Nabarro Stress Research Articles

Dislocation evolution in anisotropic deformation of GaN under nanoindentation

The exceptional performance of GaN semiconductors in lasers, wireless communication, and energy storage systems makes them crucial for future multi-functional devices. However, during the polishing of GaN wafers, abrasive particles can induce subsurface damage, compromising device performance. This study investigates dislocation loops in GaN single crystal to understand dislocation nucleation and glide under external stress. Using nanoindentation for compressive stress, we confirmed multiple slip system activation via transmission electron microscopy after pop-in. We also performed molecular dynamics to simulate the nucleation and multiplication of U-shaped dislocation loops. Furthermore, we developed a theoretical model using Peierls–Nabarro stress to quantify GaN's critical shear stress. Raman spectroscopy was also used to analyze shear stress on U-shaped loops, supporting our model. This study provides insights into GaN dislocation dynamics under mechanical stress, aiding in wafer defect evaluation during machining and offering guidance for dislocation evolution.

Microstructure-based multiscale and heterogeneous elasto-plastic properties of 2205 duplex stainless steel welded joints: Experimental and modeling

This work aims to reveal the inherent dependency between the heterogeneous microstructure induced by the metallurgical transformation during welding of 2205 duplex stainless steel (DSS) and its macroscopic and microscopic elasto-plastic properties by experiments and models. A multiscale model was developed by combining a macro-constitutive model incorporating the physically-based yield strength and a crystal plasticity model considering both the micro-mechanical properties of two phases and their realistic morphology. The results show that the heterogeneous macro-mechanical properties of base metal (BM), heat affected zone (HAZ), and weld material (WM), as well as their micro-mechanical responses related to phase morphology, ratio, and grain size, can be accurately predicted. The contribution of Peierls–Nabarro stress, solid solution strengthening, grain/phase boundary, and dislocations density on macro-yield and tensile strengths was quantified, and WM was proven to have greater yield and tensile strengths due to the greater contribution of grain refinement and solid solution strengthening. In addition, a transition mechanism of the deformation-bearing phase from austenite to ferrite with the increase of load was proved, and the banded grain boundary austenite (GBA) and its adjacent ferritic grains were prone to plastic-deformation. This work can be used as guidance of structure integrity assessment and welding process optimization for the engineering components of DSS.

Plastic deformation and strengthening mechanism of FCC/HCP nano-laminated dual-phase CoCrFeMnNi high entropy alloy

FCC-structured CoCrFeMnNi high entropy alloy (HEA) has attracted abroad interests for years because of its excellent mechanical properties, except for strength. Recent experiments have reported a kind of nano-laminated dual-phase (NLDP) FCC/HCP structure that can strengthen the HEA. However, it is still unknown why the HEA can be strengthened by this kind of NLDP structure. Here, we employ molecular dynamics simulations to study the atomistic strengthening mechanism of the NLDP HEA. Dislocation-assisted multiple plastic deformation mechanisms in both FCC and HCP single phase HEAs are observed, and amorphization is also found in the plasticity of HCP phase, which are consistent with the previous experimental characterizations. The HCP phase possesses higher strength because of its higher stacking fault energy, higher Peierls–Nabarro stress and less active dislocation slip systems. It is also found that the introduction of HCP phase can enhance the mechanical properties, including yield stress, yield strain and plastic flow stress, of the NLDP HEAs, which also show volume fraction dependence. And the phase boundary plays crucial roles in the deformation and strengthening of the NLDP HEAs. The plastic deformation of the NLDP HEAs can be divided into two stages, i.e. stage I (plasticity only appears in FCC lamella) and stage II (plasticity in both FCC and HCP lamellas). With the increase of volume fraction, the lamella thickness of FCC matrix phase decreases, leading to continuous strengthening of yield properties and flow stress of stage I because of suppressed dislocation nucleation and confined dislocation motion in FCC matrix phase by the phase boundary. While there is no monotonous relationship between the flow stresses of stage II and the increasing volume fraction of HCP phase, which can be attributed to the competitive mechanisms between strengthening effect of phase boundary on the dislocation motion in FCC phase and softening effect of phase boundary on the dislocation motion in HCP phase. The results should be helpful for understanding the underlying physical mechanism of strengthening of HEAs with NLDP structure.

Peierls–Nabarro stresses of dislocations in monoclinic cyclotetramethylene tetranitramine (β-HMX)

HMX (cyclotetramethylene tetranitramine) is an energetic material which releases substantial amounts of energy upon decomposition. The role of defects and deformation in causing reaction initiation was discussed in the literature but remains insufficiently understood. In this work, we identify, using computational methods, the slip systems which are potentially active in β-HMX and rank them in terms of their propensity for slip. To this end, we develop first a tentative ranking based on the degree of steric hindrance associated with slip. This is quantified using a geometric analog of the γ-surface. Further, we use atomistic models to compute the Peierls–Nabarro (PN) stress for the motion of dislocations in the slip systems with smallest degree of steric hindrance. A complex mechanical behavior is observed, including strong slip asymmetry, twinning and cleavage. The five systems with the lowest PN stress are and We conclude that the material has enough slip systems available for supporting a generalized state of plastic strain provided the twinning system is taken into consideration and that the resolved shear stress is at least 260 MPa.

Contribution of molecular flexibility to the elastic–plastic properties of molecular crystal α-RDX

We show in this work that the mechanical properties of molecular crystals are strongly affected by the flexibility of the constituent molecules. To this end, we explore several kinematically restrained models of the molecular crystal cyclotrimethylene trinitramine in the α phase. We evaluate the effect of gradually removing the flexibility of the molecule on various crystal-scale parameters such as the elastic constants, the lattice parameters, the thermal expansion coefficients, the stacking fault energy and the critical stress for the motion of a dislocation (the Peierls–Nabarro stress). The values of these parameters evaluated with the fully refined, fully flexible atomistic model of the crystal are taken as reference. It is observed that the elastic constants, the lattice parameters and their dependence on pressure, and the thermal expansion coefficient can be accurately predicted with models that consider the NO2 and CH2 groups rigid, and the N–N bonds and the bonds of the triazine ring inextensible. Eliminating the dihedral flexibility of the ring leads to larger errors. The model in which the entire molecule is considered rigid or is mapped to a blob leads to even larger errors. Only the fully flexible, reference model provides accurate values for the stacking fault energy and the Peierls–Nabarro critical stress. Removing any component of the molecular flexibility leads to large errors in these parameters. These results also provide guidance for the development of coarse grained models of molecular crystals.

Quantitative Analysis of Heterogeneous Microstructure and Diversified Strengthening Mechanisms in Spark Plasma Sintered Molybdenum Disilicide

Molybdenum disilicide (MoSi2) was in situ synthesized by spark plasma sintering into fully dense bulk material. Electron backscatter diffraction revealed that over the range from the center to periphery of the cylindrical sample, the microstructure is heterogeneously constituted by gradually refined MoSi2 grains and increasing amounts of Mo5Si3 impurity phase. The diversified strengthening mechanisms are primarily attributed to the Peierls–Nabarro stress in MoSi2 matrix, the Hall–Petch strengthening effect, and thermal mismatch stress between Mo5Si3 and MoSi2.

Mechanical and Optical Properties Characterization of C-Plane (0001) and M-Plane (10−10) GaN by Nanoindentation and Luminescence

ABSTRACTYield shear stress dependence on dislocation density and crystal orientation was studied in bulk GaN crystals by nanoindentation examination. The yield shear stress decreased with increasing dislocation density which is estimated by dark spot density in cathodoluminescence, and it decreased with decreasing nanoindentation strain-rate. It reached and coincided at 11.5 GPa for both quasi-static deformed c-plane (0001) and m-plane (10-10) GaN. Taking into account theoretical Peierls–Nabarro stress and yield stress for each slip system, these phenomena were concluded to be an evidence of heterogeneous mechanism associated plastic deformation in GaN crystal. Transmission electron microscopy and molecular dynamics simulation also supported the mechanism with obtained r-plane (-1012) slip line right after plastic deformation, so called pop-in event. The agreement of the experimentally obtained atomic shuffle energy with the calculated twin boundary energy suggested that the nucleation of the local metastable twin boundary along the r-plane concentrated the indentation stress, leading to an r-plane slip. This nanoindentation examination is useful for the characterization of crystalline quality because the wafer mapping of the yield shear stress coincided the photoluminescence mapping which shows increase of emission efficiency due to reduction of non-radiative recombination process by dislocation.

Fabrication and characterization of (Mg1−xFex)O (0.05≤x≤0.25) ceramics

Highly dense (Mg1−xFex)O (0.05≤x≤0.25) ceramics (relative density >98%) were prepared by ultrasonic assisted oxalate co-precipitation method together with spark plasma sintering. Mechanical measurements indicate that Young’s modulus decreases monotonously from 312 to 268.5±5.2GPa with increasing Fe content x from 0 to 0.25. In contrast, the Vickers hardness first increases from 6.8 to 8.50±0.11GPa as x increases from 0 to 0.1, and then decreases monotonously with further increasing x. The increase and decrease in the hardness with x can be mainly attributed to the solid solution strengthening and weakening of Peierls–Nabarro stress, respectively. Thermal analyses reveal that heat capacity Cp decreases (from 0.896±0.018 to 0.816±0.016(JK)−1g−1 at 323K, for instance) with increasing x (from 0.05 to 0.25) due to the lower Cp of FeO. Similarly, thermal conductivity κ of the samples decreases (say, from 24.8±1.0 to 3.7±0.1(WK)−1m−1 at 323K) with the increasing x (from 0.05 to 0.25), which could be ascribed to the decreasing melting point and increasing mean atomic weight resulting from substitution of Mg for Fe.

Intrinsic brittleness of Mo5SiB2 and alloying effect on ductility studied by first-principles calculations

The ordered intermetallic Mo5SiB2 displays a ceramic-like brittleness at the ambient temperature. The state density, charge distribution and elastic parameters were calculated by first-principles, based on the density functional method. The results indicated that the two different kinds of covalent bonds were intricately woven into the refractory phase. The improved Peierls–Nabarro stress which is caused by this kind of distribution mode makes dislocations move difficultly, resulting in intrinsic brittleness. The effects of substitutional alloying on the ductility of Mo5SiB2 were also assessed by the calculations on the elastic properties and dislocation line energy. It was shown that the metal (Nb, Tc) alloying was not to enhance effectively its toughness, but to improve the brittle-to-ductile transition temperature.

Plasticity at Absolute Zero as a Fundamental Characteristic of Dislocation Properties

Values of a plasticity characteristic H for different materials were determined by the indentation method at cryogenic temperatures. Using the linear dependence H(T) at low temperatures, the value of δH at 0 K, designated by δH(0), was obtained by the extrapolation method. Values of δH(0) for different materials, namely FCC, HCP and BCC metals, intermetallics, metallic glasses, quasicrystals, ceramics and covalent crystals, are discussed. An analytic expression for a dependence of δH(0) on the parameters of thermoactivated movement of dislocations, melting point and Young’s modulus E is obtained. It is shown that any type of hardening of a crystal and an increase in the Peierls–Nabarro stress σS(0) reduce δH(0). Only a rise in E leads to the simultaneous increase in σS(0) and δH(0). δH(0) can be considered as a dislocation plasticity in the absence of thermal vibrations of atoms and should be considered together with strength parameters as an important fundamental characteristic of dislocation properties.

Additivity of strengthening mechanisms in ultrafine grained Al–Mg–Sc alloy

In the light of unique and anomalous properties of ultrafine grained (UFG) alloys, an effort was made to develop a predictive capability of the yield strength (YS) of UFG Al–Mg–Sc alloy. UFG microstructure was introduced using friction stir processing. Microstructural characterization of grain size, dislocations, and nano-sized Al3(Sc,Zr) particles was carried out using electron backscatter diffraction and transmission electron microscope. The contribution from Peierls–Nabarro stress, solid solution strengthening, precipitation strengthening, grain boundary strengthening, and dislocation strengthening were assessed using existing strengthening models. Additivity law to predict the YS of the alloy was chosen based on the microstructural state of the alloy. The microstructural state of coarse grained and UFG alloy favored the use of linear additivity rule over Pythagorean. The use of a mixed (linear and Pythagorean) additivity rule was also carried out to assess its YS prediction capability. A difference in the range of 33–55% was observed between predicted and experimentally obtained YS for UFG alloy. The reason was related to the overprediction from grain boundary strengthening model. A smaller HP slope than the normal slope value was able to predict the YS of the UFG alloy more closely.

Strain Rate Controlled Nanoindentation Examination and Incipient Plasticity in Bulk GaN Crystal

Yield shear stress dependence on dislocation density and crystal orientation was studied in bulk GaN crystals by nanoindentation examination. The yield shear stress decreased with increasing dislocation density, and it decreased with decreasing nanoindentation strain-rate. It reached and coincided at 11.5 GPa for both quasi-static deformed c-plane (0001) and m-plane (101̄0) GaN. Taking into account theoretical Peierls–Nabarro stress and yield stress for each slip system, these phenomena were concluded to be an evidence of heterogeneous mechanism associated plastic deformation in GaN crystal. Transmission electron microscopy and molecular dynamics simulation also supported the mechanism with obtained r-plane (1̄012) slip line right after plastic deformation, so called pop-in event. The agreement of the experimentally obtained atomic shuffle energy with the calculated twin boundary energy suggested that the nucleation of the local metastable twin boundary along the r-plane concentrated the indentation stress, leading to an r-plane slip.

Effect of chromium on elastic and plastic deformation of Fe3Al intermetallics

In order to evaluate the effect of chromium concentration on the mechanical properties of single phase iron aluminum intermetallics, specifically those containing 26 at.% Al, a nano-indentation technique was used. The grain orientations of samples were revealed with the aid of Electron Backscatter Diffraction (EBSD) technique. Additionally, the surface roughness of each sample was checked after electropolishing with the aid of Atomic Force Microscopy (AFM). Finally, all nano- and micro-indents were performed within the (001) orientated grains, where the surface roughness was less than 1.06 nm on average, in all samples. The Oliver–Pharr method, with correction for pile up, was used to calculate the Young's modulus and nano-hardness for Fe3Al–xCr alloys. Additionally, the effect of Cr on the shear stress needed for homogeneous dislocation nucleation, internal friction stress of dislocation motion (Peierls–Nabarro stress) and flow stress was studied.

STUDY ON THE PEIREL–NABARRO STRESS OF IRON ALUMINIDES

The modified Peirel–Nabarro model of dislocations is not valid for the superdislocation bounded by the antiphase domain boundaries in long-range ordered intermetallic. In this work, a new Peirel–Nabarro Stress model is developed to take into account the critical resolved shear stress of antiphase domain boundary (APDB). Based on it, the Peirel–Nabarro Stress of DO3 structure and B2 structure in iron aluminides are calculated. Comparing the Peirel–Nabarro Stress of dislocations and the crystal theoretical yield strength, the results demonstrate B2-type crystal has good plasticity. It coincides with the experimental results well.

Nanoindentation study on insight of plasticity related to dislocation density and crystal orientation in GaN

Yield shear stress dependence on dislocation density and crystal orientation was studied in GaN by nanoindentation examination. The yield shear stress decreased with increasing dislocation density, and it decreased with decreasing nanoindentation strain-rate. It reached and coincided at 11.5 GPa for both quasi-static deformed c-plane and m-plane GaN. Taking into account theoretical Peierls–Nabarro stress and yield stress for each slip system, these phenomena were concluded to be an evidence of heterogeneous mechanism associated plastic deformation in GaN crystal. Transmission electron microscopy and molecular dynamics simulation also supported the mechanism with obtained r-plane dislocation line.

On Peierls Nabarro stress in Iron

It is well known that Iron is polymorphic, it exists as α Iron (ferrite), which is BCC up to 912°C after which γ Iron (austenite), which is FCC is formed, being stable up to 1394°C, at which temperature it reverts back to a BCC phase known as δ Iron (δ ferrite), which finally melts at 1538°C. Plastic deformation can be due to crystallographic slip mechanism or twinning. The slip system for FCC crystal is {111}〈110〉 in which 12 combinations exist. In BCC crystal, the slip systems are {110}〈111〉, {112}〈111〉 and {123}〈111〉 of which 48 combinations are feasible. Therefore, a FCC crystal has 12 slip systems and BCC has 48 slip systems. Several decades back, Von Mises proposed that minimum five independent slip systems are required for plastic deformation by slip to occur in crystals. From this criterion it can be argued that the BCC crystal is more ductile than FCC crystal due to the presence of more slip systems. However, experimentally it is found that FCC crystal is more ductile than BCC crystal. The scientific reason for this anomaly is attributed to the higher Peierls Nabarro stress in BCC crystals as compared to FCC crystals. In this paper, calculations are presented to calculate the Peierls Nabarro stress in α Iron and γ Iron. This Peierls Nabarro is related to the cohesive energy of the crystal.

Frank Reginald Nunes Nabarro MBE. 7 March 1916 — 20 July 2006

Frank Nabarro is remembered as one of the great pioneers who developed the theory of dislocations in solids and thereby strikingly advanced understanding of the mechanical behaviour of metals. In two crucial areas he addressed the problem of the stress required to initiate plastic flow: first in metals alloyed to produce precipitates (precipitation hardening), and second in pure crystals of all types in which dislocation movement is impeded only by the intrinsic lattice resistance (Peierls–Nabarro stress). He predicted the slow creep of crystals at high temperature by the movement of single vacancies (Nabarro–Herring creep), a universal phenomenon of great engineering importance. Working from the University of Witwatersrand he edited Dislocations in solids , a series of review articles that continue to be essential keys to the huge unruly literature on dislocations. In South Africa he had an important role not only in developing solid state physics but also in promoting educational opportunities for students of all races, particularly in planning university expansion to accommodate the much larger student numbers expected after the end of apartheid, a policy that he vehemently opposed.

Metastablity of the undissociated state of dissociated dislocations

Undissociated, metastable dislocations have been observed in various crystals in addition to stable dissociated dislocations by high-resolution transmission electron microscopy. The origin of the metastablity of the undissociated state has been discussed specifically for the dissociation into Shockley partial dislocations in fcc or hcp lattice. It is shown that the metastability is due either to a high Peierls–Nabarro stress larger than a few percent of the shear modulus of the partial dislocations and/or to the increase of the total core energy by an increase of the dangling bonds. The metastablity of undissociated dislocations in zincblende III–V compounds is concluded to be due to a contribution of the latter effect.

Glide of edge dislocations in tungsten and molybdenum

The glide of a dislocation is the fundamental process of plastic deformation in crystalline solids. For body-centered-cubic (bcc) metals, the three-fold core dissociation of screw dislocations has attracted much attention and investigation. In this paper, we present a molecular dynamics study of glide of edge dislocations in bcc tungsten and molybdenum. On the {1 1 0} glide planes, the Peierls–Nabarro stress (PNS) of the edge dislocation is found to be (3.75–5.0)×10 −4 μ and (1.25–2.50)×10 −4 μ for tungsten and molybdenum, respectively, where μ is the shear modulus. On the glide planes {1 1 2}, these two numbers are (8.13–8.75)×10 −4 μ and (3.75–5.0)×10 −4 μ along one direction, and (1.13–1.38)×10 −3 μ and (6.25–8.75)×10 −4 μ along the opposite direction. The asymmetry of the PNS on the {1 1 2} planes is attributed to the asymmetrical dislocation core structure. During the glide process on {1 1 0} planes, the three constituent planes of an edge dislocation displace in sequence. The glide on {1 1 2} planes involve two such subsequent displacements, since the core consists of two non-equivalent planes.

Predicting the mechanical properties of second period refractory transition metal alloys

Stress–strain data from single crystals of refractory transition metal alloys tested in tension and compression are analyzed to give the parameters describing dislocation motion. The athermal stress and temperature correlates well with the elemental concentration of the alloy, while the Peierls–Nabarro (P–N) stress correlates with the bonding electron concentration. An estimate of the magnitude of the P–N stress for mixed dislocations is calculated from the elastic constants and correlates well with the values derived from experiment. These correlations provide the basis for predicting the magnitude of yield stress for transition metal alloys, and two examples are given of how this procedure can be used for alloys of known yield stress.