Pile-up Model Research Articles

An analytical model of the effects of pulse pileup on the energy spectrum recorded by energy resolved photon counting x‐ray detectors

Recently, novel CdTe photon counting x-ray detectors (PCXDs) with energy discrimination capabilities have been developed. When such detectors are operated under a high x-ray flux, however, coincident pulses distort the recorded energy spectrum. These distortions are called pulse pileup effects. It is essential to compensate for these effects on the recorded energy spectrum in order to take full advantage of spectral information PCXDs provide. Such compensation can be achieved by incorporating a pileup model into the image reconstruction process for computed tomography, that is, as a part of the forward imaging process, and iteratively estimating either the imaged object or the line integrals using, e.g., a maximum likelihood approach. The aim of this study was to develop a new analytical pulse pileup model for both peak and tail pileup effects for nonparalyzable detectors. The model takes into account the following factors: The bipolar shape of the pulse, the distribution function of time intervals between random events, and the input probability density function of photon energies. The authors used Monte Carlo simulations to evaluate the model. The recorded spectra estimated by the model were in an excellent agreement with those obtained by Monte Carlo simulations for various levels of pulse pileup effects. The coefficients of variation (i.e., the root mean square difference divided by the mean of measurements) were 5.3%-10.0% for deadtime losses of 1%-50% with a polychromatic incident x-ray spectrum. The proposed pulse pileup model can predict recorded spectrum with relatively good accuracy.

Modelling of the effect of dislocation channel on intergranular microcrack nucleation in pre-irradiated austenitic stainless steels during low strain rate tensile loading

In the present article, the effect of dislocation channel on intergranular microcrack nucleation during the tensile deformation of pre-irradiated austenitic stainless steels is studied. Because several slip planes are activated within the dislocation channel, the simple dislocation pile-up model seems not well suited to predict grain boundary stress field. Finite element computations, using crystal plasticity laws and meshes including a channel of finite thickness, are also performed in order to study the effect of some microstructural characteristics on grain boundary stress field. Numerical results show that: the thickness and the length of the dislocation channel influence strongly the grain boundary normal stress field. The grain boundary orientation with respect the stress axis does not affect so much the grain boundary normal stresses close to the dislocation channel. On the contrary far away the dislocation channel, the grain boundary stress field depends on the grain boundary orientation. Based on these numerical results, an analytical model is proposed to predict grain boundary stress fields. It is valuable for large ranges of dislocation channel thickness, length as well as applied stress. Then, a macroscopic microcrack nucleation criterion is deduced based on the elastic–brittle Griffith model. The proposed criterion predicts correctly the influence of grain boundary characteristics (low-angle boundaries (LABs), non-coincident site lattice (non-CSL) high-angle boundaries (HABs), special grain boundaries (GBs)) on intergranular microcrack nucleation and the macroscopic tensile stress required for grain boundary microcrack nucleation for pre-irradiated austenitic stainless steels deformed in argon environment. The criterion based on a dislocation pile-up model (Smith and Barnby) underestimates strongly the nucleation stress. These results confirm that pile-up models are not well suited to predict microcrack nucleation stress in the case of dislocation channels impacting grain boundary. The proposed criterion is applied to the prediction of the IASCC macroscopic nucleation stress for pre-irradiated material tested in PWR environment and the predictions are discussed with respect to experimental data. Finally, the limitations of the continuum modelling are discussed.

Theoretical Research on the Dynamic Crack Propagation Velocity Based on Nonlocal Field Theories

The purpose of this paper is to discuss the nonlocal effect on dynamic crack propagation velocity. Some experimental phenomena in dynamic fracture and simulative results using molecular & atom dynamics were analyzed and discussed in this paper. The authors found that there were still some disagreements on the dynamic crack propagation velocity. Based on these researches, we introduced nonlocal field theories into the estimation of dynamic crack propagation velocity. The dynamic crack propagation velocity is affected not only by the crack instability, but by characteristic length of material. A nonlocal characteristic length parameter M is defined through a double pile-up dislocation model. According to the Mott’s research method for crack velocity in dynamic fracture and the nonlocal field theories, an approximate theoretical dynamic propagation velocity is obtained. And we conclude that the velocity is related to the combined activity of the nonlocal characteristic length parameter M, the velocity of longitudinal wave, constant k, crack length and Poisson’s ratio.

Slip Line Growth as a Critical Phenomenon

We study the growth of a slip line in a plastically deforming crystal by numerical simulation of a double-ended pileup model with a dislocation source at one end, and an absorbing wall at the other end. In the presence of defects, the pileup undergoes a continuous nonequilibrium phase transition as a function of stress, which can be characterized by finite-size scaling. We obtain a complete set of critical exponents and scaling functions that describe the spatiotemporal dynamics of the slip line. Our findings allow us to reinterpret earlier experiments on slip line kinematography as evidence of a dynamic critical phenomenon.

Effect of Grain Refinement on Tensile Properties in Fe–25Cr–1N Alloy

Nickel-free austenitic stainless steel (Fe-25Cr-1N) was produced by solution nitriding, and then the grain size was variously controlled by the two-step phase transformation method. Mechanical property of the steel with different grain size was investigated by means of tensile testing to clarify the effect of grain size on the strength, elongation and fracture surface. The as-solution-nitrided material with a coarse grain size caused brittle intergranular fracture on the way of uniform deformation before the onset of local necking deformation. As a result of grain refinement, intergranular fracture was markedly suppressed, and this leads to the enlargement of elongation as well as increase in yield strength and tensile strength. The suppression mechanism of intergranular fracture was discussed on the basis of dislocation pile-up model together with the experimental results of TEM observation for dislocation substructure in tensile-deformed materials. It was then concluded that the stress concentration at grain boundary is reduced by grain refinement, and this contributed to the suppression of intergranular fracture and enlargement of elongation in the grain-refined material.

Analysis of grain boundary effect of bulk polycrystalline materials through nanomechanical characterization

Indentation-induced deformation behaviour was characterized in nano-scale for bulk polycrystalline materials to understand strengthening factors of macroscopic properties especially for the grain boundary effect. Compared with deformation behaviour in the vicinity of single grain boundary and grain interior of an interstitial free steel, plasticity initiation occurs at a lower applied stress near the grain boundary, which means the grain boundary is an effective dislocation source. The subsequent deformation after plasticity initiation is affected by the grain boundary as a barrier to dislocation motion. Strengthening factors of matrix and grain boundaries were evaluated separately for the Fe–C tempered martensite. The contribution of grain boundaries depends on the morphology of precipitates on grain boundaries. Combining the dislocation pile-up model and the Hall–Petch relation, the existence of the film-like carbide on the grain boundary remarkably affects the locking parameter k.

Micro-bending tests: A comparison between three-dimensional discrete dislocation dynamics simulations and experiments

Discrete dislocation dynamics simulations in three dimensions are performed on micro-sized bending beams and the results are compared with experiments. A strong size dependence of the flow stress σf (or bending moment) is found. The flow stress scales approximately inversely with the beam thickness t. The simulations show that the dislocation structure exhibits pronounced pile-ups around the neutral plane of the beam. The back stress from these pile-ups on the dislocation sources is analyzed by means of an analytical pile-up model. It is shown that the scaling behavior σf∝t-1 can be explained by a combination of pile-up and source size limitation. Subsequently, the applicability of strain gradient plasticity models on micro-bending is discussed.

Fracture Toughness Enhanced by Grain Boundary Shielding in Submicron-Grained Low Carbon Steel

The enhancement of toughness at low temperatures in fine grained low carbon steel was studied, basing on a shielding theory due to dislocations and grain boundaries. Fully annealed low carbon steel was subjected to an accumulative roll bonding (ARB) process for grain refining. The grain size perpendicular to the normal direction was found to be approximately 200 nm after the ARB process. The fracture toughness of low carbon steel ARB applied was measured at 77 K by four-point bending, comparing with the fracture toughness of those without the ARB process. It was found that the value of fracture toughness at 77 K was increased by grain refining due to the ARB, indicating that the ARB process enhances toughness at low temperatures as reported in interstitial free steel and phosphorus doped interstitial free steel. It also deduces that the brittle-ductile transition (BDT) temperature shifted to a lower temperature. The enhancement of toughness and the decrease of the BDT temperature due to grain refining cannot be explained completely by the dislocation pile-up model of dislocations at grain boundaries. Quasi-two-dimensional simulations of dislocation dynamics, taking into account of crack tip shielding due to dislocations, were performed to investigate the effect of a dislocation source spacing along a crack front on the BDT. The simulation indicated that the BDT temperature is decreased by decreasing the dislocation source spacing. In addition to the simulation, the authors suggest a new concept of accommodating stress intensity at the crack tip due to grain boundaries to explain the enhancement of toughness and the decrease of the BDT temperature in fine grained materials.

Influence of image forces on the size dependence of multilayer shear strength

The dependence of the multilayer shear strength on layer thickness is investigated using a discrete dislocation pileup model accounting for image force effects. A three-layer anisotropic model, in contrast to the bilayer model used in most previous works, allows for the explicit consideration of the layer thickness and three interfaces. The image forces are calculated using an image decomposition technique. The results show that attractive image forces are sufficiently large to cause a deviation from the Hall–Petch relation as well as a peak in the relation as the thickness decreases below ∼40 nm. Furthermore, faceting of the predicted strength–thickness relation can be traced to differences between nucleation-controlled and load-controlled yielding. The predicted results agree broadly with experimental data for metallic multilayers over three length scales from 10 2 nm down to 10 0 nm.

Microstructural investigation of the volume beneath nanoindentations in copper

The deformed volume below nanoindentations in copper single crystals with a 〈 1 1 ¯ 0 〉 { 1 1 1 } orientation is investigated. Using a focused ion beam workstation, cross-sections through nanoindentations were fabricated and examined using the electron backscatter diffraction technique. Additionally a transmission electron microscopic foil through the middle of an imprint was prepared and analysed. Due to changes in the crystal orientation around and beneath the indentations the plastically deformed zone can be visualized and compared with the measured hardness values. Furthermore, the hardness data were analysed in terms of geometrically necessary dislocations using the Nix–Gao model, where a linear relationship was found for H 2 vs. 1/ h c, but with different slopes for large and shallow indentations. The orientation “micrographs” indicate that this behaviour is associated with a change in the deformation mechanism. Consequently, two models based on possible dislocation arrangements are presented and compared with the experimental findings. For large indentations a dislocation pile-up model similar to those used to explain the Hall–Petch effect is suggested, while the model for shallow imprints uses far-reaching dislocation loops to accommodate the shape change of the indenter.

Flow stress/strain rate/grain size coupling for fcc nanopolycrystals

A Hall–Petch (H–P)-type dependence is demonstrated for reciprocal activation volume measurements for nanocrystalline and conventional grain size, strengthened Ni and Cu materials, consistent with predictions derived from the dislocation pile-up model. The observed H–P dependence indicates that the shear stress for cross-slip must be involved in the full grain size regime for transmission of plastic flow at the grain boundaries of fcc metals.

Exponent for Hall–Petch behaviour of ultra-hard multilayers

Elastically inhomogeneous multilayer films are being exploited for use as ultra-hard coatings. These films exhibit a strong dependence between the compositional wavelength of the film, Λ, and the hardness, H=KΛ−a+H0 where the scaling exponent a depends on the elastic properties of the individual layers (shear moduli and Poisson ratios). The dislocation pileup model can explain this trend and form a bridge between the microscopic strength of multilayer interfaces and the macroscopic strength of the multilayer. A semianalytic solution to the pileup model of multilayer strength is presented. All parameter dependencies are solved analytically except a single dimensionless coefficient which is found from numerical simulation. The predictions are compared to data from a 2D discrete dislocation model and to experimental Cu/Ni data. Coefficients and exponents are given for some additional material systems.

Dislocation pileup as a representation of strain accumulation on a strike‐slip fault

The conventional model of strain accumulation on a vertical transform fault is a discrete screw dislocation in an elastic half‐space with the Burgers vector of the dislocation increasing at the rate of relative plate motion. It would be more realistic to replace that discrete dislocation by a dislocation distribution, presumably a pileup in which the individual dislocations are in equilibrium. The length of the pileup depends upon the applied stress and the amount of slip that has occurred at depth. I argue here that the dislocation pileup (the transition on the fault from no slip to slip at the full plate rate) occupies a substantial portion of the lithosphere thickness. A discrete dislocation at an adjustable depth can reproduce the surface deformation profile predicted by a pileup so closely that it will be difficult to distinguish between the two models. The locking depth (dislocation depth) of that discrete dislocation approximation is substantially (∼30%) larger than that (depth to top of the pileup) in the pileup model. Thus, in inverting surface deformation data using the discrete dislocation model, the locking depth in the model should not be interpreted as the true locking depth. Although dislocation pileup models should provide a good explanation of the surface deformation near the fault trace, that explanation may not be adequate at greater distances from the fault trace because approximating the expected horizontally distributed deformation at subcrustal depths by uniform slip concentrated on the fault is not justified.

Nonuniformity Effect of Surface-Nanocrystalline Materials in Nanoindentation Test

In the present research, microstructures of the surface-nanocrystalline Al alloy material are observed and measured based on the transmission electron microscopy (TEM) technique, and the corresponding mechanical behaviors are investigated experimentally and theoretically. In the experimental research, the nanoindentation test method is used, and the load and microhardness curves are measured, which strongly depend on the grain size and grain size nonuniformity. Two kinds of the nanoindentation test methods are adopted: the randomly selected loading point method and the continuous stiffness method. In the theoretical modeling, based on the microstructure characteristics of the surface-nanocrystalline Al alloy material, a dislocation pile-up model considering the grain size effect and based on the Mott theory is presented and used. The hardness-indent depth curves are predicted and modeled.

Effect of grain size from mm to nm on the flow stress and plastic deformation kinetics of Au at low homologous temperatures

Three grain size regimes were identified: Regime I ( d > ∼0.5 μm), Regime II ( d ≈ 10–500 nm) and Regime III ( d < ∼10 nm). Grain size hardening in accord with the Hall–Petch (H–P) equation occurred in Regimes I and II, grain size softening with the flow stress proportional to d −1 occurred in III. The rate-controlling mechanism in Regime I was concluded to be the intersection of dislocations, that in II grain boundary shear promoted by the pile-up of dislocations and that in III grain boundary shear without pile-ups. The transition from I to II occurred when the dislocation cell size became larger than the grain size, that from II to III when the dislocation elastic interaction spacing became larger than the grain size. The H–P behavior in Regime I is in accord with the dislocation density model, that in II with the dislocation pile-up model. The general behavior of Au is similar to that reported previously for Cu and Ag.

Electron theory based explanation and experimental study on deformation behavior of Cu/Ni multilayers

Cu/Ni multilayers with various defined thickness of Cu and Ni layers were electrodeposited on low carbon steel substrates. Hardness measurements indicated that the increase in yield strength (one-third of hardness) with a decrease of layer thickness for Cu/Ni multilayers with single layer thickness at sub-micron length scale could be described by the Hall-Petch formula of the dislocation pile-up model. In the regime of few tens to a hundred nanometers of single layer thickness, the dislocation pileup-based Hall-Petch model broke down. This could be explained quantitatively according to the criterion condition on the limit size of dislocation derived from a modified Thomas-Fermi-Dirac electron theory.

Analysis of Portevin-Le Chatelier serrations of type Bin Al–Mg

Plastic deformation of solid solutions at elevated temperatures is often accompanied by plastic instabilities due to dynamic strain aging (DSA) and dislocation interaction. The repeated breakaway of dislocations from and their recapture by (clouds of) solute atoms lead to stress serrations and localized strain (deformation bands) in strain controlled tensile tests, known as the Portevin-Le Chatelier (PLC) effect. The present work gives an analysis of stress drops for constant strain rate tensile tests on Al–(1.5–4.5) wt.% Mg in terms of their dependencies on temperature and strain rate for type B deformation bands, the unambiguous classification of which is accomplished by local strain measurements with a multizone laser extensometer. To arrive at a consistent interpretation of the thermal activation behavior of the breakaway of dislocations associated with type B serrations, a dislocation pile-up model is introduced which matches well with these as well as with earlier results on Cu–Al.

A design-centered approach in developing Al-Si-based light-weight alloys with enhanced fatigue life and strength

Material heterogeneities and discontinuities such as porosity, second phase particles, and other defects at meso/micro/nano scales, determine fatigue life, strength, and fracture behavior of aluminum castings. In order to achieve better performance of these alloys, a design-centered computer-aided renovative approach is proposed. Here, the term “design-centered” is used to distinguish the new approach from the traditional trial-and-error design approach by formulating a clear objective, offering a scientific foundation, and developing a computer-aided effective tool for the alloy development. A criterion for tailoring “child” microstructure, obtained by “parent” microstructure through statistical correlation, is proposed for the fatigue design at the initial stage. A dislocations pileup model has been developed. This dislocation model, combined with an optimization analysis, provides an analytical-based solution on a small scale for silicon particles and dendrite cells to enhance both fatigue performance and strength for pore-controlled castings. It can also be used to further tailor microstructures. In addition, a conceptual damage sensitivity map for fatigue life design is proposed. In this map there are critical pore sizes, above which fatigue life is controlled by pores; otherwise it is controlled by other mechanisms such as silicon particles and dendrite cells. In the latter case, the proposed criteria and the dislocation model are the foundations of a guideline in the design-centered approach to maximize both the fatigue life and strength of Al-Si-based light-weight alloy.

Simulation of Directional Distribution of Slip-Band Crack under Biaxial Low Cycle Fatigue

The growth of dominant crack is sometimes affected by distributed small cracks in multiaxial low cycle fatigue. Considering such a situation, crack initiation in biaxial fatigue was first modeled in this work. In modeled grains, normal directions of slip planes and slip directions on respective planes were independently given at random. A slip-band crack was presumed to be initiated along the given slip direction on the specified slip plane when the resolved shear stress in the slip direction exceeded a critical shear stress. The crack initiation life was also evaluated using a dislocation pile-up model. Simulations on crack initiation were conducted under the same loading conditions as experiments. Simulated directional distribution of initiated cracks was compared with experimental results obtained in biaxial fatigue tests using tubular specimens of pure copper, and it showed a good agreement with the experimental observation.

Effects of strain and strain-rate on the formation of the shear band in metals

An experimental study on the formation of shear bands was carried out in several metals including pure titanium (α-Ti), 304 stainless steel and aluminium (Al-7075-T6) through the compression of cylinder specimens in a Split-Hopkinson-Bar system. The shear band initiation and the distribution within the specimen were examined under different plastic strains and strain rates. The shear band forms along the maximum shear stress plane. Combining the total plastic strain and the angle between the band and the centrally symmetrical axis with the stress-strain curve, the critical strain (e c ) and stress (σ c ) for the formation of the band can be evaluated. It was found that the multiple developing shear bands were stopped in front of the grain boundary in one grain of Al-7075-T6. The present experimental results agree with Armstrong's dislocation pile-up model for the initiation for shear banding. Based on dislocation dynamics, a new constitutive equation is proposed for description of the formation of shear bands. The formation of the band is caused mainly by the strain softening from the sharp increase of mobile dislocations while the thermal softening enhances the development of the shearing process.