Dislocation Curvature Research Articles

Understanding the sigmoidal primary creep behaviour in a multi-component solid-solution Co-base alloy between 700 – 850 °C

This investigation was motivated by the lack of understanding of the role of solid solution strengthening and substructure evolution during the creep of multi-component alloys. Thus, we studied the tensile creep behaviour of a multi-component solid-solution Co-base alloy (Co41-Ni23-Cr26-W4.7-Fe2-Mn1-C0.5) at 650 – 900 °C and stress (σApplied) range: 80 – 400 MPa. Uncommon sigmoidal primary creep (SPC) was witnessed at 700–850 °C and in the intermediate stress range. The stress exponent (∼5 – 7) and activation energy (∼270 kJ/mol ≈ Qlattice) suggest that dislocation creep controls the creep rate. Transmission electron microscopy (TEM) studies revealed substructural evolution with creep strain. At 750 °C and 160 MPa, at creep strain (ε) = 0.25 %, multiple pinning points along the dislocation length were seen, and a thorough analysis confirmed them as jogs. Friction stress estimation from dislocation curvature from multiple TEM micrographs revealed that effective stress on dislocations is high (>0.6σApplied) while solid solution strengthening is nominal during creep. Overall analysis suggests that the creep rate in the multi-component alloy is recovery controlled while jogs' motion determines the SPC creep rate.

Effect of Grain Size and Temperature on Scalar Dislocation Density and Curvature–Torsion of the Crystal Lattice in Copper–Manganese Alloys during Plastic Deformation

Effect of Grain Size and Temperature on Scalar Dislocation Density and Curvature–Torsion of the Crystal Lattice in Copper–Manganese Alloys during Plastic Deformation

Statistical analysis of discrete dislocation dynamics simulations: initial structures, cross-slip and microstructure evolution

Over the past decades, discrete dislocation dynamics simulations have been shown to reliably predict the evolution of dislocation microstructures for micrometer-sized metallic samples. Such simulations provide insight into the governing deformation mechanisms and the interplay between different physical phenomena such as dislocation reactions or cross-slip. This work is focused on a detailed analysis of the influence of the cross-slip on the evolution of dislocation systems. A tailored data mining strategy using the ‘discrete-to-continuous (D2C) framework’ allows to quantify differences and to quantitatively compare dislocation structures. We analyze the quantitative effects of the cross-slip on the microstructure in the course of a tensile test and a subsequent relaxation to present the role of cross-slip in the microstructure evolution. The precision of the extracted quantitative information using D2C strongly depends on the resolution of the domain averaging. We also analyze how the resolution of the averaging influences the distribution of total dislocation density and curvature fields of the specimen. Our analyzes are important approaches for interpreting the resulting structures calculated by dislocation dynamics simulations.

Dislocation structures formation induced by thermal stress in additive manufacturing: Multiscale crystal plasticity modeling of dislocation transport

The motion of dislocations governs the plastic deformation of crystalline materials, which in turn determines the mechanical properties. The complex thermal history, large temperature gradients and high cooling rates during the process of additive manufacturing (AM) can induce high dislocation density and unique dislocation structures in the material. The origin of these dislocation structures and their stability during mechanical loading are debated. A novel temperature dependent continuum dislocation dynamics (CDD) model is developed, in which four state variables are used for each slip system representing the total dislocation density, edge and screw geometrically necessary dislocation densities and dislocation curvature. The CDD model is fully coupled with a crystal plasticity solver, which captures the plastic deformation induced by the dislocation motion. A hybrid continuous and discontinuous Galerkin formulation is developed to accurately reproduce the dynamics of highly discontinuous dislocation density fields that are typical of dislocation structures. A multiscale modeling approach is used, in which the thermally induced deformation in specific grains of a polycrystal is extracted from larger scale crystal plasticity simulations of the laser powder-bed fusion process, and is then used for single crystal scale dislocation dynamics simulations. Simulation results reveal the dynamics of dislocation structure formation in grains at different positions during laser scanning and cooling stages. The effect of the cyclic thermal stress during multi-layer AM fabrication is also investigated. The simulations provide a new perspective on the specific conditions that should be satisfied during AM process for the formation of stable dislocation structures.

A computationally efficient implementation of continuum dislocation dynamics: Formulation and application to ultrafine-grained Mg polycrystals

Continuum dislocation dynamics (CDD) represents the evolution of systems of curved and connected dislocation lines in terms of density-like field variables which include the volume density of loops (or ’curvature density’) as an additional field. Since dislocation curvature represents a spatial derivative of the underlying discrete dislocation density tensor, the curvature field evolution equation of necessity contains numerically inconvenient higher-order derivatives of the density fields. We propose a simple approximation to express curvature in terms of density fields, and demonstrate its application to a benchmark problem in deformation of Mg polycrystals.

Size scaling in bi-crystalline Cu micropillars containing a coherent twin boundary

The impact of a coherent twin boundary (CTB) on the size scaling of the shear stress in micropillar compression tests has been investigated through microcompression of bi-crystalline pillars containing a vertical CTB, as well as single-crystalline pillars in three different nominal diameters of 1, 3 and 5 µm. While both, single- and bi-crystalline pillar results follow the size scaling trend typically observed in micropillars, namely “smaller is stronger”, we could identify a size-dependent contribution of the CTB in the increase of the shear stress at 2% strain (τ2%). A probabilistic analysis was performed to quantify the magnitude of the effect and to separate the CTB contribution from the single crystal size scaling contribution of the strength increase. The CTB-related strengthening was most prominent for smaller pillars and tended to be small for larger pillar diameters. The behavior can be explained by attributing an excess dislocation curvature in the scaling law according to the double-hump dislocation line shape model for bi-crystals, which requires parallel alignment of the dislocation line and the Burgers vector at the CTB.

Effect of nanoscale amorphization on dislocation emission from a semi-elliptical surface crack tip in nanocrystalline materials

An impact analysis model is built to describe the effect of nanoscale amorphization on dislocation emission from a surface semi-elliptical crack tip in nanocrystalline materials. The nanoscale amorphization is formed by the splitting transformation of grain boundary(GB)disclinations caused by the motion of GBs. The analytical solution of the model is obtained by using the complex method, and the influence of nanoscale amorphization, dislocation emission angle, crack length, and curvature radius of surface crack tip on the critical stress intensity factor (SIF) of the first dislocation emission is investigated through numerical analysis. The numerical analysis shows that the impact of nanoscale amorphization on the critical SIF corresponding to dislocation emission depends on the dislocation emission angle, the position and the size of the nanoscale amorphous, the curvature radius, and the length of surface crack. As the curvature radius of surface crack tip and the crack length increase, the normalized critical SIF increases. When the nanoscale amorphization size is small, it has a great impact on the critical SIF for dislocation, but when the size is relatively large, the effect becomes small. The effect of the increasing strength of the nanoscale amorphization on dislocation emission from the surface crack tip is related to the distance between the nanoscale amorphization and the crack tip, and there is a critical crack-junction for which the increase of dislocation strength has little effect on dislocation emission.

Situating the Vector Density Approach Among Contemporary Continuum Theories of Dislocation Dynamics

Abstract For the past century, dislocations have been understood to be the carriers of plastic deformation in crystalline solids. However, their collective behavior is still poorly understood. Progress in understanding the collective behavior of dislocations has primarily come in one of two modes: the simulation of systems of interacting discrete dislocations and the treatment of density measures of varying complexity that are considered as continuum fields. A summary of contemporary models of continuum dislocation dynamics is presented. These include, in order of complexity, the two-dimensional statistical theory of dislocations, the field dislocation mechanics treating the total Kröner–Nye tensor, vector density approaches that treat geometrically necessary dislocations on each slip system of a crystal, and high-order theories that examine the effect of dislocation curvature and distribution over orientation. Each of theories contain common themes, including statistical closure of the kinetic dislocation transport equations and treatment of dislocation reactions such as junction formation. An emphasis is placed on how these common themes rely on closure relations obtained by analysis of discrete dislocation dynamics experiments. The outlook of these various continuum theories of dislocation motion is then discussed.

Conventional vs harmonic-structured β-Ti-25Nb-25Zr alloys: A comparative study of deformation mechanisms

Harmonic alloys processed by powder metallurgy are constituted by a core of coarse grains embedded in an interconnected small grains shell. They have attracted attention due to their excellent strength combined with large ductility, the two properties being rather antagonist from the classical metallurgy point of view. In contrast, conventional β-Ti alloys are currently vastly studied owing their excellent properties especially for biomedical applications. In the present study, we explore at the micron scale the deformation mechanisms operating both in standard and harmonic-structured β-Ti-25Nb-25Zr alloys using transmission electron microscopy (TEM). Although we show some similarities, deformation mechanisms appear significantly different due to the activation of martensitic transformation in conventional samples. The combined use of automated crystal orientation in TEM and in-situ TEM straining reveals that deformation bands nucleate and grow according to a mechanism involving both martensitic transformation and twinning. The comparison between in-situ and post-mortem experiments shows globally a good agreement and highlights a strain relaxation mechanism between martensite and twin. More importantly, a cross-glide mechanism similar to what is observed in dilute solid solutions is proposed to explain the dynamics of dislocation motion. Stress estimations derived from the observations of dislocation curvature between pinning points, show a reasonable good agreement with macroscopic values. The observation of deformation mechanisms operating both in core and shell structures of the harmonic-structured alloy allows us to propose a scenario of plastic deformation in the early stages.

Dislocation multiplication in stage II deformation of fcc multi-slip single crystals

Dislocation multiplication in plasticity research is often connected to the picture of a Frank-Read source. Although it is known that this picture is not applicable after easy glide deformation, plasticity theories often assume Frank-Read-type models for dislocation multiplication. By analyzing discrete dislocation dynamics simulations in a bulk like setting, a new view on dislocation multiplication is presented. It is observed that only two mechanisms provide a source for dislocations: cross-slip and glissile junctions. Both source mechanisms involve a change of glide system and transfer of dislocation density (line length) from the primary dislocation(s) slip system(s) to the one of the new dislocation. The motion of dislocations is found to be highly restricted by other dislocations and therefore the contribution to plastic deformation of each individual dislocation is small. Also a substantial fraction of the physical dislocation line length is annihilated by the collinear reaction, lowering dislocation storage during plastic deformation. Furthermore, multiplication events involve the loss of a substantial amount of dislocation length and curvature (sudden changes in line orientation) due to the topology changes in the dislocation network of the respective mechanisms. The findings are discussed in light of continuum dislocation theories, which currently barely account for dislocation density transfer to other systems and the limited contribution of plastic strain from individual dislocations.

Dislocation-density kinematics: a simple evolution equation for dislocation density involving movement and tilting of dislocations

Abstract

Thermodynamically consistent continuum dislocation dynamics

Dislocation based modeling of plasticity is one of the central challenges at the crossover of materials science and continuum mechanics. Developing a continuum theory of dislocations requires the solution of two long standing problems: (i) to represent dislocation kinematics in terms of a reasonable number of variables and (ii) to derive averaged descriptions of the dislocation dynamics (i.e. material laws) in terms of these variables. The kinematic problem (i) was recently solved through the introduction of continuum dislocation dynamics (CDD), which provides kinematically consistent evolution equations of dislocation alignment tensors, presuming a given average dislocation velocity (Hochrainer, T., 2015, Multipole expansion of continuum dislocations dynamics in terms of alignment tensors. Philos. Mag. 95 (12), 1321–1367). In the current paper we demonstrate how a free energy formulation may be used to solve the dynamic closure problem (ii) in CDD. We do so exemplarily for the lowest order CDD variant for curved dislocations in a single slip situation. In this case, a thermodynamically consistent average dislocation velocity is found to comprise five mesoscopic shear stress contributions. For a postulated free energy expression we identify among these stress contributions a back-stress term and a line-tension term, both of which have already been postulated for CDD. A new stress contribution occurs which is missing in earlier CDD models including the statistical continuum theory of straight parallel edge dislocations (Groma, I., Csikor, F.F., Zaiser, M., 2003. Spatial correlations and higher-order gradient terms in a continuum description of dislocation dynamics. Acta Mater. 51, 1271–1281). Furthermore, two entirely new stress contributions arise from the curvature of dislocations.

Soliton-like solutions based on geometrically nonlinear Cosserat micropolar elasticity

The Cosserat model generalises an elastic material taking into account the possible microstructure of the elements of the material continuum. In particular, within the Cosserat model the structured material point is rigid and can only experience microrotations, which is also known as micropolar elasticity. We present the geometrically nonlinear theory taking into account all possible interaction terms between the elastic and microelastic structures. This is achieved by considering the irreducible pieces of the deformation gradient and of the dislocation curvature tensor. In addition we also consider the so-called Cosserat coupling term. In this setting we seek soliton type solutions assuming small elastic displacements, however, we allow the material points to experience full rotations which are not assumed to be small. By choosing a particular ansatz we are able to reduce the system of equations to a sine–Gordon type equation which is known to have soliton solutions.

Multipole expansion of continuum dislocations dynamics in terms of alignment tensors

The development of a dislocation-based continuum theory of plasticity remains one of the central challenges of applied physics and materials science. Developing a continuum theory of dislocations requires the solution of two long-standing problems: (i) to find a faithful representation of dislocation kinematics with a reasonable number of variables and (ii) to derive averaged descriptions of the dislocation dynamics (i.e. material laws) in terms of these variables. In the current paper, we solve the first problem, i.e. we develop tensorial conservation laws for distributions of oriented lines. This is achieved through a multipole expansion of the dislocation density in terms of so-called alignment tensors containing information on the directional distribution of dislocation density and dislocation curvature. A hierarchy of evolution equations of these tensors is derived from a higher dimensional dislocation density theory. Low-order closure approximations of this hierarchy lead to continuum dislocation dynamics models of plasticity with only few internal variables. Perspectives for more refined theories and current challenges in dislocation density modelling are discussed.

Dislocation mechanisms in a zirconium alloy in the high-temperature regime: An in situ TEM investigation

Dislocation mechanisms responsible for the high-temperature mechanical properties of a Zr alloy have been investigated using in situ straining experiments between 250°C and 450°C. At 250°C and 300°C, the results show a steady and homogeneous dislocation motion in prismatic planes, with little cross-slip in the pyramidal and/or basal planes. At 350°C, the kinetics of mobile dislocations becomes very jerky and inhomogeneous, in agreement with a dynamic strain aging mechanism. Above this temperature, the motion is again steady and homogeneous. Extensive cross-slip forms super-jogs which are efficient pinning points against the glide motion. These super-jogs move by glide along the Burgers vector direction, never by climb. The glide velocity between super-jogs is linear as a function of the total driving stress (applied stress minus line-tension stress due to dislocation curvature), in agreement with the solute dragging mechanism. The origin of the stress–strain rate dependence with an exponent larger than unity is then discussed.

Gradient crystal plasticity including dislocation-based work-hardening and dislocation transport

This work aims at the formulation of a gradient crystal plasticity model which incorporates some of the latest developments in continuum dislocation theory and is, at the same time, well-suited for a three-dimensional numerical implementation. Specifically, a classical continuum crystal plasticity framework is extended by taking into account continuous dislocation density and curvature field variables which evolve according to partial differential equations (Hochrainer et al., 2014; Ebrahimi et al., 2014). These account for dislocation transport and curvature-induced line-length production and have been derived from a higher-dimensional continuum dislocation theory. The dislocation density information is used to model work hardening as a consequence of dislocation entanglement.A composite microstructure is simulated consisting of a soft elasto-plastic matrix and hard elastic inclusions. The particles are assumed to act as obstacles to dislocation motion, leading to pile-ups forming at the matrix–inclusion interface. This effect is modeled using gradient plasticity with a simplified equivalent plastic strain gradient approach (Wulfinghoff et al., 2013) which is used here in order to allow for an efficient numerical treatment of the three-dimensional numerical model. A regularized logarithmic energy is applied which is intended to approximate the higher order gradient stress of the statistical theory of Groma et al. (2003).

Internal and external stresses: In situ TEM compression of Cu bicrystals containing a twin boundary

Uniaxial compression experiments on single- and twinned bicrystalline Cu samples using conventional and scanning in situ transmission electron microscopy reveal no increase in flow stress for the bicrystals. Dislocation curvature and dislocation source size analysis combined with indenter force measurements show agreement between local internal stresses acting on the dislocations and external stresses imposed by the indenter, indicating no stress concentrations due to the twin boundary. Furthermore, the dislocation density evolution shows stochastic variations but never a complete dislocation starvation.

Continuum Dislocation Dynamics Based on the Second Order Alignment Tensor

ABSTRACTIn the current paper we present a continuum theory of dislocations based on the second-order alignment tensor in conjunction with the classical dislocation density tensor (Kröner-Nye-tensor) and a scalar dislocation curvature measure. The second-order alignment tensor is a symmetric second order tensor characterizing the orientation distribution of dislocations in elliptic form. It is closely connected to total densities of screw and edge dislocations introduced in the literature. The scalar dislocation curvature density is a conserved quantity the integral of which represents the total number of dislocations in the system. The presented evolution equations of these dislocation density measures partly parallel earlier developed theories based on screw-edge decompositions but handle line length changes and segment reorientation consistently. We demonstrate that the presented equations allow predicting the evolution of a single dislocation loop in a non-trivial velocity field.

Continuum dislocation dynamics: Towards a physical theory of crystal plasticity

The plastic deformation of metals is the result of the motion and interaction of dislocations, line defects of the crystalline structure. Continuum models of plasticity, however, remain largely phenomenological to date, usually do not consider dislocation motion, and fail when materials behavior becomes size dependent. In this work we present a novel plasticity theory based on systematic physical averages of the kinematics and dynamics of dislocation systems. We demonstrate that this theory can predict microstructure evolution and size effects in accordance with experiments and discrete dislocation simulations. The theory is based on only four internal variables per slip system and features physical boundary conditions, dislocation pile ups, dislocation curvature, dislocation multiplication and dislocation loss. The presented theory therefore marks a major step towards a physically based theory of crystal plasticity.

Grain size dependence of the strength of metals: The Hall–Petch effect does not scale as the inverse square root of grain size

The classic data in the literature for the grain size dependence of the strength in many metals are reviewed. The exponent x relating strength to grain size d−x is not often the eponymous inverse square-root relationship (known as the Hall–Petch effect), but is widely scattered from values as low as x=0.2 to values as high as x=1. These exponents for individual datasets are shown to be largely meaningless. For an ensemble of n selected datasets, the fit to the functional form ln[d]/d+const with n+1 free fitting parameters is found to be almost as good as the fit to 1/Sqrt[d]+const with 2n fitting parameters (the Hall–Petch fit). The probability that the former is the preferable fit is high. Some data sets do not agree with the ln[d]/d fit, but their deviation is readily explained on simple physical grounds. Moreover, even when they are included in the fit, statistical tests still show that the ln[d]/d form is preferable by a wide margin. The conclusion is that the Hall–Petch effect is not another size effect sui generis but is the same size effect as that observed in epitaxial thin film growth and in micromechanical testing of small specimens. Consequently we propose that grain size strengthening of metals is driven by constraints on stress and dislocation curvature according to the space available.