being investigated, which does not permit fractal plasticity mechanisms to occur. Furthermore, the relative humidity affects the number of load drops, with significantly more load drops occurring at humidity levels close to saturation. This indicates that tuning the ambient conditions may be a convenient means for using the mechanical response of this material system to simulate many different types of disordered materials. As well, this helps demonstrate how nanoparticle packings in applications may respond applied stresses under variable environments.

References
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[2] Strickland, D. J., Huang, Y.-R., Lee, D., & Gianola, D. S. (2014). PNAS, 111(51), 18167–72.
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