Difference In Elastic Constants Research Articles

Determination of elastic constants of a single-crystal topaz and their temperature dependence via sphere resonance method

Water and halogens in ocean floor sediments transported by a descending slab might play important roles in geodynamic processes. Imaging subducted sediments through seismological observations requires a thorough understanding of elastic properties of sediment origin hydrous minerals. Topaz is a sediment origin hydrous mineral, which is formed at the depth of 250–350km on a cold subducting slab. We determined elastic constants and their temperature derivatives of a natural single-crystal of topaz (Al1.97SiO4(F1.56, OH0.42)) at the temperature from 271.5 to 312.7°K by using the sphere-resonance method. Elastic constants at an ambient temperature (T=291.9°K) are C11=281.21(1)GPa, C22=346.23(9)GPa, C33=294.99(9)GPa, C44=108.49(1)GPa, C55=132.47(1)GPa, C66=130.32(1)GPa, C12=121.48(3)GPa, C13=80.94(3)GPa and C23=81.77(2)GPa. Since our sample [Al2SiO4(F1.56,OH0.42)] was relatively rich in fluorine, only small differences in elastic constants can be seen between our sample and fluorine end member. Elastic constants of OH-rich topaz should be experimentally investigated to understand the influence of F-OH substitution on elasticity of topaz. All the elastic constants decrease linearly with increasing temperature. The temperature derivatives are dC11/dT=−0.014(3)GPa/°K, dC22/dT=−0.010(7)GPa/°K, dC33/dT=−0.021(5)GPa/°K, dC44/dT=−0.011(1)GPa/°K, dC55/dT=−0.016(2)GPa/°K, dC66/dT=−0.0101(2)GPa/°K, dC12/dT=−0.0041(6)GPa/°K, dC13/dT=−0.001(2)GPa/°K and dC23/dT=−0.002(1)GPa/°K. The isotropic seismic velocities in topaz are distinctly higher than those in olivine at 10GPa and 300–1400°K. There should be a strong velocity contrast between the overlying mantle and the thin sediment-origin layer at the depth around 300km. A seismological technique like the receiver function technique should be applied to detect a thin layer of topaz in a cold subduction zone.

Micro-Brillouin Scattering Study on Composition Gradient Li Doped KTa1-xNbxO3 Wafer

The composition gradient Li doped KTa1-xNbxO3 wafer was investigated by micro-Brillouin scattering method. The positional dependence of elastic constants was clearly observed. The origin of positional dependence was discussed by the comparison with the temperature variation of elastic constants of the homogeneous crystal. The difference of elastic constants between high and low defect density regions was also studied.

Relating age and micro-architecture with apparent-level elastic constants: a micro-finite element study of female cortical bone from the anterior femoral midshaft

Homogenized elastic properties are often assumed for macro-finite element (FE) models used in orthopaedic biomechanics. The accuracy of material property assignments may have a strong effect on the ability of these models to make accurate predictions. For cortical bone, most macro-scale FE models assume isotropic elastic material behaviour and do not include variation of material properties due to bone micro-architecture. The first aim of the present study was to evaluate the variation of apparent-level (homogenized) orthotropic elastic constants of cortical bone with age and indices of bone micro-architecture. Considerable age-dependent differences in porosity were noted across the cortical thickness in previous research. The second aim of the study was to quantify the resulting differences in elastic constants between the periosteum and endosteum. Specimens were taken from the anterior femoral midshaft of 27 female donors (age 53.4 +/- 23.6 years) and micro-FE (gFE) analysis was used to derive orthotropic elastic constants. The variation of orthotropic elastic constants (Young's moduli, shear moduli, and Poisson's ratios) with various cortical bone micro-architectural indices was investigated. The ratio of canal volume to tissue volume, Ca.V/TV, analogous to porosity, was found to be the strongest predictor (r2(ave) = 0.958) of the elastic constants. Age was less predictive (r2(ave) = 0.385) than Ca.V/TV. Elastic anisotropy increased with increasing Ca.V/TV, leading to lower elastic moduli in the transverse, typically less frequently loaded, directions. Increased Ca.V/TV led to a more substantial reduction in elastic constants at the endosteal aspect than at the periosteal aspect. The results are expected to be most applicable in similar midshaft locations of long bones; specific analysis of other sites would be necessary to evaluate elastic properties elsewhere. It was concluded that Ca.V/TV was the most predictive of cortical bone elastic constants and that considerable periosteal-endosteal variations in these constants can develop with bone loss.

Influence of coherency strain and applied stress upon diffusional ferrite nucleation in austenite: Micromechanics approach

In the quest to develop ultrafine-grained ferrite steels, an external stress is often applied to control the austenite-to-ferrite transformation kinetics. To understand the role of an applied stress in diffusional ferrite nucleation, a micromechanics analysis was performed. It is well known that the austenite-to-ferrite transition is accompanied by a volume increase of up to 9% at absolute zero. The calculation due to the volume change alone shows that a coherent ferrite particle has less strain energy in the Nishiyama–Wasserman (NW) orientation than in the Kurdjumov–Sachs (KS) orientation, and preferred shapes of NW-oriented particles are disk-like, acicular, and spherical in the order. When an applied elastic stress is introduced, two interaction terms arise. The first one is an inhomogeneity term due to the difference in elastic constants between fcc and bcc Fe, and the other is an interaction term between the volume change and the applied stress. The interesting feature of the austenite elastic constants–high bulk modulus but soft shear modulus–combined with the strong elastic anisotropy of ferrite, reveals the diverse influence of applied stress upon the energetics of ferrite formation. In certain applied stress modes, both inhomogeneity and interaction energy terms are found to lower the free energy associated with the ferrite particle, promoting enhanced ferrite nucleation. As an example, coherency strain alone decreases, when compared to a strain-free case, the nucleation rate by an order of 10−21, but its interaction with an appropriate applied stress can increase the rate by a factor of about 30.

Non-Singular Stresses in Gradient Elasticity at Bi-Material Interface with Transverse Crack

Bi-material interfaces are studied with cracks that end perpendicular to the interface. As is well-known, singularities in the stresses appear when classical elasticity is used. Moreover, the nature of the singularity depends on the difference in elastic constants of the two materials. In this paper, the gradient elasticity theory of Aifantis is used to remove these singularities. This is demonstrated for a range of ratios between the two Young’s moduli.

Misfit dislocations in an annular strained film grown on a cylindrical nanopore surface

The critical thickness of a film for the formation of a misfit dislocation in the system of a strained film grown on a nanopore is studied. The influence of the ratio of the shear modulus between the film and the substrate, the misfit strain and the radius of the nanopore on the critical thickness of the film is discussed. New phenomena concerning the critical thickness of the film are obtained due to the difference in elastic constants of the film and the substrate.

Fission of a Multiphase Membrane Tube

A common mechanism for intracellular transport is the use of controlled deformations of the membrane to create spherical or tubular buds. While the basic physical properties of homogeneous membranes are relatively well known, the effects of inhomogeneities within membranes are very much an active field of study. Membrane domains enriched in certain lipids, in particular, are attracting much attention, and in this Letter we investigate the effect of such domains on the shape and fate of membrane tubes. Recent experiments have demonstrated that forced lipid phase separation can trigger tube fission, and we demonstrate how this can be understood purely from the difference in elastic constants between the domains. Moreover, the proposed model predicts time scales for fission that agree well with experimental findings.

Effect of high‐temperature protective coatings on fatigue lives of nickel‐based superalloys

This study concerns MCrAlY coatings (M is Ni, Co or both) sprayed by a vacuum plasma spraying process for protection against high‐temperature corrosion and oxidation of gas turbine components, such as turbine blades and duct segments. The effect of high‐temperature protective coatings on fatigue lives of nickel‐based superalloys were investigated at high temperature under push–pull loading and rotary bending and then compared with uncoated superalloys, such as equiaxed IN738LC, unidirectional solidified CM247LC and single‐crystal CMSX‐2. The high‐cycle fatigue lives of MCrAlY‐coated superalloys at high temperature under push–pull loading showed an inferior performance when compared with the uncoated superalloys. This was because the crack initiation site was different. The high‐cycle fatigue cracks of nickel‐based superalloys initiated at casting cavities which were exposed on the specimen surface, whereas the high‐cycle fatigue cracks of MCrAlY‐coated specimens initiated at interface defects, such as small pores and grid residue, between the MCrAlY coating and the substrate and grew into the MCrAlY coating, and then into the substrate. Similarly, the rotary bending fatigue properties of MCrAlY‐coated superalloys at high temperature showed an inferior performance when compared with the uncoated superalloys. This is because of a high stress due to the higher Young’s modulus of the MCrAlY coating (in comparison with the substrate) being induced at the MCrAlY coating surface. The crack initiation site was on the specimen surface in both cases of the nickel‐based superalloys and the MCrAlY‐coated superalloys, respectively. As a result, it was considered that, for rotary bending tests, the fatigue life reduction was due to the high stress that originated from the difference of elastic constants between the MCrAlY coating and the superalloy. Consequently, in fatigue life design it is necessary to take account of the stress levels in a coating layer.

The chemical driving force for rafting in superalloys

Until recently, all theories of the driving force for rafting have considered the compositions of the two phases to be fixed, although accepting that the rate of rafting might be controlled by diffusion. When plastic flow occurs, the difference in elastic constants becomes negligible. A high energy density builds up in the transverse {gamma} sheets, and rafting occurs by outward motion of the transverse interfaces, reducing the volume which has a high energy density. The analysis considers only the change in enthalpy between two states, one in which the two phases have the compositions which are in equilibrium in the absence of external stress, the external stress has been applied, but no diffusion has occurred, and one in which the two phases have the homogeneous compositions which are in equilibrium under the applied stress. The authors do not attempt to treat the intermediate configuration in which some diffusion has occurred, but the compositions of the phases are inhomogeneous.

A study on coherency strain and precipitate morphologyvia a discrete atom method

Morphological evolution of coherent precipitates is studied by means of a discrete atom method under a plane strain condition with a purely dilatational misfit. The method is predicated upon Hookean atomic interactions and Monte Carlo diffusion and makes no assumption of a specific precipitate shape. Precipitates having elastic constants different from those of the matrix phase are treated in both isotropic and anisotropic elastic systems. Shape evolution is examined under the condition of a constant precipitate size and an isotropic interfacial energy. The results show that in general, an elastically soft precipitate tends to have an equilibrium morphology of low symmetry such as a plate, whereas a hard particle tends to take up a shape of high symmetry such as a circle. Morphological evolution proceeds through dynamic activities of coherency-induced interfacial waves whose wavelength depends upon the difference in elastic constants, precipitate geometry, anisotropy, and diffusion temperature. Coherency-induced interfacial waves seem to be responsible for the protrusions often observed along elastically hard directions in γ′ particles of Ni-base superalloys and also to be a source for fresh ledges for growthvia the ledge mechanism. For a highly nonequilibrium precipitate, first splitting followed by coalescence appears to be a common feature in achieving its equilibrium morphology.

Elastic strain energies of sphere, plate and needle inclusions

The inclusion problem of linear isotropic elasticity is applied to discuss the orientation and shape dependencies of the elastic strain energies of sphere, plate and needle inclusions with general misfit strains. The strain-energy minimization criterion is adopted and difference in elastic constants between the inclusion and the surrounding medium is taken into account. Regardless of the types of the misfit strains, it is generally found that the plate shape is most favorable for soft inclusions. For hard inclusions, however, the minimum-energy shape depends on the signs of the three principal misfit strains, a sphere when the three misfit strains have the same sign and a needle when they have different signs. Some other new and general characteristics on the strain energies of the coherent inclusions are newly found. Stresses and strain energies of incoherent inclusions after the occurrence of plastic accommodation are also evaluated.

Coarsening of precipitate clusters in stress gradients

Several recent theoretical studies and computer simulations have shown that a misfit strain and a uniform applied stress field can induce strong spatial correlations between precipitates during coarsening of two-phase microstructures. The sign and magnitude of the misfit strain and applied stress, and the difference in elastic constants between precipitate and matrix phases influence strongly the resultant microstructure. Most two-phase structural materials, including those used as a matrix in metal-matrix composites, experience external fields that are strongly position dependent owing to, for example, the mechanical loading conditions and system geometry, the polycrystalline nature of many materials, stress concentrators and thermal mismatch between fibers and a metal matrix. The presence of non-uniform external stress fields and stress gradients is also expected to affect microstructural evolution since the elastic field affects the thermodynamic stability of the phases. The scope of this work is to illustrate by computer simulation how such stress gradients can affect microstructural evolution during coarsening.

Elastic States of Inhomogeneous Spheroidal Inclusions

Eshelby’s inclusion problem of linear isotropic elasticity is applied to evaluate the elastic strain energies of general spheroidal inclusions with various misfit strains. The strain-energy minimization criterion is adopted and difference in elastic constants between the inclusion and the surrounding matrix is taken into account. It is found that not necessarily one of the three representative shapes of the spheroid, i.e., plate, sphere and needle, but also more general oblate and prolate spheroids can be the most favorable shape to minimize the strain energy. In addition, some features and characteristics involved in the inclusion problem are newly found.

Anisotropic elasticity study of the critical thickness of an epilayer on a substrate with different elastic constants

The critical thickness of an epilayer on a substrate with different elastic constants is investigated by following Stroh’s treatment of anisotropic elasticity [Philos. Mag. 3, 625 (1958)]. A closed formula is derived to calculate the critical thickness and an exact solution may involve numerical evaluation of the equation. The results indicate that the self-energy of the dislocation is controlled by the soft phase between the epilayer and the substrate, while the interaction energy depends only on the elastic constants of the thin film. It is easier for a dislocation to be formed if the substrate is softer than the film, and consequently the critical thickness is smaller. On the other hand, a soft epilayer can have a large thickness without any mismatch dislocation. Explicit equations are given here for the {100}, {110}, and {111} epitaxial planes. The system of a GexSi1−x epilayer on a Si substrate was taken as an example to demonstrate the influence of the difference in elastic constants on the critical thickness. Even though the difference between the elastic constants of the epilayer and the substrate is not very large, ignoring this difference can cause a relative error over 20% in calculation of the critical thickness. For this system, a simplified equation yields sufficiently accurate results.

Triaxial stress distribution in a textured titanium nitride coating

The elastic constants and residual stresses of a physical vapor deposition, highly (111) textured TiN coating were measured using X-ray diffraction Cu Kα and Cr Kα radiation. No significant difference in elastic constants was found on the (222) and (333)–(511) planes which were determined from sampling depths of 1.8 μm and 5.3 μm respectively. The depth profiles of triaxial residual stresses in the coating were characterized using Cr Kα radiation. Within the coating, the principal stresses in the coating plane exhibit a symmetric biaxial state, varying with the coating depth from 2400 to 3000 MPa in compression, whereas the principal stress normal to the coating surface is between 107 and 650 MPa in tension. A steep stress gradient is observed at the near-surface region in the coating.

Borehole flexural modes in anisotropic formations

A perturbation method of solution is an efficient way of analyzing elastic wave propagation along a borehole in anisotropic formations. The perturbation model allows us to calculate changes in the modal dispersion curves caused by the differences in elastic constants between the anisotropic formation of interest and a reference, or unperturbed, isotropic formation. The equivalent isotropic constants in the reference formation are obtained from the appropriate compressional‐and shear‐wave velocities for the selected propagation and polarization directions of the flexural mode. This choice of the unperturbed solution means that the required perturbation is minimal, resulting in enhanced accuracy of the perturbed solution. Computational results are presented for the dispersion curves of borehole flexural waves in a transversely isotropic (TI) formation as a function of borehole deviation from the TI symmetry axis. In addition, radial distributions of displacement and stress fields associated with the flexural wave are obtained as a function of frequency. These provide qualitative information on the radial depth of investigation with flexural wave logging. The flexural wave excitation function is a measure of the energy that a source converts to flexural motion. We deduce an expression for the flexural wave excitation and show that its bandlimited characteristic is influenced by both the borehole diameter and formation parameters. From the dispersion curves and excitation functions, we can compute the flexural waveforms caused by a dipole source with arbitrary orientation in the borehole. In the numerical computations, we have used the unperturbed mode shapes for an equivalent isotropic medium together with the perturbed dispersion relations caused by the formation anisotropy.

A comparison between calculated and observed elastically induced precipitate shape transitions in a Cu-2 At. Pct Co alloy

Elastic and interfacial energy calculations have been performed for misfitting spherical-, cuboidal-, octahedral-, and tetrakaidecahal-shaped precipitates in a face-centered cubic (fcc) matrix ac- counting to first order for the difference in elastic constants between precipitate and matrix phases. The energy calculations predict a number of possible precipitate shape transitions with increasing particle size, depending upon the material parameters and anisotropy of the interfacial energy. Predicted shape transitions were compared with experimentally observed transitions of coherent, Co-rich precipitates in a low volume fraction Cu-2 at. pct Co alloy. The predicted octahedron-to-tetrakaidecahedron transition was not observed. Rather, the corners of the octa- hedra were observed to become progressively more rounded with increasing particle size. Pos- sible sources of disagreement between calculations and experiment are discussed.

The force on a screw dislocation due to a series of layers of alternating shear modulus

The force on a screw dislocation lying parallel to a series of interfaces separating materials of differing shear moduli is evaluated using the method of images. On the basis of this calculation, suitable materials systems that are able to repel dislocations by virtue of their larger elastic constants are suggested. By using such materials as components of strained-layer superlattices, it may be possible to divert dislocations into an interface using both the strain field and the difference in elastic constants. This work will have relevance for highly lattice-mismatched systems (e.g. GaAs-on-Si) in which 'dislocation filtering' is required.

Precipitate shape transitions during coarsening under uniaxial stress

Precipitate shape transitions of elastically misfitting inclusions in the presence of a uniaxial stress field are examined for cubic materials using simple bifurcation theory. Both size-induced shape transitions that occur during growth of the precipitate under zero or constant stress and stress-induced shape transitions resulting from changes in the external stress field at constant precipitate volume and misfit are identified. Using elastic fields valid for small differences in elastic constants between precipitate and matrix, a dimensionless stress parameter and precipitate volume are identified from a Landau-type expansion that indicate the type and nature of permissible shape transitions. These dimensionless parameters incorporate the interfacial energy density, difference in elastic constants between precipitate and matrix, precipitate misfit, precipitate volume and external stress field into simple algebraic relationships. They indicate under what combination of material parameters a shape transition might be expected and whether the transition is continuous or discontinuous. Results of the model are applied using material parameters from a nickel-based alloy.

The tensile deformation and fracture of Al-“Saffil” metal-matrix composites

In this paper we have studied the tensile deformation and fracture of aluminium alloy composites containing “Saffil” δ-Al2O3 fibres. The Bauschinger effect has been used to measure internal stress and shows that the tensile behaviour of these materials is determined by the development of these stresses due to the plastic flow of the matrix, differences in elastic constants of the two phases and residual thermal stresses developed during fabrication. Good agreement is obtained with the theoretical predictions derived in an earlier paper. Fracture is observed to occur by the growth of cracks from failed fibres until a crack large enough to nucleate catastrophic failure is formed.