Cauchy Relation Research Articles

Elastic and thermoelastic properties of synthetic Ca2MgSi2O7 (�kermanite) and Ca2ZnSi2O7 (hardystonite)

Elastic and thermoelastic constants of large single crystals of Ca2MgSi2O7 and Ca2ZnSi2O7 have been derived from ultrasonic resonance frequencies of plane-parallel plates and their shift upon variation of temperature, respectively. In addition, coefficients of thermal expansion and dielectric constants were determined. Both species possess quite similar properties. As observed in other isotypic magnesium and zinc compounds, the mean elastic stiffness and the deviation from the Cauchy relations are significantly larger in the zinc compound, due to a covalent contribution of the Zn–O bond. Positive thermoelastic constants T44 and T66 in Ca2MgSi2O7 allow temperature-independent ultrasonic generators and oscillators to be manufactured.

Influence of quadrupolar deformability on phonon dispersion in silver and copper halides

The quadrupolar deformability (QD) shell model is extended to calculate the phonon dispersion relation of silver and copper halides (AgCl, AgBr, AgI, CuCl, and CuBr). The QD is derived explicitly from the virtual d– s excitation of Ag and Cu ions induced by the crystal-field change of neighboring anions. The eight parameters (four short-range force constants, four shell parameters) are so economically determined that the QD shell model well reproduces the experimental phonon dispersion relations of these compounds. The QD shell model leads to the remarkable deviation from the Cauchy relation for elastic constants. We found that this QD effect is stronger in AgCl and AgBr than in AgI, CuCl and CuBr.

Elastic constants of noble-gas crystals under pressure and the cauchy relations

The equations of state are solved and the elastic constants responsible for the propagation of sound in strongly compressed crystals of noble gases are calculated in the next-to-nearest neighbor approximation using the interatomic potential proposed by the authors. The results of calculations are in satisfactory agreement with the experimental data. Slightly worse agreement is achieved in calculating the shear modulus ℬ44. The validity of the Cauchy relation for krypton is confirmed in the experiment. On this basis, the inference is drawn that the interatomic interaction in the krypton crystal has a central character.

Equations in Stresses for Two- and Three-Dimensional Dynamic Problems of Thermoelasticity in Spherical Coordinates

The basic system of differential equations for the six components of the dynamic stress tensor is deduced in a spherical coordinate system by using the equations of motion, Hooke's law, Cauchy relations, and Saint-Venant conditions of compatibility of strains. The obtained system of differential equations is reduced (without introducing additional potential functions) to a system of wave equations used for the successive evaluation of the first invariant and unknown components of the stress tensor. We also present the corresponding systems of wave equations for the axisymmetric, polar, and centrally symmetric problems of thermoelasticity.

The Cauchy relations in linear elasticity theory

In linear elasticity, we decompose the elasticity tensor into two irreducible pieces with 15 and 6 independent components, respectively. The vanishing of the piece with 6 independent components corresponds to the Cauchy relations. Thus, for the first time, we recognize the group-theoretical underpinning of the Cauchy relations.

Systematics of elasticity: Ab initio study in B1-type alkaline earth oxides

The elastic properties and their pressure dependence of four B1-type alkaline earth oxides, MgO, CaO, SrO, and BaO, are calculated using the ab initio full-potential linear muffin-tin-orbital (FP-LMTO) generalized gradient approximated (GGA) method to elucidate their systematics. The calculated results agree quite well with the comparable experimental data. The large pressure dependence of c11 and c44 of MgO observed over 25 GPa is not predicted as well as the previous local-density approximation (LDA) calculations. It is inferred that the high-pressure measurement of elastic constant is quite sensitive to nonhydrostaticity. The deviation from the Cauchy relation and the elastic anisotropy are investigated. It is found that the interatomic interaction in SrO is nearest to the two-body force and the many-body contribution is largest in MgO. The elastic anisotropy in SrO and BaO are almost the same and only MgO has a large positive anisotropy under low pressure. The normalized elastic constants cij′ are introduced for more detail analysis. It is found that while there is no substantial difference in c11′ and c12′ of the four oxides, only c44′ and its pressure dependence vary systematically. The difference in the Cauchy deviation and elastic anisotropy of alkaline earth oxides under pressure is characterized by c44.

Derivations of Generalized Cauchy's Relations Based on Atomic Configurations.

Microscopic expressions of generalized Cauchy's relations for higher-order stresses and heat flux are disscused on the basis of the equation of motion of atom. A macroscopic solid is modeled as an assembly of atoms, and the concept of the mesodomain is introduced in order to describe macroscopic quantities with microscopic motions of atoms. Furthermore, a mesoscopic tetrahedral element which is composed of a number of atoms is introduced in the mesodomain. Higher-order moments such as bimoment do not satisfy the equilibrium of the tetrahedron. The microscopic expression of generalized Cauchy's relation for the higher-order stresses can be derived from the first-order tensor product of equation of motion of the atom and position vector. On the other hand, the microscopic description of generalized Cauchy's relation for the heat flux can also be obtained from the summation of higher -order tensor product of motion of atom and position vectors.

Elasticity of MgO and a primary pressure scale to 55 GPa.

We have studied the elasticity and pressure-density equation of state of MgO in diamond cells to 55 GPa and have doubled the previous pressure limit of accurate elasticity determinations for crystals. Integrating single-crystal velocity data from Brillouin scattering measurements and density data from polycrystalline x-ray diffraction, we obtained the three principal elastic tensor elements (C(11), C(12), and C(44)) and various secondary elasticity parameters, including single-crystal elastic anisotropy, Cauchy relation, aggregate sound velocities, and Poisson's ratio, as functions of pressure. The present study also provides a direct determination of pressure without recourse to any prior pressure standard, thus creating a primary pressure scale. The commonly used ruby fluorescence pressure scale has thus been improved to 1% accuracy by the new MgO scale.

Mechanism for negative poisson ratios over the alpha- beta transition of cristobalite, SiO2: A molecular-dynamics study

The adiabatic elastic constants ( C(ij)) of cristobalite have been evaluated successfully over the temperature range of 300-1800 K using the molecular-dynamics method with a fluctuation formula. Cristobalite shows a negative Poisson ratio over this temperature range. However, the mechanisms differ between the alpha and beta phases. In the cubic beta phase, C44 exhibits a value extremely close to C11 rather than C12, in contrast to the Cauchy relation. This predicts a remarkable property that the longitudinal and transverse velocities coincide for the acoustic waves propagating along the [100] direction.

Single-crystal elasticity of MgO at high pressure

The single-crystal elastic properties of MgO were measured by Brillouin scattering in diamond anvil cell to 18.6 GPa. These measurements demonstrate that quasihydrostatic stress conditions can be obtained in solid pressure medium by thermal annealing, and establish the accuracy with which elastic moduli can be measured over this extensive pressure range (\ifmmode\pm\else\textpm\fi{}5% in ${\mathrm{dC}}_{\mathrm{ij}}/dP).$ The acoustic anisotropy of MgO decreases with pressure, with MgO becoming isotropic at \ensuremath{\sim}21.5 GPa. Deviations from noncentral interatomic forces, as measured by the Cauchy relations, increase steadily with pressure.

The high-pressure, single-crystal elasticity of pyrope, grossular, and andradite

High-pressure Brillouin scattering experiments were conducted on three natural, singlecrystal garnets with near end-member compositions: pyrope (Mg 3 Al 2 Si 3 O 12 ), grossular (Ca 3 Al 2 Si 3 O 12 ), and andradite (Ca 3 Fe 3+ 2 Si 3 O 12 ). Acoustic velocities of the samples were measured from the samples at pressures to 10 GPa in a diamond-anvil cell with a methanolethanol- water pressure medium. Elastic stiffness constants (C ij ) and their pressure dependencies (∂C ij /∂P) were calculated, as were aggregate elastic moduli (K S and G) and their respective pressure derivatives. The Cauchy relations changed with pressure for both grossular and andradite, suggesting that assuming these are constant is not valid for extrapolation to high pressures for these and possibly other minerals. The variation in ∂KS /∂P and ∂G/∂P was sensitive to composition and required a complex mixing model for extrapolation of end-member elasticity values from garnets of intermediate compositions.

On anisotropic invariants of a symmetric tensor: crystal classes, quasi–crystal classes and others

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Xiao H. 1998On anisotropic invariants of a symmetric tensor: crystal classes, quasi–crystal classes and othersProc. R. Soc. Lond. A.4541217–1240http://doi.org/10.1098/rspa.1998.0203SectionRestricted accessOn anisotropic invariants of a symmetric tensor: crystal classes, quasi–crystal classes and others H. Xiao H. Xiao Institute of Mechanics, Ruhr–University Bochum, D–44780 Bochum, GermanyDepartment of Mathematics, Peking University, Beijing 100871, People's Republic of China, Google Scholar Find this author on PubMed Search for more papers by this author H. Xiao H. Xiao Institute of Mechanics, Ruhr–University Bochum, D–44780 Bochum, GermanyDepartment of Mathematics, Peking University, Beijing 100871, People's Republic of China, Google Scholar Find this author on PubMed Search for more papers by this author Published:08 April 1998https://doi.org/10.1098/rspa.1998.0203AbstractThis paper is concerned with invariants of a symmetric second–order tensor relative to crystal classes and quasicrystal classes and some non–continuous infinite orthogonal subgroups. Simple irreducible functional bases in unified forms, each of which consists of eight polynomial invariants only, are presented for all kinds of finite subgroups of the maximal transverse isotropy group, D∞ h, except the subgroups of the orthotropy group D2h. Results are also provided for other kinds of orthogonal subgroups including icosahedral quasicrystal classes, etc. For the trigonal, tetragonal and hexagonal crystal classes without two–fold axis, the results given here are even more compact than the corresponding results recently derived by this author (Xiao 1996a). For all non–crystal classes except the transverse isotropy groups, the presented results are the first ones. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Itin Y (2016) Quadratic Invariants of the Elasticity Tensor, Journal of Elasticity, 10.1007/s10659-016-9569-2, 125:1, (39-62), Online publication date: 1-Oct-2016. Wright T (2014) Bootstrap Elasticity III: Minimal Nonlinear Constitutive Representation for Cubic Materials, Journal of Elasticity, 10.1007/s10659-014-9507-0, 120:1, (109-119), Online publication date: 1-Jun-2015. Itin Y and Hehl F (2013) The constitutive tensor of linear elasticity: Its decompositions, Cauchy relations, null Lagrangians, and wave propagation, Journal of Mathematical Physics, 10.1063/1.4801859, 54:4, (042903), Online publication date: 1-Apr-2013. Clayton J (2012) Towards a nonlinear elastic representation of finite compression and instability of boron carbide ceramic, Philosophical Magazine, 10.1080/14786435.2012.682171, 92:23, (2860-2893), Online publication date: 11-Aug-2012. Wright T (2011) Bootstrap Elasticity: From Linear to Nonlinear Constitutive Representations, Journal of Elasticity, 10.1007/s10659-011-9367-9, 109:1, (35-49), Online publication date: 1-Sep-2012. Xiao H, Bruhns O and Meyers A (2006) On isotropic extension of anisotropic constitutive functions via structural tensors, ZAMM, 10.1002/zamm.200410226, 86:2, (151-161), Online publication date: 16-Feb-2006. Xiao H, Bruhns O and Meyers A (2006) Elastoplasticity beyond small deformations, Acta Mechanica, 10.1007/s00707-005-0282-7, 182:1-2, (31-111), Online publication date: 1-Mar-2006. Moakher M and Norris A (2006) The Closest Elastic Tensor of Arbitrary Symmetry to an Elasticity Tensor of Lower Symmetry, Journal of Elasticity, 10.1007/s10659-006-9082-0, 85:3, (215-263), Online publication date: 26-Oct-2006. Bruhns O, Xiao H and Meyers A (2003) Some basic issues in traditional Eulerian formulations of finite elastoplasticity, International Journal of Plasticity, 10.1016/S0749-6419(03)00047-0, 19:11, (2007-2026), Online publication date: 1-Nov-2003. Xiao H, Bruhns O and Meyers A (1999) On anisotropic invariants of N symmetric second-order tensors: crystal classes and quasi–crystal classes as subgroups of the cylindrical group D, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 455:1986, (1993-2020), Online publication date: 8-Jun-1999.Xiao H (1998) On scalar–, vector– and second–order tensor–valued anisotropic functions of vectors and second–order tensors relative to all kinds of subgroups of the transverse isotropy group Cinfinityh, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 356:1749, (3087-3122), Online publication date: 15-Dec-1998. This Issue08 April 1998Volume 454Issue 1972 Article InformationDOI:https://doi.org/10.1098/rspa.1998.0203Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online08/04/1998Published in print08/04/1998 License: Citations and impact Keywordsanisotropysymmetric tensorinvariantscrystal classesirreducible functional basesquasi–crystal classes

Breathing shell model in molecular dynamics simulation: Application to MgO and CaO

Molecular dynamics (MD) simulation is used to calculate the elastic constants of both MgO and CaO at zero pressure, and their temperature dependences, as well as the temperature–pressure–volume equation of states of the two oxides. The interionic potential is taken to be the sum of pairwise additive Coulomb, van der Waals, and repulsive interactions. In order to account for the observed large departures from the Cauchy relation of the elastic constants of the two oxides, the breathing shell model (BSM) is introduced in MD simulation, in which the repulsive radii of O ions are allowed to deform isotropically under the effects of other ions in the crystal, with each core and breathing shell being linked by a harmonic spring with force constant k. Required energy parameters, including k, were derived empirically to reproduce the observed molar volumes and elastic constants of the two oxides at ambient conditions, and their temperature dependences as accurately as possible. The MD simulation with BSM is very satisfactory in reproducing accurately not only the observed large Cauchy violations,but also the measured molar volumes and individual elastic constants of the two oxides over wide temperature ranges, at zero pressure. In addition the MD simulation reproduces accurately the measured volume compression data of CaO up to 60 GPa at 300 K. Here we present the MD simulated temperature–pressure–volume equation of state of CaO as a useful internal pressure calibration standard at high temperatures and high pressures.

Numerical simulation of crack growth in an isotropic solid with randomized internal cohesive bonds

A virtual internal bond (VIB) model with randomized cohesive interactions between material particles is proposed as an integration of continuum models with cohesive surfaces and atomistic models with interatomic bonding. This approach differs from an atomistic model in that a phenomenological “cohesive force law” is assumed to act between “material particles” which are not necessarily atoms; it also differs from a cohesive surface model in that, rather than imposing a cohesive law along a prescribed set of discrete surfaces, a randomized network of cohesive bonds is statistically incorporated into the constitutive law of the material via the Cauchy-Born rule, i.e., by equating the strain energy function on the continuum level to the potential energy stored in the cohesive bonds due to an imposed deformation. This work is motivated by the notion that materials exhibit multiscale cohesive behaviors ranging from interatomic bonding to macroscopic ductile failure. It is shown that the linear elastic behavior of the VIB model is isotropic and obeys the Cauchy relation; the instantaneous elastic properties under equibiaxial stretching are transversely isotropic, with all the in-plane components of the material tangent moduli vanishing at the cohesive stress limit; the instantaneous properties under equitriaxial stretching are isotropic with a finite strain modulus. We demonstrate through two preliminary numerical examples that the VIB model can be applied in direct simulation of crack growth without a presumed fracture criterion. The prospect of this type of approach in numerical simulations of fracture seems to be highly promising.

The stress recovery procedure for the finite element method

The idea of the proposed stress recovery procedure has been explained, peculiarities of its realization have been demonstrated on the example of the plane problem with small deformations, and generalization of this procedure to problems with finite deformations has been carried out. The proposed procedure allows us to construct the stress field for the problems with small and finite deformations without numerical differentiation of the displacement field obtained by the finite element method. Firstly, in accordance with this procedure the distributed forces are defined on three internal surfaces (in general case) within the body. This problem is reduced to solving the systems of linear algebraic equations of low-dimensions with symmetrical and positive definite band matrices. It is one of the advantages of the proposed procedure in contrast with other methods of the stress field construction. Then, using the Cauchy relation, the components of the stress tensor or displacement vector increment derivative are computed for the nodes of the finite element grid.

THE EFFECT OF THREE-BODY FORCES ON THE VOLUME DEPENDENCE OF THE ELECTRONIC POLARIZABILITY OF ALKALI HALIDES

Violation of the Cauchy relation between the elastic constants for the alkali halide crystals clearly indicates the presence of multi-body forces. Generalizations of the shell model with three-body forces is usually used to describe the physical properties depending on lattice vibrations in these crystals. In the shell model, the electronic polarizability is given by the sum of the free-state polarizabilities and a pure crystalline term which depends on the potential. In this work, we propose a model for the three-body potential. This model is used with the breathing version of the shell model to calculate the volume dependence of the electronic polarizability. The obtained results are compared with available experimental data, and good agreement is found. For completeness, we also compare our results with other theoretical predictions. According to the present model, the sign of the Cauchy relation is the result of competition between the three-body and breathing terms. © 1997 Published by Elsevier Science Ltd

Density-functional theory of surfacelike elasticity of nematic liquid crystals

Based on the density-functional theory, we show that the splay-bend surfacelike bulk elastic constant ${\mathrm{K}}_{13}$ is an artifact of the phenomenological as well as the gradient expansion construction of the curvature elastic free energy of nematic liquid crystals, while the saddle-splay constant ${\mathrm{K}}_{24}$ is real and approximately obeys an extended Nehring-Saupe relation ${\mathrm{K}}_{24}$ \ensuremath{\geqslant}(${\mathrm{K}}_{11}$ -${\mathrm{K}}_{22}$ )/2, with ${\mathrm{K}}_{11}$ and ${\mathrm{K}}_{22}$ being the splay and twist Frank constants, respectively. The result ${\mathrm{K}}_{13}$ =0 automatically resolves the Oldano-Barbero pathology, which inevitably accompanies a nonzero ${\mathrm{K}}_{13}$ , and gives a concrete rationale for the conventional approaches in the continuum theory disregarding the surfacelike elasticity. The source of an apparently nonzero ${\mathrm{K}}_{13}$ in previous microscopic theories is discussed in detail and is demonstrated to be a result of an inconsistent use of a nonlocal-to-local mapping of the elastic free-energy functional at the boundary. The absence of ${\mathrm{K}}_{13}$ can be regarded as a type of Cauchy relation in the nematic continuum theory in the sense that it is not directly rooted in any of the macroscopic symmetries existing in the nematic phase, but is a general consequence of the particular algebraic form of the nonlocal free-energy term from which ${\mathrm{K}}_{13}$ derives; its linearity in the distortion amplitude and the symmetry of the relevant direct correlation function with respect to the permutation of a molecular pair leads always to the vanishing ${\mathrm{K}}_{13}$ . In this respect, ${\mathrm{K}}_{13}$ =0 applies not only to nematic liquid crystals but also to a more general class of phases such as cholesteric liquid crystals, whose structure can be viewed as a weak modulation of a translationally invariant phase. We finally consider the elastic description of nematic liquid crystals in the presence of real interfaces. The present formulation allows a straightforward decomposition of the elastic excess free energy into the bulk contribution and the interfacial excess in the Gibbs sense. The bulk part yields the bulk Oseen-Frank elastic free-energy density along with the ${\mathrm{K}}_{24}$ term evaluated at the Gibbs dividing surface as an unambiguous local quantity. The interfacial excess, when gradient expanded, reduces to the surface free-energy density comprised of the anchoring energy, surface-excess Oseen-Frank elastic energy, ${\mathrm{K}}_{24}$ -like term, and elastic free-energy contributions reflecting the broken symmetry at the interface. The ${\mathrm{K}}_{24}$ -like term is formally similar to the bulk ${\mathrm{K}}_{24}$ term, but is no longer an intrinsic property of bulk nematic liquid crystal, as it depends also on the nature of the medium with which the nematic liquid crystal is in contact.

Model of a layered plate considering nonideal contact between layers

We propose a mathematical model for doing calculations for layered plates, allowing for both rigid and sliding contact in the presence of frictional forces between the sliding layers. The model takes into account the distribution of tangential and normal displacements across the thickness of the sliding layered stack, and also the distribution of transverse normal stresses. The strain tensor is obtained using the Cauchy relations; the stress tensor is obtained based on Hooke's law. Tne Lagrange variational principle allows us to obtain the resolvent system of differential equations and the corresponding boundary conditions. The spatial model for deformation of a layered plate has a number of special features compared with familiar models. The system of differential equations has operators no higher than second order. It is described relative to displacements on the faces of the stack. This is convenient in solving problems involving sliding of layers with and without friction.

SELF-CONSISTENT THEORY OF ELASTIC PROPERTIES OF STRONGLY ANHARMONIC CRYSTALS II: PROPERTIES OF HEAVY RARE GAS SOLIDS

The method developed in the preceding papers has been used to calculate the full set of equilibrium thermodynamic properties of solid Ar, Kr and Xe at normal pressure. We have used various pairwise interatomic potentials taking into account the three-body Axilrod–Teller forces. For Ar and Kr the multiparameter potentials proposed by Barker et al. provide the best fit to available experimental data, and for Xe the old Lennard–Jones potential does this. Deviations from the Cauchy relations are discussed. Temperature dependencies of the stability coefficients and anisotropy have received much attention. In the unstable point that is about 1.35–1.45 times larger than the experimental melting temperature of each crystal, the isothermal bulk modulus goes to zero (and the thermal expansion coefficient and the isobaric specific heat tend to infinity). Other stability coefficients remain finite and positive.

Some static and vibrational properties of aluminium by a local pseudopotential

The pseudopotential investigation of various properties of aluminium is studied in terms of the real-space sum of Born-von Karman central-force constants. A simple local model potential depending on a single parameter has been used to investigate the ion-ion interaction, interatomic force constants, dynamical elastic constants, bulk modulus, deviation from Cauchy relation, Poisson's ratio, Young's modulus, propagation velocity of elastic waves, phonon dispersion curves and degree of elastic anisotropy. The effect of exchange and correlation on conduction electrons is also considered separately by using various forms of the dielectric (screening) functions. Overall good agreement with the available theoretical and experimental data is achieved in the present work.