Dislocation Pile-up Theory Research Articles

Modelling needle domains with dislocations: A dislocation pile-up model and comparison with synchrotron experiment

The shape and strain field of a needle domain in a barium titanate single crystal are modelled using a distribution of dislocations and line charges. The arrangement of these dislocations and charges is a result of the balance of modified Peach–Koehler forces acting among the dislocations and a lattice friction assumed to act at each dislocation site. Based on measurements of needle shape by synchrotron X-ray diffraction, dislocation pile-up theory is used to compute the distribution of discrete dislocations along the needle and hence estimate the lattice friction. It is found that the lattice friction in this model is proportional to the opening angle of a wedge-shape needle domain and consistent with the observed magnitude of stress required to mobilize needle domains. The microstrain distribution around an a-a needle domain tip, obtained from X-ray diffraction measurement, is further used to test the dislocation model, with a similar pattern and magnitude of strains identified in the model and the experiment.

Correlating dislocation structures to basal and prismatic slip bands near grain boundaries in tensile-tested Mg–4Al using a multiscale electron microscopy approach

Dislocation structures where basal and prismatic slip bands meet grain boundaries in tensile-tested Mg with 4 wt% Al (Mg–4Al) were studied using a multiscale electron microscopy approach to explore the assumptions made for the dislocation pileup theory: a) slip bands consist of dislocation arrays piling up near grain boundaries, and b) stress concentration in the adjacent grain is solely caused by dislocation pileup. After post-testing electron backscatter diffraction (EBSD) scans, focused-ion-beam (FIB) lift-out specimens were prepared from regions of interest so that the specimen plane is parallel to the bulk sample surface to allow for direct correlation between mesoscale and microscale characterization. High dislocation density and the dislocation arrays in the slip bands corroborated with the first assumption. Evidence of plastic deformation in all grains, however, showed that the second assumption was false. In addition, dislocation structures including crystal rotation boundaries and <c+a> dislocations dissociating to the basal plane provide additional stress relief mechanisms near grain boundaries that are not accounted for in many crystal plasticity models, which requires auxiliary models to more accurately predict the stress and strain distribution in a microstructure.

Study on dynamic mechanical properties and constitutive model description of Inconel718

In this study, the effects of strain rate and temperature on the flow stress of Inconel718 were analyzed by Split Hopkinson Pressure Bar (SHPB) experiment and quasi-static compression experiment. The classical JC constitutive model was established by combining the quasi-static compression experiment with the SHPB experiment. According to the effects of different grain sizes and [Formula: see text] phase on dislocation pile-up, the dislocation pile-up theory was introduced to modify the JC constitutive model. The modified constitutive model was compiled in FORTRAN language, and VUMAT user material subroutine was called and secondary development was carried out to establish the polycrystalline simulation model with different grain sizes. The uniaxial tensile and compression simulation process of polycrystal with different grain sizes was performed. Through comparing the simulation results with the experimental data. The correlation coefficient R, between the simulation and experimental values, is 0.97,981, and the average relative error is only 3.72%. The accuracy of the modified constitutive model was verified.

A novel constitutive model for Ti–6Al–4V alloy based on dislocation pile-up theory

ABSTRACTThe mechanical properties of the titanium alloy Ti–6Al–4V, which vary with the specimen size under different temperatures, are studied through the Split Hopkinson Pressure Bar (SHPB) test and the quasi-static tensile test to determine the parameters for the classical Johnson-Cook (JC) constitutive model. Based on the dislocation pile-up theory, the classical JC constitutive model is modified by adding a grain strain term Δσ to consider the influence of grain size. The SHPB and tensile tests are analysed using a finite element method simulation. Compared with the experimental results, the simulation results based on the modified JC model exhibit a much higher calculation accuracy than that of the classical JC model.

The Analysis of Contact Fatigue Behavior in Ni-Base Single Crystal Superalloy Turbine Blade Rabbet

Based on the dislocation pile-up theory and the crystal plastic theory, a rate-dependent crystallographic plastic finite element method (FEM) was used to analyze the stress distribution in the contact zone of single crystal turbine blade rabbet. The FEM results show that the maximum stress and the maximum resolved shear stress location are in the upper edge of the first tooth contact area of the rabbet. The surface crack initiates in the edge and grows as the zigzag wave. The deflected angle of the plane defined by maximum resolved shear stress gradient direction and the upper edge of the first tooth contact area of the rabbet with respect to the Z axis is 35°. The fracture occurs along the {-1-11} plane. Fracture behavior of rabbet/groove modeling specimens made of nickel-based single crystal superalloys was studied by corresponding contact fatigue experiments. The initial place and developing direction of the fatigue crack observed from experiments are found to be in good agreement with the predicted results based FEM.

Comparison between Pile-Up Singularities and Stress Fields Induced by Thin Slip Bands. Application to the Prediction of Grain Boundary Microcrack Nucleation

Slip localization is widely observed in metallic polycrystals after tensile deformation, cyclic deformation or pre-irradiation followed by tensile deformation. Such strong deformation localized in thin slip bands induces local stress concentrations in the quasi-elastic matrix around, at the intersections between slip bands (SBs) and grain boundaries (GBs) where microcrack initiation is often observed. Since the work of Stroh, such stress fields have been mostly modeled using the dislocation pile-up theory which leads to stress singularities similar to the LEFM ones. The Griffith criterion has then been widely applied, leading usually to strong underestimations of the macroscopic stress to GB crack initiation. In fact, slip band thickness is finite: 20nm-1000nm depending on material, temperature and loading conditions. Then, many slip planes are plastically activated through the thickness, and not only one single atomic plane. To evaluate more realistic stress fields, numerous crystalline finite element (FE) computations have been carried out using microstructure inputs (slip band aspect ratio, crystal and GB orientation...). A strong influence of slip band thickness close to the slip band corner has been highlighted, which is not accounted for by the pile-up theory. But far away, the thickness has a negligible effect and the predicted stress fields are close to the one predicted by the pile-up theory. Closed-form expressions are deduced from the numerous FE computation results allowing a straightforward prediction of GB stress fields. Slip band plasticity parameters, such as length and thickness, as well as crystal orientation, GB plane and remote stress are taken into account. The dependence with respect to the various parameters can be understood in the framework of matching expansions usually applied to cracks with V notches of finite thickness. As the exponent of the GB stress close-field is only about one-half of the pile-up or LEFM crack one, the Griffith criterion may not be used for GB microcrack prediction in case of finite thickness. That is why finite crack fracture mechanics is used together with both energy and stress criteria. Taking into account SB finite thickness, t&gt;0, leads to predicted remote stresses to GB microcrack initiation three to six times lower than the ones predicted using the to pile-up theory, in agreement with experimental data.

Prediction of grain boundary stress fields and microcrack initiation induced by slip band impingement

Slip localization is widely observed in metallic polycrystals undergoing cyclic deformation or post-irradiation tensile deformation, whatever their crystallographic structure. Hence, strong strain localization occurs in thin slip bands (SBs) inducing by the way local stress concentrations at their intersections with grain boundaries (GBs). Many GB stress field formulae based on the dislocation pile-up theory have been proposed since the pionnering work of Stroh and others. These allow the use of the Griffith criterion for prediction GB fracture initiation. However, recent observations show that assuming that slip is localized on a single atomic plane leads to unrealistic results. In fact, a large number of slip planes are plastically activated and then finite slip band thickness should be accounted for. Numerous crystalline finite element (FE) computations have been carried out using considering a slip bands with low critical resolved shear stress embedded in an elastic matrix. The computed GB normal and shear stress fields: are considerable lower than the pile-up ones and exhibit strong dependency on the slip band thickness close to the SB corner but are in fair agreement with the solution predicted by the pile-up theory far away Since the pile-up theory leads to the overestimation of the local GB stress fields, the main goal of the current paper is to perform analytical model of GB stress components based upon FE calculations. The effect of various parameters can be understood in the framework of matching asymptotic expansions which is usually applied to cracks with V notches of finite thickness. Finally, the predicted remote stresses to GB fracture in pre-irradiated austenitic stainless steels subjected to tensile loading in various environment are compared to experimental data and the pile-up based predictions.

Influence of plastic slip localization on grain boundary stress fields and microcrack nucleation

Slip localization is widely observed in metallic polycrystals after tensile deformation, cyclic deformation (persistent slip bands) or pre-irradiation followed by tensile deformation (channels). Such strong deformation localized in thin slip bands induces local stress concentrations in the quasi-elastic matrix around, at the intersections between slip bands and grain boundaries where microcracks are often observed.Since the work of Stroh, such stress fields have been modeled using the dislocation pile-up theory which leads to stress singularities similar to the LEFM ones. The Griffith criterion has then been widely applied, leading usually to strong underestimations of the macroscopic stress for microcrack nucleation.In fact, slip band thickness is finite: 50–1000nm depending on material, temperature and loading conditions. Then, many slip planes are plastically activated through the thickness. Stress fields have probably been overestimated using the pile-up theory which assumes that all dislocations are located on the same atomic plane. To evaluate more realistic stress fields, crystalline finite element (FE) computations are carried out using microstructure inputs (slip band aspect ratio and spacing). Slip bands (low critical resolved shear stress) are embedded in an elastic matrix. The following results are obtained concerning grain boundary normal stress fields:–strong influence of slip band thickness close to the slip band corner, which is not accounted for by the pile-up theory. But far away, the thickness has a negligible effect and the predicted stress fields are close to the one predicted by the pile-up theory,–analytical formulae are deduced from the numerous FE computation results which allows the prediction of surface/bulk slips as well as grain boundary stress fields. Slip band plasticity parameters, slip band length and thickness, Schmid factor and remote stress are taken into account. The dependence with respect to the various parameters can be understood in the framework of matching expansions usually applied to cracks with V notches of finite thickness,–as the exponent of the GB stress close-field is lower than the pile-up or crack one, that is 0.5, the Griffith criterion may not be used for GB microcrack prediction in case of finite thickness. That is why finite crack fracture mechanics is used together with both energy and stress criteria,–the pile-up theory leads to large underestimation of the critical remote stress leading to GB microcrack nucleation measured in the case of pre-irradiated austenitic stainless steels subjected to tensile loading in inert environment, probably because of the overestimation of the local GB stress field. And the critical remote stress computed using the proposed modeling of slip bands of finite thickness is much closer to the experimental values.

Analytic treatment of metallic multilayer strength at all length scales: Influence of dislocation sources

Metallic multilayers exhibit a very pronounced size effect where the mechanical strength depends on the layer thickness. Traditionally the Hall–Petch relation is used to account for the size effect. However, rigorous application of dislocation pileup theory predicts significant deviation from the Hall–Petch relation due to elastic inhomogeneity, discreteness of dislocations and dislocation source operation. Elastic inhomogeneity leads to anomalous scaling where the scaling exponent deviates from 1/2 of the classical Hall–Petch relation. The discrete dislocation effect is properly accounted for by a piecewise approach that can be applied at all length scales. In this article, a key step in the formulation is taken: the dislocation source characteristics are taken into consideration. Thus, all the three effects are accounted for. Analytic formulas linking yield stress to microscopic interface strength, dislocation source activation stress and other easily obtainable parameters (the Burgers vector, the elastic constants of constituent materials, crystal structure and layer thickness) are provided for all length scales. The model is then applied to Cu/Ni multilayers and the predicted strength is compared with experimental data.

Strength of metallic multilayers at all length scales from analytic theory of discrete dislocation pileups

Metallic multilayers can be used as ultra-high strength coatings. They exhibit a very strong Hall–Petch-like size-effect where the mechanical strength depends on the layer thickness. This trend suggests that dislocation pileup theory can be used to predict the strength of multilayers from fundamental and microscopic material parameters. At large length scales, the behavior of multilayers can be described by a scaling law. At small length scales, the effect of discrete dislocations becomes important, and large deviation from the scaling law occurs. A complete analytic model should apply at all length scales and properly account for this dislocation discreteness effect. Such a model is proposed here. The layer thickness of multilayers are divided into four length-scale regimes, and simple analytic formulas are given for both the regime length scales and multilayer strength in each regime. The model is applied to Cu/Ni multilayers and the predicted strength is compared to experimental data. Furthermore, the predicted polycrystalline multilayer deformation map is presented.

Study on the Fatigue Crack Initiation Life Under Spherical Contact

In case of contact fatigue, the accurate calculation of surface tractions and subsurface stress is essential to the predication of crack initiation life. Surface tractions influencing shear stress amplitude have been obtained by contact analysis based on influence function. Subsurface stress has been obtained by using rectangular patch solutions. In this study, to simulate asperity contact under sliding condition, the tip of asperity was simulated by sphere and to calculate crack initiation life in the substrate, dislocation pileup theory was used.

Scale dependence of submicron polycrystals due to configurational entropy

It is well known that polycrystals exhibit abnormal Hall-Petch dependence when their grain sizes fall into the submicron range. A plausible explanation is that the scale dependence of the yield stress of submicron polycrystals is mediated not only by the internal energy of the dislocation patterns, but also by their configurational entropy. Two-level dislocation arrangements are explored which naturally embed a grain size. Detailed calculations of the internal energy and the configurational entropy are facilitated by the dislocation pileup theory and the Ashby model of geometric necessary dislocations. It is shown that the present model is capable of interpreting the abnormal Hall-Petch dependence for grain sizes in the submicron range.

A dislocation density approximation for the flow stress—grain size relation of polycrystals

Based on the dislocation pile-up theory and the Ashby model of polycrystalline deformation, an expression for flow stress was derived, which shows that the flow stress depends on the densities of statistically stored dislocation ρ s and geometrically necessary dislocation ρ g. The contribution of ρ g to the flow stress is related to the grain boundary region fraction 1 − x which was derived to be the functions of ρ s and ρ g. This expression does not predict a precise Hall—Petch relation. If ρ s dominates, the expression reduces to a linear σ− d −1 relation: if ρ g dominates, the expression reduces to a linear σ− d −1/2 relation, i.e. the Hall—Petch relation. The expression was applied to calculate the σ− d −1/2 curves of three typical polycrystalline materials, aluminium, copper and brass in the literatures and very good agreements between the calculations and experiments were obtained. By incorporating the experimental data, the contributions of ρ s and ρ g to flow stress was analysed and the Hall—Petch parameters were discussed in terms of ρ s and ρ g.

Effect of grain size on mechanical properties of nanocrystalline materials

The possibility of a dislocation mechanism in the deformation process of nanocrystalline materials is reviewed and analyzed. The present theoretical calculation, by taking the anisotropic characteristic of crystallographic symmetry and different choices of critical shear strength into account, results in a reasonable limit in grain size for applying dislocation pile-up theory to nanocrystalline materials. The deviation from the Hall—Petch relationship is rationalized in terms of a small number dislocation pile-up mechanism. A composite model is proposed to evaluate the strength of nanocrystalline materials. It is shown that this model can be used for interpreting the various cases observed in Hall—Petch studies. An analytical expression for assessing the creep rate of nanocrystalline materials by a diffusion mechanism, including triple line diffusion, is derived. It is predicted that the creep rate due to triple line diffusion will exhibit a stronger grain size dependence than that due to grain boundary diffusion.

On the role of non-equilibrium grain-boundary structure in the yield and flow stress of polycrystals

Abstract At the beginning of the plastic deformation of polycrystals, dislocations are mainly accumulated in grain boundaries (GBs), forming extrinsic grain-boundary dislocations (EGBDs). EGBDs create long-range stress fields which in turn considerably affect the dislocation glide in grains. On the basis of dislocation pile-up theory, the role of these stresses in the deformation behaviour of polycrystals is studied. Two types of internal stress induced by non-equilibrium GBs are considered: the stresses of triple-line disclination dipoles and those of disordered EGBD arrays. It is shown that these two kinds of stress, which differ in their scales, contribute to the flow stress in different ways. The former, scaling with the grain size, leads to an increase in the parameter σ0(η) in the Hall-Petch relation, while the latter, scaling with the average EGBD spacing, increases the parameter k(η). These parameters are calculated for several metals and good agreement with experimental data is obtained.

Hall-Petch Analysis of Yield, Flow and Fracturing

AbstractThe explanation for the grain size dependence of the polycrystal yield stress or cleavage stress of steel, investigated in complementary studies by Hall and Petch, is rooted in dislocation pile-up theory first described by Eshelby, Frank and Nabarro just a step away from previous theoretical work on the stress concentrating properties of cracks. Beginning from Cottrelllocking of dislocations at grain boundaries by carbon that is responsible for the pronounced yield point behavior and the grain size dependence of the yield strength of steel, the same concepts have been shown to apply for the complete stress/strain behavior of steel and other materials. Other mechanical properties such as fatigue strength show a similar dependence on grain size. A notable application of such grain size based considerations has been to the ductile-brittle transition and fracture toughness properties of steel and related materials — because refinement of grain size both strengthens a material and improves its fracture toughness. Early results for the polycrystal hardness dependence on grain size gave evidence that the strength benefits of grain size refinement ought to extend to ultrafine grain sizes.

Micromechanics Modeling of Crack Initiation Under Contact Fatigue

Using dislocation pileup theory, a model is given for the prediction of crack initiation life under contact fatigue. Near surface crack initiation is investigated by introducing the sliding contact boundary condition. Crack initiation originated at the surface and substrate are treated as extreme cases. The new model physically explains how a surface crack can be initiated and shows that the surface crack initiation life should be shorter than the subsurface crack initiation life under the same stress amplitude conditions. A discussion is given about the influence of residual stress, hardness, temperature, irreversibility of the plastic deformation, as well as other parameters that affect the crack initiation life. Preliminary comparisons show that the new model agrees well with the experimental observations of surface and near surface crack initiation.