Independent Elastic Parameters Research Articles

Контактная задача для трансверсально изотропного слоя с трением

It is investigated the spatial contact problem of asymmetric interaction of two punches on a transversally isotropic layer with friction forces taken into account in an unknown contact domain. The planes of isotropy are parallel to the layer faces. The friction forces are taken into account along one coordinate axis. The lower layer face is subjected to sliding support. The layer material is characterized by five independent elastic parameters. The problem is reduced to an integral equation with respect to the contact pressure with the help of a double Fourier transformation together with the Coulomb law. The anisotropy degree is characterized be three dimensionless parameters arising in the kernel of the integral equation, two of which satisfy characteristic equation. In particular cases, the integral equation coincides with those well-known for the corresponding contact problems with friction for the isotropic layer and half-space. The B.A. Galanov method is used for numerical solutions. A system of the integral equation and integral inequality is considered. A rectangle is taken which a priori contains the unknown contact domain. By introducing special nonlinear operators, the system is reduced to only one nonlinear equation of the Hammerstein type which can be solved by the successive approximations method. The contact domain is determined by nodes at which the function required is positive. The punches are taken in the form of asymmetric elliptic paraboloids. The method allows us to investigate percolation, i.e. the process of junction of the discrete contact domains due to increasing the forces applied and settlements of the punches. Calculations are made for different friction coefficients, materials and relative thicknesses of the transversally isotropic layer.

Single-test evaluation of directional elastic properties of anisotropic structured materials

When the elastic properties of structured materials become direction-dependent, the number of their descriptors increases. For example, in two-dimensions, the anisotropic behavior of materials is described by up to 6 independent elastic stiffness parameters, as opposed to only 2 needed for isotropic materials. Such high number of parameters expands the design space of structured materials and leads to unusual phenomena, such as materials that can shear under uniaxial compression. However, an increased number of properties descriptors and the coupling between shear and normal deformations render the experimental evaluation of material properties more challenging. In this paper, we propose a methodology based on the virtual fields method to identify six separate stiffness tensor parameters of two-dimensional anisotropic structured materials using just one tension test, thus eliminating the need for multiple experiments, as it is typical in traditional methods. The approach requires no stress data and uses full-field displacement data and global force data. We show the accuracy of our method using synthetic data generated from finite element simulations as well as experimental data from additively manufactured specimens.

Analysis of the angular influence in the spatial study of mechanical displacements in highly anisotropic media

Tumors with anisotropy in their structural properties experience unpredictable and asymmetric growth along certain directions that can vary over time during the tumor progression. An early recognition of such anisotropy could provide information about their malignity, facilitating early diagnostics/prognostic. However, it is difficult to be observed by classic echography. Diverse studies of the literature show 3D images of elastic properties in media with anisotropy in a certain direction, but are transversely isotropic. However, for highly anisotropy media this assumption is no longer valid and requires specific measurements of shear velocities in the 3D directions. These media with 21 independent elastic material parameters and no symmetry plane. Techniques using ultrafast focused sound scanners present difficulties in covering immediate regions around the focused area and specific information for different incident beam orientations to obtain complete 3D spatial information, which is relevant in these types of media without planes of symmetry. It is the case of malignant tumors. The current paper presents a flexible spatial study of mechanical displacements induced in highly anisotropic soft media by low frequency shaker (including printed inorganic, pseudo-organic grids and cow fiber tissues respectively). Homogeneous and heterogeneous media without planes of symmetry have been tested in the experiments by using speckle ultrasonic interferometry. Velocities associated to the induced low frequency-mechanical displacements have been analyzed at a point of interest within the sample from different angles of propagation, covering a 3D angular space. These measurements show a high spatial-dependence of results, demonstrating the relevance of making spatial angular scans to derive parameters related to the anisotropy. This technique allows detection of anisotropies in small areas of analysis, even if they are weak.

Ab Initio Calculation of Li-Sn System: Unraveling New Phases of Superconducting Materials with Increasing Compression

Stoichiometry, crystal compound, electronic attributes and superconductivity of compressed lithium-tin composites have been thoroughly studied using quantum mechanical genetic algorithm approach and the first principles computations based on density functional theory. Our simulations at moderate pressure (5-20 GPa) predict a complex convex hull diagram, with the following stable Li‑rich phases: I4/mmm-Li6Sn2, -Li7Sn2, -Li5Sn2, Ama2-Li4Sn2, -Li5Sn2, -Li6Sn2, C2/m-Li4Sn1, P21/m-Li6Sn2, -Li7Sn2 and Cmcm-Li4Sn2. Careful examination at their independent elastic parameters reveals sufficient mechanical stability in them. These phases are metallic system, with reasonably high electron concentration near to Fermi level or N(EF) that ranges from 0.6 to 2.4 states/eV cell. It is also interesting for us to observe soft modes and steep-flat energy bands at Fermi levels of Li6Sn2 structures which are stable throughout the pressure range. These features are prerequisites for superconducting behavior. Linear response function with Gaussian and tetrahedron methods reveals satisfactory superconducting transition temperature Tc (3.1 ~ 6.6 K) and Tc (2.1 ~ 2.4 K), respectively. Structural transition results for based elements Li and Sn agree well with literature thus signifying reliable prediction of intermediate phases.

Isogeometric dynamic analysis of shells based on the nonlinear micropolar theory

Studying the large-amplitude forced vibration characteristics of micropolar shell-type structures is the main objective of this paper. In order to obtain the full geometrically nonlinear model of micropolar theory, the micromorphic seven-parameter shell kinematic is first developed. Via a nonlinear micro-motion map, the extracted formulation is fitted for the micropolar continuum. As a result, possessing three stress–strain fields with eighteen independent elastic parameters of a linear material, the proposed micropolar shell theory is able to capture the inhomogeneity and size-effects. On the basis of Hamilton’s principle, the variational form of governing equations of motion is determined. The finite element isogeometric solution approach is then utilized to produce the geometry and approximate the displacement field in mid-surface area of shell. This would be helpful for resolving the locking and instability issues of the traditional low-order standard finite element shell models. By discretizing the period of harmonic vibration, a numerical method is also implemented in time domain. To demonstrate the microstructural effects, dynamic behavior of micropolar plate- and shell-type structures are investigated in several parametric studies. Since the classical strain and stress tensors are retained in present micropolar elasticity, both the large macro- and micro-deformations are captured for the first time.

Mechanical Response of PbSSe, PbSTe Ternary and PbSnSTe Quaternary Alloys at High Pressure

Property of the semiconductors under high pressure is investigated by the density functional theory and paralleled to the foretelling of the linear elasticity theory. In addition, ternary alloys of PbSxSe1-x and PbSxTe1-x lattice matching PbS substrate for x = 0.5 compositions are studied. Furthermore, quaternary alloys PbxSn1-xSyTe1-y lattice matching PbS substrate for x and y = 0.5 compositions are studied. The six independent elastic parameters (Cij) are also calculated. Meanwhile, the results data are analyzed in high pressure. The mechanical response of all alloys to pressures 0, 50, and 100 kbar increases progress to decrease in (Cij) in separate rates. The rapprochement between the calculated results and the available published data for these alloys demonstrate that they had worthy accordance at zero pressure and the results at high pressure may be required as an acceptable reference.

Reconciling elasticity tensor constraints from mineral physics and seismological observations: applications to the Earth’s inner core

SUMMARY This study establishes the proper framework in which to compare seismic observations with mineral physics constraints for studies of the inner core by determining how the elements of the elasticity tensor are sampled by the normal modes of the Earth. The obtained mapping between the elements of the elasticity tensor and the seismic wave speeds shows that the choice of averaging scheme used to calculate isotropic properties is crucial to understand the composition of the inner core, especially for comparison with the shear wave speed such as that provided in PREM. For example, the appropriate shear wave speed calculated for an Fe-Ni-Si hcp alloy at inner-core conditions differs from the shear wave speed obtained by taking a Reuss average by as much as $27\, {\rm per\, cent}$. It is also shown for the first time that by combining the isotropic observations based upon normal-mode characteristic frequencies and anisotropic parameters from their splitting, the five independent elastic parameters (A, C, F, L and N) that fully describe a transversely isotropic inner core can be uniquely constrained. The elastic values based upon a variety of mode-splitting studies are reported, and the differences between models from various research groups are shown to be relatively small ($\lt 10\, {\rm per\, cent}$). Additionally, an analogous body-wave methodology is developed to approximately estimate the five independent elastic constants from observations of compressional wave traveltime anomalies. The body-wave observations are utilized to consider the depth dependence of inner-core anisotropy, in particular, the structure of the innermost inner core. Finally, we demonstrate that substantial errors may result when attempting to relate seismically observed P and S wave speeds from Debye velocities obtained through nuclear resonant inelastic X-ray scattering. The results of these experiments should be compared directly with the Debye velocity calculated from seismically constrained elastic constants. This manuscript provides a new set of formulae and values of seismic observations of the inner core that can be easily compared against mineral physics constraints for better understanding of the inner-core composition.

An Equivalent Modeling Method for Honeycomb Sandwich Structure Based on Orthogonal Anisotropic Solid Element

An equivalent modeling method for honeycomb sandwich structure is presented in this paper. Honeycomb core is regarded as an interlayer and orthogonal anisotropic solid elements are used to model it, while the panels of honeycomb sandwich structure are represented by shell elements. This method not only controls model size and ensures computational efficiency, but also solves the problem that two-dimensional model cannot represent the internal stress distribution and local deformation. Based on the orthogonal anisotropy of honeycomb and the actual cellular size, 9 independent elastic parameters of the interlayer are given, so that the physical properties of the interlayer are described completely. In the example, the displacement errors under typical static loadcases are less than 3.12% and the frequency errors of the first six orders are less than 4.07%, compared with the precise model. A modal tapping test was carried out on a payload mounting panel with honeycomb sandwich structure. By comparing the test data with the analysis data of the equivalent model, it was shown that the frequency errors of the first six orders were all within 5%, and the analysis modes were consistent with the experimental fitting modes, which further verified the validity of the equivalent method.

Modeling atomic force microscopy and shell mechanical properties estimation of coated microbubbles.

We present an extensive comparison with experimental data of our theoretical/numerical model for the static response of coated microbubbles (MBs) subject to compression from an atomic force microscope (afm). The mechanics of the MB's coating is described in the context of elastic thin shell theory. The encapsulated fluid is treated as compressible/incompressible pertaining to a gas/liquid, while the thinning of the liquid film between the MB and the afm cantilever is modeled via introduction of an interaction potential and the resulting disjoining pressure. As the external force increases, the experimental force-deformation (f-d) curves of MBs covered with polymer have an initial linear response (Reissner regime), followed by a non-linear curved downwards response (Pogorelov regime) where buckling takes place. On the other hand, the f-d curve for MBs covered with lipid monolayers initially follows the Reissner regime, but buckling is bypassed to a curved upwards regime where internal gas pressure dominates. The elastic properties, namely Young's modulus and shell thickness, for MB's covered with polymer can be estimated by combining the buckling point and the slope of the Reissner regime or the slopes of Reissner and Pogorelov regimes. Comparison of the present model with afm f-d curves for polymer shows satisfactory agreement. The area dilatation and bending moduli are shown to be the appropriate independent elastic parameters of MBs covered with phospholipid monolayers and are estimated by combination of the transition from Reissner to pressure dominated regime. Simulations and experiments in this case are in excellent agreement.

Investigation on anisotropy of mechanical properties of Callovo-Oxfordian claystone

Anisotropic mechanical properties of the sedimentary Callovo-Oxfordian clay rock were investigated with triaxial compression and extension, cyclic compression, and creep tests. The tested cylindrical samples differ in bedding orientation. The compression was accomplished by increasing axial stress at constant radial stress. The extension was carried out vice versa: radial stress was increased while axial stress was kept constant. Moreover, extension was also achieved by simultaneously increasing radial stress and decreasing axial stress under constant mean stress. Five independent elastic parameters were obtained at different deviatoric stresses during cyclic loading-unloading compressions. The deformation and strength were determined in relation to the load direction inclined to bedding and in relation to the load paths. The claystone exhibits elastoplastic stress-strain behaviour with volume compaction until failure. The strength is dependent on both load path and load direction to the bedding. For a given major load direction, the compressive strength is higher than the extensional one and the lowest strength occurs by extension at constant mean stress. In each load path, the maximum strength values are reached parallel and normal to the bedding, while the minimum occurs at the load orientation of about 45° during compression and of about 30° during extension. The elastic stiffness increases with deviatoric loading due to hardening. The deformability is time dependent.

Stress Measurements for an In Situ Stimulation Experiment in Crystalline Rock: Integration of Induced Seismicity, Stress Relief and Hydraulic Methods

An extensive campaign to characterize rock stresses on the decameter scale was carried out in three 18–24 m long boreholes drilled from a tunnel in foliated granite at the Grimsel Test Site, Switzerland. The survey combined stress relief methods with hydrofracturing (HF) tests and concomitant monitoring of induced seismicity. Hydrofracture traces at the borehole wall were visualized with impression packer tests. The microseismic clouds indicate sub-vertical south-dipping HFs. Initial inversion of the overcoring strains with an isotropic rock model yielded stress tensors that disagreed with the HF and microseismic results. The discrepancy was eliminated using a transversely isotropic rock model, parametrized by a novel method that used numerical modelling of the in situ biaxial cell data to determine the requisite five independent elastic parameters. The results show that stress is reasonably uniform in the rock volume that lies to the south of a shear zone that cuts the NNW of the study volume. Stress in this volume is considered to be unperturbed by structures, and has principal stress magnitudes of 13.1–14.4 MPa for σ1, 9.2–10.2 MPa for σ2, and 8.6–9.7 MPa for σ3 with σ1 plunging to the east at 30–40°. To the NNW of the uniform stress regime, the minimum principal stress declines and the principal axes rotate as the shear zone is approached. The stress perturbation is clearly associated with the shear zone, and may reflect the presence of more fragmented rock acting as a compliant inclusion, or remnant stresses arising from slip on the shear zone in the past.

First-principle calculations of structural, electronic, optical, elastic and thermal properties of $$\hbox {MgXAs}_{2}$$ MgXAs 2 ( $$\hbox {X}=\hbox {Si}, \hbox {Ge}$$ X = Si , Ge ) compounds

First-principle calculations on the structural, electronic, optical, elastic and thermal properties of the chalcopyrite $$\hbox {MgXAs}_{2}$$ ( $$\hbox {X}=\hbox {Si}, \hbox {Ge}$$ ) have been performed within the density functional theory (DFT) using the full-potential linearized augmented plane wave (FP-LAPW) method. The obtained equilibrium structural parameters are in good agreement with the available experimental data and theoretical results. The calculated band structures reveal a direct energy band gap for the interested compounds. The predicted band gaps using the modified Becke–Johnson (mBJ) exchange approximation are in fairly good agreement with the experimental data. The optical constants such as the dielectric function, refractive index, and the extinction coefficient are calculated and analysed. The independent elastic parameters namely, $$C_{11}$$ , $$C_{12}$$ , $$C_{13}$$ , $$C_{33}$$ , $$C_{44}$$ and $$C_{66 }$$ are evaluated. The effects of temperature and pressure on some macroscopic properties of $$\hbox {MgSiAs}_{2}$$ and $$\hbox {MgGeAs}_{2}$$ are predicted using the quasiharmonic Debye model in which the lattice vibrations are taken into account.

Ray tracing of turning wave in elliptically anisotropic media with an irregular surface

Seismic ray tracing in anisotropic media with irregular surface is crucial for the exploration of the fine crustal structure. Elliptically anisotropic medium is the type of anisotropic media with only four independent elastic parameters. Usually, this medium can be described by only the vertical phase velocity and the horizontal phase velocity for seismic wave propagation. Model parameterization in this study is described by flexible triangular grids, which is beneficial for the description of irregular surface with high degree of approximation. Both the vertical and horizontal phase velocities are defined in the triangular grids, respectively, which are used for the description of phase velocity distribution everywhere in the model by linear interpolation. We develop a shooting ray tracing method of turning wave in the elliptically anisotropic media with irregular surface. Runge-Kutta method is applied to solve the partial differential equation of seismic ray in elliptically anisotropic media. Linearly modified method is used for adjusting emergent phase angles in the shooting scheme. Numerical tests demonstrate that ray paths coincide well with analytical trajectories in transversely homogeneous elliptically anisotropic media. Seismic ray tracing results in transversely inhomogeneous elliptically anisotropic media demonstrate that our method is effective for further first-arrival tomography in elliptically anisotropic media with an irregular surface.

Anisotropic surface-wave characterization of granular media

In unconsolidated granular media, the state of stress has a major effect on the elastic properties and wave velocities. Effective media modeling of granular packs shows that a vertically oriented uniaxial state of stress induces vertical transverse isotropy (VTI), characterized by five independent elastic parameters. The Rayleigh wave velocity is sensitive to four of these five VTI parameters. We have developed an anisotropic Rayleigh wave inversion that solves for four of the five independent VTI elastic parameters using anisotropic Thomson-Haskell matrix equations that can be used to determine the degree of anisotropy and the state of stress in granular media. The proposed method incorporates an initial model search guided by rock-physics models and dispersion curve inversion that uses a gradient-based solver. We carried out the inversion on a synthetic example and a real data set in which we inverted for in situ elastic moduli of unconsolidated sands at very low effective stress. The synthetic data example results showed that we are able to correctly estimate the granular media’s effective elastic moduli. When the inversion was carried out with real data, the results indicate that the sand at very low effective stress is elastically anisotropic and that it is stiffer along the vertical axis than the horizontal axis. Furthermore, our results are in agreement with past studies related to granular media.

A Constitutive Model for Soft Clays Incorporating Elastic and Plastic Cross-Anisotropy.

Natural clays exhibit a significant degree of anisotropy in their fabric, which initially is derived from the shape of the clay platelets, deposition process and one-dimensional consolidation. Various authors have proposed anisotropic elastoplastic models involving an inclined yield surface to reproduce anisotropic behavior of plastic nature. This paper presents a novel constitutive model for soft structured clays that includes anisotropic behavior both of elastic and plastic nature. The new model incorporates stress-dependent cross-anisotropic elastic behavior within the yield surface using three independent elastic parameters because natural clays exhibit cross-anisotropic (or transversely isotropic) behavior after deposition and consolidation. Thus, the model only incorporates an additional variable with a clear physical meaning, namely the ratio between horizontal and vertical stiffnesses, which can be analytically obtained from conventional laboratory tests. The model does not consider evolution of elastic anisotropy, but laboratory results show that large strains are necessary to cause noticeable changes in elastic anisotropic behavior. The model is able to capture initial non-vertical effective stress paths for undrained triaxial tests and to predict deviatoric strains during isotropic loading or unloading.

High-pressure phases of AlxIn1−xN compounds: First principles calculations

Novel high pressure metastable phases within Al-In-N system have been comprehensively explored via evolutionary algorithm USPEX, under both variable and fixed composition schemes. At 2.5 GPa, only Al7In2N9 (Cm) and Al2In2N4 (I4¯2d) are identified. Candidate structures at 5.0 GPa expand to orthorhombic Al4In2N6 (Cmc21), two tetragonal Al2In2N4 (I4¯2d) plus AlIN4N5 (I4¯) and trigonal AlIn5N6 (R3). Increasing the pressure beyond 5.0 GPa appears to upset the stability of AlxIn1-xN. Using ab initio calculations based on systematic density functional theory, we also outline the electronic, lattice, mechanical and thermodynamic properties of these phases. Band structure, PDOS and Bader analyses show that these alloys are direct band gap semiconductor with certain degree of covalent nature in an ionic bonding. Corrected energy gap ranges from 0.771 eV to 4.995 eV. Mechanical and dynamical stabilities of abovementioned alloys are further revealed by the computations of independent elastic parameters and phonon dispersion curves. All the studied species demonstrate considerable elastic anisotropy and moderate hardness. Their simulated Vickers hardness ranges between 13.3 GPa and 21.6 GPa. Through quasi harmonic Debye model, composition dependence of thermal influenced macroscopic properties is quantitatively analyzed as well.

Interaction of punches on orthotropic half-space

An integral equation of the three-dimensional contact problem for an orthotropic half-space (9 independent elastic parameters in Hooke’s law) is obtained where its kernel does not include integrals, but it depends on the solution of a characteristic binary cubic. The interaction between two identical symmetrically embedded punches is considered for the case of the elliptic paraboloids. Galanov’s method of nonlinear boundary integral equations is used for solving the problem with an unknown contact domain that makes it possible to determine simultaneously the contact domain and the contact pressure. The exact solution to one elliptical punch is used for debugging the computer program. Contact pressures, contact zones and pressing forces are calculated for various orthotropic materials at the specified settlement, base forms of the punches, and relative distances between the punches. The orthotropic body model is applicable for describing lots of materials which are in-demand in the machinery and industry: sulfur, Rochelle salt, wolframite, barite, and various wood species.

On anelliptic approximations for qP velocities in transversely isotropic and orthorhombic media

Anelliptic approximations for phase and group velocities of qP waves in transversely isotropic (TI) media have been widely applied in various seismic data processing and imaging tasks. We have revisited previously proposed approximations and suggested two improvements. The first improvement involves finding an empirical connection between anelliptic parameters along different fitting axes based on laboratory measurements of anisotropy of rock samples of different types. The relationship between anelliptic parameters observed was strongly linear suggesting a novel set of anisotropic parameters suitable for the study of qP-wave signatures. The second improvement involves suggesting a new functional form for the anelliptic parameter term to achieve better fitting along the horizontal axis. These modifications led to improved three-parameter and four-parameter approximations for phase and group velocities of qP-waves in TI media. In several model comparisons, the new three-parameter approximations appeared to be more accurate than previous approximations with the same number of parameters. These modifications also served as a foundation for an extension to orthorhombic media, where qP velocities involved nine independent elastic parameters. As determined by previous researchers, qP-wave propagation in orthorhombic media could be adequately approximated using just six combinations of those nine parameters. We have developed novel six-parameter approximations for phase and group velocities for qP-waves in orthorhombic media. The orthorhombic phase-velocity approximation provides a more accurate alternative to previously known approximations and can find applications in full-wave modeling, imaging, and inversion. The group-velocity approximation is also highly accurate and can find applications in ray tracing and velocity analysis.

Waveform inversion in triclinic anisotropic media—a resolution study

Seismic full waveform inversion (FWI) has been applied to simple elastic problems with certain symmetries, such as isotropic, transverse isotropic or vertical transversely isotropic media. In this study, the FWI concept is extended to the most general anisotropic case with 21 independent elastic material parameters and no symmetry plane (triclinic). Beside a short description of the 3-D finite-difference scheme to solve the forward problem and the FWI optimization algorithm, we present a sensitivity study for a simple anisotropic medium. This test problem consists of a homogenous triclinic anisotropic full space, which contains 21 spatially separated spheres. In each sphere one component of the elastic tensor deviates by 5 per cent from the background medium. The resolution of the different spheres, ambiguities between the different elastic parameters, as well as the effect of the acquisition geometry can be systematically investigated. Due to the high computational costs of the triclinic forward problem a few compromises have to be made regarding the acquisition geometries. Point sources are replaced by plane wave sources which lead to a limitation of incidence angles and therefore a strong decrease in resolution of the nondiagonal elastic tensor components. It is shown that, despite these limitations, a tomographic acquisition geometry would be able to resolve to some extent a monoclinic symmetry via FWI. Restricting the acquisition geometries (e.g. VSP combined with reflection seismic or reflection seismic only) significantly reduces the number of resolvable tensor elements in strict dependence of the covered incidence angles.

First principles study of the structural, electronic, optical, elastic and thermodynamic properties of CdXAs2 (X=Si, Ge and Sn)

The first principles calculations were performed by the linearized augmented plane wave (LAPW) method as implemented in the WIEN2K code within the density functional theory to obtain the structural, electronic and optical properties for CdXAs2 (X=Si, Ge and Sn) in the body centered tetragonal (BCT) phase. Optical features, such as dielectric functions, refractive indices, extinction coefficient, optical reflectivity, absorption coefficients, and optical conductivities, were calculated for photon energies up to 40eV. The six independent elastic parameters (C11, C12, C13, C33, C44 and C66) were evaluated and thermodynamic calculations within the quasi-harmonic approximation were used to give an accurate description of the pressure–temperature dependence of the thermal-expansion coefficient, bulk modulus, specific heat, Debye temperature, entropy and Grüneisen parameters. Based on the semi-empirical relation, we have determined the hardness of the materials for the first time at different temperature and pressure.