Analysis Of Stress-strain Curves Research Articles

Characterisation of Mechanical Properties, Damage Progression and Fracture Modes of Recycled Aluminium Alloys AA6061 Reinforced Alumina Oxide

Characterisation of the deformation behaviour of recycled aluminium alloy reinforced alumina oxide in terms of mechanical properties, damage progression and fracture mode is investigated via the Uniaxial Tensile Test in this manuscript. The tests are carried out at room temperature at strain rates of 6X10-3s-1, 6X10-2s-1, and 6X10-1s-1. Stress-strain curves and microstructural analysis are used to examine tensile behaviour, damage progression, and fracture mechanism. The primary form of aluminium alloy AA6061 was compared to the stress-strain curve analysis. It has been observed that when the strain rate increases, the flow stress increases. Field-Emission Scanning Electron Microscopy (FESEM) is used to analyse the damage progression, including fracture mode and continued with a close review on voids quantity and size using Image J Tooling Software. The results show that increasing strain rate increases the number of voids. In general, the recycled AA6061 reinforced Alumina Oxide exhibits strain-rate dependency behaviour. In terms of the fracture mode, an apparent necking develops around the fracture area in primary specimen. Necking, on the other hand, is less apparent in the recycled specimen. Due to alumina characteristics (ceramic), which promote fracture when necking begins to occur. The ductility of the recycled specimen is degraded after going through the recycling operations. With an increase in strain rate, the flow stress and damage evolution become more pronounced. This work contributes valuable information that can be established in various engineering applications significantly to improve the existing recycling process.

Machine learning-based constitutive models for cement-grouted coal specimens under shearing

Cement-based grouting has been widely used in mining engineering; its constitutive law has not been comprehensively studied. In this study, a novel constitutive law of cement-grouted coal specimens (CGCS) was developed using hybrid machine learning (ML) algorithms. Shear tests were performed on CGCS for the analysis of stress-strain curves and the preparation of the dataset. To maintain the interpretation of the trained ML models, regression tree (RT) was used as the main technique. The effect of maximum RT depth (Max_depth) on its performance was studied, and the hyperparameters of RT were tuned using the genetic algorithm (GA). The RT performance was also compared with ensemble learning techniques. The optimum correlation coefficient on the training set was determined as 0.835, 0.946, 0.981, and 0.985 for RT models with Max_depth = 3, 5, 7, and 9, respectively. The overall correlation coefficient was over 0.9 when the Max_depth ≥ 5, indicating that the constitutive law of CGCS can be well described. However, the failure type of CGCS could not be captured using the trained RT models. Random forest was found to be the optimum algorithm for the constitutive modeling of CGCS, while RT with the Max_depth = 3 performed the worst.

Analysis of Stress-Strain Curves to Predict Dynamic Recrystallization During Hot Deformation of M300 Grade Maraging Steel

18Ni-2000 MPa maraging steel (M300 grade) is extensively used for performance critical aerospace applications where strength-toughness balance is the basis for material selection. The key to achieve the desired mechanical properties in the final product is to process the material under conditions that will impart an equiaxed, fine grained microstructure. This in turn is controlled by the selected deformation processing conditions such as strain, strain rate and temperature. Therefore, hot deformation behavior of the material is usually studied under wide temperature, and strain rate ranges to map the microstructural evolution with specimens deformed to a given strain. However, the effect of strain on the microstructural evolution, its effect on the initiation and termination of dynamic recrystallization are essential to obtain balanced mechanical properties. In this study, analysis of stress-strain curves was conducted with an intention to obtain fine prior austenite grain (PAG) size through dynamic recrystallization (DRX), and the same was verified experimentally from the microstructures evolved through hot isothermal compression tests on cylindrical specimens subjected to different strains. A single peak type DRX curve was identified for analysis from the high temperature stress-strain curves. The theoretically determined critical strain was used to experimentally verify the initiation of DRX (DRXI) and transition from DRX dominant region to grain growth dominant region (DRXT). Hot isothermal compression tests have been conducted at T=1100°C and e = 0.1s−1 and obtained PAG size of 7.68 µm in the specimen deformed to theoretically determined optimum strain of 0.6, thereby validating the used models.

Materials Characterisation and Fracture Mechanism of Recycled Aluminium Alloy AA6061 Reinforced with Alumina Oxide

This manuscript characterises the tensile behaviour and fracture mechanism of recycled aluminium alloy AA6061 reinforced with alumina oxide undergoing finite strain deformation of a uniaxial tensile test. The tests are conducted at three different strain rates of 6×10−3s−1, 6×10−2s−1, and 6×10−1s−1, at room temperature. The specimens are primarily examined in terms of stress-strain curves and microstructural analysis. The stress strain curve analysis is compared with the primary aluminium alloy AA6061. In general, the flow stress of both materials increases with increasing strain rate. The microstructures are investigated using Field-Emission Scanning Electron Microscopy (FESEM). The analysis on the deformed specimens showed that the deformation is driven by a ductile fracture mechanism via voids nucleation, and sensitive to the strain rate changes. The results of this work provide valuable information to improve the existing recycling process before an appropriate engineering application can be established in relevant areas.

The Domain of Portevin-Le Chatelier Effect in Al-3Mg Alloy

An Al-3Mg (wt. %) alloy was studied after equal channel angular pressing and subsequent cold rolling. The mechanical behavior of the alloy in the temperature range from 223 K to 373 K (from –50°C to 125°C) at strain rates 2.1×10–1 – 5.2×10–5 s–1 was investigated. The analysis of stress-strain curves was performed to determine the conditions of manifestation of the Portevin – Le Chatelier (PLC) effect in investigated alloy. The deformation curve at a temperature of 298 K (25°C) and a strain rate of 1×10–3 s–1 is characterized by instability of plastic flow in contrast to the deformation curves obtained under other studied strain rate/temperature conditions. Stress oscillations at the necking stage were observed at high temperatures (>323 K (50°C)) and lower strain rates (1×10–4 s–1 and 5.2×10–5 s–1) forming the left border of the PLC effect domain. In general, deformation curves are characterized by the absence of stress serrations during the uniform elongation.

Segregated Network Polymer Composites with High Electrical Conductivity and Well Mechanical Properties based on PVC, P(VDF-TFE), UHMWPE, and rGO.

The formation of a segregated network structure (wittingly uneven distribution of a filler) is one of the most promising strategies for the fabrication of electrically conductive polymer composites at present. However, the simultaneous achievement of high values of electrical conductivity with the retention of well mechanical properties within this approach remains a great challenge. Here, by means of X-ray photoelectron spectra (XPS), near-edge X-ray absorption fine structure (NEXAFS) spectra, scanning electron microscopy (SEM), dielectric spectroscopy, and compression engineering stress–strain curve analysis, we have studied the effect of a segregated network structure on the electrical conductivity and mechanical properties of a set of polymer composites. The composites were prepared by applying graphene oxide (GO) with ultralarge basal plane size (up to 150 μm) onto the surface of polymer powder particles, namely, poly(vinyl chloride) (PVC), poly(vinylidene fluoride-co-tetrafluoroethylene) (P(VDF-TFE)), and ultrahigh-molecular-weight poly(ethylene) (UHMWPE) with the subsequent GO reduction and composite hot pressing. A strong dependence of the segregated network polymer composites’ physical properties on the polymer matrix was demonstrated. Particularly, 12 orders of magnitude rise of the polymers’ electrical conductivity up to 0.7 S/m was found upon the incorporation of the reduced GO (rGO). A 17% increase in the P(VDF-TFE) elastic modulus filled by 1 wt % of rGO was observed. Fracture strength of PVC/rGO at 0.5 wt % content of the filler was demonstrated to decrease by fourfold. At the same time, the change in strength was not significant for P(VDF-TFE) and UHMWPE composites in comparison with pure polymers. Our results show a promise to accelerate the development of new composites for energy applications, such as metal-free supercapacitor plates and current collectors of lithium-ion batteries, bipolar plates of proton-exchange membrane fuel cells, antistatic elements of various electronic devices, etc.

The Application of Dynamic Shock Mechanics Test in Engineering Blasting

With the continuous advancement of China's infrastructure construction to the west, according to the geographic situation in the southwest region, such as mountainous areas and complex terrain, the road construction process is inevitably accompanied by earth and rock blasting. To improve the quality and safety of the project, this paper addresses the problems of land and rock blasting faced in the construction of mountain road projects, taking the research of rock dynamic mechanics test as the starting point, and using a combination of theoretical analysis and experimental research methods. The specific research content includes the following parts: dynamic impact compression test (SHPB), dynamic splitting tensile test, and stress-strain curve analysis of the test results, which provides the theoretical basis and numerical parameters for the numerical simulation of future engineering blasting.

The Effects of Hot-Pack Coating Materials on the Pack Rolling Process and Microstructural Characteristics during Ti-46Al-8Nb Sheet Fabrication.

The effects of the package materials on the hot workability and stress-strain characteristics of high-Nb TiAl alloy with a nominal composition of Ti-46Al-8Nb (in at.%) were systematically studied via “sandwich structure” hot compression. TiAl sheet fabrication was conducted by hot pack rolling, and the microstructural characteristics and deformation mechanisms were investigated. Based on the analysis of compressed samples and stress-strain curves, the stainless steel/TiAl structure showed better deformation compatibility with homogeneous deformation and decreasing resistance. However, severe interfacial reactions were inevitable. Meanwhile, for the titanium alloy/TiAl structure, few interfacial reactions happened, but wavy deformation and high resistance complicated the compression process. Finally, a package structure with an outer stainless steel isolation layer and inner titanium alloy was determined for the pack rolling process. A TiAl sheet with no crack defects was obtained with 80% reduction. The pack-rolled TiAl sheet took on alternate microstructure of the grain-boundary Al-enriched ribbons and elongated lamellar colonies ribbons. The grain-boundary recrystallized α2 phase, lumpy γ phase, and massive α2/γ lamellae could be observed, which led to the scatter microstructure. The microstructural characteristics mainly resulted from the solute segregations of as-cast Ti-46Al-8Nb alloys, which triggered the local flow softening and deformation incompatibility during hot pack rolling.

Interface and crack propagation of cement-based composites with sulfonated asphalt and plasma-treated rock asphalt

Sulfonated asphalt and natural rock asphalt were used to toughen oil well cement paste (OWCP). The difference of micro-morphology, surface functional groups and heat resistance of two kinds of asphalt was studied. Rock asphalt was treated by low-temperature plasma, and the mechanism of improving hydrophilicity was further analyzed and explained. The toughness of cement-based composites with asphalt was analyzed from triaxial stress-strain curves. Interface characteristics between sulfonated asphalt or rock asphalt and cement paste were gained. The crack propagation in cement paste was discussed systematically by using SEM microscopic observation. The results show that the thermal stability of the rock asphalt is higher than that of the sulfonated asphalt. XPS analysis results show that oxygen-containing functional groups, such as C–OH, CO, and C–O appear on the surface of rock asphalt treated by oxygen plasma. The surface free energy of plasma-treated rock asphalt was increased from 27.05 mJ·m−2 to 53.29 mJ·m−2, which indicates the improvement of hydrophilicity of the surface of rock asphalt. Triaxial stress-strain curve analysis results show that 3% content of sulfonated asphalt and treated rock asphalt can improve the toughness of oil well cement paste. A good interfacial transition zone with few micropores is found between the cement hydration products and sulfonated asphalt or treated rock asphalt by using SEM microscopic observation. SEM observation of crack propagation behavior in cement-based composites with asphalt indicated that sulfonated asphalt and plasma-treated rock asphalt improved the toughness of oil well cement by deflecting, bending and bridging the cracks.

Effect of Temperature of Severe Plastic Deformation on Mechanical Properties of High Entropy Alloy CoCrFeNiMn

High entropy alloy (HEA) CoCrFeNiMn was produced by arc melting of the components in high-purity argon atmosphere with consequent multiple homogenization annealing. The disc-shaped samples with diameter 10 mm and thickness of ~ 1 mm were produced from the ingots obtained. These samples were subjected to severe plastic deformation by high pressure torsion (HPT) in Bridgman anvil at a hydrostatic pressure of 6 GPa and at temperature 77 K. Plungers have been rotated for 5 times at a speed of 0.2 rot/min that allows to produce uniform nanocrystalline structural state with average grain size of less than 100 nm. Mechanical tests have been provided under conditions of uniaxial compression of rectangular samples with size 1.3×0.6×0.6 mm3. The samples were cut from the discs after HPT at a distance of 3 mm form disc centre. The analysis of stress-strain curves have been made in the temperature range of 300-4.2 K for the obtained nanostructured state. It was found that yield stress value monotonically increasing from 1.44 GPa to 2.48 GPa while the temperature decrease from 300 K to 4.2 K, which is typical for thermally activated character of plastic deformation. Anomalous decrease in yield strength values in comparison with the same values for nanostructured HEA after HPT at 300 K was established in all the temperature range (300-4.2 K) for the structural state after HPT at 77 K. The conducted analysis have been shown that the observed anomalous behaviour of yield strength during active deformation is conditioned by peculiarities of microsturcture appearing after cryodeformation by HPT at 77 K, in particular by formation of martensite phase with hcp lattice and connected with this decrease in dislocation density. It was shown that peculiarities of microstructure after HPT at 77 K effect considerably not only on strength of the alloy in local areas, i.e. its microhardness value, but also on the acting stresses responsible for the plastic deformation process under conditions of active deformation of nanocrystalline HEA CoCrFeMnNi.

The effects of water content and external incident energy on coal dynamic behaviour

Water content and incident energy are important factors in influencing the mechanical properties and stress field of coal seam, which are closely related to the dynamic disaster intensity. In this study, after doing some static compression tests in different water content, comprehension dynamic compressive tests on pre-stressed coal specimens under different water contents and incident energy levels were conducted using a modified Split Hopkinson Pressure Bar test system. The effects of water contents and incident energy levels on dynamic behavior of the studied coal were evaluated by analysis of dynamic stress-strain curves, inspection of the occurred damage and in-situ test on the mine site. For detail analysis, some indices including peak strength, elastic modulus, dissipation energy, impact energy index, and failure characteristics were employed. The results indicated that dynamic strength and elastic modulus are negatively correlated with the water content and positively correlated with the incident energy. Two types of the dynamic failure behavior including incident energy-dominated failure and incident energy-induced failure were analyzed based on dissipation energy and stress-strain curve types. The coal burst occurrence possibility was proposed according to the dissipation energy density. Finally, based on laboratory test results and field engineering practices, the dynamic disaster mechanisms and their respective prevention concepts were further discussed.

Flow behavior and processing maps of high-strength low-alloy steel during hot compression

Deformation behavior of a new kind of high-strength low-alloy steel was investigated by isothermal compression tests at temperatures of 800–1200 °C and strain rates of 0.01–10 s−1. The flow behavior of the steel was characterized through the analysis of true stress-strain curves. A relevant constitutive equation was established, and proved to be suitable by comparing experimental and calculated data under various deformation conditions. Processing maps based on the dynamic material model were constructed with strains of 0.3, 0.5 and 0.7. The maps showed that the optimal hot-working temperature range of the new steel was 1050–1200 °C, while the optimal strain rate range was 0.018–0.562 s−1, which was consistent with the microstructure of the deformed specimens. Dislocations and substructures under different deformation conditions were observed and analyzed by TEM. The results indicated that there were two kinds of DRX nucleation mechanisms during hot deformation.

Mechanical evaluation of aggregate gradation to characterize load carrying capacity and rutting resistance of asphalt mixtures

The gradation and mechanical properties of aggregates impressively affect the load carrying capacity and rutting resistance of asphalt mixtures. They are able to provide some fundamental parameters that are linked to mixture performance. In order to accurately detect critical controlled sizes and obtain more fundamental parameters that can characterize load carrying capacity and rutting resistance of asphalt mixtures, this paper aims to develop a mechanics-based approach to evaluate aggregate gradation, which contains the theoretical packing analysis and discrete element simulation.An extended theoretical morphology framework is developed by checking the local and global interlock of graded aggregates in the theoretical packing analysis. The sieve size range of primary structure which has an interactive interlock can be determined in the local interlock check model. The relationship between the transferred force and the induced force is proposed in the densest and loosest packing arrangements, respectively. The morphological parameters of disruption factor and weighted average size for the global scale are also proposed in theoretical packing analysis.Then the discrete element method is used to validate the theoretical framework. The advanced gradation input algorithm procedure is established in the discrete element simulation to simulate the compression test of graded aggregates. The mechanical parameters of contact force, contact points and stress–strain curves are extracted from the simulation results.The results show that the discrete element method can validate the theoretical framework and output the performance related parameters. The contact force analysis shows that the aggregates retaining on sieve sizes of 2.36 and 4.75 mm provide more than 50% contribution to resist load, and the aggregates retaining on sieve sizes of 1.18, 0.6 and 0.3 mm provide more than 50% contribution to stabilize the structure. The coordination number analysis suggests that the gradation with more fine aggregates might lead to a greater number of voids but smaller air-void size when the asphalt content and porosity remain constant. The stress–strain curve analysis shows that the modulus and secondary strain are highly related to the rutting performance of an asphalt mixture.

Elastic modulus and stress-strain curve analysis of a tungsten mine waste alkali-activated concrete

In the paper the compressive strength, the elastic modulus and the stress-strain curve of an alkaliactivated concrete were studied. A tungsten mine waste mud (TMWM), aggregate (also from the tungsten mine), glass waste and metakaolin were used as raw materials. Sodium silicate and sodium hydroxide were used as activators. First, TMWM chemical composition was determined by scanning electron microscopyenergy dispersive spectroscopy (SEM-EDS). The maximum particle size was 18mm. Two cubes with side dimension of 15cm were prepared from the mixture. Samples were cured at 60°C for 24 hours. A concrete mixer, vibration table and an oven were used in the process. After the curing process, cubes were cut into seven prisms and one cube with the dimensions 15x7.4x7.4cm and 7.4cm respectively. After 28 days, the laboratory tests were performed. During the compressive strength tests, the displacements were also recorded which allowed drawing the stress-strain curve of the samples. The compressive strength ranged from 17.27 to 28.84MPa. The elastic modulus was calculated by four different standards: ASTM, LNEC and European standard. The elastic modulus ranged from 2.48 to 7.49GPa what showed that the material is more elastic than ordinary Portland cement concrete.

Developing coal burst propensity index method for Australian coal mines

Coal burst is the violent failure of overstressed coal, and it is often accompanied by sound, coal ejection and seismic events. It is subsequently recognized as a serious safety risk of Australia after double fatalities coal burst happened at Austar Coal Mine. Considering the increasing trend of coal burst severity and frequency with mining depth, it is an urgent task to develop the coal burst risk assessment methods for Australia underground coal mines. Coal burst propensity index method is a widely used method of burst risk evaluation of coal as it is summed up from the coal burst research and practice of many countries. This paper presents the experimental and theoretical research of coal burst propensity index method for coal burst risk assessment in Australia. The definition of four indexes including elastic strain energy index (WET), bursting energy index (KE), dynamic failure time (DT) and uniaxial compression strength (RC) is introduced in the first part. Then, the standard laboratory test process and test parameter of coal burst propensity index is presented. DT test is conducted with 0.3 mm/min displacement control loading rate while other test is with 0.5 mm/min. Besides, modified data processing and risk classification method of test are proposed. Differentiate analysis of stress-strain curve is adopted in the data processing of DT and KE index. A four level risk classification form of burst risk is recommended for Australian underground coal mines. Finally, two likely improvement methods of WET test, including volumetric strain indicator method and theoretical calculation method, are discussed.

Shale brittleness evaluation based on energy balance analysis of stress-strain curves

Brittleness is a key mechanical characteristic of rock and critical to various engineering practices particularly to design efficient fracturing stimulation of unconventional shale gas reservoirs. Variety of brittleness definitions and evaluation criteria have been proposed to describe rock failure behavior; however, their suitability and reliability have yet to be verified. In this paper, the existing brittleness indices were reviewed in detail and their applicability to reflect rock brittle characteristics were then investigated. The analysis indicated that most of expressions cannot describe brittle failure characteristics of rock efficiently and either partially or thoroughly fail to correctly address the impact of stress state on rock brittleness. Therefore, a new brittleness evaluation method was proposed based on promising energy transformation analysis of the complete stress-strain behavior of rock under compression. The new criterion is capable of dealing with the plastic energy dissipation mechanisms in the pre-peak region and the extent and rate to which rock strength degrades during macroscopic fracturing in the post-peak stage. Besides, the proposed brittleness index describes the entire transition of rock failure behavior of class I, from completely plastic to absolutely brittle states, in a continuous scale corresponding to 0 to 1. The applicability and outperformance of the proposed brittleness index were then verified against uniaxial and triaxial compression test results of some carbonaceous lithofacies from the Garau shale gas play within the Lurestan Basin, Iran. Samples under compression revealed various deformation and failure characteristics transitioned from brittle to semi-brittle behaviors with increasing confining pressure. Application of the brittleness indices to the tested rock specimens clearly demonstrated that the developed energy-based expression is able to rigorously capture the relative brittleness sequences of lithofacies and evolution trends under application of confinement, whereas further proved lack of either sufficient physical basis or appropriate formulation strategy in the existing indices. Accordingly, the new evaluation method seems to provide reliable estimates of rock brittleness in various field practices even though further verification is necessary. Finally, the important relation between microstructural carbonaceous texture and rock brittleness was discussed.

Investigation of the tensile behavior of treated flax fibre bio-composites at ambient humidity

The high specific strength and stiffness and excellent damping behavior promote the use of flax fibre composites in the construction sector. However, the limited compatibility between raw flax fibres and bio-epoxy resin often leads to composites with mechanical properties well below their theoretical capacities. Moreover, the incomplete understanding of the intrinsic non-linear mechanical behavior of flax fibre composites forces the application of larger safety factors. In this study, three chemical treatments are applied to improve the interphase properties and gain further insight in the longitudinal tensile behavior of flax fibre bio-epoxy composites. Both alkali and APS treatment result in a threefold improvement in the transverse flexural strength to ∼30MPa. Both treatments additionally shift the weakest link to the elementary fibre interphase strength. Analysis of the longitudinal tensile stress-strain curves of UD flax fibre composites results in a tri-linear shape that is preserved after treatment.

Constitutive Equations for 6061 Aluminum Alloy at Elevated Temperature

Isothermal compression tests were conducted on 6061 aluminum alloy using a Gleeble-3500 thermal simulator under constant strain rates and at deformation temperatures ranging from 623 to 773K, up to a 60% height reduction of the sample. The high temperature deformation behavior of 6061 aluminum alloy was characterized based on an analysis of the stress-strain curves. A set of constitutive equations for 6061 aluminum alloy were proposed by employing an Arrhenius-type equation. Material constants, A, n and activation energy Q were found to be functions of strain. The equations revealed the dependence of flow stress on strain, strain rate and temperature. In order to evaluate the accuracy of the deformation constitutive equations, the mean errors of flow stress between the experimental data and predicted results were plotted. The results showed that the predicted data agreed well with the experimental stress-strain curves.

Strain-induced ferrite formation and its effect on mechanical properties of a dual phase steel produced using laboratory simulated strip casting

Thermo-mechanical processing of a strip cast dual phase (DP) steel was carried out using a Gleeble thermo-mechanical simulator. The effect of deformation temperatures in the range from 1050 to 700 °C on the microstructure evolution was investigated using optical, scanning and transmission electron microscopy along with electron backscattering diffraction (EBSD). Strain-induced ferrite (SIF) formation was observed following austenite deformation (∼0.41 reduction) in the 800–700 °C temperature range, leading to a ferrite grain refinement down to 3.1 ± 2.3 μm. A novel segmentation procedure was applied to separate selected EBSD maps into polygonal ferrite, SIF and second phase regions (bainite/martensite). Following this, the microtexture, misorientation angle distribution and the deviation from the Kurdjumov–Sachs and Nishiyama–Wasserman orientation relationships of each microstructure constituent were analysed. Based on iso-work modelling analysis of tensile stress-strain curves, the SIF was found to enhance strength with a slight decrease in ductility compared to polygonal ferrite. The tensile mechanical properties after deformation at 750 °C reached the level of DP 600 produced in industry, highlighting the potential to manufacture DP steels via the strip casting technique.

Dynamic Recrystallization Kinetics of Cu-0.36Cr-0.03Zr Alloy During Hot Compression

Based on compression tests in the temperature range of 800-950 °C and the strain rate range of 0.001-20 s−1 on a Gleeble-3500 thermomechanical simulator, the dynamic recrystallization (DRX) kinetics of Cu-0.36Cr-0.03Zr alloy is developed by further analysis of true stress-strain curves and by investigating microstructures under different deformation conditions. The kinetics of dynamic recrystallization is expressed as the modified JMAK model, and effects of varying strain rate and temperature on the model parameters, including the critical strain, peak strain and materials constant k d , are investigated. The complete DRX grain size is described as a power law function of Zener-Hollomon parameter (Z), and it agrees well with the experimental data.