Frank-Read Source Research Articles

Stress-based model on work hardening and softening of materials at large strains: corrugation process of sheet

In this article, a stress-based model is proposed to investigate the strength evolution during severe plastic deformation (SPD) or large strains. In this model, the work hardening mechanisms are described by Frank-Read sources, while the cross-slip and climb processes are considered as the mechanisms for work softening phenomenon. Within all SPD processes, one of corrugation processes such as constrained groove pressing is chosen to assess the validity of the model predictions. The model predictions are in agreement with the earlier reports and the experimental results achieved in this study.

Initial dislocation structures in 3-D discrete dislocation dynamics and their influence on microscale plasticity

Realistic dislocation network topologies were generated by relaxing an initially pinning point free dislocation loop structure using three-dimensional discrete dislocation dynamics simulations. Traction-free finite-sized samples were used. Subsequently, these equilibrated structures were subjected to tensile loading and their mechanical behavior was investigated with respect to the initial configuration. A strong mechanical size effect was found. The flow stress at 0.2% plastic deformation scales with specimen size with an exponent between −0.6 and −0.9, depending on the initial structure and size regime. During relaxation, a mechanism, also favored by cross-slip, is identified which leads to rather stable pinning points. These pinning points are comparable to those of the isolated Frank–Read sources often used as a starting configuration in previous discrete dislocation dynamics simulations. These nodes act as quite stable dislocation sources, which can be activated multiple times. The influence of this source mechanism on the mechanical properties of small-scale specimens is discussed.

902 様々な寸法の結晶粒中における転位ループ放出速度と作用応力に関する3次元転位動力学シミュレーション(OS9.材料の組織・強度に関するマルチスケールアナリシス(1),オーガナイズドセッション)

Incubation time for the emission of a dislocation loop from a Frank-Read type dislocation source in crystal grains of various sizes are studied by three dimensional dislocation dynamics simulations. Results show that there are two regimes where stress exponent is as large as 500 in one regime and quite small in the other. Comparison of results with various values for mobility of dislocations shows that the mobility does not contribute to the critical resolved shear stress (CRSS) for emission of dislocatin loops but it contributes to the time needed to emit dislocation loops when the applied shear stress is sufficiently larger than the CRSS.

Interaction between dislocations in bcc iron at high temperature

When a crystal deforms plastically, sources within, such as the Frank–Read source, emit dislocations, which then glide in response to the applied stress. As the dislocations move away from the source, they may encounter an obstacle, for example a grain boundary, impurity atom or locked dislocation, which they cannot overcome. As more dislocations are emitted, they ’pile up’ near the obstacle, until their own stress fields acting back on the source prevent more dislocations from being produced, unless the external applied stress is increased. The properties of these pile-ups strongly influence the deformation of the crystal as a whole giving rise to the Hall–Petch effect [E. Hall, Proc. Phys. Soc. B 64 (1951) 747–753, N. Petch, J. Iron and Steel Institute 173 (1953) 25–28.] relating yield strength to grain size. In this paper we investigate how the observed strong variation of elastic constants as functions of temperature affects the strength of interactions between dislocations, and mechanisms of plastic deformation of iron at elevated temperatures. We find that the observed severe softening of the tetragonal shear modulus C′ at high temperature gives rise to a drastic reduction in the repulsion between parallel like edge dislocations, and hence to a greatly increased number of dislocations in pile-ups, especially those in the 〈1 0 0〉(0 0 1) configuration. The associated increase in plastic strain will lead in turn to a conspicuous reduction in tensile strength as C′ falls. The phenomenon is inherently anisotropic, and this work underlines the importance of including anisotropic elastic effects when modelling Fe at high temperatures.

A dislocation-dynamics-based derivation of the Frank–Read source characteristics for discrete dislocation plasticity

In this paper, the main characteristics of Frank–Read (F–R) sources used in a mechanism-based discrete dislocation plasticity (M-DDP) analysis are estimated by employing a recently developed non-singular continuum elastic theory of dislocations. The critical nucleation stress, τnuc, is determined more accurately because the dislocation core effects are considered precisely by atomistically-informing the dislocation dynamics simulations. The nucleation time is calculated and compared with the previous predictions. The dependence of the drag coefficient of dislocations on dislocation line orientation, which affects the nucleation time and also the shape of the emitted dislocation loop, is considered. In M-DDP simulations, τnuc used for sources is calculated based on the assumption of an infinite domain. In reality, however, the critical nucleation stress is affected by other F–R sources. It is proposed in this paper that the critical nucleation stress should be modified by considering the effects of other dislocation sources. To this end, τnuc should be determined for an F–R source in a finite cell with periodic boundary conditions.

Growth mechanisms, polytypism, and real structure of kaolinite microcrystals

The mechanisms of growth of kaolinite microcrystals (0.1–5.0 μm in size) at deposits related to the cluvial weathering crust, as well as to the low-temperature and medium-temperature hydrothermal processes of transformations of minerals in different rocks in Russia, Kazakhstan, Ukraine, Czechia, Vietnam, India, Cuba, and Madagascar, are investigated using transmission electron microscopy and vacuum decoration with gold. It is established that kaolinite microcrystals grow according to two mechanisms: the mechanism of periodic formation of two-dimensional nuclei and the mechanism of spiral growth. The spiral growth of kaolinite microcrystals is dominant and occurs on steps of screw dislocations that differ in sign and magnitude of the Burgers vector along the c axis. The layered growth of kaolinite originates from a widespread source in the form of a step between polar (+ and −) dislocations, i.e., a growth analogue of the Frank-Read dislocation source. The density of growth screw dislocations varies over a wide range and can be as high as ∼109 cm−2. Layered stepped kaolinite growth pyramids for all mechanisms of growth on the (001) face of kaolinite exhibit the main features of the triclinic 1Tc and real structures of this mineral.

A transmission electron microscopy study of experimentally deformed quartzite with different degrees of doping

TEM study of Heavitree quartzite deformed at high temperatures and a pressure of 1.5 GPa with three different preparations: water-added, vacuum dried, and sodium doped show a wide range of dislocation microstructure. The vacuum dried sample has a very heterogeneous dislocation distribution varying from almost no dislocations to very high densities. Dislocation nucleation occurs by cross-slip mechanisms, such as Frank-Read sources and Orowan loops around hard inclusions. Shear bands develop in the basal plane composed of two closely spaced shear planes. Dislocations have straight segments and they are aligned along directions of dense packing, suggesting strong Peierls stress control due to the crystal structure. No voids, bubbles or dislocation walls where observed in the vacuum dried sample. The sodium-doped sample had a homogeneous high dislocation density. The water-added sample has a significantly lower, relatively uniform dislocation density. Bubbles associated with dislocations are seen in every grain. Dislocations were either present in loops around bubbles or as straight dislocations with small bubbles spaced along the dislocation line. Some of the long straight dislocations are aligned parallel to the c-axis. Sub-grains where very frequent, sometimes form cells of about 2 μm in diameter with straight edges. Analysis of dislocations show 1/3<a> and [c] dislocations in sub-grain walls and free 1/3<a> dislocations on prism planes. There is clear evidence for dislocation climb with sub-grain walls, dislocation cells, dislocation junctions and dislocation debris of small loops. The major difference between vacuum dried and water-added samples are the homogeneity of the microstructure and evidence for climb in the water-added sample. Glide is clearly difficult, with a high Peierls stress in the vacuum dried sample as shown by areas of very low and very high dislocation density and the crystallographic control of dislocation line direction. The sodium-doped sample indicates that nucleation was easy by the high homogenous dislocation density, but no climb recovery has taken place. The density is high and the dislocations are strongly interacting causing tangles; no evidence for crystallographic control can be observed at these levels of strain. The activation energy for creep in Heavitree quartzite decreases with inferred water content of the specimens form 185 kJ mol −1 for vacuum dried to 151 for 0.4 wt% water-added samples. Analysis of diffusion data for oxygen under hydrothermal conditions and inferred diffusion data for the hydrogarnet defect and dislocation velocity suggests activation energies for these processes are similar to the activation energy for the dislocation creep of Heavitree quartzite and other quartz aggregates. For data from the previously published experiments with the highest stress resolution there is a correlation between the A pre-factor in the power law creep equation and the activation energy for creep. It is speculated that the correlation may be due to variable hydrogarnet defect concentrations.

Numerical Modeling of Scale Dependent Mechanical Properties of Metal Polycrystals

Movement of dislocations and their interaction with grain boundaries during yielding process of metal polycrystals are numerically simulated . Simulation results show that when the size of Frank-Read source is close to grain diameter, the resolved shear stress needed to emit a dislocation loop increases sharply with decreasing grain size. This effect is introduced into the critical resolved shear stress, which is used in the continuum mechanics-based crystal plasticity analysis. Results of the analyses of polycrystal models show distinct increases of yield stress for specimens with smaller mean grain diameter. Evolution of geometrically necessary dislocations in fine- and coarse-grained polycrystals are discussed.

A comment on the role of Frank–Read sources in plasticity of nanomaterials

Dislocation generation by a Frank–Read source was considered for the case of nanocrystalline metallic materials. It was shown that thermally activated operation of this mechanism, which was proposed in the literature as an explanation for the observed grain size dependence of the strain rate sensitivity (SRS) of the flow stress, is only possible in a very limited range of the source lengths and stresses. By comparing the calculated SRS with the experimental data for copper, it was further demonstrated that the thermally activated Frank–Read mechanism can be discarded as a process underlying the observed strain rate sensitivity data.

Etch Pit Observation of Dislocations in Early Deformation of Cu-Al Alloy and Pure Mg Polycrystals

In order to make clear the micro-yielding mechanisms of polycrystalline metals including twins, the movement of dislocations in the surface grains of Cu-6.8at%Al alloy and pure Mg polycrystals during the early stages of deformation was directly observed by using etch pit technique. The fresh dislocations multiply from the Frank-Read sources within the grains, and pile up against the twin and grain boundaries of two kinds of specimens. The pile-up dislocations on the primary and/or secondary slip planes are also confirmed in Cu-6.8at%Al alloys. Especially during the compressive loading for pure Mg, the occurrence of deformation twins is remarkable with an increase of strain rate, whereas the distribution of fresh dislocations tends to decrease in the surface grains. The present results suggest that the effect of twin boundaries on micro-yielding is almost equivalent for that of grain boundaries, which act as barriers to moving dislocations even in the pre-yield deformation.

Three-dimensional dislocation dynamics simulation of the influence of sample size on the stress–strain behavior of fcc single-crystalline pillars

The size-dependent flow stress in uniformly loaded pillars has been modeled using a discrete dislocation dynamics tool. Starting from Frank-Read sources of given length and random orientation, the simulated flow stress at 0.15% plastic strain shows a clear size effect, similar to experimental findings for larger strains. The scattering of the simulated stress–strain curves decreases with increasing sample size, which reflects that plasticity of small scale samples is very sensitive to the underlying dislocation microstructure.

An increase in the rate of plastic deformation under the effect of ultrasound

The multiplication of dislocations by a Frank-Read source upon a simultaneous effect of constant and harmonic components of external stress has been studied using computer-aided mathematical simulation (the harmonic component of the stress was by an order of magnitude less than the constant component.) It has been found that the number of dislocation loops that have been formed in a time unit depends on the frequency of the harmonic component. A maximum was observed at ultrasonic frequencies. The number of the loops generated can increase by several times. It has been shown that the appearance of a maximum is caused by the fact that the time of loop formation corresponds to the period of ultrasonic oscillations. On this ground, we propose an explanation to the phenomenon of ultrasound-induced plasticization known in the physics of plasticity.

Multiscale modelling of size effect in fcc crystals: discrete dislocation dynamics and dislocation-based gradient plasticity

A multiscale approach to modelling size effect in crystal plasticity is presented. At the microscale, discrete dislocation dynamics (DD) coupled with finite element (FE) analysis allows the rigorous treatment of a broad range of micro-plasticity problems with minimal phenomenological assumptions. At the macroscale, a gradient crystal plasticity model, which incorporates scale-dependence by introducing the density of geometrically necessary dislocations (GNDs) in the expression of mean glide path length, is used. As a case study, bending of micro-sized single crystal beams is considered and the correspondence between the predictions of both models is made. In its current framework, the macroscale model did not capture the experimentally observed effect of specimen size on the initial yield stress. With this effect naturally captured in the corresponding DD analysis, the absence of a density-independent size effect in the expression for the strength of slip systems was concluded. In an independent work on the tensile loading of micrometer-sized polycrystals 1, a size effect, physically rooted in the size and location of Frank–Read sources (FRS) relative to grain boundaries, was identified. This effect can be generalized in the context of dislocation–interface interactions, typically missing to one degree or another, in current gradient crystal plasticity models and can, in principle, be used to understand the initial yield size-dependence in single crystal bending identified through DD analysis.

Frank Read sources in semiconductors

Dislocations in semiconductors can start to multiply at Frank Read sources. Since in the pre-yield region they are aligned along the 〈1 1 0〉 directions, it is possible to calculate the energy of a bow-out analytically and to derive an expression for the stress necessary to activate a Frank Read source. For the hexagonal loops emitted, there exists a metastable equilibrium configuration, and the stress dependence of its magnitude and shape has been determined.

A discrete dislocation analysis of strengthening in bilayer thin films

A two-dimensional (2D) discrete dislocation plasticity framework, which incorporates some three-dimensional mechanisms through constitutive additions, is used to analyse the response to uniaxial tension of nanoscale bilayer thin films. Frank–Read sources, modelled as junction dipoles in 2D, act as sources of dislocations. Infinite, homogeneous medium fields of the discrete dislocations are superposed with a non-singular complementary field that enforces the boundary conditions and accounts for image stresses arising from the difference in elastic properties between the layers. The resulting boundary value problem is solved using the finite element method. Analysis has been carried out for fully coherent bilayer Al/Cu and Cu/Ni films oriented for double slip. The analysis accounts for the effects of three key mechanisms: resistance to dislocation nucleation and motion due to elastic modulus mismatch (e.g. Koehler barrier); single-dislocation bow-out within layers (Orowan process) and slip blocking at interfaces (Hall–Petch mechanism). The relative importance of each mechanism is studied as a function of the bilayer thickness. The results indicate a significant strengthening with decreasing bilayer thickness. Conclusions are drawn regarding the possible causes of the observed strengthening.

A multiscale approach for modeling scale-dependent yield stress in polycrystalline metals

Modeling of scale-dependent characteristics of mechanical properties of metal polycrystals is studied using both discrete dislocation dynamics and continuum crystal plasticity. The initial movements of dislocation arc emitted from a Frank-Read type dislocation source and bounded by surrounding grain boundaries are examined by dislocation dynamics analyses system and we find the minimum resolved shear stress for the FR source to emit at least one closed loop. When the grain size is large enough compared to the size of FR source, the minimum resolved shear stress levels off to a certain value, but when the grain size is close to the size of the FR source, the minimum resolved shear stress shows a sharp increase. These results are modeled into the expression of the critical resolved shear stress of slip systems and continuum mechanics based crystal plasticity analyses of six-grained polycrystal models are made. Results of the crystal plasticity analyses show a distinct increase of macro- and microscopic yield stress for specimens with smaller mean grain diameter. Scale-dependent characteristics of the yield stress and its relation to some control parameters are discussed.

Dislocation transport using an explicit Galerkin/least-squares formulation

An explicit Galerkin/least-squares formulation is introduced for a quasilinear transport equation in field dislocation mechanics (FDM) and applied to the study of the kinematics of dislocation density evolution in the following physical contexts: annihilation of dislocations, expansion of a polygonal dislocation loop and simulation of a Frank–Read source. Stability analysis is carried out for the corresponding linear one-dimensional (1D) case. The formulation reduces to the Lax–Wendroff finite difference scheme for the 1D equation when equal weighting is used for the Galerkin and least-squares terms and the shape functions are linear. This conditionally stable method leads to a symmetric well-conditioned system of equations with constant coefficients, making it attractive for large-scale problems.It is shown that the transport equation, in the contexts mentioned above simplifies to the Hamilton–Jacobi equations governing geometrical optics and level-set methods. The weak solutions to these equations are not unique, and the numerical method is able to capture solutions corresponding to shock as well as rarefraction waves by appropriate algorithmic modifications.

Plastic deformation of freestanding thin films: Experiments and modeling

Experimental measurements and computational results for the evolution of plastic deformation in freestanding thin films are compared. In the experiments, the stress–strain response of two sets of Cu films is determined in the plane-strain bulge test. One set of samples consists of electroplated Cu films, while the other set is sputter-deposited. Unpassivated films, films passivated on one side and films passivated on both sides are considered. The calculations are carried out within a two-dimensional plane strain framework with the dislocations modeled as line singularities in an isotropic elastic solid. The film is modeled by a unit cell consisting of eight grains, each of which has three slip systems. The film is initially free of dislocations which then nucleate from a specified distribution of Frank–Read sources. The grain boundaries and any film-passivation layer interfaces are taken to be impenetrable to dislocations. Both the experiments and the computations show: (i) a flow strength for the passivated films that is greater than for the unpassivated films and (ii) hysteresis and a Bauschinger effect that increases with increasing pre-strain for passivated films, while for unpassivated films hysteresis and a Bauschinger effect are small or absent. Furthermore, the experimental measurements and computational results for the 0.2% offset yield strength stress, and the evolution of hysteresis and of the Bauschinger effect are in good quantitative agreement.

Size effects in uniaxial deformation of single and polycrystals: a discrete dislocation plasticity analysis

The effect of specimen size on the uniaxial deformation response of planar single crystals and polycrystals is investigated using discrete dislocation plasticity. The dislocations are all of edge character and modelled as line singularities in a linear elastic material. The lattice resistance to dislocation motion, dislocation nucleation, dislocation interaction with obstacles and dislocation annihilation are incorporated through a set of constitutive rules. Grain boundaries are modelled as impenetrable to dislocations. Two types of polycrystalline materials are considered: one that only has grains with a single orientation while the other has a checker-board arrangement of two types of grains which are rotated 90° with respect to each other. The single crystals display a strong size dependence with the flow strength increasing with decreasing specimen size. In sufficiently small single crystal specimens, the nucleation rate of the dislocations is approximately equal to the rate at which the dislocations exit the specimens so that below a critical specimen size the flow strength is set by the strength of the initially present Frank–Read sources. On the other hand, grain boundaries acting as barriers to plastic deformation in polycrystalline specimens of the same size lead to a more diffuse deformation pattern and to a nearly size-independent response.

Discrete Dislocation Dynamics Simulation of Interfacial Dislocation Network in Gamma/Gamma-Prime Microstructure of Ni-based Superalloys

A simple back force model is proposed for a dislocation cutting into γ � precipitate, taking the work for making and recovering an anti-phase boundary (APB) into account. The first dislocation, or a leading partial of a superdislocation, is acted upon by a back force whose magnitude is equal to the APB energy. The second dis- location, or a trailing partial of a superdislocation, is at- tracted by the APB with a force of the same magnitude. The model is encoded in the 3D discrete dislocation dy- namics (DDD) code and applied to the cutting behavior of dislocations at a γ/γ � interface covered by an interfa- cial dislocation network. Dislocations are generated from Frank-Read sources and approach the interface. The first dislocation piles up at the interface not by the stress field of the network but by the back force against making an APB. The second dislocation, however, stands off from the interface by the stress field of the first dislocation and the dislocation network. The finer mesh of the network, the further the second dislocation piles up. These two dislocations cut into the precipitate forming a superdis- location under the force from follow-on dislocations. It is also clarified that the penetration takes place from the interspace of the network.