Analytical solution for the effective elastic properties of rocks with the tilted penny-shaped cracks in the transversely isotropic background

Seismic prediction of cracks is of great significance in many disciplines,for which the rock physics model is indispensable.However,up to now,multitudinous analytical models focus primarily on the cracked rock with the isotropic background,while the explicit model for the cracked rock with the anisotropic background is rarely investigated in spite of such case being often encountered in the earth.Hence,we first studied dependences of the crack opening displacement tensors on the crack dip angle in the coordinate systems formed by symmetry planes of the crack and the background anisotropy,respectively,by forty groups of numerical experiments.Based on the conclusion from the experiments,the analytical solution was derived for the effective elastic properties of the rock with the inclined penny-shaped cracks in the transversely isotropic background.Further,we comprehensively analyzed,according to the developed model,effects of the crack dip angle,background anisotropy,filling fluid and crack density on the effective elastic properties of the cracked rock.The analysis results indicate that the dip angle and background anisotropy can significantly either enhance or weaken the anisotropy degrees of the P-and SH-wave velocities,whereas they have relatively small effects on the SV-wave velocity anisotropy.Moreover,the filling fluid can increase the stiffness coefficients related to the compressional modulus by reducing crack compliance parameters,while its effects on shear coefficients depend on the crack dip angle.The increasing crack density reduces velocities of the dry rock,and decreasing rates of the velocities are affected by the crack dip angle.By comparing with exact numerical results and experimental data,it was demonstrated that the proposed model can achieve high-precision estimations of stiffness coefficients.Moreover,the assumption of the weakly anisotropic background results in the consistency between the proposed model and Hudson's published theory for the orthorhombic rock.

First principles investigation of binary intermetallics in Mg-Al-Ca-Sn alloy:Stability,electronic structures,elastic properties and thermodynamic properties

The structural stability, electronic structures, elastic properties and thermodynamic properties of the main binary phases Mg_（17）Al_（12）, Al_2Ca, Mg_2 Sn and Mg_2 Ca in Mg-Al-Ca-Sn alloy were determined from the first-principles calculation. The calculated lattice parameters are in good agreement with the experimental and literature values. The calculated heats of formation and cohesive energies show that Al_2Ca has the strongest alloying ability and structural stability. The densities of states（DOS）, Mulliken electron occupation number, metallicity and charge density difference of these compounds are given. The elastic constants of Mg_（17）Al_（12）, Al_2Ca, Mg_2 Sn and Mg_2 Ca phases are calculated, and the bulk moduli, shear moduli, elastic moduli and Poisson ratio are derived. The calculations of thermodynamic properties show that the Gibbs free energies of Al_2Ca and Mg_2 Sn are lower than that of Mg_（17）Al_（12）, which indicates that Al_2Ca and Mg_2 Sn are more stable than Mg_（17）Al_（12） phase. Hence, the heat resistance of Mg-Al-based alloys can be improved by adding Ca and Sn additions.

Computation of elastic properties of 3D digital cores from the Longmaxi shale

The dependence of elastic moduli of shales on the mineralogy and microstructure of shales is important for the prediction of sweet spots and shale gas production. Based on 3D digital images of the microstructure of Longmaxi black shale samples using X-ray CT, we built detailed 3D digital images of cores with porosity properties and mineral contents. Next, we used finite-element （FE） methods to derive the elastic properties of the samples. The FE method can accurately model the shale mineralogy. Particular attention is paid to the derived elastic properties and their dependence on porosity and kerogen. The elastic moduli generally decrease with increasing porosity and kerogen, and there is a critical porosity （0.75） and kerogen content （ca. ≤3%） over which the elastic moduli decrease rapidly and slowly, respectively. The derived elastic moduli of gas- and oil-saturated digital cores differ little probably because of the low porosity （4.5%） of the Longmaxi black shale. Clearly, the numerical experiments demonstrated the feasibility of combining microstructure images of shale samples with elastic moduli calculations to predict shale properties.

A first-principles study on elastic properties and stability of Ti_xV_(1-x)C multiple carbide

The structure,stability and elastic properties of di-transition-metal carbides TixV1-xC were investigated by using the first-principles with a pseudopotential plane-waves method.The results show that the equilibrium lattice constants of TixV1-xC show a nearly linear reduction with increasing addition of V.The elastic properties of TixV1-xC are varied by doping with V.The bulk modulus of Ti0.5V0.5C is larger than that of pure TiC,as well as Ti0.5V0.5C has the largest C44 among TixV1-xC（0≤x≤1）,indicating that Ti0.5V0.5C has higher hardness than pure TiC.However,Ti0.5V0.5C presents brittleness based on the analysis of ductile/brittle behavior.The Ti0.5V0.5C carbide has the lowest formation energy,indicating that Ti0.5V0.5C is more stable than all other alloys.

First principles study of electronic structure, chemical bonding and elastic properties of BiOCuS

The electronic structures, chemical bonding and elastic properties of the tetragonal phase BiOCuS were investigated by using density-functional theory （DFT） within generalized gradient approximation （GGA）. The calculated energy band structures show that the tetragonal phase BiOCuS is an indirect semiconductor with the calculated band gap of about 0.503 eV. The density of states （DOS） and the partial density of states （PDOS） calculations show that the DOS near the Fermi level is mainly from the Cu-3d state. Population analysis suggests that the chemical bonding in BiOCuS has predominantly ionic character with mixed covalent-ionic character. Basic physical properties, such as lattice constant, bulk modulus, shear modulus, elastic constants, were calculated. The elastic modulus and Poisson ratio were also predicted. The results show that tetragonal phase BiOCuS is mechanically stable and behaves in a ductile manner.

First principles calculation of electronic structure, chemical bonding and elastic properties of ultra-incompressible Re_2P

The electronic structures, chemical bonding and elastic properties of the Co2P-type structure phase ultra-incompressible Re2P （orthorhombic phase） were investigated by density-functional theory （DFT） within generalized gradient approximation （GGA）. The calculated energy band structures show that the orthorhombic structure phase Re2P is metallic material. The density of state （DOS） and the partial density of state （PDOS） calculations show that the DOS near the Fermi level is mainly from the Re-5d state. Population analysis suggests that the chemical bonding in Re2P has predominantly covalent character with mixed covalent-ionic character. Basic physical properties, such as lattice constant, bulk modulus, shear modulus, and elastic constants Cij, were calculated. The elastic modulus and Poisson ratio were also predicted. The results show that the Co2P-type structure phase Re2P is mechanically stable and behaves in a brittle manner.

Chemical bonding and elastic properties of quaternary arsenide oxides YZnAsO and LaZnAsO investigated by first principles

The structural parameters, chemical bonding and elastic properties of the tetragonal phase quaternary arsenide oxides YZnAsO and LaZnAsO were investigated by using density-functional theory （DFT） within generalized gradient approximation （GGA）. The GGA calculated structural parameters are in agreement with the experimental results. Population analysis suggests that the chemical bonding in YZnAsO and LaZnAsO can be classified as a mixture of ionic and covalent characteristic. Single-crystal elastic constants were calculated and the polycrystalline elastic modules were estimated according to Voigt, Reuss and Hill＇s approximations （VRH）. The result shows that both YZnAsO and LaZnAsO are relatively soft materials exhibiting ductile behavior. The calculated polycrystalline elastic anisotropy result shows that LaZnAsO is more anisotropy in compressibility and YZnAsO is more anisotropy in shear.

First-principles calculation of structural and elastic properties of Pd_(3-x)Rh_xV alloys

The structural stability, electronic and elastic properties of Pd3-xRhxV alloys with L12 and D022 structures were investigated theoretically by the first-principles calculations. The results reveal that with the increase of Rh content, the unit cell volume of Pd3-xRhxV alloys with L12 and D022 structures decreases, and the structure of Pd3-xRhxV alloys tends to transform from D022 to L12. The elastic parameters such as elastic constants, bulk modulus, shear modulus, elastic modulus, and Poisson ratio, were calculated and discussed in details. Electronic structures were also computed to reveal the underlying mechanism for the stability and elastic properties.

Phase stability,elastic properties and electronic structures of Mg-Y intermetallics from first-principles calculations

The phase stability,elastic properties and electronic structures of three typical Mg-Y intermetallics including Mg_(24)Y_(5),Mg_(2)Y and MgY are systematically investigated using first-principles calculations based on density functional theory.The optimized structural parameters including lattice constants and atomic coordinates are in good agreement with experimental values.The calculated cohesive energies and formation enthalpies show that either phase stability or alloying ability of the three intermetallics is gradually enhanced with increasing Y content.The single-crystal elastic constants C_(ij) of Mg-Y intermetallics are also calculated,and the bulk modulus B,shear modulus G,Young's modulus E,Poisson ratio v and anisotropy factor A of polycrystalline materials are derived.It is suggested that the resistances to volume and shear deformation as well as the stiffness of the three intermetallics are raised with increasing Y content.Besides,these intermetallics all exhibit ductile characteristics,and they are isotropic in compression but anisotropic to a certain degree in shear and stiffness.Comparatively,Mg_(24)Y_(5) presents a relatively higher ductility,while MgY has a relatively stronger anisotropy in shear and stiffness.Further analysis of electronic structures indicates that the phase stability of Mg-Y intermetallics is closely related with their bonding electrons numbers below Fermi level.Namely,the more bonding electrons number below Fermi level corresponds to the higher structural stability of Mg-Y intermetallics.

A first-principles study on electronic structure and elastic properties of Al_4Sr, Mg_2Sr and Mg_(23)Sr_6 phases

The electronic structures and mechanical properties of Al4Sr, Mg2Sr and Mg23Sr6 phases were determined by the use of first-principles calculations. The calculated heat of formation and cohesive energy indicate that Al4Sr has the strongest alloying ability as well as the highest structural stability. The elastic parameters were calculated, and then the bulk modulus, shear modulus, elastic modulus and Poisson ratio were derived. The ductility and plasticity were discussed. The results show that Al4Sr and Mg2Sr phases both are ductile, on the contrary, Mg23Sr6 is brittle, and among the three phases, Mg2Sr is a phase with the best plasticity.

Electronic structures and elastic properties of monolayer and bilayer transition metal dichalcogenides MX_2(M= Mo,W;X= O,S,Se,Te):A comparative first-principles study

First-principle calculations with different exchange-correlation functionals, including LDA, PBE, and vd W-DF functional in the form of opt B88-vd W, have been performed to investigate the electronic and elastic properties of twodimensional transition metal dichalcogenides（TMDCs） with the formula of MX2（M = Mo, W; X = O, S, Se, Te） in both monolayer and bilayer structures. The calculated band structures show a direct band gap for monolayer TMDCs at the K point except for MoO2 and WO2. When the monolayers are stacked into a bilayer, the reduced indirect band gaps are found except for bilayer WTe2, in which the direct gap is still present at the K point. The calculated in-plane Young moduli are comparable to that of graphene, which promises possible application of TMDCs in future flexible and stretchable electronic devices. We also evaluated the performance of different functionals including LDA, PBE, and opt B88-vd W in describing elastic moduli of TMDCs and found that LDA seems to be the most qualified method. Moreover, our calculations suggest that the Young moduli for bilayers are insensitive to stacking orders and the mechanical coupling between monolayers seems to be negligible.

Thermodynamics and elastic properties of Ta from first-principles calculations

Within the framework of the quasiharmonic approximation, the thermodynamics and elastic properties of Ta, including phonon density of states （DOS）, equation of state, linear thermal expansion coefficient, entropy, enthalpy, heat capacity, elastic constants, bulk modulus, shear modulus, Young＇s modulus, microhardness, and sound velocity, are studied using the first-principles projector-augmented wave method. The vibrational contribution to Helmholtz free energy is evaluated from the first-principles phonon DOS and the Debye model. The thermal electronic contribution to Helmholtz free energy is estimated from the integration over the electronic DOS. By comparing the experimental results with the calculation results from the first-principles and the Debye model, it is found that the thermodynamic properties of Ta are depicted well by the first-principles. The elastic properties of Ta from the first-principles are consistent with the available experimental data.

First-principles study of structural stability and elastic properties of MgPd_(3) and its hydride

Theoretical study of structural stability and elastic properties ofα-andβ-MgPd_(3)intermetallic compounds as well as their hydrides have been carried out based on density functional theory.The results indicateα-MgPd_(3)is more stable thanβphase with increased stability in their hydrides.The calculated elastic constants ofα-MgPd_(3)are overall larger thanβphase.After hydrogenation,the elastic constants are enlarged.And the elastic moduli exhibit similar tendency.The anisotropy ofα-MgPd_(3)is larger thanβphase,and the hydrides demonstrate larger anisotropy.Their ductility follows the order ofα-MgPd_(3)H_(0.5)<α-MgPd_(3)<β-MgPd_(3)H<β-MgPd_(3).Compared withβphase,higher Debye temperature ofα-MgPd_(3)implies stronger covalent interaction,and the Debye temperature of hydrides increases slightly.The electronic structures demonstrate that the Pd-Pd interaction is stronger than Pd-Mg,and Pd-H bonds play a significant role in the phase stability and elastic properties of hydrides.

Theoretical investigation of sulfur defects on structural, electronic,and elastic properties of ZnSe semiconductor

The structural, electronic, and elastic properties of ZnSe1-xSx for the zinc blende structures have been studied by using the density functional theory. The calculations were performed using the plane wave pseudopotential method, as implemented in Quantum ESPRESSO. The exchange-correlation potential is treated with the local density approximation pz-LDA for these properties. Moreover, LDA＋U approximation is employed to treat the ＂d＂ orbital electrons properly. A comparative study of the band gap calculated within both LDA and LDA＋U schemes is presented. The analysis of results show considerable improvement in the calculation of band gap. The inclusion of compositional disorder increases the values of all elastic constants. In this study, it is found that elastic constants C11, C12, and C44 are mainly influenced by the compositional disorder. The obtained results are in good agreement with literature.

First-principles calculations for elastic properties of rutile TiO_2 under pressure

This paper studies the equilibrium structure parameters and the dependences of the elastic properties on pressure for rutile TiO2 by using the Cambridge Serial Total Energy Package （CASTEP） program in the frame of density functional theory. The obtained equilibrium structure parameters, bulk modulus B0 and its pressure derivative B′0 are in good agreement with experiments and the theoretical results. The six independent elastic constants of rutile TiO2 under pressure are theoretically investigated for the first time. It is found that, as pressure increases, the elastic constants C11, C33, C66, C12 and C13 increase, The variation of elastic constant C44 is not obvious and the anisotropy will weaken.

Elastic properties of Nb-based alloys by using the density functional theory

A first-principles density functional approach is used to study the electronic and the elastic properties of Nb15X （X = Ti, Zr, Hf, V, Ta, Cr, Mo, and W） alloys. The elastic constants cn and c12, the shear modulus CI, and the elastic modulus E（lOO） are found to exhibit similar tendencies, each as a function of valence electron number per atom （EPA）, while c44 seems unclear. Both cu and c12 of Nb15X alloys increase monotonically with the increase of EPA. The C/ and E000） also show similar tendencies. The elastic constants （except c44） increase slightly when alloying with neighbours of a higher d-transition series. Our results are supported by the bonding density distribution. When solute atoms change from Ti（Zr, Hf） to V（Ta） then to Cr（Mo, W）, the bonding electron density between the central solute atom and its first neighbouring Nb atoms is increased and becomes more anisotropic, which indicates the strong interaction and thus enhances the elastic properties of Nb-Cr（Mo, W） alloys. Under uniaxial {100） tensile loading, alloyed elements with less （more） valence electrons decrease （increase） the ideal tensile strength.

Three-dimensional analysis of a faulted CO2 reservoir using an Eshelby-Mori-Tanaka approach to rock elastic properties and fault permeability

This work develops a three-dimensional(3D) multiscale model to analyze a complex carbon dioxide(CO_2) faulted reservoir that includes some key geologic features of the San Andreas and nearby faults southwest of the Kimberlina site.The model uses the STOMP-CO2 code for flow modeling that is coupled to the ABAQUS~ finite element package for geomechanical analysis.A 3D ABAQUS~ finite element model is developed that contains a large number of 3D solid elements with two nearly parallel faults whose damage zones and cores are discretized using the same continuum elements.Five zones with different mineral compositions are considered:shale,sandstone,fault damaged sandstone,fault damaged shale,and fault core.Rocks' elastic properties that govern their poroelastic behavior are modeled by an Eshelby-Mori-Tanaka approach(EMTA).which can account for up to 15 mineral phases.The permeability of fault damage zones affected by crack density and orientations is also predicted by an EMTA formulation.A STOMP-CO2 grid that exactly maps the ABAQUS~ finite element model is built for coupled hydromechanical analyses.Simulations of the reservoir assuming three different crack pattern situations(including crack volume fraction and orientation) for the fault damage zones are performed to predict the potential leakage of CO_2 due to cracks that enhance the permeability of the fault damage zones.The results illustrate the important effect of the crack orientation on fault permeability that can lead to substantial leakage along the fault attained by the expansion of the CO_2 plume.Potential hydraulic fracture and tendency for the faults to slip are also examined and discussed in terms of stress distributions and geomechanical properties.

First-principles Investigation of the Structural, Electronic and Elastic Properties of Al_2Ca and Al_4Sr Phases in Mg-Al-Ca(Sr) Alloy

First-principles calculations have been carried out to investigate the structural stabilities, electronic structures and elastic properties of Mg17Al12, Al2Ca and Al4Sr phases. The optimized structural parameters are in good agreement with the experimental and other theoretical values. The calculated formation enthalpies and cohesive energies show that Al2Ca has the strongest alloying ability, and Al4Sr has the highest structural stability. The densities of states （DOS）, Mulliken electronic populations, and electronic charge density difference are obtained to reveal the underlying mechanism of structural stability. The bulk modulus, shear modulus, Young＇s modulus, and Poisson＇s ratio are estimated from the calculated elastic constants. The mechanical properties of these phases are further analyzed and discussed. The Gibbs free energy and Debye temperature are also calculated and discussed.

Crystal structures and elastic properties of Ti(Cu,Pt)2 and Ti(Cu,Pt)3 phases

This paper focused on the crystal structures of two new ternary phases,Ti(Cu,Pt)2 and Ti(Cu,Pt)3,which were studied by X-ray powder diffraction data using Rietveld method.Electron probe microanalysis was used for sample composition examination.Elastic properties of these phases were further measured by nano-indentation,and meanwhile calculated with first-principle(FP)calculations.It is found that the crystal structure of Ti(Cu,Pt)2 is of orthorhombic cell space group Amm2(No.38)with structural prototype of VAu2.The resolved structure of Ti(Cu,Pt)3 is of tetragonal Al Pt3 type,belonging to the space group P4/mmm(No.123).The nano-indentation measurement and FP calculations show that the elastic modulus of Ti(Cu,Pt)2 increases firstly then decreases with Pt content,whereas that of Ti(Cu,Pt)3almost linearly increases with Pt content.

Density functional theory investigation on lattice dynamics,elastic properties and origin of vanished magnetism in Heusler compounds CoMnVZ(Z=Al,Ga)

The lattice dynamics,elastic properties and the origin of vanished magnetism in equiatomic quaternary Heusler compounds CoMnVZ(Z=Al,Ga)are investigated by first principle calculations in this work.Due to the similar constituent atoms in CoMnVAl and CoMnVGa compounds,they are both stable in LiMgPdSn-type structure with comparable lattice size,phonon dispersions and electronic structures.Comparatively,we find that CoMnVAl is more structurally stable than CoMnVGa.Meanwhile,the increased covalent bonding component in CoMnVAl enhances its mechanical strength and Vickers hardness,which leads to better comprehensive mechanical properties than those of CoMnVGa.Practically and importantly,structural and chemical compatibilities at the interface make non-magnetic semiconductor CoMnVAl and magnetic topological semimetals Co2MnAl/Ga more suitable to be grown in heterostructures.Owing to atomic preferential occupation in CoMnVAl/Ga,the localized atoms Mn occupy C(0.5,0.5,0.5)Wyckoff site rather than B(0.25,0.25,0.25)and D(0.75,0.75,0.75)Wyckoff sites in LiMgPdSn-type structure,which results in symmetric band filling and consequently drives them to be non-magnetic.Correspondingly,by tuning localized atoms Mn to occupy B(0.25,0.25,0.25)or/and D(0.75,0.75,0.75)Wyckoff sites in off-stoichiometric Co-Mn-V-Al/Ga compounds and keeping the total valence electrons as 24,newly compensated ferrimagnetic compounds are theoretically achieved.We hope that our work will provide more choices for spintronic applications.