Hardening Rate Research Articles

An Investigation on unstable flow during hot deformation of Inconel X750 superalloy using 3D processing map and shear band formation criterion

In this study, deformation efficiency and instability criteria were employed to delineate the unstable flow regions for Inconel X750 superalloy through high temperature deformation. High temperature stress–strain data attained from isothermal hot compression experiments in the temperature range of 950-1150 °C and strain rates between 0.0001 and 1s⁻¹ to identify safe and un-safe deformation domains in terms of an integrated 3D processing maps. These domains were validated through detailed microstructure observation and material tendency to shear band formation in terms of the competition between work hardening and strain rate sensitivity i.e. flow localization parameter. The superimposed effect of microstructure features like dynamic precipitation, dynamic recovery (DRV), and dynamic recrystallization (DRX) with destructive shear banding were responsible for controlling the material behaviour. Based on the TEM images, it was observed that material revealed stable flow at lower strain rates while adiabatic shear band and flow localization occurred at higher strain rates due to activation of slip systems and cell formation.

The influence of structure dispersity on the intensity of hardening during tensile testing of pearlitic steel

During uniform and concentrated stretching of thin‑plate pearlite steel, a comparison of the structural dependence of strength and plastic properties, indicators of hardening intensity in the form of the degree of relative strength increase and the rate of deformation hardening, was carried out. Increased structural sensitivity is characteristic of the fracture stress, uniform relative contraction, and the degree of relative strength increase. It is recommended to evaluate the contribution of the deformation hardening intensity to the formation of the ductility of pearlitic steel by the parameters of the relative strength increase.

Two-way coupled modeling of dislocation substructure sensitive crystal plasticity and hydrogen diffusion at the crack tip of FCC single crystals

Dislocation substructure-sensitive crystal plasticity (DSS-CP) modeling accounts for the evolution of mesoscale structures using dislocation-based parameters informed by experiments and computation at various lower length scales. To a first-order approximation, DSS-CP model parameters are affected by hydrogen (H) concentration, accounting for both H-dependent yield strength and strain hardening rate. This H-affected DSS-CP model is two-way coupled with H-diffusion to explore both effects of plastic deformation on H-diffusion and effects of H on yield strength and strain hardening in the DSS-CP model. Crack tip simulations are performed for face-centered cubic (FCC) metals under monotonic loading conditions with and without H. Enhanced maximum plastic deformation in the vicinity of the crack tip (i.e., localization or intensification of plastic strain) and crack tip opening displacement (CTOD) are predicted in the presence of H, consistent with experimental observations. In spite of increased initial strength due to H, subsequent reduction of the rate of strain hardening in the presence of H is shown to enhance localization of crack tip plasticity. Furthermore, this modeling framework predicts that higher H-diffusivity (leading to a larger H-affected zone) will enhance the crack tip plasticity, making use of the two-way coupling algorithm implemented in this work. On the other hand, we find that the H-sensitivity of crack tip strain localization response, based only on modification of model parameters, is too weak to explain typical experimental observations. This points to the need to develop more advanced DSS-CP constitutive relations that consider highly complex dislocation interactions with point defects.

Experimental and theoretical analysis of strain hardening behavior of friction-stir-welded Al-4.2Zn-3.1Mg aluminium alloy

The strain hardening behavior of friction stir welded AA7039 (Al-4.2Zn-3.1 Mg) alloy was explored in terms of dislocation density, crystallite size, hardening capacity, hardening exponent, and strain hardening rate. Strain hardening behavior was analyzed for base metal and friction stir welded specimens. Four different models were used to calculate the dislocation density viz., Scherrer model, uniform strain model (USM), uniform stress distribution model (USDM), and uniform strain energy distribution model (USDEM). Maximum dislocation density was obtained in base metal for all models (maximum for USM, 3.9 × 10 15 m − 2 ) with a minimum crystallite size of 15.81 nm. The low dislocation density in FSW samples is due to the higher heat input in the FSW process. Strain hardening stages were shown by Kocks–Mecking type plots. Lower strain hardening capacity was obtained for BM, i.e. 0.427 as compared to FSW samples, due to the higher dislocation density of BM. It was determined that there is an inverse relationship between the strain hardening capacity and the strain hardening exponent. Strain hardening rate was higher for BM than FSW specimens, and similar results were obtained from mathematical relations that indicate the adequacy of the results. The ultimate tensile strength was 2.5% higher than the BM at a rotational speed of 1325 rpm, 35 mm/min of welding speed and 1.65° tilt angle. The maximum strength friction stir-welded specimen showed a more ductile fracture surface than the BM because the grain boundary phase disappeared or broke up in the thermomechanical affected zone.

Effect and mechanism of phase change lightweight aggregate on temperature control and crack resistance in high-strength mass concrete

Under temperature stress, high-strength mass concrete is brittle due to its high total heat of hydration. In this study, a control mixture of C55 high-strength mass concrete meeting the performance requirements was prepared. The effects of different functional aggregates on the thermal properties, such as setting and hardening rate, peak temperature, and temperature rise and fall rate in the early stage of cement hydration, as well as the effects of functional lightweight aggregates on the workability, mechanical properties, and deformation properties of concrete, were investigated in this paper using two types of aggregates with thermal insulation and phase change energy storage functions. The study shows that, in addition to having some early strength impacts and temperature management capabilities, thermal storage coarse lightweight aggregate (PCCLA) and thermal insulation fine lightweight aggregate (TIFLA) can both increase the workability and crack resistance of concrete. When it comes to temperature regulation, thermal storage coarse aggregate outperforms thermal insulation functional aggregate significantly. Thermal insulation fine aggregate can improve the rate of setting and hardening, while coarse thermal storage aggregate may work as in-situ self-heating curing in the early stages due to its phase change energy storage. This quickens the rate of setting and hardening and increases the resistivity of the set period, which in turn increases the rate of drying shrinkage.

Deformation-induced multi-step TRIP/TWIP(fcc, hcp) behaviors in metastable Fe55Mn25Co10Cr10 high-entropy alloy at ambient and cryogenic temperatures

The mechanical performance and multi-step TRIP/TWIP behaviors in a metastable Fe55Mn25Co10Cr10 were systematically investigated at room temperature (RT) and 77 K. The deformation behaviors and mechanically induced ε-martensite, fcc twin and hcp twin bands in Fe55Mn25Co10Cr10 were studied based on the thermodynamics and nanostructural analyses. Deformation-induced nanostructural evolutions were found to vary greatly as the temperture decreases from RT to 77K. Deformation induced ε-martensite and fcc deformation twins (fcc) carrying more strain developed at RT. On the other hand, bi-directional ε-martensite bands were favored to accommodate the increased strain at 77K instead of fcc deformation twins because of the increased stability of hcp phase. Complicated hcp ε-martensite twins developed adjacent to the crossed region of two ε-martensitec bands at 77K. The tensile stress developed perpendicular to the original hcp ε-martensite band because of its intrinsic smaller c/a ratio (1.615–1.620) than the ideal ratio (1.633) in hcp ε-martensite, inducing {10Ī2} twins with (0001) basal plane approximately perpendicular (86.0°) to that of the original ε-martensite band. Hcp twinning in the ε-martensite band at 77K is suggested to be driven by the structural and electronic relaxations associated with the c/a distortion of the hcp ε-martensite bands. Excellent strength-ductility combinations were also obtained at RT (861MPa/73.2 %) and 77 K (1.42GPa/61.8 %) because of the enhanced hardening rates due to TRIP/TWIP(fcc) at RT and/or TRIP/TWIP(hcp) at 77 K in this study.

Effectiveness of thermoviscoplastic material models in predicting the thermomechanical behavior of rolled homogenous armor steel

This paper studies the dynamic and elevated temperature behavior of rolled homogeneous armor steel, which is crucial for applications such as blast, impact, crash, and ballistic resistance. The deformation behavior is analyzed through tensile tests. The tensile tests of RHA steel are conducted at quasi-static 10−4s−1≤εṗ≤10−1s−1 to dynamic 10−1s−1≤εṗ≤36.439s−1 strain-rates and at high temperatures ranging from room temperature (RT,27°C) to 500°C. Phenomenological Johnson-Cook (JC) models and semi-physical Rusinek-Klepaczko (RK) material models are employed in finite element (FE) simulations of tensile tests. The JC failure model, with a small modification, is used along with the material models to simulate the loss of load-bearing capacity of the component. All necessary material parameters are extracted from the test data. The material models and the damage model are integrated into ABAQUS CAE FE software through a user-defined material subroutine (UMAT). The simulated outcomes acquired from diverse material models are validated with relevant experimental findings, and the effectiveness of each material model is evaluated through qualitative and quantitative assessments. Observations reveal limitations in the existing models to accurately replicate material behavior under tension, particularly in the dynamic strain rate, and elevated temperature range. Consequently, a modified version of the JC model (MJC) is explored to better account for the impact of temperature on strain hardening and strain rate hardening. The proposed modification in the material model significantly enhances the predictive accuracy of FE simulations of tensile deformation across the entire spectrum of studied strain rates and temperatures compared to the original JC and RK models.

Low-cycle fatigue property of Inconel 617 alloy at 700 °C: Mechanisms of cyclic deformation, precipitation and cracking behavior

Effects of strain amplitude ranging from 0.2 % to 1.0 % on the low-cycle fatigue properties of Inconel 617 alloy at 700 °C were studied. Much attention was paid to reveal the physical mechanisms of the cyclic deformation, γ′ phase precipitate and cracking behaviors based on the methods of flow stress partitioning and microstructure evolution characterization. Results showed that the material demonstrated a two-slope cyclic hardening behavior with different hardening rates. The strain amplitude of 0.4 % was found to be the threshold value in terms of both the precipitation behavior and dynamic strain aging (DSA). The diameter of γ′ phase precipitate was decreased from 28 nm to 12 nm when the strain amplitude increased from 0.2 % to 0.4 % and became disappeared at the larger strain amplitudes. The DSA activity characterized by the DSA strain range and maximum stress drop got significantly intensified with the increasing strain amplitude, about five times larger at 1.0 % than that at 0.4 %. Further, the mutual interaction between precipitation behavior and DSA was thoroughly analyzed. The number of crack initiation sites was increased with the increasing strain amplitude, and the dominant damage mechanism responsible for the crack propagation behavior was elucidated.

Surface characteristics evolution under the controllable impact cycles of an interrupted discrete waterjet

The interrupted discrete waterjet (DWJ) offers advantages in surface treatment due to its exceptional load characteristics and high controllability. However, the surface characteristics of materials under controllable DWJ parameters have not been fully elucidated. This study delves into the surface morphology, roughness parameters, microhardness, residual stress, and distribution of crystal characteristics in both the depth and radial directions under 2400–36,000 impact cycles of DWJ through peening experiments and material characterization. The findings indicate that the surface morphology is influenced by high-frequency roughness components with limited impact cycles, while low-frequency waviness components gradually take precedence with more impact cycles. As impact cycles increase, certain roughness parameters such as Sq, S5p, S5v, and S10z initially rise until 21,600 cycles, after which they fluctuate within a specific range. Furthermore, DWJ peening leads to an increase in maximum hardness with impact cycles, with the depth of peak value transitioning from the surface layer to the subsurface. The residual compressive stress initially increases and then decreases with impact cycles. Under the experimental conditions, the maximum hardness reaches 268.1HV0.3, a 49.95 % increase from the original state, and the peak residual compressive stress reaches 297.7 MPa with a 310.2 % increase. As impact cycles continue to rise, a balance between material removal and surface hardening is achieved, resulting in the stabilization of maximum hardness and compressive stress as erosion strength grows. Following DWJ peening, there is an increase in the percentage of low-angle grain boundaries, along with enhanced grain refinement, increased geometrically necessary dislocation (GND) density, and the emergence of new phases. As the impact cycles increase, the effects of DWJ peening on crystal grains become more pronounced. Additionally, it has been observed that the lateral jetting influence extends further horizontally than in the depth direction of the incoming jet. It is considered that cyclic hardening and high strain rate effects (1.2 × 104) play significant roles in material hardening and failure.

The Impacts of Red Mud on Road Performance and Aging Resistance of Asphalt Mixture

Abstract Red mud as an asphalt modifier has been investigated in some studies, but few studies are further concerned about the effect of red mud on performance of asphalt mixture. In this study, the impact of red mud on road performance and aging endurance of asphalt mixtures was evaluated. Primarily, the optimal asphalt content was determined based on mechanical strength and volumetric properties. Then, the performance evaluations, including Marshall stability, low-temperature splitting, freeze-thaw splitting, indirect tensile fatigue, and thermal-oxidative aging, were executed. The outcomes demonstrate that in comparison with the base asphalt mixture, red mud asphalt mixture exhibits worse high-temperature stability and moisture resistance, and superior temperature shrinkage cracking resistance and fatigue performance. After the surface of red mud has been modified, the adhesion between asphalt mastics and aggregate is strengthened, and the distribution of red mud in asphalt mixture becomes more uniform, which intensifies the rigidity and road performance (except for low-temperature cracking resistance) of asphalt mixture. Furthermore, since red mud absorbs a portion of asphalt content, it can hinder the physical hardening and oxidation rate of asphalt mixture. The anti-aging ability of asphalt mixture is further reinforced by the incorporation of organic red mud.

Large-deformation elastic hardening of a structured graphene kirigami nano-spring with re-entrant honeycombs

The programmable nature of graphene kirigami enables it highly applicable in the development of stretchable nanodevices. In this study, we employed molecular dynamics simulations to investigate the large deformation behavior of a structured graphene kirigami nano-spring (GKNS) with re-entrant honeycombs. During a uniaxial test, the elastic deformation can be categorized into two stages based on the applied load level. Stage I exhibits a maximal strain (εYstart) exceeding 50 % under low loading conditions, while stage II demonstrates a strain (εYend) surpassing 70 % at a high hardening rate. Both εYstart and εYend are greater than those observed in GKNS with transversal periodic boundary conditions. Furthermore, increasing the length or decreasing the width of bars within GKNS with cells of identical shape leads to an increase in both εYstart and εYend values. Ambient temperature has minimal impact on εYstart and εYend however, higher temperatures result in lower hardening rates due to weakened carbon–carbon bonds within GKNS structures. Additionally, variations in cell/void arrangement within structured GKNS affect both εYstart and εYend values, whereas widening the GKNS structure leads to reduced hardening rates. The findings presented herein provide valuable insights for designing nanodevices that necessitate significant deformations.

Virtual Sensor for Estimating the Strain-Hardening Rate of Austenitic Stainless Steels Using a Machine Learning Approach

This study introduces a Multiple Linear Regression (MLR) model that functions as a virtual sensor for estimating the strain-hardening rate of austenitic stainless steels, represented by the Hardening Rate of Hot rolled and annealed Stainless steel sheet (HRHS) parameter. The model correlates tensile strength (Rm) with cold thickness reduction and chemical composition, evidencing a robust linear relationship with an R-coefficient above 0.9800 for most samples. Key variables influencing the HRHS value include Cr, Mo, Si, Ni, and Nb, with the MLR model achieving a correlation coefficient of 0.9983. The Leave-One-Out Cross-Validation confirms the model’s generalization for test examples, consistently yielding high R-values and low mean squared errors. Additionally, a simplified HRHS version is proposed for instances where complete chemical analyses are not feasible, offering a practical alternative with minimal error increase. The research demonstrates the potential of linear regression as a virtual sensor linking cold strain hardening to chemical composition, providing a cost-effective tool for assessing strain hardening behaviour across various austenitic grades. The HRHS parameter significantly aids in the understanding and optimization of steel behaviour during cold forming, offering valuable insights for the design of new steel grades and processing conditions.

Autochthonous lactic acid bacteria in gluten-free sourdoughs produce nutritional and technological improvements in quinoa and buckwheat breads

Gluten-free (GF) baked goods available in the market are often poor in nutrients, such as protein, dietary fibre, micronutrients and bioactive compounds, and have low technological quality. Therefore, it is important to develop GF products with nutrient-rich flours, like quinoa (Q) and buckwheat (BW). Sourdough (SD) technology generates desirable nutritional and technological effects to improve GF bread. The objective of this work was to evaluate the effect of four autochthonous strains of lactic acid bacteria used as sourdough starters on the nutritional and technological quality of GF breads made with non-conventional flours. Lactiplantibacillus plantarum T3, Limosilactobacillus fermentum T4, Pediococcus acidilactici 26 and Pediococcus pentosaceus 82 were isolated in Q and BW flours. Nutritional and technological properties of flours, SD and breads were evaluated. In Q SD and breads, there was a 90% increase of polyphenols, 36% of ferulic acid and 131% in antioxidant capacity with respect to control compared to BW. In Q BW SD and breads, there were higher amounts of free amino acids and lower phytic acid content compared to SD controls. The incorporation of SD increased the specific volume, reduced the initial firmness of the crumb and the rate of hardening, produced a darkening of the crust; the crumbs of the breads were spongier, with a greater fraction of air and quantity of alveoli. Breads with Q SD presented better technological quality than breads with BW SD. Lim. fermentum T4 and P. pentosaceus 82 produced SD and breads with better technological quality.

Running-in Period During Sliding Wear of Austenitic Steels

The running-in period during dry sliding wear might determine the evolution to steady-state wear behaviour. To this end, the running-in period during sliding wear of austenitic stainless steel, AISI 316L stainless steel, and Hadfield steel were studied through the testing pin (flat-ended)-on-disk configuration. The effects of the normal load, sliding speed, and alloy type were assessed, and the specific wear rate and strain hardening characteristics were determined. The wear rate was correlated with wear mechanism, friction coefficient, hardening, and roughness to characterize the changes occurring during the running-in period. These changes could influence the responses of these materials to wear during the steady-state period. The stabilization of the specific wear rate and hardness was noted to align with the end of the running-in period.Graphical

Dislocation loops and hardening of silicon ion irradiated W and W-3Re alloy at 400 °C and 550 °C

Si2+ irradiation with the dose of 33.8–100 dpa was used to study the dislocation loops and hardening in W and W-3Re alloy at 400 °C and 550 °C using transmission electron microscope (TEM) and nano-indentation tests. When irradiated at 550 °C, larger dislocation loops and numerous dislocation network formed with the dose increased from 33.8 dpa to 100 dpa for both materials. The density of dislocation loops might be saturated between 33.8–100 dpa for W and saturated at ∼53.8 dpa for W-3Re alloy. Compared with pure W, the mean size of dislocation loop was smaller in W-3Re alloy when irradiated to 33.8 dpa for both temperatures. Meanwhile, it was found that the irradiation hardening rate of W was larger than that of W-3Re alloy under the same irradiation conditions, indicating that the addition of 3 wt.% Re element could indeed decrease the irradiation hardening rate of W. The hardening rate might be saturated after ∼53.8 dpa irradiation for W-3Re alloy, which might be related to the saturated loop densities.

High Strain Rate Hardening of Metallic Cellular Metamaterials

AbstractStrain rate hardening caused by the changed deformation mode is a fascinating phenomenon in cellular metamaterials where the material’s stiffness and energy absorption capabilities increase as the strain rate increases. This unique behaviour is attributed to a combination of micro-inertia effects, base material’s strain rate hardening and inertia effects. At high strain rates, the metamaterial’s inertia influences its deformation response, which changes to shock mode. This work briefly presents the geometry and fabrication of different metallic metamaterials. Then, it evaluates their mechanical response at different strain rates, ranging from quasi-static to intermediate dynamic and shock, determined by experimental and computational investigation. The three deformation modes can be separated into two critical loading velocities, unique for each metamaterial, which are also presented and compared in this work for various metamaterials. The investigations show that the deformation mode change in metallic metamaterials depends on their porosity. The critical velocities separating the deformation modes decrease with increasing porosity, i.e., decreased density of the metamaterial results in reduced critical loading velocities. The shock deformation mode in cellular metamaterials is thus attainable at much lower loading velocities than in homogeneous (nonporous) materials.

Effects of strain rate on dynamic deformation behavior and microstructure evolution of Fe-Mn-Cr-N austenitic stainless steel

In this work, the effects of strain rate on dynamic hot deformation behavior and microstructure evolution of Fe-Mn-Cr-N steel were studied under quasi-static and dynamic loading conditions using a Gleeble-3800 thermal-mechanical simulator and Split-Hopkinson pressure bar (SHPB) system. The stress-strain curves show that the yield and flow stresses are both significantly dependent on strain rate, also the strain rate sensitivity increases from 0.050 at a low strain rate (0.001–0.1 s−1) to 1.507 at a high strain rate (4500–5500 s−1). At a low strain rate (0.001–0.1 s−1), the microstructure evolution is dominated by dislocation slip and dynamic recovery (DRV). When strain rates are increased to 2500–3500 s−1, dislocation slip and mechanical twining together coordinate plastic deformation. When the strain rate was 4500–5500 s−1, the dynamic recrystallization (DRX) based on twinning formed in the deformation bands, and its fraction increased with the increasing strain rate due to adiabatic temperature rise (ΔT). The high micro-hardness value was obtained mainly because of the strain rate hardening (2500–3500 s−1), but the decrease at the strain rate of 4500–5500 s−1 is primarily attributed to recrystallization softening after deformation. The modified Johnson-Cook (J-C) model considering adiabatic temperature rise agrees with experimental results. It can be used to predict the plastic deformation behavior of Fe-Mn-Cr-N steel under high strain rate loading.

Effective characterization for the dynamic indentation and plastic parameters acquisition of metals

As a fundamental test technique for the mechanical properties of materials, indentation shows broad application prospects. Compared to the well-studied quasi-static indentation, relatively few and incomplete studies on the dynamic indentation test and characterization have been performed. In this work, a dynamic indentation test technique based on the split Hopkinson pressure bar system is developed, and the corresponding “three-wave method” is proposed to effectively acquire the time-resolved dynamic indentation load and depth. Systematic indentation tests under different loading rates and indenter cone angles are conducted on typical metal materials such as Cu, α-Fe, and α-Ti. Based on the dynamic indentation contact stiffness analysis and indentation theory the indentation hardness under different loading conditions are determined. A procedure for plastic parameter inversion of metal materials, including strain hardening and strain rate strengthening terms based on dynamic indentation, is provided. The effectiveness is verified by the traditional compression test results, providing a new method for characterizing the dynamic mechanical properties of metal materials.

Effect of H+ pre-irradiation on irradiation hardening and microstructure evolution of FeCoCrNiAl0.3 alloy with He2+ irradiation

FeCoCrNiAlx-based high entropy alloys (HEAs) exhibit excellent high-temperature strength, good radiation tolerance and corrosion resistance. However, studies on light-ion irradiation damage are relatively lacking. The influence of hydrogen and helium effect on HEAs deserves further study, given their complex interaction during different sequence irradiation at different irradiation conditions. In the present study, FeCoCrNiAl0.3 HEA was irradiated under two different modes at 723 K, one is single He2+ and the other is pre-iradiation of H+ followed by He2+. Microstructure evolution and irradiation hardening were studied by transmission electron microscopy and nanoindentation technique. There is no distinct difference of the irradiation hardening rate between sequential H+-He2+ and single He2+ irradiated samples. A large number of irradiation-induced nano-defects were observed after pre-irradiation of H+, which act as sinks to trap the subsequent helium atoms in the H+-He2+ irradiated sample. Fine helium bubbles and dislocation loops with high density dispersed in both irradiated samples. Compared with single He2+ irradiation, pre-irradiation of H+ resulted in a decrease in bubble density and an increase in loop density in H+-He2+ irradiated sample, but had little effect on their size. It is speculated that the critical concentration of bubble formation originating from H-He-V clusters is supposed to be higher than that of HeV clusters at elevated temperatures in FeCoCrNiAl0.3 alloy, leading to the aggregation and redistribution of He atoms at the H-He-V clusters rather than the formation of stable bubble nucleus after succeeding He2+ irradiation. The influence of pre-irradiation of H+ on FeCoCrNiAl0.3 alloy on the formation of helium bubbles and the relationship between bubble, loop and irradiation hardening are discussed.

Unravelling the existence of asymmetric bubbles in viscoelastic fluids

We study the motion and deformation of a single bubble rising inside a cylindrical container filled with a viscoelastic material. We solve numerically the mass and momentum balances along with the constitutive equation for the viscoelastic stresses, without assuming axial symmetry to allow the growth of three-dimensional disturbances. Hence, we may predict the emergence of the notorious knife-edge shape of the bubble, which is a result of a purely elastic instability triggered in the locality of the trailing edge. Our results compare well with existing experiments. We visualize, for the first time to the best of our knowledge, the flow kinematics and dynamics that arise downstream of the bubble. We propose two quantities, one kinematical and one geometrical, for the determination of the onset of the instability. We demonstrate that extension-rate thinning in the constitutive law is necessary for the emergence of the instability. Moreover, our results indicate that increasing (a) the deformability of the bubble and (b) the initial extension rate hardening of the viscoelastic material, prior to thinning, triggers the instability earlier. These novel findings help us formulate and propose a mechanism that controls the onset of the instability and explain why the knife-edge shape is not encountered as frequently.