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.

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.

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.

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.

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.

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.

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.

Density-functional theory study of the effect of pressure on the elastic properties of CaB_6

Under high pressure, the long believed single-phase material CaB6 was latterly discovered to have a new phase tI56. Based on the density-functional theory, the pressure effects on the structural and elastic properties of CaB6 are obtained. The calculated bulk, shear, and Young’s moduli of the recently synthesized high pressure phase tI56-CaB6 are larger than those of the low pressure phase. Moreover, the high pressure phase of CaB6 has ductile behaviors, and its ductility increases with the increase of pressure. On the contrary, the calculated results indicate that the low pressure phase of CaB6 is brittle. The calculated Debye temperature indicates that the thermal conductivity of CaB6 is not very good. Furthermore, based on the Christoffel equation, the slowness surface of the acoustic waves is obtained.

Structural, magnetic, electronic, and elastic properties of face-centered cubic PuH_x (x= 2, 3):GGA (LSDA) + U + SO

We perform first-principles calculations to investigate the structural, magnetic, electronic, and mechanical properties of face-centered cubic （fcc） Pull2 and fcc Pull3 using the full potential linearized augmented plane wave method （FP- LAPW） with the generalized gradient approximation （GGA） and the local spin density approximation （LSDA） taking account of both relativistic and strong correlation effects. The optimized lattice constant a0 = 5.371 A for fcc Pull2 and a0 = 5.343 A for fcc PuH3 calculated in the GGA ＋ sp （spin polarization） ＋ U （Hubbard parameter） ＋ SO （spin-orbit coupling） scheme are in good agreement with the experimental data. The ground state of fcc PuH3 is found to be slightly ferromagnetic. Our results indicate that fcc PuH2 is a metal while fcc PuH3 is a semiconductor with a band gap about 0.35 eV. We note that the SO and the strong correlation between localized Pu 5f electrons are responsible for the band gap of fcc PuH3. The bonds for PuH2 have mainly covalent character while there are covalent bonds in addition to apparent ionicity bonds for PuH3. We also predict the elastic constants of fcc PuH2 and fcc PuH3, which were not observed in the previous experiments.

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.

Elastic properties and electronic structures of lanthanide hexaborides

The structural, elastic, and electronic properties of a series of lanthanide hexaborides（Ln B6） have been investigated by performing ab initio calculations based on the density functional theory using the Vienna ab initio simulation package.The calculated lattice and elastic constants of Ln B6 are in good agreement with the available experimental data and other theoretical results. The polycrystalline Young＇s modulus, shear modulus, the ratio of bulk to shear modulus B/G, Poisson＇s ratios, Zener anisotropy factors, as well as the Debye temperature are calculated, and all of the properties display some regularity with increasing atomic number of lanthanide atoms, whereas anomalies are observed for Eu B6 and Yb B6. In addition, detailed electronic structure calculations are carried out to shed light on the peculiar elastic properties of Ln B6.The total density of states demonstrates the existence of a pseudogap and indicates lower structure stability of Eu B6 and Yb B6 compared with others.

The stabilities, electronic structures and elastic properties of Rb-As systems

The structural, electronic and elastic properties of Rb As systems （RbAs in NaP, LiAs and AuCu structures, RbAs2 in the MgCu2 structure, Rb3As in NaaAs, Cu3P and Li3Bi structures, and Rb5As4 in the A5B4 structure） are investigated with the generalized gradient approximation in tile frame of density functional theory. The lattice parameters, cohesive energies, formation energies, bulk moduli and the first derivatives of the bulk moduli （to fit Murnaghan＇s equation of state） of the considered structures are calculated and reasonable agreement is obtained. In addition, the phase transition pressures are also predicted. The electronic band structures, the partial densities of states corresponding to the band structures and the charge density distributions are presented and analysed. The second-order elastic constants based on the stress-strain method and other related quantities such as Young＇s modulus, the shear modulus, Poisson＇s ratio, sound velocities, the Debye temperature and shear anisotropy factors are also estimated.

Phase Transition and Elastic Properties of NbN under Hydrostatic Pressure

First-principles pseudopotential calculations are performed to investigate the phase transition and elastic properties of niobium nitrides （NbN）. The lattice parameters a0 and c0/a0, elastic constants Cu, bulk modulus B0, and the pressure derivative of bulk modulus B0＇ are calculated. The results are in good agreement with numerous experimental and theoretical data. The enthalpy calculations predict that NbN undergoes phase transition from NaCl-type to NiAs-type structure at 13.4 GPa with a volume collapse of about 4.0% and from AsNi-type to CW-type structure at 26.5 GPa with a volume collapse of about 7.0%. Among the four types of structures, CW-type is the most stable structure. The elastic properties are analyzed on the basis of the calculated elastic constants. Isotropic wave velocities and anisotropic elasticity of NbN are studied in detail. The longitudinal and shear-wave velocities, Vr, Vs and V increase with increasing pressure, respectively. The Debye temperature OD increases monotonically with increasing pressure except for NiAs-type structure. Both the longitudinal velocity and the shear-wave velocity increase with pressure for wave vector along all the propagation directions, except for VTA（[100]） and VTA[001]（[110]） with NaCl structure and VTA[001]（[100]） with the other three types of structures.

Ab initio study of the electronic structure and elastic properties of Al_5C_3N

We investigate the electronic structure, chemical bonding and elastic properties of the hexagonal aluminum carbonitride, Al5C3N, by ab initio calculations. Al5C3N is a semiconductor with a narrow indirect gap of 0.81 eV. The valence bands below the Fermi level （EF） originate from the hybridized Al p-C p and A1 p-N p states. The calculated bulk and Young＇s moduli are 201 GPa and 292 GPa, which are slightly lower than those of Ti3SiC2. The values of the bulk-to-shear-modulus and bulk-modulus-to-c44 are 1.73 and 1.97, respectively, which are higher than those of Ti2AlC and Ti2AlN, indicating that Al5C3N is a ductile ceramic.

First-principles analysis of the structural, electronic, and elastic properties of cubic organic-inorganic perovskite HC(NH_2)_2PbI_3

The structural, electronic, and elastic properties of cubic HC（NH2）2PbI3 perovskite are investigated by density functional theory using the Tkatchenko-Scheffler pairwise dispersion scheme. Our relaxed lattice parameters are in agreement with experimental data. The hydrogen bonding between NH2 and I ions is found to have a crucial role in FAPbI3 stability. The first calculated band structure shows that HC（NH2）2PbI3 has a direct bandgap （1.02 eV） at R-point, lower than the bandgap （1.53 eV） of CH3NH3PbI3. The calculated density of states reveals that the strong hybridization of s（Pb）-p（I） orbital in valence band maximum plays an important role in the structural stability. The photo-generated effective electron mass and hole mass at R-point along the R-Γ and R-M directions are estimated to be smaller：me^＊=0.06m0 and mh^＊=0.08m0 respectively, which are consistent with the values experimentally observed from long range photocarrier transport. The elastic properties are also investigated for the first time, which shows that HC（NH2）2PbI3 is mechanically stable and ductile and has weaker strength of the average chemical bond. This work sheds light on the understanding of applications of HC（NH2）2PbI3 as the perovskite in a planar-heterojunction solar cell light absorber fabricated on flexible polymer substrates.

Elastic properties of CaCO3 high pressure phases from first principles

Elastic properties of three high pressure polymorphs of CaCO_3 are investigated based on first principles calculations.The calculations are conducted at 0 GPa–40 GPa for aragonite, 40 GPa–65 GPa for post-aragonite, and 65 GPa–150 GPa for the P2_1/c-h-CaCO_3 structure, respectively. By fitting the third-order Birch–Murnaghan equation of state（EOS）, the values of bulk modulus K_0 and pressure derivative K~＇_0 are 66.09 GPa and 4.64 for aragonite, 81.93 GPa and 4.49 for post-aragonite, and 56.55 GPa and 5.40 for P2_1/c-h-CaCO_3, respectively, which are in good agreement with previous experimental and theoretical data. Elastic constants, wave velocities, and wave velocity anisotropies of the three highpressure CaCO_3 phases are obtained. Post-aragonite exhibits 25.90%–32.10% V_P anisotropy and 74.34%–104.30% V_S splitting anisotropy, and P2_1/c-h-CaCO_3 shows 22.30%–25.40% V_Panisotropy and 42.81%–48.00% V_S splitting anisotropy in the calculated pressure range. Compared with major minerals of the lower mantle, CaCO_3 high pressure polymorphs have low isotropic wave velocity and high wave velocity anisotropies. These results are important for understanding the deep carbon cycle and seismic wave velocity structure in the lower mantle.

Characterisation of the high-pressure structural transition and elastic properties in boron arsenic

This paper carries out the First principles calculation of the crystal structures （zinc blende （B3） and rocksalt （B1）） and phase transition of boron arsenic （BAs） based on the density-functional theory. Using the relation between enthalpy and pressure, it finds that the transition phase from the B3 structural to the B] structural occurs at the pressure of l13.42GPa. Then the elastic constants Cll, C12, C44, bulk modulus, shear modulus, Young modulus, anisotropy factor, Kleinman parameter and Poisson ratio are discussed in detail for two polymorphs of BAs. The results of the structural parameters and elastic properties in B3 structure are in good agreement with the available theoretical and experimental values.