Components Of Elastic Stiffness Tensor Research Articles

Elastic Coefficients of β-HMX as Functions of Pressure and Temperature from Molecular Dynamics

The isothermal second-order elastic stiffness tensor and isotropic moduli of β-1,3,5,7- tetranitro-1,3,5,7-tetrazoctane (β-HMX) were calculated, using the P21/n space group convention, from molecular dynamics for hydrostatic pressures ranging from 10−4 to 30 GPa and temperatures ranging from 300 to 1100 K using a validated all-atom flexible-molecule force field. The elastic stiffness tensor components were calculated as derivatives of the Cauchy stress tensor components with respect to linear strain components. These derivatives were evaluated numerically by imposing small, prescribed finite strains on the equilibrated β-HMX crystal at a given pressure and temperature and using the equilibrium stress tensors of the strained cells to obtain the derivatives of stress with respect to strain. For a fixed temperature, the elastic coefficients increase substantially with increasing pressure, whereas, for a fixed pressure, the elastic coefficients decrease as temperature increases, in accordance with physical expectations. Comparisons to previous experimental and computational results are provided where possible.

Wave dispersion in nonlocal anisotropic macro/nanoplates made of functionally graded materials

By means of a higher-order shear deformation theory (HSDT) coupled with the nonlocal elasticity theory, this work is attempted to elucidate the size-dependent behavior of functionally graded (FG) Anisotropic macro/nanoplates. In the HSDT, the displacement field is modeled as a combination of an exponential formula and a trigonometric function. This type of theory has only five unknowns but can have precise results in various conditions for thick and thin plates. By employing Hamilton’s principle, the governing equations are derived for FG anisotropic macro/nanoplates. In the case of FG anisotropic materials, all components of elastic stiffness tensor are involved, and the material gradation is considered to be arbitrary in the thickness direction. In order to attain the analytical dispersion relations, an eigenvalue problem is solved. The model’s validation is evaluated by comparing authors’ results with other papers available in the open literature. The influences of nonlocal parameter, exponential factor, geometrical shape, and wave number on the circular frequencies and phase velocities are inspected. From the knowledge of authors, it is the first time that the small-scale influences for FG anisotropic macro/nanoplates based on the present higher-order shear deformation theory are investigated.

Pure elastic stiffness of sand represented by response envelopes derived from cyclic triaxial tests with local strain measurements

The elastic stiffness of a fine sand at small to moderate strains ( $$\varepsilon \le 2 \times 10^{-4}$$ ) has been studied based on cyclic triaxial tests on cube-shaped samples with local strain measurements. Three samples with different densities (between very loose and very dense) were tested at up to 21 average stresses in succession (variation of mean pressure and stress ratio). At each average stress cycles were applied in six different directions, as a result of a simultaneous sinusoidal oscillation of the axial and lateral effective stress. Ten preconditioning cycles with larger strain amplitudes $$\varepsilon ^{\mathrm{ampl}}= 3 \times 10^{-4}$$ per polarization were intended to induce a shakedown. An almost purely elastic response was then observed during the 5 subsequent cycles per polarization with lower strain amplitudes $$\varepsilon ^{\mathrm{ampl}}= 2 \times 10^{-4}$$ . The stress and strain paths measured during the latter ones were used to derive the components of the incremental elastic stiffness tensor $${\mathsf{E}}= \{E_{PP}, E_{PQ}, E_{QP}, E_{QQ}\}$$ at a given average stress. The analysis of $${\mathsf{E}}$$ is based on all 60 reversal points and the corresponding stress and strain points lying in a certain distance $$\Delta \varepsilon$$ (e.g. $$10^{-4}$$ ) from the reversals. A graphical presentation of $${\mathsf{E}}$$ by means of response envelopes is shown. They are used to calibrate the hyperelastic stiffness incorporated into a new hypoplastic constitutive model.

Experimental and Theoretical Study of Water Retention Effects on Elastic Properties of Opalinus Shale

Understanding of shales elastic properties behaviour with saturation changes is important for geological storage of nuclear waste, CO2 sequestration as well as for development of conventional and unconventional shale oil and gas reservoirs. Existing data describing effects of saturation on elastic properties of shales are sparse and contradictory. To improve understanding of the effects of changing water content on elastic properties in shales, we conduct an experimental study on Opalinus shale samples. We measure vertical and horizontal ultrasonic P- and S-wave velocities of the same set of samples with controlled water content. The measured velocities are used to calculate components of elastic stiffness tensor in the shale at different saturations assuming its vertical transverse isotropy. Obtained results show increasing C11 and decreasing C33 with drying of the samples. Moreover, we observe 80% and 60% increase of shear moduli C44 and C66, respectively, with reduction of the water content from 5.5% weight in the preserved state to 0.3%. Conventional rock physics models are not designed to explain the observed dynamics. Here we perform a theoretical investigation of the influence on the shale moduli of following factors: (1) mechanical softening of the rock with the decrease of water saturation; (2) shrinkage of clay leading to reduction of porosity; (3) chemical hardening of clay particles; and (4) enhancing stiffness of contacts due to removing of water between clay particles.

Temperature dependent elastic properties of γ-phase U – 8 wt% Mo

Polycrystalline elastic moduli and stiffness tensor components of γ-phase U – 8 wt% Mo have been determined by resonant ultrasound spectroscopy in the temperature range of 25–650 °C. The ambient temperature elastic properties are compared to results measured via other experimental methods and show reasonable agreement, though there is considerable variation of these properties within the literature at both the U – 8 wt% Mo composition and as a function of Mo concentration. The Young's modulus of U – 8 wt% Mo measured in this study decreases steadily with temperature at a rate that is slower than trends previously observed at similar Mo concentrations, though the difference is not statistically significant. This first measurement of the temperature dependent elastic stiffness tensor of a polycrystalline U-Mo alloy clarifies that the behavior of the Young's modulus is due to a strongly weakening C11 polycrystalline stiffness tensor component, along with milder decreases in C12 and C44. The unique partially auxetic properties recently predicted for single-crystalline U-Mo are discussed in regard to their possible impact on the polycrystalline behavior of the alloy.

Low-temperature phase transition in γ-glycine single crystal. Pyroelectric, piezoelectric, dielectric and elastic properties

Temperature changes in the pyroelectric, piezoelectric, elastic and dielectric properties of γ-glycine crystals were studied in the range 100 ÷ 385 K. The pyroelectric coefficient increases monotonically in this temperature range and its value at RT was compared with that of other crystals having glycine molecules. A big maximum in the d14 component of piezoelectric tensor compared by maximum in attenuation of the resonant face-shear mode were observed at 189 K. The components of the elastic stiffness tensor and other components of the piezoelectric tensor show anomalies at this temperature. The components of electromechanical coupling coefficient determined indicate that γ-glycine is a weak piezoelectric. The real and imaginary part of the dielectric constant measured in the direction perpendicular to the trigonal axis show the relaxation anomalies much before 198 K and the activation energies were calculated. These anomalies were interpreted as a result of changes in the NH3+ vibrations through electron-phonon coupling of the so called “dynamical transition”. The anomalies of dielectric constant ε*11 and piezoelectric tensor component d14 taking place at 335 K are associated with an increase in ac conductivity caused by charge transfer of protons.

Measurements of elastic and electrical properties of an unconventional organic shale under differential loading

We have developed an experimental approach to simultaneously measure the stress dependence of ultrasonic wave velocities at 1 MHz, and therefore the components of the undrained elastic stiffness tensor, as well as the components of the complex conductivity tensor in the frequency range from 100 mHz to 10 Hz. We performed the experiments on a cylindrical core sample from the Haynesville Formation (porosity of approximately 0.08, bound water excluded, and clay content, mostly illite, approximately 30–40 wt%). We performed experiments under controlled confining and pore fluid pressures, achieving differential pressure states representative of autochthonous reservoir conditions. Directional measurements were made using independent acquisition arrays (piezoelectric crystals and nonpolarizing electrodes) distributed azimuthally on the core sample external surface, the bedding plane being along the axis of the cylindrical core sample. Ultrasonic waveforms were recorded on a high-resolution oscilloscope, and complex impedance spectra were recorded with a four-electrode acquisition system using an impedance meter with precision of 0.1 mrad. Experiments were repeated under drained and undrained conditions, over loading and unloading sequences for fully water-saturated conditions. Measurements show strong stress dependence on ultrasonic and complex conductivity measurements, which can be ascribed to the opening and closing of cracks within the samples. The data were used to estimate the anisotropic electrical and elastic effective pressure coefficients of the core sample, resulting in effective stress coefficients smaller than one for both elastic and electrical properties. These effective stress coefficients were also smaller for the electrical and acoustic properties once the cracks have closed. The anisotropy ratio for the components of the complex conductivity tensor was on the order of 30, whereas it was only 2 for the compressional components of the stiffness tensor ([Formula: see text]).

A homogenisation-based continuum damage mechanics model for cyclic damage in 3D composites

AbstractThis paper develops a 3D homogenisation based continuum damage mechanics (HCDM) model for fibre-reinforced composites undergoing micromechanical damage under cyclic loading. Micromechanical damage in a representative volume element (RVE) of the material occurs by fibre-matrix interfacial debonding, which is incorporated in the model through a hysteretic bilinear cohesive zone model. The proposed model expresses a damage evolution surface in the strain space in the principal damage co-ordinate system or PDCS. PDCS enables the model to account for the effect of non-proportional load history. The material constitutive law involves a fourth order orthotropic tensor with stiffness characterised as a macroscopic internal variable. Cyclic damage parameters are introduced in the monotonic HCDM model to describe the material degradation due to fatigue. Three dimensional damage in composites is accounted for through functional forms of the fourth order damage tensor in terms of components of macroscopic strain and elastic stiffness tensor. The HCDM model parameters are calibrated from homogenisation of micromechanical solutions of the RVE for a few representative cyclic strain histories. The proposed model is validated by comparing results of the HCDM model with pure micromechanical analysis results followed by homogenisation. Finally, the potential of cyclic HCDM model as a design tool is demonstrated through macro-micro analysis of cyclic damage progression in composite structures.

Ultrasonic study of the elastic and nonlinear acoustic properties of single-crystal triglycine sulphate in the ferroelectric and paraelectric phases

Abstract An experimental ultrasonic study has been made of the elastic and nonlinear acoustic behaviour of single-crystal triglycine sulphate (TGS) in both the ferroelectric and paraelectric phases. The ultrasonic wave velocities for 12 modes of vibration propagated along the [100], [010], [001] and [101] directions have been measured as functions of temperature in the range 293-338 K, which covers the Curie temperature T C, and hydrostatic pressure up to 0.2 GPa at selected temperatures in the same temperature range. The results provide ten out of the 13 elastic stiffness tensor components C ij of TGS (the exceptions being C 12, C 23 and C 25) and their hydrostatic-pressure derivatives as a function of temperature. Marked anomalies are observed in the C ij (T) in the vicinity of T c, with that for C 33 being the most pronounced. The elastic stiffness tensor component C 22, which determines the temperature dependence of the velocity of longitudinal ultrasonic waves propagated along the ferroelectric axis,...

Acoustic Vibrational Properties and Fractal Bond Connectivity of Praseodymium Doped Glasses

The velocities of longitudinal and shear ultrasonic waves propagated in the (Pr2O3)x(P2O5)1-x glass system, where x is the mole fraction of Pr2O3 and (1 - x) is the mole fraction of P2O5, have been measured as functions of temperature and hydrostatic pressure. The temperature dependencies of the second order elastic stiffness tensor components (SOEC) CS IJ , which have been determined from the velocitydata between 10 and 300 K, show no evidence of phonon mode softening throughout the whole temperature range. The elastic stiffnesses increased monotonically, the usual behaviour associated with the effect of the phonon anharmonicityof atomic vibration. At low temperatures, strong phonon interactions with two-level systems have been observed. The ultrasonic wave attenuation of longitudinal and shear waves is dominated bya broad acoustic loss peak whose height and peak position are frequencydependent. This behaviour is consistent with the presence of thermally activated structural relaxation of the two-level systems in these glasses. The fractal bond connectivity of these glasses, obtained from the elastic stiffnesses determined from ultrasonic wave velocities, has a value between 2.32 to 2.55, indicating that their connectivitytends towards having a threedimensional character. The hydrostatic pressure dependencies of longitudinal ultrasonic waves show a slight increase with pressure. As a consequence, the hydrostatic pressure derivatives ( CS11/ P)P=0 of the elastic stiffness CS11/ and (BS/P)P=0 of the bulk modulus BS of (Pr2O3)x(P2O5)1-x glasses are positive. The bulk modulus increases with pressure, and thus these glasses stiffen under pressure, which is associated with the normal elastic behaviour. The GrÜneisen parameter approach has been used to quantifythe vibrational anharmonicityof the long-wavelength acoustic phonons in these glasses.

High-pressure ultrasonic study of monocrystalline Cr in the antiferromagnetic and paramagnetic phases

Abstract An experimental ultrasonic study has been made of the elastic and nonlinear acoustic behaviour of single-crystal Cr in both the antiferromagnetic (AFM) spin-density wave (SDW) and paramagnetic (PM) phases. The velocities of the ultrasonic waves propagated along the [100] and [110] directions have been measured as a function of hydrostatic pressure up to 0.2 GPa at fixed temperatures in the range from room temperature up to 400 K, which covers the Néel temperature (TN). The results provide complete sets of the elastic-stiffness tensor components and their pressure derivatives as a function of temperature. The longitudinal acoustic-mode Griineisen parameters are positive at room temperature, but they become negative in the range from about 300 K to TN indicating longitudinal-mode softening due to a strong magnetoelastic interaction between the SDW and long-wavelength longitudinal acoustic phonons: the mode softening is much enhanced just below the Nkel point. The Grüneisen gamma γs{N[110]U[110]} corresponding to the shear C′ mode increases markedly with increasing temperature up towards T N. The mode gamma γs{N[100]U(001)} of the shear C44 mode has a small positive value and is essentially temperature independent throughout the AFM and PM phases. The vibrational anharmonicities of the longitudinal C 11 and CL and the shear C′ modes are greatly influenced by the magnetoelastic interaction between the SDW and long-wavelength acoustic phonons. The longitudinal acoustic-mode Grüneisen parameters for Cr in the PM phase above the Néel point have large positive values, which decrease substantially with increasing temperature; those for both shear modes are also positive and have values appropriate for a paramagnetic material in marked contrast to the soft mode behaviour in the antiferromagnetic spin-density wave state.

Non-linear acoustic properties and acoustic-mode vibrational anharmonicity of 18R martensite Cu-Zn-Al shape-memory alloy

To shed further light on the dynamics of the martensitic transformation in shape-memory alloys, measurements have been made of the effects of hydrostatic pressure on the velocities of 15 different pure ultrasonic modes propagated at room temperature in a single-crystal alloy in its 18R martensitic phase. Although the point symmetry group of the 18R martensite is monoclinic, the elastic behaviour can be reasonably described in terms of an orthorhombic structure; the further monoclinicity beyond such an orthorhombic representation is very small and can be neglected to a first approximation. The results obtained for the nine independent second-order elastic stiffness tensor components in the 18R structure and their hydrostatic-pressure derivatives have been used to calculate the mode Grüneisen parameters , which quantify the vibrational anharmonicity of long-wavelength acoustic phonons. Results are compared with those computed from TOEC data measured on a -phase Cu-Zn-Al single crystal with a similar composition. At room temperature, about 28 K below the martensitic transition temperature (= 323 K), the longitudinal mode have normal, positive values; however, those associated with some shear modes have negative values. A pronounced minimum has been found in the Grüneisen parameter of the shear mode propagated along the [111] direction with polarization vector. It is concluded that the lattice stability against this particular shear mode decreases as the sample approaches the martensitic transition.

Ultrasonic study of the temperature and pressure dependences of the elastic properties of a single-crystal Heusler structure Cu41Mn20Al39 alloy

Pulse-echo-overlap measurements of ultrasonic wave velocity have been used to determine the elastic-stiffness tensor components Cu and the adiabatic bulk modulus, BS, of a ferromagnetic Heusler structure Cu41Mn20Al39 at % alloy single crystal as functions of temperature in the range 14–300 K and hydrostatic pressure up to 0.2 GPa at room temperature. At 295 K the elastic stiffnesses are: C11=133 GPa, C44=92 GPa, C′ (=(C11−C12)/2)=17 GPa, C12=99 GPa, CL(=C11+C44−C′)=205 GPa, and BS (=C11−4C′/3)=106 GPa. Cu41Mn20Al39 is a comparatively soft material elastically because its elastic properties are influenced strongly by magnetoelastic effects. The results of measurements of the effects of hydrostatic pressure on the ultrasonic wave velocity have been used to obtain the hydrostatic-pressure derivatives of the elastic – stiffness tensor components. At 295 K (∂C11/∂P)P=0, (∂C44/∂P)P=0, (∂C′/∂P)P=0, (∂C12/∂P)P=0, (∂CL/∂P)P=0, and (∂BS/∂P)P=0 are 5.0±0.1, 3.0±0.1, 1.0±0.2, 3.0±0.3, 7.7±0.4 and 3.7±0.4, respectively. Application of hydrostatic pressure does not induce acoustic-mode softening: the pressure derivatives (∂CIJ/∂P)P=0 and (∂BS/∂P)P=0 and the acoustic-mode Gruneisen parameters are positive. An interesting feature of the non-linear acoustic behaviour of this alloy is that the value obtained for (∂C′/∂P)P=0, associated with the softer shear mode propagated along the [1 1 0] direction and polarized along the [1 1 0] direction, is small in comparison with those of the other shear and longitudinal modes. The Gruneisen parameter of this mode, and hence its vibrational anharmonicity, is much larger than those of the other long-wavelength acoustic phonon modes. © 1998 Kluwer Academic Publishers

Ultrasonic study of the temperature and pressure dependences of the elastic propertiesof fcc Co-Mn alloy single crystals

To search for ramifications of magnetovolume effects in the magnetoelasticity of Co-Mn alloys, the elastic and nonlinear acoustic properties of single crystals with compositions ${\mathrm{Co}}_{75}$${\mathrm{Mn}}_{25}$, ${\mathrm{Co}}_{68}$${\mathrm{Mn}}_{32}$, ${\mathrm{Co}}_{52}$${\mathrm{Mn}}_{48}$, and ${\mathrm{Co}}_{46}$${\mathrm{Mn}}_{54}$, which have different magnetic configurations, have been determined ultrasonically. Pulse-echo-overlap measurements of the velocities of ultrasonic modes have been used to determine the three independent elastic-stiffness tensor components ${\mathrm{C}}_{\mathrm{IJ}}$ and the adiabatic bulk modulus ${\mathrm{B}}^{\mathrm{S}}$, as a function of temperature in the range from 4.2 up to about 800 K, and the pressure derivatives (\ensuremath{\partial}${\mathrm{C}}_{\mathrm{IJ}}$/\ensuremath{\partial}P${)}_{\mathrm{P}\mathrm{=}0}$ and (\ensuremath{\partial}${\mathrm{B}}^{\mathrm{S}}$/\ensuremath{\partial}P${)}_{\mathrm{P}\mathrm{=}0}$ at room temperature. These measurements of the elastic and nonlinear acoustic behavior of fcc Co-Mn alloys assist understanding of the structural and magnetic phase stability in this alloy system. Two of the alloys studied (${\mathrm{Co}}_{52}$${\mathrm{Mn}}_{48}$ and ${\mathrm{Co}}_{46}$${\mathrm{Mn}}_{54}$) are antiferromagnetic at room temperature; their elastic-stiffness tensor components and bulk moduli are affected by the onset of antiferromagnetic ordering and strong magnetoelastic coupling in the antiferromagnetic phase. The ${\mathrm{Co}}_{68}$${\mathrm{Mn}}_{32}$ alloy, with composition centered between the superantiferromagnetic and superparamagnetic regions in the phase diagram, does not undergo any transition. The temperature dependences of the elastic stiffnesses associated with the three independent ultrasonic modes propagated in the austenite and martensite phases of the ferromagnetic ${\mathrm{Co}}_{75}$${\mathrm{Mn}}_{25}$ alloy show that the \ensuremath{\epsilon}\ensuremath{\rightarrow}\ensuremath{\gamma} transition is strongly first order; the absence of long-wavelength acoustic-mode softening shows that its driving mechanism is not elastic in origin. Application of pressure does not induce acoustic mode softening for any of these alloys: the pressure derivatives (\ensuremath{\partial}${\mathrm{C}}_{\mathrm{IJ}}$/\ensuremath{\partial}P${)}_{\mathrm{P}\mathrm{=}0}$ and (\ensuremath{\partial}${\mathrm{B}}^{\mathrm{S}}$/\ensuremath{\partial}P${)}_{\mathrm{P}\mathrm{=}0}$ are positive. The long-wavelength longitudinal-acoustic-mode Gr\uneisen parameters of antiferromagnetic ${\mathrm{Co}}_{46}$${\mathrm{Mn}}_{54}$ show the unusual feature of being smaller than those for the shear waves: magnetovolume contributions to vibrational anharmonicity are stronger for longitudinal than for shear waves. The longitudinal-mode gamma ${\ensuremath{\gamma}}_{\mathrm{L}[110]}$ of this ${\mathrm{Co}}_{46}$${\mathrm{Mn}}_{54}$ alloy passes through a maximum value of 7.0 at 388 K: the vibrational anharmonicity of this mode is markedly enhanced in the vicinity of the N\'eel temperature.

Ultrasonic study of the elastic and nonlinear acoustic properties of the paramagnetic Cr-5 at.% V single-crystal alloy

Abstract To assist understanding the effects of pressure on the magnetoelastic interactions in antiferromagnetic Cr alloys, the elastic and nonlinear acoustic properties of single-crystal Cr-5 at. %V alloy, which is paramagnetic at all temperatures, have been studied by the ultrasonic pulse-echo-overlap technique. Measurements of the hydrostatic-pressure dependences of the velocities of ultrasonic modes, propagated along the [100] and [110] directions, have been used to obtain the hydrostatic-pressure derivatives (∂CIJ∂P)P=0 of the elastic-stiffness tensor components CIJ . At 296 K, (∂CIJ/∂P}P=0 , (∂CL/∂P)P=0 , (∂C44/∂P)P=0 , (∂C1/∂P)P=0 and (∂B2/∂P)P=0 are 9·1 ± 0·1, 8·7 ± 0·1, 1·0 ± 0.02, 1·74 ± 0·02 and 6·8 ± 0·1 respectively. To establish the vibrational anharmonicity of the long-wavelength acoustic modes, the results obtained for CIJ and (∂CIJ/∂P)P=0 have been used to calculate the corresponding Gruneisen parameters. The mean long-wavelength acoustic Gruneisen parameter γd(= 1·43) is similar to the t...

Elastic and anelastic properties, vibrational anharmonicity, and fractal bond connectivity of superionic glasses.

To quantify the thermally activated relaxations of the mobile silver ions in superionic silver phosphosulphate ~Ag2SO4!x~AgPO3!(12x) and phosphosulphide ~Ag2S!x~AgPO3!(12x) glasses, the broad attenuation peaks reported previously have been analyzed in terms of a Gaussian-type energy distribution. The parameters obtained are used to determine the influence of thermally activated relaxation processes on the temperature dependence of the ultrasonic wave velocity measured between 1.5 and 300 K. After subtraction of the relaxation effects together with those due to anharmonic interactions, another contribution to the temperature dependence of the ultrasonic velocity remains below 100 K, which follows a linear temperature dependence—as predicted by the soft-potential model ~SPM! for relaxation of soft harmonic oscillators. The soft HO relaxation contribution to the ultrasonic velocity temperature dependences of silver phosphate-based glasses has a similar magnitude to those determined previously for lanthanide metaphosphate glasses. While the silver phosphate-based glasses have skeletons that are comprised of long chains of phosphate ions, the lanthanide metaphosphate glasses are close to having a three-dimensional structure. The agreement of the excess contribution to the temperature dependence of the ultrasonic velocity with the predictions of the SPM for both types of glass, in spite of their complete differences in structure, is further evidence for the universal applicability of the soft-potential model. To determine the vibrational anharmonicity of the long wavelength acoustic modes in superionic glasses, the hydrostatic pressure derivatives of the second-order elastic stiffness tensor components have been measured for these ~Ag2SO4!x~AgPO3!(12x), ~Ag2S!x~AgPO3!(12x), vitreous AgPO3 and also for silver iodide molybdate ~AgI!0.75~Ag2MoO4!0.25. To examine further the effects of vibrational anharmonicity, the thermal expansions of the vitreous phosphates have also been measured. The linear thermal expansion coefficient becomes anomalously negative at lower temperatures for the ~Ag2SO4!x~AgPO3!(12x) and ~Ag2S!x~AgPO3!(12x) glasses. The wide variations found between the elastic and nonlinear acoustic properties of superionic silver phosphate, molybdate, and borate glasses stem from differences in the bonding and connectivities of the glass skeletons.

Anisotropy of the elastic and nonlinear acoustic properties of dense textured Bi2Sr2CaCu2O8+y.

A description of the anisotropy of elastic and nonlinear acoustic properties of dense (95% of theoretical), highly textured, sinter-forged, ceramic ${\mathrm{Bi}}_{2}$${\mathrm{Sr}}_{2}$${\mathrm{CaCu}}_{2}$${\mathrm{O}}_{8+\mathit{y}}$ (Bi-2:2:1:2) high-${\mathit{T}}_{\mathit{c}}$ superconductors is provided from measurements of the velocities of longitudinal and shear ultrasonic waves as functions of temperature and hydrostatic pressure. A high proportion of the grains are aligned preferentially with the c axis along the forging direction in this material, while it is essentially isotropic in the plane normal to the forging axis. A complete set of the elastic stiffness tensor components has been determined as a function of temperature from 10 to 290 K, treating this material as if it had cylindrical symmetry. The results are used to compute the orientation dependences of the Young's modulus and Poisson's ratio which are compared with those estimated for single crystal Bi-2:2:1:2. The main finding is a marked anisotropy of the elastic stiffness moduli and their dependences on pressure. The elastic and nonlinear acoustic properties of the textured and single-crystal Bi-2:2:1:2 comply with the behavior expected for a layerlike material with rather weak interlayer binding forces: ${\mathit{C}}_{11}$g${\mathit{C}}_{33}$, the linear compressibility and thermal expansion along to the c axis, i.e., normal to the layers, being substantially larger than in the ab plane. The hydrostatic pressure derivative (\ensuremath{\partial}${\mathit{C}}_{33}$/\ensuremath{\partial}P${)}_{\mathit{P}=0}$(=21.3) of the elastic stiffness tensor component, which corresponds to a longitudinal wave propagated along the c axis, is larger than that (\ensuremath{\partial}${\mathit{C}}_{11}$/\ensuremath{\partial}P${)}_{\mathit{P}=0}$(=15.1), for such a wave propagated in the ab plane. The anisotropies of the linear compression and the acoustic mode Gr\uneisen parameters are also typical of a layerlike material.

Contrasting effects of lanthanum and samarium modifiers on the elastic and non-linear acoustic properties of phosphate glasses

While samarium phosphate (Sm2O3) x (P2O5)(1−x) glasses are known to show pronounced long-wavelength acoustic-mode softening, their lanthanum (La2O3) x (P2O5)(1−x) analogues do not. To investigate experimentally this different influence of lanthanide modifiers, a comparative study has been made of the effects of temperature and pressure on the velocities of ultrasonic waves propagated in binary ((La2O3) x (P2O5)(1−x) and (Sm2O3) x (P2O5)(1−x)) and ternary ((La2O3) x (Sm2O3) y (P2O5)(1−x−y)) phosphate glasses with compositions near to that corresponding to the metaphosphate (R2O3)0.25(P2O5)0.75. For each glass the second-order elastic stiffness tensor componentsC IJ S continue to increase down to 10 K in a manner consistent with ultrasonic interactions with two-level systems. Measurements of the effects of hydrostatic pressure on the ultrasonic wave velocities have been used to determine the hydrostatic pressure derivatives (∂C IJ S /∂P)T,P=0 of the second-order elastic stiffness tensor components and (∂B 0 S /∂P)T,P=0 of the bulk modulusB 0 S at room temperature (293 K). For the ternary glasses (∂C 11 S /∂P)T,P=0, (∂C 44 S /∂P)T,P=0 and (∂B 0 S /∂P)T,P=0 are small but positive; these glasses stiffen under pressure. Replacement of samarium by lanthanum in the ternary glasses negates the acoustic-mode softening. Possible sources of the different effects of lanthanum and samarium modifiers on the non-linear acoustic properties of metaphosphate glasses are discussed.

Elastic Behaviour of Terbium Metaphosphate Glasses Under High Pressures

The elastic and nonlinear acoustic vibrational properties of terbium metaphosphate glasses (Tb2O3)x(P2O5)1?x with x = 0�226,0�247,0�263 and 0�271 (x is the mole fraction) have been determined from measurements of the effects of temperature, hydrostatic pressure, and uniaxial stress on' ultrasonic wave velocity. At temperatures below about 140 K, the elastic stiffness of' (Tb2O3)x(P2O5)1?x glasses becomes anomalously dependent upon temperature, a behaviour usually associated with interactions between acoustic phonons and two-level systems. Except for the (Tb2O3)0�271(P205)0�729 glass, the hydrostatic pressure derivatives of the elastic stiffness and also of the bulk modulus BS of terbium metaphosphate glasses are small and negative. The third-order elastic stiffness tensor components CIJK of the (Tb2O3)0�247(P2O5)0�753 glass between 77 K and 400 K have also been determined. At room temperature, C112, C123 and C144 are positive while C111, C155 and C456 are negative. Both longitudinal and shear acoustic mode Gr�neisen parameters are small and negative: the application of pressure softens the long-wavelength acoustic phonon mode frequencies. The mode softening is enhanced as the temperature is reduced.

Optical and ultrasonic properties of europium phosphate glasses

To provide an overall picture of the vibrational properties of phosphate glasses containing high concentrations of europium, measurements have been made of their ultrasonic wave velocities and attenuation, optical absorption spectra and laser-induced fluorescence. To examine their valence state, the fluorescence of the glasses has been examined. The spectra do not show any obvious sign of divalent europium ions, only trivalent europium ion fluorescence being observed. Room-temperature absorption spectra of these glasses also provide evidence of only the absorption bands of trivalent europium. The elastic stiffnessesC11 andC44 continue to increase down to low temperatures and the ultrasonic attenuation is characterized by a broad peak, properties which are consistent with thermally activated relaxations of two-level systems. The longitudinal and shear ultrasonic wave velocities decrease under pressure; the hydrostatic pressure derivatives (∂C11/∂P)T, P=0 and (∂C44/∂P)T, P=0 of the elastic stiffness tensor componentsCIJ and (∂B/∂P)T, P=0 of the bulk modulus,B0, are negative. When compressed, the europium phosphate glasses, like their samarium analogues, show the interesting property of becoming easier to squeeze. Measurements, using a pulse superposition technique, of the effect of uniaxial stress on ultrasonic wave velocities have been used to determine the temperature dependences of the third-order elastic stiffness tensor components of (Eu2O3)0.186(P2O5)0.814 and (Eu2O3)0.20(P2O5)0.80 glasses between 77 and 400 K. The uniaxial and hydrostatic pressure dependences of the elastic constants quantify the cubic term in the strain Hamiltonian and the vibrational anharmonicity of the long-wavelength phonons of these glasses. The acoustic mode Gruneisen parameters are negative: application of pressure induces a decrease in the long-wavelength acoustic phonon mode frequencies. As the temperature is reduced the pressure-induced mode softening becomes enhanced. The hydrostatic pressure derivative (∂C11/∂P)T, P=0 is larger than (∂C44/∂P)T, P=0 over the whole temperature range, and the longitudinal acoustic mode Gruneisen parameter ∥γL∥ is larger than that ∥γS∥ of the shear wave: the longitudinal mode softens more with pressure than the shear mode. Murnaghan's equation-of-state is used to determine the compressionV(P)/V0.