Variation Of Elastic Constants Research Articles

Variations of Elastic Constants on Al-Si Alloys by Solid-solutioning under High Pressure

Supersaturated solid-solution alloys of the Al-Si system up to 15at%Si were prepared by means of high pressure and high temperature techniques. A cylindrical specimen with a size of φ 6×6 mm for the solid-solutioning treatment was held under the pressure of 5.4 GPa and the temperature of 600°C for 10 hours, and then quenched to room temperature. The elastic constants of thus obtained specimens were measured by a three-dimensional resonance method. Young's modulus E and shear modulus G of the solid solutions were quite smaller than those of the two phase alloys, and they had lower values depending on Si-contents. For example, E=77.1 GPa of the two phase Al-10at%Si alloy dropped to E=62.0 GPa of the solid solution. Density variations of the specimens were also discussed.

Variations of elastic constants on Al-Si alloys by solid-solutioning under high pressure.

Supersaturated solid-solutioning alloys of the Al-Si system up to 15at%Si were prepared by means of high pressure and high temperature techniques. A cylindrical specimen with a size of φ6×6 mm for the solid-solutioning treatment was preserved under the pressure of 5.4 GPa and the temperature of 600°C for 10 hours, and then quenched to room temperature. The elastic constants of thus obtained specimens were measured by a three dimensional resonance method. The Young's modulus E and shear modulus G of the solid solutions were very smaller than that of the two phase alloys, and also were varied to lower values with Si-contents. For example E=77.1 GPa of the two phase Al-10at%Si alloy was fallen to E=62.0 GPa of the solid solution. Density variations of the specimens were also discussed.

Fragmentation of rock by geometrical simulation of crack motion—I: Independent planar cracks

We employ numerical simulation and photographic reproduction of a manifold of cracks to study crack motion under stress, leading to fracturation. In our model (which is two-dimensional) the rock contains one planar crack, or a number of such cracks which may be randomly distributed in the medium, or else may be a set of radial cracks (such as would arise near a cylindrical borehole). The detonation is simulated by a time and stress varying wave emerging from one (or several) boreholes. The sense (tensile or compressive) and magnitude of the instantaneous pressure at each crack-tip determine the motion of the crack, in particular its direction. The finiteness of the rock is taken care of by the presence of a set of image-boreholes. Alternatively, a regular array of boreholes is “fired”, as in bench-blasting of rocks. Our technique of visual representation illustrates the motion of the cracks as the pressure waves progress in the rock and exhibits the development of “damage” up to the stage of fracture. One sees clearly the effect of spalling (i.e. formation of tensile waves by reflection of compressive waves from the boundary) on crack growth and the bending of cracks as a boundary is approached. In the vicinity of the borehole the strongly compressive wave causes in the initial stages of the detonation a violent zig-zagging of the cracks; this results in the observed crushed-zone near the borehole. Effects on crack-growth of boundaries, time-delays and variations in elastic constants of the medium are examined.

Van der waals' forces in face centred cubic cobalt

AbstractFor analysing the role played by the Van der‐Waals' forces in the phonon dispersion of metals, four different models are developed for the face‐centered configuration. It is found that these forces enhance the efficacy of the models. The model parameters, calculated by the knowledge of the measured frequency, are free from the errors originating from the variation of elastic constants with temperature and the nature of the forces dominant in elastic behaviour. The least attended metal, face centered cubic cobalt, is chosen for the present analysis.

Low-temperature elastic-constant anomalies in Fe-Cr-Ni-Mn alloys

By measuring velocities of longitudinal and shear waves we determined accurately the temperature variation of elastic constants — Young modulus, shear modulus, bulk modulus, Poisson ratio — of a polycrystalline austenitic steel: Fe-5Cr-26Mn. Versus temperature, near 335 K, this alloy exhibits an anomalous elastic-constant transition. In similar alloys, a magnetic-susceptibility cusp occurs at the transition temperature, T c,. The bulk modulus shows a maximum at a higher temperature. Below T c, the Poisson ratio behaves anomalously, increasing with decreasing temperature.

The variation of elastic constants of nickel-iron single crystal alloys from 78.76 TO 300 K

The temperature dependences of the three independent adiabatic elastic stiffness coefficients c 11, C 12 and C 44 of four Ni-Fe single crystal alloys with compositions from 50 to 90 wt.% of Ni have been studied by the ultrasonic pulse superposition technique from 78.76 to 300 K. The elastic constants of the alloys exhibit normal temperature dependencies in the range of temperature investigated. A good fit of the experimental data to Lakkad's phenomenological model for the temperature dependence of elastic constants was obtained.

複合材料積層設計における面内弾性特性の変動

Two important problems are discussed in this paper with respect to the material design of composite laminates. One of them is the selection of optimal constitution of laminates from the viewpoint of the statistical variation of orientation angle. The other is concerned with the problem of the degree of accuracy of lay-up fabrication process.The variation sensitivity of in-plane elastic properties was studied statistically. The effect of the variation of orientation angle on effective engineering elastic constants is investigated by the partial derivative and Monte Carlo simulation technique. In addition, the effect of number of plies on standard deviation of in-plane elastic constants was studied. These sensitivity studies are summarized as follows.(1) The difference in constitution of laminates considerably affects the sensitivity of the variation of effective engineering elastic constants to the orientation angle. Therefore, from these equivalent constitutions, one can select the optimal constitution taking account of statistical sensitivity of in-plane elastic constants.(2) The procedure to evaluate the influence of the number of plies on the standard deviation of elastic constants was presented.(3) The proposed evaluation procedure suggests a method how to fabricate the composites having the variation of elastic constants less than a required value.

Elastic constants of bismuth polycrystal containing tin, lead, antimony and tellurium impurities

Investigations of ultrasonic velocity and internal friction are carried out in bismuth polycrystals containing Sn, Pb, Sb and Te impurities in the concentration range of 0–1 atomic percent using the composite oscillator technique, and elastic constants are estimated from velocity and density data. The variations of elastic constants are interpreted in terms of changes of lattice parameter, valency of the impurity, the electron-atom ratio, atomic size and Fermi energy. It is observed that the elastic constants are less affected in Bi-Te alloys compared with Bi-Sb, Bi-Pb and Bi-Sn alloys in the above concentration range.

Temperature variation of elastic constants of pyrope-almandine garnets.

Elastic constants of three natural pyrope-almandine garnets (Fe/(Mg+Fe)=0.18, 0.48 and 0.58) were measured by an RPR method in the temperature range between -190° and 80°C. The temperature derivative of bulk modulus ∂Ks/∂T does not depend much on the composition, but those of shear constants ∂C44/∂T and∂Cs/∂T [Cs=(C11-C12)/2] increase considerably with increase in the Fe-content. The temperature derivatives of the elastic constants (kbar/deg) of pyrope and almandine at room temperature are determined by extrapolation: From the measured elastic parameters of garnet, the material of the high velocity layer (Vp=8.6km/s and Vs=4.9km/s) in the oceanic lithosphere is interpreted as eclogite or garnet peridotite constituted dominantly of garnet.

Orbit-lattice interaction in Γ8quartets. II. A rigorous test for various models of the interaction for Er3+:MgO and Dy3+:CaF2

For pt.I see ibid., vol.9, p.3819 (1976). The authors discuss the experimental results on the effect of uniaxial stress on the EPR of the Gamma 8 ground states. Baker and Currell (in pt.I) used the data to calculate eight of the nine unknown parameters of the orbit-lattice Hamiltonian HOL. Such a large number of parameters can be measured only for the few systems which exhibit EPR in a Gamma 8 state. Hence, to make quantitative calculations involving HOL it is imperative to be able to express HOL in terms of fewer parameters. The data of Baker and Currell have been used to make a rigorous test of two models for reducing the number of independent parameters of HOL: (1) the generalised electrostatic model of Buisson and Borg (1970) and (2) the superposition model of Newman (1971). These models have also been modified to relax the relationship between HOL and the static crystal field, and to make allowance for local variations of elastic constants and other local ionic displacements. Within the context of a pure LS coupling approximation for the two ions, neither the simple nor the modified form of either model is adequate to describe HOL.

Variations in the mechanical properties of glass induced by high-pressure phase change

The elastic property of As 2Se 3 glass has been determined under high pressure from the data of the sound wave velocities and density. The sound velocities in As 2Se 3 glass have been measured under hydrostatic pressure up to 20 kb by means of ultrasonic interferometry. Sound velocities for both the longitudinal and shear mode, bulk modulus and shear modulus were found to increase linearly with pressure up to 8 kb. Above 9 kb the pressure derivatives diminished. The acoustic Grüneisen parameter shows an abrupt decrease at about 9 kb. Comparisons were made among the observed values of the volume ratio V/ V 0, the isothermal bulk modulus K T and the pressure derivative K′ T , and predictions based upon the Murnaghan equation or the lattice theory equation of state for solids. It is concluded that the abrupt changes in the parameters are probably due to a phase transition in glass at 9–10 kb. The clear resolution of the bulk modulus on both sides of the transition enables one to compute all the mechanical properties of the high-pressure phase, using the law of corresponding states. The decrease of K′ T from 7.86 to 6 in the high-pressure phase change is consistent with the change of density from 4.60 to 4.75. A modified equation of state has been proposed to describe the elastic property of glass. This equation includes a variable parameter C which reflects the volume change in glass due to bond angle change with pressure. By this equation the pressure dependence of C can also explain the elastic anomaly of silica glass as well as the variation of elastic constants of As 2Se 3 glass across the phase transition pressure.

Temperature variation of elastic constants of single-crystal forsterite between -190.DEG. and 400.DEG.C.

Elastic constants of forsterite Mg2SiO4 were measured by the rectangular parallelepiped resonance (RPR) method at temperatures between -190° and 400°C. Elastic constant Cij(Mbar) and their temperature derivatives ∂Cij/∂T (kbar/deg) are:The adiabatic bulk modulus Ks=1.292Mbar (25°C) and its temperature derivatives ∂Ks/∂T=-0.160kbar/deg (20°-400°C) are obtained by the Voigt-Reuss-Hill scheme.In the temperature range from 100 to 1000K, the Gruneisen constant calculated from the measured bulk modulus is virtually constant at 1.1±0.1, while the Gruneisen-Anderson parameter δs, defined as -(∂ In Ks/∂T)/α(α: the coefficient of volume thermal expansion), decreases with increase of temperature, but tends to approach a constant value of 4.0±0.1 above 500K.

Singularities in Nematics: The Effect of Elastic Constant Variations

Abstract In this paper we report typical calculations of the variations in the deformation energy, interaction energy and orientational pattern of + 1, - 1, +½ and -½ singularities in the neighbourhood of a nematic-smectic transition in CBOOA. We find the interaction between +½ and -½ singularities to increase much more rapidly than that between + 1 and - 1 as the smectic point is reached, indicating that the former have a greater probability of annihilating each other near the transition.

Elastic constants of mixed crystals of GaAs xP 1 − x

Expressions for elastic constants of mixed crystals of GaAs x P 1P 1 − x as a function of composition have been derived on the random element isodisplacement model using the long wave method. Calculations have been made on the assumptions (i) that the second-neighbour As-P interaction is the average of the As-As and P-P interactions, or (ii) the As-P interaction is that obtained from the infrared reflectivity data. The variation of elastic constants with concentration is almost linear in the first case; but in the second case it shows a well-defined maximum. The Debye temperature also shows a non-linear variation with composition in the second case, though the deviation from linearity is less pronounced than with the elastic constants.

Variation of elastic constants of partially ordered Cu3Au alloy with temperature

Variation of elastic constants of partially ordered Cu3Au alloy with temperature

Surface acoustic waves in strained media

Variations of elastic constants and density with strain and the introduction of stresses can significantly affect the propagation velocity of surface acoustic waves. Digital computation using iterative methods is employed to obtain the variations of propagation velocity with strain. It is found that while the individual contributions of the initial stresses, changes in elastic constants, and changes in density with strain can be considerable, their combined effect in some materials may be reduced by mutual cancellation. Computed results for YX quartz YZ LiNbO3 are compared with experimentally measured values.

Free torsional vibrations of an elastic cylinder with laminated periodic structure

The theory of torsional vibrations of a circular cylinder, with a periodic variation of elastic constants and density normal to the axis of the cylinder, is developed in terms of Floquet waves. Floquet waves are quasi-periodic waves, whose amplitude profile has the same periodicity as that of the material and repeats with the periodicity of the cell. Using Floquet's theory, the dispersion spectrum is obtained for time-harmonic waves propagating in a laminated cylinder with periodic structure. It is shown that the dispersion spectrum has a band structure, consisting of passing bands and stopping bands. Motion in the case of grazing incidence, and motion at the end of the zones is discussed. It is also shown that as the radius of the cylinder tends to infinity, the torsional waves in a circular cylinder degenerate to SH-waves in laminated plates.

Ultrasonic studies on bismuth-antimony alloys

Investigations of ultrasonic velocity and absorption are carried out on bismuth-antimony alloys with a concentration of 0-60 at%antimony using the composite oscillator method and the pulse method respectively. The elastic constants are estimated from velocity and density measurements. Anomalous variations of elastic constants and ultrasonic attenuation are observed at a certain critical concentration region, and these are interpreted in terms of changes in lattice parameter and carrier concentration. Studies of the temperature variation of Young's modulus and ultrasonic attenuation are also carried out at some of the concentrations. The observed anomalies in ultrasonic properties confirm the peculiar electronic properties of these alloys in that concentration range.

Elastic Properties of Metals and Alloys, I. Iron, Nickel, and Iron-Nickel Alloys

A comprehensive compilation is given of elastic properties of iron-nickel alloys. When sufficient data exist, preferred values are recommended. This compilation covers, besides pure iron and pure nickel, the entire binary composition range, both b.c.c. and f.c.c. phases. Elastic constants included are: Young's modulus, shear modulus, bulk modulus (reciprocal compressibility), Poisson's ratio, and single-crystal elastic stiffnesses, both second-order and higher-order. Data are compiled for variation of elastic constants with composition, temperature, pressure, magnetic field, mechanical deformation, annealing, and crystallographic transitions. An overview is given from the vantage points of the electron theory of metals, elasticity theory, and crystallographic theory. Also included are discussions of iso-thermal and adiabatic elastic constants, interrelationships among engineering elastic constants, computation of the latter from single-crystal elastic stiffnesses, and similar topics. Where key data have not been measured, they were generated if possible from existing data using standard formulae. Other gaps, both theoretical and experimental, in the elastic properties of iron-nickel alloys are indicated. A few theoretical results are included where experimental data are nonexistent or scarce. A semantic scheme is proposed for distinguishing elastic constants of solids.

Temperature variations in elastic constants and Debye temperature of single crystals of a KBr-KI solid solution

Temperature variations in elastic constants and Debye temperature of single crystals of a KBr-KI solid solution