First principles study on the charge density and the bulk modulus of the transition metals and their carbides and nitrides

A first principles study of the electronic properties and bulk modulus （B0） of the fcc and bcc transition metals, transition metal carbides and nitrides is presented. The calculations were performed by plane-wave pseudopotential method in the framework of the density functional theory with local density approximation. The density of states and the valence charge densities of these solids are plotted. The results show that B0 does not vary monotonically when the number of the valence d electrons increases. B0 reaches a maximum and then decreases for each of the four sorts of solids. It is related to the occupation of the bonding and anti-bonding states in the solid. The value of the valence charge density at the midpoint between the two nearest metal atoms tends to be proportional to B0.

INTERFACE EFFECT ON THE EFFECTIVE BULK MODULUS OF A PARTICLE-REINFORCED COMPOSITE

Classical micromechanical methods for calculating the effective moduli of a heteroge- neous material are generalized to include the interface(surface)effect.By using Hashin's Composite Sphere Assemblage(CSA)model,a new expression of the bulk modulus for a particle-reinforced com- posite is derived.It is emphasized that the present study is within the finite-deformation framework such that the effective properties are not influenced by the interface stress itself solely,but influenced by the change of the interface stress due to changes of the shape and size of the interface.Hence some inadequacies in previous papers are pointed out.

Theoretical Model of Dynamic Bulk Modulus for Aerated Hydraulic Fluid

Existing models of bulk modulus for aerated hydraulic fluids primarily focus on the effects of pressure and air fraction,whereas the effect of temperature on bulk modulus is disregarded.Based on the lumped parameter method and the full cavitation model,combined with the improved Henry’s law and the air polytropic course equation,a theoretical model of dynamic bulk modulus for an aerated hydraulic fluid is derived.The effects of system pressure,air fraction,and temperature on bulk modulus are investigated using the controlled variable method.The results show that the dynamic bulk modulus of the aerated hydraulic fluid is inconsistent during the compression process.At the same pressure point,the dynamic bulk modulus during expansion is higher than that during compression.Under the same initial air faction and pressure changing period,a higher temperature results in a lower dynamic bulk modulus.When the pressure is lower,the dynamic bulk modulus of each temperature point is more similar to each other.By comparing the theoretical results with the actual dynamic bulk modulus of the Shell Tellus S ISO32 standard air-containing oil,the goodness-of-fit between the theoretical model and experimental value at three temperatures is 0.9726,0.9732,and 0.9675,which validates the theoretical model.In this study,a calculation model of dynamic bulk modulus that considers temperature factors is proposed.It predicts the dynamic bulk modulus of aerated hydraulic fluids at different temperatures and provides a theoretical basis for improving the analytical model of bulk modulus.

Isothermal bulk modulus and its first pressure derivative of NaCl at high pressure and high temperature

The isothermal bulk modulus and its first pressure derivative of NaCl are investigated using the classical molecular dynamics method and the quasi-harmonic Debye model. To ensure faithful molecular dynamics simulations, two types of potentials, the shell-model （SM） potential and the two-body rigid-ion Born-Mayer-Huggins-FumiqTosi （BMHFT） potential, are fully tested. Compared with the SM potential based simulation, the molecular dynamics simulation with the BMHFT potential is very successful in reproducing accurately the measured bulk modulus of NaCl. Particular attention is paid to the prediction of the isothermal bulk modulus and its first pressure derivative using the reliable potential and to the comparison of the SM and the BMHFT potentials based molecular dynamics simulations with the quasi-harmonic Debye model. The properties of NaCl in the pressure range of 0-30 GPa at temperatures up to the melting temperature of 1050 K are investigated.

Improved material descriptors for bulk modulus in intermetallic compounds via machine learning

Bulk modulus is an important mechanical property in the optimal design and selection of intermetallic compounds.In this study,bulk modulus datasets of intermetallic compounds were collected,and the features affecting the bulk modulus of intermetallics were screened via feature engineering.Three features B_(cal),dB_(avg),and TIE(corresponding to calculated bulk modulus,mean bulk modulus,and third ionization energy,respectively)were found to be the dominant factors influencing bulk modulus and can be extended to other multi-component alloys.Particularly,we predicted the bulk modulus with an accuracy of 95%using surrogate machine learning models with the selected features,and these features were also demonstrated to be effective for high-entropy alloys.Moreover,symbolic regression provided an expression for the relationship between bulk modulus and the screened features.The machine learning models provide a new approach for optimizing and predicting the bulk moduli of intermetallic compounds.

A novel relationship between elastic modulus and void ratio associated with principal stress for coral calcareous sand

Elastic moduli,e.g.shear modulus G and bulk modulus K,are important parameters of geotechnical materials,which are not only the indices for the evaluation of the deformation ability of soils but also the important basic parameters for the development of the constitutive models of geotechnical materials.In this study,a series of triaxial loading-unloading-reloading shear tests and isotropic loading-unloadingreloading tests are conducted to study several typical mechanical properties of coral calcareous sand(CCS),and the void ratio evolution during loading,unloading and reloading.The test results show that the stress-strain curves during multiple unloading processes are almost parallel,and their slopes are much greater than the deformation modulus at the initial stage of loading.The relationship between the confining pressure and the volumetric strain can be defined approximately by a hyperbolic equation under the condition of monotonic loading of confining pressure.Under the condition of confining pressure unloading,the evolution of void ratio is linear in the e-lnp0 plane,and these lines are a series of almost parallel lines if there are multiple processes of unloading.Based on the experimental results,it is found that the modified Hardin formulae for the elastic modulus estimation have a significant deviation from the tested values for CCS.Based on the experimental results,it is proposed that the elastic modulus of soils should be determined by the intersection line of two spatial surfaces in the G/K-e-p’/pa space(pa:atmosphere pressure).“Ye formulation”is further proposed for the estimation of the elastic modulus of CCS.This new estimation formulation for soil elastic modulus would provide a new method to accurately describe the mechanical behavior of granular soils.

Bulk moduli of two-dimensional Yukawa solids and liquids obtained from periodic compressions

Langevin dynamical simulations are performed to determine the bulk modulus in twodimensional(2D) dusty plasmas from uniform periodic radial compressions. The bulk modulus is calculated directly from its physical definition of the ratio of the internal pressure/stress to the volume strain. Under various conditions, the bulk moduli obtained agree with the previous theoretical derivations from completely different approaches. It is found that the bulk moduli of2D Yukawa solids and liquids are almost independent of the system temperature and the external compressional frequency.

Trends in the band-gap pressure coefficients and bulk moduli in different structures of ZnGa_2S_4,ZnGa-2Se_4 and ZnGa-2Te_4

We have performed a first-principles investigation for the family of compounds ZnGa2X4 （X = S, Se, Te）. The properties of two possible structures, defect chalcopyrite and defect famatinite are both calculated. We reveal that ZnGa2S4 and ZnGa2Se4 have direct band gaps, while ZnGa2Te4 has an indirect band gap. The local density approximation band gaps are found to be very different in two structures, while the lattice parameters and bulk moduli are similar. We extend Cohen＇s empirical formula for zinc-blende compounds to this family of compounds. The pressure coefficients are calculated and metallization pressures are discussed. We find that agi remains fairly constant when thegroup-V/element X is varied in ZnGa2X4（Ⅱ-Ⅲ2-Ⅵ4）.

Calculation of Bulk Moduli of Carbon Nitride/metal Nitride Composites

ased on Marvin L. Cohen′s empirical approach, a simple model of calculation of bulk moduli of carbon nitride/metal nitride composites is shown. The calculated bulk modulus of the crystalline carbon nitride/TiN composite coating is comparable with that of cBN and diamond. This model predicts that the modulus of the composite is between the moduli of the two components.

Prediction of a superhard material of ReN_4 with a high shear modulus

Using first-principles calculations, this paper systematically investigates the structural, elastic, and electronic properties of ReN4. The calculated positive eigenvalues of the elastic constant matrix show that the orthorhombic Pbca structure of ReN4 is elastically stable. The calculated band structure indicates that ReN4 is metallic. Compared with the synthesized superhard material WB4, it finds that ReN4 exhibits larger bulk and shear moduli as well as a smaller Poisson＇s ratio. In addition, the elastic constant c44 of ReN4 is larger than all the known 5d transition metal nitrides and borides. This combination of properties makes it an ideal candidate for a superhard material.

Rock physics model for velocity–pressure relations and its application to shale pore pressure estimation

Acoustic wave velocity has been commonly utilized to predict subsurface geopressure using empirical relations.Acoustic wave velocity is, however, affected by many factors. To estimate pore pressure accurately, we here propose to use elastic rock physics models to understand and analyze quantitatively the various contributions from these different factors affecting wave velocity. We report a closed-form relationship between the frame flexibility factor(γ) in a rock physics model and differential pressure, which presents the major control of pressure on elastic properties such as bulk modulus and compressional wave velocity. For a gas-bearing shale with abundant micro-cracks and fractures, its bulk modulus is much lower at abnormally high pore pressure(high γ values) where thin cracks and flat pores are open than that at normal hydrostatic pressure(low γ values) where pores are more rounded on average. The developed relations between bulk modulus and differential pressure have been successfully applied to the Upper Ordovician Wufeng and Lower Silurian Longmaxi formations in the Dingshan area of the Sichuan Basin to map the three-dimensional spatial distribution of pore pressure in the shale, integrating core, log and seismic data. The estimated results agree well with field measurements. Pressure coefficient is positively correlated to gas content. The relations and methods reported here could be useful for hydrocarbon exploration, production, and drilling safety in both unconventional and conventional fields.

First principle study on the elastic and thermodynamic properties of TiB_2 crystal under high temperature

This paper predicts the elastic and thermodynamic characteristics of TiB2 crystal through the method of density functional theory within the generalized gradient approximation （GGA）. The five independent elastic constants （Cij）, the bulk modulus （B0）, the dependence of bulk modulus （B0） on temperature T and pressure P and the coefficient of thermal expansion （αL） at various temperatures have been evaluated and discussed. According to calculation, the bulk modulus will increase with increasing pressure while decrease with the increasing temperature. The coefficient of thermal expansion is consistent with the famous Griineisen＇s law when the temperature is not too high. The obtained results agree well with the experimental and other theoretical results.

First-principle study on the physical properties of ultra-incompressible ReB_2

The elastic and physical characteristics of ReB2 crystal have been predicted through a method of density functional theory within the generalized gradient approximation （GGA）. Five independent elastic constants are C11=662 GPa, C12=150 GPa, C13=146 GPa, C33=1090 GPa and Caa=263 GPa. The bulk modulus （B）, shear modulus （G）, Young＇s modulus （E）, Poisson＇s ratio （γ） and the ratio of linear com- pressibility coefficient along the a- and c-axis crystal direction （Ka/Kc） are 356 GPa, 305 GPa, 711 GPa, 0.167 and 1.758, respectively. In addition, the dependence of bulk modulus （B） on temperature （T） and pressure （p） as well as the coefficient of thermal expansion （αL） at various temperatures are evaluated and discussed. The coefficient of thermal expansion is consistent with the famous Grüneisen＇s law when the temperature is less than 1500 K. Our results agree well with the other experimental results.

Compression behavior and phase transition of β-Si_3N_4 under high pressure

The compressibility and pressure-induced phase transition of β-Si3N4 were investigated by using an angle dispersive x-ray diffraction technique in a diamond anvil cell at room temperature. Rietveld refinements of the x-ray powder diffraction data verified that the hexagonal structure（with space group P63/m, Z = 2 formulas per unit cell） β-Si3N4 remained stable under high pressure up to 37 GPa. Upon increasing pressure, β-Si3 N4 transformed to δ-Si3N4 at about 41 GPa. The initial β-Si3N4 was recovered as the pressure was released to ambient pressure, implying that the observed pressureinduced phase transformation was reversible. The pressure–volume data of β-Si3N4 was fitted by the third-order Birch–Murnaghan equation of state, which yielded a bulk modulus K0= 273（2） GPa with its pressure derivative K0= 4（fixed）and K0= 278（2） GPa with K 0= 5. Furthermore, the compressibility of the unit cell axes（a and c-axes） for the β-Si3N4 demonstrated an anisotropic property with increasing pressure.

Unreacted equation of states of typical energetic materials under static compression:A review

The unreacted equation of state（EOS） of energetic materials is an important thermodynamic relationship to characterize their high pressure behaviors and has practical importance. The previous experimental and theoretical works on the equation of state of several energetic materials including nitromethane, 1,3,5-trinitrohexahydro-1,3,5-triazine（RDX）,1,3,5,7-tetranitro-1,3,5,7-tetrazacyclooctane（HMX）, hexanitrostilbene（HNS）, hexanitrohexaazaisowurtzitane（HNIW or CL-20）, pentaerythritol tetranitrate（PETN）, 2,6-diamino-3,5-dinitropyrazine-1-oxide（LLM-105）, triamino-trinitrobenzene（TATB）, 1,1-diamino-2,2-dinitroethene（DADNE or FOX-7）, and trinitrotoluene（TNT） are reviewed in this paper. The EOS determined from hydrostatic and non-hydrostatic compressions are discussed and compared. The theoretical results based on ab initio calculations are summarized and compared with the experimental data.

Pressure-induced phase transition in silicon nitride material

The equilibrium crystal structures,lattice parameters,elastic constants,and elastic moduli of the polymorphs α-,β-,and γ-Si3N4,have been calculated by first-principles method.β-Si3N4 is ductile in nature and has an ionic bonding.γSi3N4 is found to be a brittle material and has covalent chemical bonds,especially at high pressures.The phase boundary of the β→γ transition is obtained and a positive slope is found.This indicates that at higher temperatures it requires higher pressures to synthesize γ-Si3N4.On the other hand,the α→γ phase boundary can be described as P = 14.37198＋ 3.27 × 10？3T-7.83911 × 10？7T2-3.13552 × 10？10T3.The phase transition from α-to γ-Si3N4 occurs at 16.1 GPa and 1700 K.Then,the dependencies of bulk modulus,heat capacity,and thermal expansion on the pressure P are obtained in the ranges of 0 GPa-30 GPa and 0 K-2000 K.Significant features in these properties are observed at high temperatures.It turns out that the thermal expansion of γ-Si3N4 is larger than that of α-Si3N4 over wide pressure and temperature ranges.The evolutions of the heat capacity with temperature for the Si3N4 polymorphs are close to each other,which are important for possible applications of Si3N4.

Characteristics of fluid substitution in porous rocks

Analysis of the effect of changes in fluid properties of rocks on the compressional-wave velocity VP and shear-wave velocity Vs is very important for understanding the rock physical properties, especially in oilfield exploration and development. The fluid substitution process was analyzed by using ultrasonic measurement and theoretical calculations. The results showed that the effect of fluid substitution on the rock elastic modulus was mainly controlled by fluid properties, saturation, and confining pressure. For a rock with specific properties and porosity, the result of theoretical prediction for fluid substitution accorded with the experimental result under high confining pressure （higher than 60 MPa for our experimental data）, but failed to describe the trend of experimental result under low confining pressure and VP predicted by Gassmann＇s equation was higher than that measured by experiment. A higher porosity resulted in stronger sensitivity of the bulk modulus of saturated rocks to the change of fluid properties.

ANALYTICAL SOLUTIONS FOR ELASTOSTATIC PROBLEMS OF PARTICLE-AND FIBER-REINFORCED COMPOSITES WITH INHOMOGENEOUS INTERPHASES

By transforming the governing equations for displacement components into Riccati equations, analytical solutions for displacements, strains and stresses for Representive Volume Elements (RVEs) of particle_ and fiber_reinforced composites containing inhomo geneous interphases were obtained. The analytical solutions derived here are new and general for power_law variations of the elastic moduli of the inhomogeneous interphases. Given a power exponent, analytical expressions for the bulk moduli of the composites with inho mogeneous interphases can be obtained. By changing the power exponent and the coefficients of the power terms, the solutions derived here can be applied to inhomogeneous interphases with many different property profiles. The results show that the modulus variation and the thickness of the inhomogeneous interphase have great effect on the bulk moduli of the composites. The particle will exhibit a sort of “size effect”, if there is an interphase.

Determination of elastic moduli of composite medium containing bimaterial matrix and non-uniform inclusion concentrations

Reservoir porous rocks usually consist of more than two types of matrix materials, forming a randomly heterogeneous material. The determination of the bulk modulus of such a medium is critical to the elastic wave dispersion and attenuation. The elastic moduli for a simple matrix-inclusion model are theoretically analyzed. Most of the efforts assume a uniform inclusion concentration throughout the whole single-material matrix. However, the assumption is too strict in real-world rocks. A model is developed to estimate the moduli of a heterogeneous bimaterial skeleton, i.e., the host matrix and the patchy matrix. The elastic moduli, density, and permeability of the patchy matrix differ from those of the surrounding host matrix material. Both the matrices contain dispersed particle inclusions with different concentrations. By setting the elastic constant and density of the particles to be zero, a double-porosity medium is obtained. The bulk moduli for the whole system are derived with a multi-level effective modulus method based on Hashin＇s work. The proposed model improves the elastic modulus calculation of reservoir rocks, and is used to predict the kerogen content based on the wave velocity measured in laboratory. The results show pretty good consistency between the inversed total organic carbon and the measured total organic carbon for two sets of rock samples.

Topological Design of Microstructures of Materials Containing Multiple Phases of Distinct Poisson’s Ratios

A methodology for achieving the maximum bulk or shear modulus in an elastic composite composed of two isotropic phases with distinct Poisson’s ratios is proposed.A topology optimization algorithm is developed which is capable of finding microstructures with extreme properties very close to theoretical upper bounds.The effective mechanical properties of the designed composite are determined by a numerical homogenization technique.The sensitivities with respect to design variables are derived by simultaneously interpolating Young’smodulus and Poisson’s ratio using different parameters.The so-called solid isotropicmaterial with penalizationmethod is developed to establish the optimization formulation.Maximum bulk or shearmodulus is considered as the objective function,and the volume fraction of constituent phases is taken as constraints.Themethod ofmoving asymptotes is applied to update the design variables.Several 3D numerical examples are presented to demonstrate the effectiveness of the proposed structural optimization method.The effects of key parameters such as Poisson’s ratios and volume fractions of constituent phase on the final designs are investigated.A series of novel microstructures are obtained fromthe proposed approach.It is found that the optimized bulk and shearmoduli of all the studied composites are very close to the Hashin-Shtrikman-Walpole bounds.