Dislocation Solution Research Articles

Strong pinning of triple junction migration for robust high strain nanostructures

ABSTRACTThe universality of a key recovery mechanism: triple junction migration in high strain nanostructures is revealed herein. This migration is the only means to uniformly coarsen deformed lamellar microstructures. Migration of medium to high angle geometrically necessary boundaries at triple junctions is resisted by strong pinning phenomena. Pinning by low angle dislocation boundaries is the novel mechanism that greatly adds to the solute drag of these higher angle boundaries during migration at triple junctions. Solutes furthermore cause a significant increase in the dislocation density of the low angle boundaries formed during deformation and thus greatly enhance the observed pinning. Boundary pinning by dislocation boundaries and solute drag is analysed for deformed Ni of different purities via in and ex situ electron microscopy. A kinetic model is utilised to obtain activation energies that quantitatively demonstrate the strength of this pinning. A new strategy for achieving robust nanostructured metals is developed based on solute and dislocation pinning of triple junction migration – a universal recovery mechanism in deformed lamellar microstructures.

Rate sensitivity and work-hardening behavior of an advanced Ti-Al-N alloy under uniaxial tensile loading

In the present work, titanium alloys doped with different amounts of impurity solutes were studied as regards their strain rate sensitivity, work-hardening and fracture manner. The experimental results revealed a transition of both the strain hardening behavior and fracture manner with increasing the impurity content. Namely, the low impurity content Ti exhibited features of intergranular crack, whereas the high impurity content Ti showed transgranular cracking instead. In addition, the physical activation volumes were found to reduce with the impurity content, which is attributed to both dislocation pinning and solute drag effects. A constitutive model was proposed to understand the mechanical behavior of the materials. The highlights of the model are of separating deforming grains into plastic and elastic groups and of giving them different properties. To describe the volume of grains changing from elastic to plastic deformation with strain, a controlling function was applied based on an empirical analysis. It turns out that the proposed model predicted stress-strain curves in well agreement with the experimental results.

Percutaneous endobutton fixation of acute acromioclavicular joint injuries and lateral clavicle fractures

IntroductionThis paper describes a novel technique developed by the senior author to address acute acromioclavicular joint (ACJ) dislocations and certain distal clavicle fractures. MethodsThe procedure employs a four strand, single tunnel, double endobutton repair performed entirely percutaneously, without any arthroscopic guidance or deep surgical dissection. ResultsWe present the preliminary results from our series of 6 consecutive patients performed over a period of 18 months. The mean length of surgery was 36min (range 32–40) and the mean correction of coracoclavicular (CC) distance achieved was 12.6 mm (range 10.3–14.1). There was no restriction of movement in any of the patients post-operatively and their average QuickDASH scores at final follow-up was 4.2 (range 0–6.8). ConclusionResults in the present series were at least comparable to those for other techniques, validating percutaneous treatment as a solution for acute ACJ dislocations.

Anti-plane stress analysis of a half-plane with multiple cracks by distributed dislocation technique in nonlocal elasticity

The effective role of a distributed dislocation technique accompanied by a nonlocal elasticity model has been demonstrated for the crack problem in a half-plane. The dislocation solution is employed to model and analyze the anti-plane crack problem for nonlocal elasticity using the distributed dislocation technique. The solution of dislocation in the half-plane has been extracted through the solution of dislocation in an infinite plane by the image method. The dislocation solution has been utilized to formulate integral equations for dislocation density functions on the surface of a smooth crack embedded in the half-plane under anti-plane loads. The integral equations are of the Cauchy singular type, and have been solved numerically. Multiple cracks with different configurations have been solved; results demonstrate that the nonlocal theory predicts a certain stress value in the crack tip.

Concentrated sources on the interface of diffusive bimaterial media: The steady-state deformation

While the fundamental solutions of concentrated forces and dislocations in purely elastic media are available for various applications, solutions in the corresponding diffusive materials have seldom been derived due to the involved complexity. In this paper, in terms of the cylindrical system of vector functions and under the assumption of steady-state deformation, we derive analytically the elastic and diffusive fields induced by the concentrated forces and dislocations which are located on the interface of a transversely isotropic and diffusive bimaterial space. Solution of the concentrated diffusive source has been also derived. These solutions can further be reduced to the corresponding isotropic bimaterial case and the transversely isotropic full-space case. Based on the newly derived solutions, field quantities induced by different concentrated dislocation sources are presented numerically to demonstrate the effect of the diffusive coefficients, material heterogeneity, and dislocation types on the induced fields.

A mesh‐independent method for planar three‐dimensional crack growth in finite domains

SummaryThe discrete crack mechanics (DCM) method is a dislocation‐based crack modeling technique where cracks are constructed using Volterra dislocation loops. The method allows for the natural introduction of displacement discontinuities, avoiding numerically expensive techniques. Mesh dependence in existing computational modeling of crack growth is eliminated by utilizing a superposition procedure. The elastic field of cracks in finite bodies is separated into two parts: the infinite‐medium solution of discrete dislocations and an finite element method solution of a correction problem that satisfies external boundary conditions. In the DCM, a crack is represented by a dislocation array with a fixed outer loop determining the crack tip position encompassing additional concentric loops free to expand or contract. Solving for the equilibrium positions of the inner loops gives the crack shape and stress field. The equation of motion governing the crack tip is developed for quasi‐static growth problems. Convergence and accuracy of the DCM method are verified with two‐ and three‐dimensional problems with well‐known solutions. Crack growth is simulated under load and displacement (rotation) control. In the latter case, a semicircular surface crack in a bent prismatic beam is shown to change shape as it propagates inward, stopping as the imposed rotation is accommodated.

Multiple cylindrical interface cracks in FGM coated cylinders under torsional transient loading

In the present study, the problem of multiple cylindricalinterfacecracks between a homogeneouscircular cylinder and its functionally graded materials (FGM) coating under torsional impact loading is investigated. First, by using the Laplace and complex Fourier transforms, the solution for Somigliana-type dynamic rotational ring dislocation in a cylinder and its coating is obtained. Next, the distributed dislocation technique is employed to derive a set of Cauchy singular integral equations for multiple cylindricalinterface cracks situated between the isotropic cylinder and its FGM coating. These integral equations are solved by Erdogan’s collocation method; the dislocation densities on the faces of the cracks are obtained, and the results are used to determine dynamic stress intensity factors (DSIFs). Several examples are provided to study the influences of material nonhomogeneity constant, the FGM layer thickness and crack geometry configuration on the DSIFs at the tips of cracks and, the interaction between the cracks.

Quantifying the grain boundary segregation strengthening induced by post-ECAP aging in an Al-5Cu alloy

Hardening on annealing (HOA) has been frequently observed in nanostructured metals and alloys. For nanostructured materials obtained by severe plastic deformation (SPD), HOA has been attributed to the reduction of dislocation sources within grains and grain boundary relaxation during annealing. In the present work, it is shown that when a bimodal grain structured (a mixture of micron-sized and ultrafine grains) Al-5Cu alloy prepared by equal channel angular pressing (ECAP) was subjected to post-ECAP natural and artificial aging treatments, the alloy shows a completely different precipitation behavior with an accelerated precipitation kinetics. No coherent θ'' or semi-coherent θ' precipitates form in the bulk of grains, while a large fraction of stable incoherent θ precipitates form along high angle boundaries. After artificial aging at low temperatures for a short time, a significant improvement of both ultimate tensile strength and uniform elongation was achieved without sacrificing the yield strength. A systematic microstructure characterization by EBSD, TEM and APT has been carried out to investigate the evolution of grain size, dislocation density and solid solution level of Cu as well as the precipitation of Al-Cu precipitates during natural and artificial aging treatments. A quantitative evaluation of different supposed strengthening mechanisms revealed that the segregation of Cu elements at grain boundaries plays a more important role than grain boundary relaxation and the dislocation source-limited strengthening to compensate the yield strength reduction caused by the decrease in dislocation density and solute content of Cu in solid solution.

Absence of dynamic strain aging in an additively manufactured nickel-base superalloy

Dynamic strain aging (DSA), observed macroscopically as serrated plastic flow, has long been seen in nickel-base superalloys when plastically deformed at elevated temperatures. Here we report the absence of DSA in Inconel 625 made by additive manufacturing (AM) at temperatures and strain rates where DSA is present in its conventionally processed counterpart. This absence is attributed to the unique AM microstructure of finely dispersed secondary phases (carbides, N-rich phases, and Laves phase) and textured grains. Based on experimental observations, we propose a dislocation-arrest model to elucidate the criterion for DSA to occur or to be absent as a competition between dislocation pipe diffusion and carbide–carbon reactions. With in situ neutron diffraction studies of lattice strain evolution, our findings provide a new perspective for mesoscale understanding of dislocation–solute interactions and their impact on work-hardening behaviors in high-temperature alloys, and have important implications for tailoring thermomechanical properties by microstructure control via AM.

From Atomistic Model to the Peierls–Nabarro Model with $${\gamma}$$ γ -surface for Dislocations

The Peierls-Nabarro (PN) model for dislocations is a hybrid model that incorporates the atomistic information of the dislocation core structure into the continuum theory. In this paper, we study the convergence from a full atomistic model to the PN model with $\gamma$-surface for the dislocation in a bilayer system (e.g. bilayer graphene). We prove that the displacement field of and the total energy of the dislocation solution of the PN model are asymptotically close to those of the full atomistic model. Our work can be considered as a generalization of the analysis of the convergence from atomistic model to Cauchy-Born rule for crystals without defects in the literature.

Factors controlling ambient and high temperature yield strength of ferritic stainless steel susceptible to intermetallic phase formation

The microstructure evolution in Ti-Nb dual stabilized ferritic steel at high service temperature was simulated with heat treatments at 600 °C for up to 120 h. Thermodynamic calculations indicated that, in addition to conventional MX type carbides and nitrides containing Nb and Ti, heat treatment at high temperature can promote the formation of intermetallic Laves, Chi (χ) and sigma (σ) phases. During the heat treatment, Laves (FeSi2NbMo) and σ (FeCrMo) phases formed. The effect of the intermetallic phases on the ambient and high temperature yield strength (σy) was investigated through a comprehensive breakdown of the mechanisms contributing to strengthening, i.e. grain boundary, dislocation, precipitation and solid solution strengthening, the last two of which are influenced by the precipitation of the Laves and σ phases. On the basis of a regression analysis of atomic radius and shear modulus misfit with respect to Fe, the solid solution strengthening coefficient of Nb in α-Fe was predicted to be 16 MPa/at%. The coefficient was experimentally validated by measuring the yield strengths for two conditions with different amounts of Nb in solution. The present value is considered more reasonable than the 4320 MPa/wt% (approximately 7187 MPa/at%) presented earlier in the literature. Heat treatment at 600 °C raised the yield strength at ambient temperature, due to intermetallic precipitation strengthening. However, the opposite was true regarding the high temperature yield strength, which suggests that the effects of precipitation strengthening are relatively smaller and solid solution strengthening greater at elevated temperatures.

Analysis of nonlinear elastic aspects of precursor attenuation in shock-compressed metallic crystals

Unsteady characteristics of shock waves in metals, for example elastic precursor decay, have often eluded a complete model description. Historic continuum elastic-plastic theories tend to require excessive initial dislocation densities in order to match experimental observations. Studies incorporating superposition of linear elastodynamic solutions for dislocations, either in analytical solutions or in discrete numerical simulations, omit nonlinear elastic effects and only consider effects of defects immediately at the elastic shock front. Prior analytical treatments of hydrodynamic attenuation consider nonlinearity manifesting as effects of stress gradients or particle velocity gradients immediately behind the front. The present analysis seeks to augment the predictive precursor decay equation from linear elastodynamics to account for elastic nonlinearity and dislocation nucleation in the wake of the precursor shock. A complete solution for the precursor magnitude at a material point is shown to require consideration of the prior history of dislocation generation and the entire flow field behind the elastic shock up to that material point. However, introduction of reasonable simplifying assumptions and basic models for dislocation generation and glide resistance enable derivation of a mathematically tractable relation for precursor decay. This relation is a nonlinear first-order ordinary differential equation that, in non-dimensional form, contains only one scalar parameter controlling the rate of dislocation generation behind the shock. Model predictions that include nonlinear effects are shown to provide a better match to experimental data for three metals. Effects of nonlinearity are shown to emerge early, but not immediately, in the shock attenuation process, and these effects increase in prominence with increasing impact stress.

Direct prediction of the solute softening-to-hardening transition in W–Re alloys using stochastic simulations of screw dislocation motion

Interactions among dislocations and solute atoms are the basis of several important processes in metal plasticity. In body-centered cubic (bcc) metals and alloys, low-temperature plastic flow is controlled by screw dislocation glide, which is known to take place by the nucleation and sideward relaxation of kink pairs across two consecutive Peierls valleys. In alloys, dislocations and solutes affect each other’s kinetics via long-range stress field coupling and short-range inelastic interactions. It is known that in certain substitutional bcc alloys a transition from solute softening to solute hardening is observed at a critical concentration. In this paper, we develop a kinetic Monte Carlo model of screw dislocation glide and solute diffusion in substitutional W–Re alloys. We find that dislocation kinetics is governed by two competing mechanisms. At low solute concentrations, nucleation is enhanced by the softening of the Peierls stress, which dominates over the elastic repulsion of Re atoms on kinks. This trend is reversed at higher concentrations, resulting in a minimum in the flow stress that is concentration and temperature dependent. This minimum marks the transition from solute softening to hardening, which is found to be in reasonable agreement with experiments.

Long term creep closure of salt cavities

We quantify the long term creep closure of salt cavities under in-situ conditions. We use an incompressible Carreau viscosity model to treat combined dislocation and pressure solution creep of rock salt. We build three material models based on the published compilations of experimental data. Analytical expressions are given for the maximum closure velocity under combined pressure and shear loads in the far field. A non-linear finite element model is used to study the apparent viscosity around a circular hole. In the low stress regime, depending on the grain size, creep rate can increase several orders of magnitude by the action of pressure solution. The influence of grain size dominates over all other parameters such as temperature or dislocation creep parameters. In general, pressure solution is dominant in cold fine-grained salts under low loads, while dislocation creep is dominant in warm coarse-grained salts under high loads. In the dislocation creep regime, the hole closure rate is n (stress exponent) times more sensitive to the far field shear stress than to the pressure differences. Significant hole closure can be reached in less than one day in very fine-grained salts under in-situ loads in the order of 1MPa. A thorough understanding of salt grain size and water availability is needed to properly assess the relative contribution of dislocation and pressure solution creep in hole closure. A direct relationship relating grain size with deformation mechanism cannot be established in general. The closure of a cavity in a typical salt having a 10mm grain size can either be dominated by pressure solution or dislocation creep depending on the temperature and salt type. Best practices to prevent hole closure include increasing hole pressure, and avoiding, when possible, the salt layers which are known to be fined-grained and under high deviatoric stress.

Modelling the static stress–strain state around the fan-structure in the shear rupture head

• Mathematical model of an equilibrium fan between elastic half-planes is constructed. • Length of the fan is evaluated based on the singular solution for edge dislocation. • Potential mechanisms of fan-structure formation in a fault are considered. • Original method of superposition of dislocations is suggested. • Fields of stresses and displacements around the fan-structure are computed. The mathematical model of an equilibrium fan-structure in the interface between two elastic blocks, simulating the shear rupture head in a hard rock under high confining pressure, is constructed. The stress–strain state far from the fan-structure is analyzed with the help of a solution of the problem on edge dislocation. The fan length is estimated using this solution. The model of formation of two oppositely directed fans due to the localized action of tangential stress, which pushes two edge dislocations with antiparallel Burgers vectors, is proposed. In complete formulation, the problem on an equilibrium fan-structure in the interface between infinite elastic half-planes is analyzed by means of original method of superposition of dislocations, leading to two nonlinear integral equations in the fan zone. To solve them numerically, the method of successive approximations is applied. Based on this method, fields of stresses and displacements around the equilibrium fan modelling of a deep-seated shear rupture in the seismogenic zone of the Earth’s crust are computed. Such fields can be used, when setting the initial data in the analysis of dynamics of the fan-shaped mechanism.

Defects in Static Elasticity of Quasicrystals

A review on mathematical elasticity of quasicrystals is given. In this review, the focus is on various defects of quasicrystals. Dislocation and crack are two classes of typical topological defects, while their existence has great influence on the mechanical behavior of quasicrystals. The analytic and numerical solutions of dislocations and crack in quasicrystals are the core of the static and dynamic elasticity theory, and this paper gives a comprehensive review on the solutions for dislocations and crack with different configurations in different various important quasicrystalline systems. We review some results in linear elasticity of quasicrystals, referring to different boundary value problems. We also add some new achievements.

Torsion analysis of infinite hollow cylinders of functionally graded materials weakened by multiple axisymmetric cracks

In this paper, mode III stress intensity factor for the multiple axisymmetric cracks in the infinite hollow cylinder made by functionally graded materials base on the dislocation technique is formulated. The shear modulus of the material of the cylinder is assumed to vary with the radial coordinate by a power law. It is assumed that the outer surface of the hollow cylinder is under the action of two distributed self-equilibrating lateral shear tractions and inner surface of the cylinder is free of stress. The solution of an axisymmetric rotational Somigliana ring dislocation in a functionally graded materials infinite hollow cylinder is obtained by means of a separation of variables technique. Next, the distributed dislocation method is used to formulate integral equations for a system of coaxial axisymmetric cracks, including annular, inner edge and outer edge cracks. The ensuing equations are of the Cauchy singular type and have been solved numerically to determine stress intensity factor. Numerical examples are provided to show the effects of material properties and crack type and location on the stress intensity factors and also to study the interaction of multiple cracks.

The microstructure and mechanical properties of Pb alloying layer on Al using surface alloying by high-current pulsed electron beam

In this research, lead (Pb) alloying layer was introduced on the aluminum (Al) substrate to improve the surface properties after surface alloying by high-current pulsed electron beam (HCPEB). The microstructure and phase analysis of the Pb alloying layer was investigated by x-ray diffraction (XRD), scanning electron microscopy (SEM) and transmission electron microscopy (TEM). The properties of microhardness and wear resistance were also studied. The results show that Pb particles were dissolved into Al substrate and an alloying layer of about 15 µm in thickness was formed after 35-pulsed irradiation. Meanwhile, the refinement and uniform distribution of Pb particles occurred with the increasing number of pulses, as well as the solid solution and structure defects such as dislocation walls and dislocation loops. Besides, some Pb particles exhibited cube-cube orientation relationship with the Al matrix. The surface hardening is mainly owing to the fine grain, dispersion, dislocation and solid solute strengthening. The enhanced wear property is attributed to the improved hardness and the addition of Pb element to form lubricious tribolayer.

Analysis of an orthotropic circular bar weakened by multiple radial cracks under torsional transient loading

In this research, the problem of a circular orthotropic bar with multiple cracks under torsional transient loading is investigated. The solution to the problem weakened by a Volterra-type screw dislocation is first obtained with the aid of the Laplace and finite Fourier sine transform. The bar is considered to be subjected to a torsional transient loading. The solution is obtained for stress fields and displacement in the domain under consideration. Then, the dislocation solution is used to derive a set of Cauchy singular integral equations for analysis of the circular bar containing several cracks. The solution to these equations is used to analyze the torsional rigidity of the bar and also the mode III stress intensity factors at the crack tips. Finally, several examples of a single and multiple cracks are presented.

Effects of microstructure and crystallography on mechanical properties of cold-rolled SAE1078 pearlitic steel

The evolution of the microstructure and crystallography in SAE1078 pearlitic steel sheets under different cold-rolling reductions of up to 90% were quantified using transmission electron microscopy (TEM), electron backscatter diffraction (EBSD), and X-ray diffraction (XRD). The mechanical properties were determined by tensile testing at room temperature. TEM analysis showed that the pearlite structure was obviously refined with the interlamellar spacing decreasing to about 57nm at the rolling reduction of 90%. EBSD investigations indicated that the ferrite exhibited a {001} <110> texture in the 90% cold-rolled pearlitic steel. The dislocations were mainly concentrated during cold rolling between the 10% and 70% reduction ratios as the average kernel average misorientation (KAM) angle increased from 0.75° to 1.20°. XRD examination revealed that a transformation from bcc to bct crystal structure of ferrite occurred at 90% rolling reduction due to the supersaturation of carbon. Significant augmentation in the ultimate tensile strength during cold rolling results from the boundary, dislocation, and solid solution strengthening mechanisms.