Dislocation avalanches

Last updated December 29, 2023

Dislocation avalanches are rapid discrete events during plastic deformation, in which defects are reorganized collectively. This intermittent flow behavior has been observed in microcrystals, whereas macroscopic plasticity appears as a smooth process. Intermittent plastic flow has been observed in several different systems. In AlMg Alloys, interaction between solute and dislocations can cause sudden jump during dynamic strain aging.^[1] In metallic glass, it can be observed via shear banding with stress localization;^[2] and single crystal plasticity, it shows up as slip burst.^[3] However, analysis of the events with orders-magnitude difference in sizes with different crystallographic structure reveals power-law scaling between the number of events and their magnitude, or scale-free flow.^[4]

This microscopic instability of plasticity can have profound consequences on mechanical behavior of microcrystals. The increased relative size of the fluctuations makes it difficult to control the plastic forming process.^[5] Moreover, at small specimen sizes the yield stress is not well defined by the 0.2% plastic strain criterion anymore, since this value varies specimen by specimen.^[6]

Similar intermittent effects has been studied in many completely different systems, including intermittency of energy dissipation in magnetism (Barkhausen effect), superconductivity, earthquakes, and friction.^[7]

Background

Macroscopic plasticity are well-described by continuum model. Dislocations motions are characterized by an average velocity

\tau =\rho b{\bar {v}}

which is known as Orowan's equation. However, this approach completely fails to account for well-known intermittent deformation phenomena such as the spatial localization of dislocation flow into "slip bands"^[8](also known as Lüders band) and the temporal fluctuations in stress-strain curves (the Portevin–Le Chatelier effect first reported in the 1920s).^[9]^[1]

Experimental Approach

Although evidence of intermittent flow behavior is long known and studied, it is not until past two decades that a quantitative understanding of the phenomenon is developed with help of novel experimental techniques.

Acoustic emission

Acoustic emission (AE) is used to record the crackling noise from deforming crystals.^[10]^[11] The amplitudes of the acoustic signals can be related to the area swept by the fast-moving dislocations and hence to the energy dissipated during deformation events. The result shows that cracking noise is not smooth, with no specific energy scale. Effect of grain structure for "supercritical" flow has been studied in polycrystalline ice.^[12]

Direct mechanical measurement

Recent developments in small scale mechanical testing, with sub-nm resolution in displacement and sub-μN resolution in force, now allow to directly study discrete events in stress and strain. Currently, the most prominent method is a miniaturized compression experiment, where a nanoindenter equipped with a flat indentation tip is used. Equipped with in-situ techniques in combination with Transmission electron microscopy, Scanning electron microscopy, and micro-diffraction methods, this nanomechanical testing method can give us rich detail in nanoscale plasticity instabilities in real time.

One potential concern in nanomechanical measurement is: how fast can the system respond? Can indentation tip remain contact with the sample and track the deformation? Since dislocation velocity is strongly effected by stress, velocity can be many orders different in different systems. Also, multiscale nature of dislocation avalanche event gives dislocation velocity a large range. For example, single dislocations have been shown to move at speeds of ~10 ms⁻¹ in pure Cu, but dislocation groups moved with ~10⁻⁶ ms⁻¹ in Cu-0.5%Al. The opposite is found for iron, where dislocation groups are found to move six orders of magnitude faster in a FeSi-alloy than individual dislocations in pure iron.

To resolve this issue, Sparks et al. has designed an experiment to measure first fracture of Si beam and compare with theoretical prediction to determine the respond speed of the system.^[13] In addition to regular compression experiments, in-situ electrical contact resistance measurements (ECR) were performed. During these in-situ tests, a constant voltage was applied during the deformation experiment to record current evolution during intermittent plastic flow. Result shows that indentation tip remains contact with the sample throughout experiments, which proves the respond speed is fast enough.

Theoretical analysis and simulations

Avalanche strain distributions have the general form^[5]

P(s)=Cs^{-\tau }exp[-(s/s_{0})^{2}]

where C is a normalization constant, t is a scaling exponent, and s₀ is the characteristic strain of the largest avalanches.

Dislocation dynamic simulations have shown that $\tau$ can be sometimes close to 1.5, but also, many times higher exponents are observed, with values that may even approach 3 in special circumstances.^[14]^[15] While traditional mean-field theory predictions suggest the value of 1.5,^[16]^[17] more advanced mean-field investigations have demonstrated that larger exponents can be caused by non-trivial, but prevalent mechanisms in crystal plasticity that suppress mobile dislocation propulations.^[18]^[19]

Effect of crystal structure on dislocation avalanches

In FCC crystals, scaled velocity shows a main peak in distribution with relatively smooth curve, which is expected from theory except for some disagreement at low velocity.^[20] However, in BCC crystal, distribution of scaled velocity is broader and much more dispersed.^[21] The result also shows that scaled velocity in BCC is a lot slower than FCC, which is not predicted by mean field theory. A possible explanation this discrepancy is based on different moving speed of edge and screw dislocations in two type of crystals. In FCC crystals, two parts of dislocation move at same velocity, resulting in smooth averaged avalanches profile; whereas in BCC crystals, edge components move fast and escape rapidly, while screw parts propagate slowly, which drag the overall velocity. Based on this explanation, we will also expect a direction dependence of avalanche events in HCP crystals, which currently lack in experimental data.

Related Research Articles

In physics and materials science, plasticity is the ability of a solid material to undergo permanent deformation, a non-reversible change of shape in response to applied forces. For example, a solid piece of metal being bent or pounded into a new shape displays plasticity as permanent changes occur within the material itself. In engineering, the transition from elastic behavior to plastic behavior is known as yielding.

$<span class="mw-page-title-main">Electron backscatter diffraction</span> Scanning electron microscopy technique$

Electron backscatter diffraction (EBSD) is a scanning electron microscopy (SEM) technique used to study the crystallographic structure of materials. EBSD is carried out in a scanning electron microscope equipped with an EBSD detector comprising at least a phosphorescent screen, a compact lens and a low-light camera. In the microscope an incident beam of electrons hits a tilted sample. As backscattered electrons leave the sample, they interact with the atoms and are both elastically diffracted and lose energy, leaving the sample at various scattering angles before reaching the phosphor screen forming Kikuchi patterns (EBSPs). The EBSD spatial resolution depends on many factors, including the nature of the material under study and the sample preparation. They can be indexed to provide information about the material's grain structure, grain orientation, and phase at the micro-scale. EBSD is used for impurities and defect studies, plastic deformation, and statistical analysis for average misorientation, grain size, and crystallographic texture. EBSD can also be combined with energy-dispersive X-ray spectroscopy (EDS), cathodoluminescence (CL), and wavelength-dispersive X-ray spectroscopy (WDS) for advanced phase identification and materials discovery.

<span class="mw-page-title-main">Crystal twinning</span> Two separate crystals sharing some of the same crystal lattice points in a symmetrical manner

Crystal twinning occurs when two or more adjacent crystals of the same mineral are oriented so that they share some of the same crystal lattice points in a symmetrical manner. The result is an intergrowth of two separate crystals that are tightly bonded to each other. The surface along which the lattice points are shared in twinned crystals is called a composition surface or twin plane.

<span class="mw-page-title-main">Lüders band</span> Bands of plastic deformation in metals

Lüders bands, is type of slip bands in metals or stretcher-strain marks which are formed due to localized bands of plastic deformation in metals experiencing tensile stresses, common to low-carbon steels and certain Al-Mg alloys. First reported by Guillaume Piobert, and later by W. Lüders, the mechanism that stimulates their appearance is known as dynamic strain aging, or the inhibition of dislocation motion by interstitial atoms, around which "atmospheres" or "zones" naturally congregate.

The Portevin–Le Chatelier (PLC) effect describes a serrated stress–strain curve or jerky flow, which some materials exhibit as they undergo plastic deformation, specifically inhomogeneous deformation. This effect has been long associated with dynamic strain aging or the competition between diffusing solutes pinning dislocations and dislocations breaking free of this stoppage.

Zirconium alloys are solid solutions of zirconium or other metals, a common subgroup having the trade mark Zircaloy. Zirconium has very low absorption cross-section of thermal neutrons, high hardness, ductility and corrosion resistance. One of the main uses of zirconium alloys is in nuclear technology, as cladding of fuel rods in nuclear reactors, especially water reactors. A typical composition of nuclear-grade zirconium alloys is more than 95 weight percent zirconium and less than 2% of tin, niobium, iron, chromium, nickel and other metals, which are added to improve mechanical properties and corrosion resistance.

In materials science, slip is the large displacement of one part of a crystal relative to another part along crystallographic planes and directions. Slip occurs by the passage of dislocations on close/packed planes, which are planes containing the greatest number of atoms per area and in close-packed directions. Close-packed planes are known as slip or glide planes. A slip system describes the set of symmetrically identical slip planes and associated family of slip directions for which dislocation motion can easily occur and lead to plastic deformation. The magnitude and direction of slip are represented by the Burgers vector, $b$ .

A shear band is a narrow zone of intense shearing strain, usually of plastic nature, developing during severe deformation of ductile materials. As an example, a soil specimen is shown in Fig. 1, after an axialsymmetric compression test. Initially the sample was cylindrical in shape and, since symmetry was tried to be preserved during the test, the cylindrical shape was maintained for a while during the test and the deformation was homogeneous, but at extreme loading two X-shaped shear bands had formed and the subsequent deformation was strongly localized.

Pseudoelasticity, sometimes called superelasticity, is an elastic (reversible) response to an applied stress, caused by a phase transformation between the austenitic and martensitic phases of a crystal. It is exhibited in shape-memory alloys.

A cryogenic treatment is the process of treating workpieces to cryogenic temperatures in order to remove residual stresses and improve wear resistance in steels and other metal alloys, such as aluminum. In addition to seeking enhanced stress relief and stabilization, or wear resistance, cryogenic treatment is also sought for its ability to improve corrosion resistance by precipitating micro-fine eta carbides, which can be measured before and after in a part using a quantimet.

In geology, a deformation mechanism is a process occurring at a microscopic scale that is responsible for changes in a material's internal structure, shape and volume. The process involves planar discontinuity and/or displacement of atoms from their original position within a crystal lattice structure. These small changes are preserved in various microstructures of materials such as rocks, metals and plastics, and can be studied in depth using optical or digital microscopy.

Severe plastic deformation (SPD) is a generic term describing a group of metalworking techniques involving very large strains typically involving a complex stress state or high shear, resulting in a high defect density and equiaxed "ultrafine" grain (UFG) size or nanocrystalline (NC) structure.

Dynamic strain aging (DSA) for materials science is an instability in plastic flow of materials, associated with interaction between moving dislocations and diffusing solutes. Although sometimes dynamic strain aging is used interchangeably with the Portevin–Le Chatelier effect (or serrated yielding), dynamic strain aging refers specifically to the microscopic mechanism that induces the Portevin–Le Chatelier effect. This strengthening mechanism is related to solid-solution strengthening and has been observed in a variety of fcc and bcc substitutional and interstitial alloys, metalloids like silicon, and ordered intermetallics within specific ranges of temperature and strain rate.

In continuum mechanics, ratcheting, or ratchetting, also known as cyclic creep, is a behavior in which plastic deformation accumulates due to cyclic mechanical or thermal stress.

Crystal plasticity is a mesoscale computational technique that takes into account crystallographic anisotropy in modelling the mechanical behaviour of polycrystalline materials. The technique has typically been used to study deformation through the process of slip, however, there are some flavors of crystal plasticity that can incorporate other deformation mechanisms like twinning and phase transformations. Crystal plasticity is used to obtain the relationship between stress and strain that also captures the underlying physics at the crystal level. Hence, it can be used to predict not just the stress-strain response of a material, but also the texture evolution, micromechanical field distributions, and regions of strain localisation. The two widely used formulations of crystal plasticity are the one based on the finite element method known as Crystal Plasticity Finite Element Method (CPFEM), which is developed based on the finite strain formulation for the mechanics, and a spectral formulation which is more computationally efficient due to the fast Fourier transform, but is based on the small strain formulation for the mechanics.

Electroplasticity, describes the enhanced plastic behavior of a solid material under the application of an electric field. This electric field could be internal, resulting in current flow in conducting materials, or external. The effect of electric field on mechanical properties ranges from simply enhancing existing plasticity, such as reducing the flow stress in already ductile metals, to promoting plasticity in otherwise brittle ceramics. The exact mechanisms that control electroplasticity vary based on the material and the exact conditions. Enhancing the plasticity of materials is of great practical interest as plastic deformation provides an efficient way of transforming raw materials into final products. The use of electroplasticity to improve processing of materials is known as electrically assisted manufacturing.

Indentation plastometry is the idea of using an indentation-based procedure to obtain (bulk) mechanical properties in the form of stress-strain relationships in the plastic regime. Since indentation is a much easier and more convenient procedure than conventional tensile testing, with far greater potential for mapping of spatial variations, this is an attractive concept.

Slip bands or stretcher-strain marks are localized bands of plastic deformation in metals experiencing stresses. Formation of slip bands indicates a concentrated unidirectional slip on certain planes causing a stress concentration. Typically, slip bands induce surface steps and a stress concentration which can be a crack nucleation site. Slip bands extend until impinged by a boundary, and the generated stress from dislocations pile-up against that boundary will either stop or transmit the operating slip depending on its (mis)orientation.

Angus J Wilkinson is a professor of materials science based at University of Oxford. He is a specialist in micromechanics, electron microscopy and crystal plasticity. He assists in overseeing the MicroMechanics group while focusing on the fundamentals of material deformation. He developed the HR-EBSD method for mapping stress and dislocation density at high spatial resolution used at the micron scale in mechanical testing and micro-cantilevers to extract data on mechanical properties that are relevant to materials engineering.

Dierk Raabe is a German materials scientist and researcher, who has contributed significantly to the field of materials science. He is a professor at RWTH Aachen University and director of the Max Planck Institute for Iron Research in Düsseldorf. He is the recipient of the 2004 Leibniz Prize, and the 2022 Acta Materialia's Gold Medal. He also received the honorary doctorate of the Norwegian University of Science and Technology.

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