Dimensional Dislocation Dynamics Research Articles

Modeling viscosity of (Mg,Fe)O at lowermost mantle conditions

The viscosity of the lower mantle results from the rheological behavior of its two main constituent minerals, aluminous (Mg,Fe)SiO3 bridgmanite and (Mg,Fe)O ferropericlase. Understanding the transport properties of lower mantle aggregates is of primary importance in geophysics and it is a challenging task, due to the extreme time-varying conditions to which such aggregates are subjected. In particular, viscosity is a crucial transport property that can vary over several orders of magnitude. It thus has a first-order control on the structure and dynamics of the mantle.Here we focus on the creep behavior of (Mg,Fe)O at the bottom of the lower mantle, where the presence of thermo-chemical anomalies such as ultralow-velocity zones (ULVZ) may significantly alter the viscosity contrast characterizing this region. Two different iron concentrations of (Mg1–xFex)O are considered: one mirroring the average composition of ferropericlase throughout most of the lower mantle (x = 0.20) and another representing a candidate magnesiowüstite component of ULVZs near the base of the mantle (x = 0.84). The investigated pressure-temperature conditions span from 120 GPa and 2800 K, corresponding to the average geotherm at this depth, to core-mantle boundary conditions of 135 GPa and 3800 K.In this study, dislocation creep of (Mg,Fe)O is investigated by dislocation dynamics (DD) simulations, a modeling tool which considers the collective motion and interactions of dislocations. To model their behavior, a 2.5 dimensional dislocation dynamics approach is employed. Within this method, both glide and climb mechanisms can be taken into account, and the interplay of these features results in a steady-state condition. This allows the retrieval of the creep strain rates at different temperatures, pressures, applied stresses and iron concentrations across the (Mg,Fe)O solid solution, providing information on the viscosity for these materials.A particularly low viscosity is obtained for magnesiowüstite with respect to ferropericlase, the difference being around 10 orders of magnitude. Thus, the final section of this work is devoted to the assessment of the dynamic implications of such a weak phase within ULVZs, in terms of the viscosity contrast with respect to the surrounding lowermost mantle.

An Orowan precipitate strengthening equation for mechanical twinning in Mg

This study develops an Orowan precipitate hardening equation applicable to twin propagation. It is derived from a series of two dimensional dislocation dynamics simulations of a twin tip impinging upon a line of obstacles. When the number of twinning dislocations present is low, the Orowan stress increases with the number of twinning dislocations in the twin. When the number of twinning dislocations is high, the trailing dislocations do not partake in the critical initial bypass event, instead, they multiply the stress on those leading dislocations that do. The result is that bypass takes place at lower stresses when more twinning dislocations are present. A double super dislocation model provides a good description of the phenomenon. It is seen that particle size exerts a stronger role than in the conventional Orowan hardening equation and that there exists a number of twinning dislocations for which the Orowan stress is greatest. For alloy design using non-shearable precipitates, the basic objective of decreasing the inter-particle spacing on the twin plane remains most efficacious.

Strain hardening in 2D discrete dislocation dynamics simulations: A new ‘2.5D’ algorithm

The two-dimensional discrete dislocation dynamics (2D DD) method, consisting of parallel straight edge dislocations gliding on independent slip systems in a plane strain model of a crystal, is often used to study complicated boundary value problems in crystal plasticity. However, the absence of truly three dimensional mechanisms such as junction formation means that forest hardening cannot be modeled, unless additional so-called ‘2.5D’ constitutive rules are prescribed for short-range dislocation interactions. Here, results from three dimensional dislocation dynamics (3D DD) simulations in an FCC material are used to define new constitutive rules for short-range interactions and junction formation between dislocations on intersecting slip systems in 2D. The mutual strengthening effect of junctions on preexisting obstacles, such as precipitates or grain boundaries, is also accounted for in the model. The new ‘2.5D’ DD model, with no arbitrary adjustable parameters beyond those obtained from lower scale simulation methods, is shown to predict athermal hardening rates, differences in flow behavior for single and multiple slip, and latent hardening ratios. All these phenomena are well-established in the plasticity of crystals and quantitative results predicted by the model are in good agreement with experimental observations.

The mechanisms governing the activation of dislocation sources in aluminum at different strain rates

This article examines the time to activate Frank–Read sources in response to macroscopic strain rates ranging from 101s−1 to 1010s−1 in aluminium under athermal conditions. We develop analytical models of the bowing of a pinned dislocation segment as well as numerical simulations of three dimensional dislocation dynamics. We find that the strain rate has a direct influence on both the activation time and the source strength of Frank–Read sources at strain rates up to 106s−1, and the source strength increases in almost direct proportion to the strain rate. This contributes to the increase in the yield stress of materials at these strain rates. Above 106s−1, the speed of the bowing segments reaches values that exceed the domain of validity of the linear viscous drag law, and the drag law is modified to account for inertial effects on the motion of the dislocation. As a result the activation times of Frank–Read sources reach a finite limit at strain rates greater than 108s−1, suggesting that Frank–Read sources are unable to operate before homogeneous nucleation relaxes elastic stresses at the higher strain rates of shock loading. Elastodynamic calculations are carried out to compare the contributions of Frank–Read sources and homogeneous nucleation of dislocations to plastic relaxation. We find that at strain rates of 5×107s−1 homogeneous nucleation becomes the dominant generation mechanism.

Stochastic behaviors in plastic deformation of face-centered cubic micropillars governed by surface nucleation and truncated source operation

Three dimensional dislocation dynamics (DD) simulations are performed to investigate the governing mechanism of size dependent plastic deformation in submicron face-centered cubic (fcc) micropillars under uniaxial loading. Based on previous atomistic simulations, we introduce an algorithm for dislocation nucleation at the free surface as a function of stress and temperature in the DD simulation. The simulation results show stochastic behaviors in agreement with experimental observations, and reveal that dislocation nucleation at the free surface is the dominant mechanism of plastic flow in small pillars with diameters less than 200nm, while the operation of truncated dislocation sources is the governing mechanism in large pillars with diameters exceeding 1μm. In between, both mechanisms come into play in a stochastic way.

Enabling microstructural changes of FCC/BCC alloys in 2D dislocation dynamics

Dimension reduction procedure is the recipe to represent defects in two dimensional dislocation dynamics according to the changes in the geometrical properties of the defects triggered by different conditions such as radiation, high temperature, or pressure. In the present study, this procedure is extended to incorporate further features related to the presence of defects with a special focus on face-centered cubic/body-centered cubic alloys used for diverse engineering purposes. In order to reflect the microstructural state of the alloy on the computational cell of two dimensional dislocation dynamics, the distribution of the multi-type defects over slip lines is implemented by using corresponding strength and line spacing for each type of defect. Additionally, a simple recursive incremental relation is set to count the loop accumulation on the precipitates. In the case of continuous resistance against the motion of edge dislocations on the slip lines, an expression of friction is introduced to see its contribution on the yield strength. Each new property is applied independently on a different material by using experimental information about defect properties and grain sizes under the condition of plain strain deformation: both constant and dynamically increasing obstacle strength for precipitate coarsening in prime-aged and heat-treated copper-chromium-zirconium, internal friction in tantalum-2.5tungsten, and mixed hardening due to the presence of precipitates and prismatic loops in irradiated oxide dispersion strengthened EUROFER with 0.3% yttria.

Dimension reduction of defect properties for application in 2D dislocation dynamics

In this study, a dimension reduction procedure of defect properties is proposed together with a two dimensional dislocation dynamics framework in order to simulate tensile response of the materials at different levels of external conditions such as radiation. This procedure delivers ordered pairs of strength and line density according to the changes in the geometrical properties of the defects. Plain strain deformation of irradiated oxygen-free high conductivity copper is investigated by using experimental information about the evolution of stacking-fault tetrahedra at the irradiation doses of 0–0.2dpa.

Singularity-free dislocation dynamics with strain gradient elasticity

The singular nature of the elastic fields produced by dislocations presents conceptual challenges and computational difficulties in the implementation of discrete dislocation-based models of plasticity. In the context of classical elasticity, attempts to regularize the elastic fields of discrete dislocations encounter intrinsic difficulties. On the other hand, in gradient elasticity, the issue of singularity can be removed at the outset and smooth elastic fields of dislocations are available. In this work we consider theoretical and numerical aspects of the non-singular theory of discrete dislocation loops in gradient elasticity of Helmholtz type, with interest in its applications to three dimensional dislocation dynamics (DD) simulations. The gradient solution is developed and compared to its singular and non-singular counterparts in classical elasticity using the unified framework of eigenstrain theory. The fundamental equations of curved dislocation theory are given as non-singular line integrals suitable for numerical implementation using fast one-dimensional quadrature. These include expressions for the interaction energy between two dislocation loops and the line integral form of the generalized solid angle associated with dislocations having a spread core. The single characteristic length scale of Helmholtz elasticity is determined from independent molecular statics (MS) calculations. The gradient solution is implemented numerically within our variational formulation of DD, with several examples illustrating the viability of the non-singular solution. The displacement field around a dislocation loop is shown to be smooth, and the loop self-energy non-divergent, as expected from atomic configurations of crystalline materials. The loop nucleation energy barrier and its dependence on the applied shear stress are computed and shown to be in good agreement with atomistic calculations. DD simulations of Lomer–Cottrell junctions in Al show that the strength of the junction and its configuration are easily obtained, without ad-hoc regularization of the singular fields. Numerical convergence studies related to the implementation of the non-singular theory in DD are presented.

Modeling effects of initial dislocation microstructure on the yield strength in FCC single crystal Cu with dislocation dynamics

Three dimensional dislocation dynamics (3D-DD) method was used to study how initial dislocation microstructure, with varying dislocation density, length and distribution, affects the yield strength in face-centered cubic (FCC) single crystal Cu. Initial dislocations are generated for three average lengths for each one of the seven dislocation densities ranging from 0.1–5×1012m/m3.They are randomly distributed on slip planes. The uniaxial tension simulation results for two crystallographic loading orientations, [100] and [111], all show that the yield strength has a well-defined linear relationship with the average length of these dislocations, except for very low dislocation densities where small number of dislocations lead to a stochastic behavior. This indicates that dislocation-density based crystal plasticity formulations, including the square-root relationship between flow stress and dislocation density, only work when dislocation densities/numbers are sufficiently large so that a statistically accurate description of the microstructure can be obtainable.

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.

Fracture toughness of layered structures: Embrittlement due to confinement of plasticity

The fracture toughness of a layered composite material is analyzed employing a combined two dimensional dislocation dynamics (DD)–cohesive zone (CZ) model. The fracture mechanism of an elastic–plastic (ductile) material sandwiched within purely elastic layers approaches ideally brittle behaviour with decreasing layer thickness. We investigate the influence of different constitutive parameters concerning dislocation plasticity as well as the effect of cohesive strength of the ductile material on the scaling of fracture toughness with layer thickness. For a constant layer thickness, the results of the numerical model are consistent with the expectation that fracture toughness decreases with increasing yield strength, but increases with the cohesive strength of the material. The scaling behaviour of the fracture toughness with layer thickness depends on these material parameters, but also on the dislocation microstructure in the vicinity of the crack tip. Strain localization due to easy dislocation generation right at the crack tip improves toughness in thin layers and leads to a jump-like increase of fracture toughness with layer thickness. However, the fracture toughness for films that are thick enough to exhibit bulk behaviour proves to be higher when the distribution of dislocations is more homogeneous, because in this case the crack grows in a stable fashion over some distance.

705 3 次元離散転位法を用いた転位源モデルと転位間相互作用の解析

Three dimensional dislocation dynamics methodology (3D-DD) is much helpful to investigate complicated physical nonlinear process during plastic deformation of materials. A dislocation loop is divided into straight discrete dislocation segments and their evolution are driven by the elastic interaction and the applied force as the long-term effect and geometric local rules as the short-term effect. In this study, the stress-strain relation in the whole system is estimated by the plastic strain based on the evolution of the 3-D discrete dislocations.

Calculation of stress—strain curves by using 2 dimensional dislocation dynamics

A method is presented to derive stress-strain curves by using 2 dimensional dislocation dynamics. The simulation treats plastic deformation of single grains, in which one or more slip systems are active. The involved dislocations are regarded as infinite straight line defects which are embedded in an otherwise isotropic linear elastic medium. As the model is two dimensional only edge dislocations are incorporated. The calculation of the local stressfield considers the long range elastic stress contribution of all dislocations within the grain, of the externally imposed stressfields and of the short range stressfield imposed by large angle grain boundaries. Due to the net local value of the stressfield the dislocations may either glide or climb. Dislocation multiplication, annihilation and reactions are taken into account. Thermal activation is considered. The simulations allow to compute the evolution of the dislocation distribution, the stressfield and the plastic strain during a simulated deformation experiment. The latter quantities are used for the calculation of stress-strain curves.

Experimental Investigation and Numerical Simulation of theCorrelation of Recovery and Texture in Bcc Metals and Alloys

Depending on the strain and temperature regime examined recovery of bcc metals during hot rolling or annealing subsequent to cold deformation often leads to the preservation of certain deformation texture components. A broad variety of mesoscopic and macroscopic texture data from both hot rolled and cold rolled and annealed bcc metals (Fe, Ta, Mo, Nb) and alloys (low‐carbon steels, ferritic stainless steels, transformer steels) is re‐examined with respect to such phenomena. The data suggest that two conditions promote strong recovery of certain bcc deformation texture components (e.g. {001} <110>), namely, the absence of kinetic instabilities, and the absence of thermodynamic and mechanic instabilities. The first case applies if the grains do not tend to generate large local misorientations in their interiors during plastic deformation. The second case applies if the stored dislocation density is low. Both features depend on the grain orientation. The experimental observations are complemented by three types of numerical simulations. The influence of the number of Burgers vectors equally involved during preceding deformation on the kinetics of static recovery is numerically studied using 2 dimensional dislocation dynamics. The orientation dependent inclination to generate local misorientations in the grain interiors is simulated using a Taylor type approach with different constraints on either side of the grain. The influence of the orientation dependence of recovery on the final annealing texture is simulated using a 3 dimensional cellular automation. Experiment and simulation show that both the recovery kinetics and the tendency to form kinetic instabilities considerably depend on the orientation. The latter effect seems to be essential for the prevalence of either RX or RC.