Variation Of Shear Modulus Research Articles

Propagation of shock waves at the surface of heterogeneous soil grounds

AbstractThe paper deals with the propagation of shock waves at the surface of soils. Heterogeneity and damping are introduced into analytical half‐space solutions. The suggested model explains two phenomena, often observed with shock propagation in actual soils, that differ from the behaviour of the homogeneous half‐space: the pronounced decay of the disturbances with distance and the elongation of the disturbance into a train of waves. The effects of heterogeneity and damping are discussed quantitatively.The response of footings on heterogeneous soils has been investigated by several authors. Awojobi4 considered the Gibson soil in which the shear modulus increases linearly with depth. Luco5 and Gazetas and Roesset6 investigated a multi‐layered soil, the shear modulus being constant within each layer. Gazetas7, using a technique suggested by Gupta8 extended this method to layers with linearly varying shear modulus. Little work is available on the propagation of waves in heterogeneous bodies. Some results concerning the modes and the mode shapes in heterogeneous soils were reported by Ewing, Jardetzky and Press9 and Bath.10 The modes have to be superposed in an appropriate way to obtain the displacement field at the surface. This has been approximately achieved by the finite element formulations of Lysmer, 11, Lysmer and Waas12 and Waas.13 Auersch14 applied this method to a homogeneous layer on a rigid base. He found some dispersion of the Rayleigh wave within a narrow frequency range. Finite elements combined with discrete Laplace transforms, however, consume much computer time. Rao and Goda15 and Rao16 calculated surface vibrations of a half‐space with exponentially varying shear modulus and density. Their method is similar to Lamb'S1 procedure for the homogeneous half‐space. Only one mode–the Rayleigh wave–occurs in their heterogeneous half‐space. The examples show the considerable effect of heterogeneity on wave propagation. In the present paper, more general variations of the shear modulus are considered.

Determination of non-linear dynamic properties of carbon-filled rubbers

A forced non-resonance test method is described for determining the dynamic mechanical properties of polymeric materials over wide ranges of strain and frequency. The use of this method for carrying out studies on carbon-filled rubbers is illustrated by results which demonstrate the variation of the dynamic shear modulus and damping factor of a tyre tread material with dynamic strain amplitude, frequency and temperature. Procedures are discussed for the analysis and presentation of such data. Two methods are described for the determination of loss factor, and results from these are compared in order to assess the validity of phase angle measurements on non-linear materials. Brief reference is made to dynamic testing under compressive and combined compression and shear modes of deformation. The prediction of performance under this combined loading situation from experimental data obtained in shear is demonstrated.

Vibration of a homogeneous half‐space of arbitrary Poisson's ratio interrupted by a frictionless plane

AbstractThe work presents an exact expression for the Hankel transform of the surface displacement of a rigid circular body performing harmonic vertical vibrations on a homogeneous elastic half‐space of arbitrary Poissn'S ratio interrupted at some depth by a frictionless horizontal plane. The limiting case of the uninterrupted half‐space is recovered in each of the extreme cases when the frictionless plane coincides with the surface or sinks to an infinite depth. Furthermore, the static case is derived in the limiting case of zero frequency and it is shown that the result agrees with the limiting case of zero gradient shear modulus variation for the static case of the incompressible non‐homogeneous half‐space.It is also established that Poisson'S ratio effect is linear, and the important proposition is again sustained that only surface shear modulus dominates vibrations on elastic media whereas any other factor such as the interruption of the frictionless plane as in the present case, merely constitutes a negligible secondary parameter.

Experiences in the use of composite material for a wing skin

Experiences in using composite skin material on an aeroelastic reserach wing used in flight flutter testing are described. Significant variations in skin shear modulus due to stress and temperature were encountered with the original fiberglass laminate skin designed to minimize wing torsional stiffness. These variations, along with the sensitivity of wing torsional stiffness to the skin-to-frame attachment method, complicated the structural model yirbration mode predictions. A wing skin redesign with different fiber orientation and a reduction in the amount of skin-to-frame bonding resulted in more predictable modal characteristics without sacrificing design objectives. Design and modeling considerations for future applications are discussed.

Giant magnetoelastic effects in amorphous ribbons by exciting standing torsional waves

We have found that transverse field annealed ribbons of Metglas 2605 SC exhibit shear modulus variations, Δμ effect, larger than 50 and transverse coupling coefficient about 0.99. Excited in standing torsional wave the peak resonant frequency has been measured as a function of the applied tensile stress when a constant dc current was flowing through the ribbon. A simplified model has been developed in order to explain the appearance of such giant values, one order of magnitude larger than any other previously reported. It has been found that the symmetrical distribution of shear strain concerning the longitudinal and transverse ribbon axes is the main source of the magnetoelastic effects enhancement.

Vertical vibration on a compound half-space of a homogeneous stratum with Gibson subsoil

An exact expression is derived for the displacement transform of a rigid circular bodyperforming harmonic vertical vibration on a compound half-space comprising a homogeneous stratum of arbitrary Poisson's ratio with a frictionless bed and a Gibson subsoil which is an incompressible half-space with shear modulus increasing linearly with depth. The limiting case of either a Gibson half-space or a homogeneous half-space is recovered asthe frictionless bed approaches the surface or is pushed down to an infinite depth. Similarly, the static case is derived in the limit when the frequency of vibration tends to zero from which the further limiting case of a static homogeneous half-space interrupted bya frictionless horizontal plane is derived as the gradient of the Gibson shear modulus variation approaches zero. A practical result is confirmed: that the reduction of the stiflness of the soil due to the interruption of the frictionless bed is counteracted by the increasing rigidity of the Gibson subsoil so that, for a given depth of the frictionless bed. there exists a rate of increase of shear modulus of the Gibson subsoil which makesthecompound half-space behave as if it were a homogeneous elastic half-space. Numerical evaluation of the governing function for the Hankel transform of the surface displacement shows that the complete counteracting effect is provided by a variation which, produced upwards. gives a zero shear modulus at the surface like the basic Gibson soil of the Winklermodel. All the results show that surface shear modulus is still the dominant parameter andthat both the frictionless bed and non-homogeneity of the Gibson subsoil have only compensating secondary effects. Une expression exacte est donnée pour la transturmation par déplacement d'un corps circulaire rigide qui produit des vibrations verticales harmoniques sur un demi-espace compose comprenant une couche homogene ayant un coefticient de Poisson arbitraire avec une base sans frottement et un sous-sol Gibsonien qui est un demi-espace incompressible dont le module de cisaillement s'accroit linéairement avec la profondeur. Le cas limite d'un demi-espace Gibsonien ou d'un demi-espace homogéne est retrouvé lorsque la base sans frottement s'approche de la surface ou biens'enfonce jusqu'á une profondeur infinie. De méme, le cas statique est trouvé à la hmite lorsque la fréquence de vibration tendvers zéro, d'ou le cas limite supplémentaire d'un demi-espace homogene statique interrompu par un plan horizontal sans frottement trouvé lorsque legradient de la variation dans le module de cisaillement Gibsonien tend vers zéro. Un résultat pratique est confirmé la réduction de la rigiditédu sol due a I'interruption de la base sans frottement est contrebalanctée par la rigidité croissante du sous-sol Gibsonien. II s'ensuit que pour une pro-fondeur donnée de la base sans frottement il existe une vitesse d'accroissement du module de cisaillement du sous-sol Gibsonien, pour laquelle le demi-espace compose sé comporte comme s'il était un demi-espace homogene élastique. L'evaluation numérique de la fonction dominante pour la transformation type Hankel du dtplacement superficiel indique que I'effet opposant complet est fourni parune variation qui produit dans le cas de prolongement vers le haut un module de cisaillement zero a la surface, tout comme le sol Gibsonien basique du modtle de Winkler. Tous lesrésultats indiquent que le module de cisaillement superficiel reste le paramétre le plus important, tandis que la base sans frottement et la non-homogénéité du sous-sol Gibsonien néxercent que des effets secondaries compensateurs.

Regularization of optimal design problems for bars and plates, part 1

In this paper, we consider a number of optimal design problems for elastic bars and plates. The material characteristics of rigidity of an elastic nonhomogeneous medium are taken as the control variables. A linear functional of the solutions to the equilibrium boundary-value problem is minimized under additional restrictions upon the control variables. Special variations of the control within a narrow strip provide a necessary condition for a strong local minimum (Weierstrass test). This necessary condition can be used for the detailed analysis of the following problems: bar of extremal torsional rigidity; optimal distribution of isotropic material with variable shear modulus within a plate; and optimal orientation of principal axes of elasticity in an orthotropic plate. For all of these cases, the stationary solutions violate the Weierstrass test and therefore are not optimal. This is because, in the course of the approximation of the optimal solution, the curve dividing zones occupied by materials with different rigidities displays rapid oscillations sweeping over a two-dimensional region. Within this region, the material behaves as a composite medium assembled of materials of the initial class.

Forced torsional properties of PMR composites with varying nadic ester concentrations and processing histories

AbstractPMR polyimide resin was prepared from 4,4′‐methylenedianiline (MDA), the dimethyl ester of 3,3′,4,4′‐ benzophenonetetracarboxylic acid (BTDE) and the monomethyl ester of 5‐norbornene‐2,3‐dicarboxylic acid (NE). The NE group serves as a chain terminator and crosslinking site. PMR/Celion 6000 composites were fabricated from resins having varying NE concentrations using two molding processes, and the laminates characterized in forced torsion. Glass transition temperatures (Tg) of 360–390°C were observed in the crosslinked resins, as compared with the literature value of 284°C reported for the uncrosslinked systemTg did not decrease with decreasing NE concentrations over the range from 2.0 to 1.25 moles. Stoichiometry, within the range studied, showed little influence on shear properties; however, a 25% variation in matrix shear modulus with processing was observed. The G12 values determined in forced torsion were in excellent agreement with those reported from tensile tests of ±45° laminates. A branching and possible secondary crosslink mechanism is proposed based on dynamic mechanical behavior and infrared spectra of the composites.

Shear modulus gradients in adhesive interfaces as determined by means of ultrasonic Rayleigh waves

Rayleigh surface waves at ultrasonic frequency have been utilized to determine shear modulus variations in the vicinity of adhesive-adherend interfaces. The measurement technique is that of acoustic critical-angle reflectivity conducted at sequentially exposed surfaces of adhesive through the interfacial accommodation zone toward the bondline. Basic theory is presented for determining shear modulus from measured Rayleigh critical angles at adhesive surfaces. Laboratory ultrasonic Rayleigh wave apparatus and its evaluation are described. Experimental procedures for sample preparation and measurement are discussed. Results of Rayleigh critical-angle reflectivity measurements on two different types of adhesives display a decided gradient in shear modulus through the interfacial region. Reference is made to other applications of the ultrasonic Rayleigh wave approach in studying the shear properties near the bondline of adhesive materials.

HEAT‐INDUCED GELATION OF MYOSIN: FACTORS OF pH AND SALT CONCENTRATIONS

ABSTRACTThe heat‐induced gelation of myosin was optimally developed at temperatures between 60° and 70°C and at pH 6.0 as studied quantitatively by measuring shear modulus of myosin in 0.6M KCl. No difference was observed in the gelation profile, when KCl was replaced by NaCl. The shear modulus of the gel formed at 65°C and pH 6.0 increased proportionally to the 1.8 power of myosin concentration. Although the heat gelling ability of myosin as measured by the shear modulus did not increase with increasing salt concentration from 0.4–1.0M, irrespective of storage time at 0°C of the stock myosin solution (0.6M KCl) or suspension (0.1M KCl), it varied drastically with time of storage at salt concentrations between 0.1 and 0.3. This variation in shear modulus at O.1–0.3M KCl was apparent due to the storage conditions. These changes in the salt concentration dependence of the heat gelation of myosin corresponded well with the changes in solubility of myosin during storage. Studies on selected physico‐chemical properties of the original stock myosin during storage indicated gradual increase in viscosity with little inactivation of ATPase as well as very slight decrease in sulfhydryl content, suggesting the spontaneous transformation (aggregation) of myosin molecules to a less soluble state.

Strike slip faulting in a downward varying crust

Both laboratory measurements and in situ velocity data show that (with the exception of anomalous low‐velocity zones) the rigidity μ of the upper crust increases monotonically with depth. Fits to these data show that this variation can be approximated quite reasonably with μ(x) = μ∞(1 ‐ θe −x/x). Best fit to the data gives 0.45 ≤ θ ≤ 0.95 and x0, the characteristic depth, approximately equal to 2.5 km. To determine the effect it produced on faulting characteristics, this rigidity profile was used in conjunction with the theory for a screw dislocation in an inhomogeneous elastic medium (Barnett, 1972). A model for a very long strike slip fault in a crust with a downward varying shear modulus was developed. The free‐surface deformation and fault stress change were calculated for typical faulting events and compared with the uniform half‐space model. Results were obtained for faults which broke the surface and faults which were completely buried. For surface faults the free‐surface displacements and strains were practically identical for both the variable and uniform modulus cases, but the accompanying stress drop for the variable modulus crust was substantially less than the uniform crust. By contrast, for the buried faults the stress changes were practically identical for crustal models, while the free‐surface displacements were substantially larger for the variable modulus model. Further, the corresponding surface strains were amplified by a factor of 2 or more depending on the depth of the source. By a Poisson's ratio effect this amplification would also increase the magnitude of the tilt. These results suggest that the rigidity change with depth will, in general, modify faulting characteristics enough to effect conclusions inferred from data inversions assuming a uniform half space.

Vertical and torsional impedances for radially inhomogeneous viscoelastic soil layers

A study of the dynamic impedances of thin, infinite viscoelastic soil layers with a circular hole is made considering a continuous, ramp-like radial variation for the shear modulus of the material. Both vertically and torsionally excited systems are examined. The impedances are evaluated over wide ranges of the parameters involved and compared with those obtained both for homogeneous layers, and for inhomogeneous layers with discontinuous variations in shear modulus. In addition, a simple approximate method of analysis based on the Galerkin approach is presented for vertically excited layers, and it is used to assess the parameters that control their high-frequency response.

機能的印象材の物性に関する基礎的研究

At present time, the functional impression materials have been willingly used for performing the tissue treatment and taking the dynamic impression. However, the physical and mechanical properties of these materials in oral cavity are not elucidated sufficiently as yet. So the author measured the variation of the coefficient of viscosity and torsional shear modulus with time of the mixture mixed in various ratios at 23℃ and 37℃ for five commercial functional impression materials (Hi-soft, Hydrocast, Tri-plastic, Visco-gel, Tempo). In addition, the swelling of powder to liquid was examined by microscope and the water absorption of the functional impression material (powder-liquid mixture) was investigated by using gas chromatograph. The results obtained were as follows : 1. The working time of Hydro-cast, Hi-soft and Visco-gel was longer than that of Tri-plastic and Tempo, and the variation of torsional shear modulus and the coefficient of viscosity within working time were smaller. 2. The working time of the materials used was varied by mixing temperature and mixing ratio (P/L). When the temperature of mixture rose and ratio (P/L) increased, the working time was shortend. 3. The torsional shear modulus at 60 min from start of mix was 49.7×10^4dyne/cm^2 for Tri-plastic, 41.5×10^4dyne/cm^2 for Visco-gel, 22.4×10^4dyne/cm^2 for Hydro-cast, 21.0×10^4dyne/cm^2 for Tempo and 17.5×10^4dyne/cm^2 for Hi-soft. 4. The materials used in this investigation lost ethanol and absorbed water by soaking in distilled water.

Viscoelastic properties of elastomeric impression materials

The variation of torsional shear modulus with time for several commercial elastomeric impression materials has been determined. The effects of mixing proportions and temperature on rheological properties are presented. The relevance of these properties to the actual use of these materials are discussed.

Antiplane crack problems for an inhomogeneous elastic material

This paper examines some antiplane crack problems in an elastic material with variable shear modulus. An expression for the energy release rate is obtained for each of the cracks considered and a comparison made with the energy release rate for the corresponding crack in a homogeneous material.

The elastic properties of (Mg x Fe1?x )O solid solutions

The velocities of compressional (V p) and shear (V s) waves in five well-characterized polycrystalline aggregates of (Mg x Fe1−x )O (0.85≧x>0.23) have been measured as functions of hydrostatic pressure (to 6 kbar) using the techniques of ultrasonic pulse transmission and super-position. Both velocities decrease systematically with increasing iron content. The elastic properties of stoichiometric FeO are inferred by inverting the solid solution data using a model in which the single crystal compliances (s ij) vary linearly with the volume fraction of either end member. This model, along with identification of the aggregate moduli with the arithmetic average of the Hashin-Shtrikman bounds, may be used to describe the variation of both bulk (K s) and shear modulus (μs) with composition for polycrystalline aggregates of phases of arbitrary symmetry. For the case of bulk modulus of an isotropic aggregate of crystals of cubic symmetry, the present model yields the formula first proposed by Liu (1968). The isotropic elastic moduli of FeO thus obtained (K s =1.82±0.05 Mbar, μ s =0.59±0.02 Mbar) are consistent with the wustite data of Mizutani et al. (1972) but inconsistent with bulk moduli derived from static compression data by Mao et al. (1969) and Rosenhauer et al. (1976). An alternative description of the variation of μ with x based on consideration of the solid solutions as two-phase aggregates is also presented. The observed differences in elastic moduli between MgO (K s=1.63 Mbar, μs=1.31 Mbar) and FeO (K s=1.82 Mbar, μs=0.59 Mbar) are consistent with the effects of Mg/Fe substitution in other oxide and silicate crystal structures.

High-frequency torsional oscillations—II: The Reissner-Sagocci problem at high frequencies

The Reissner-Sagocci Problem at high-frequencies for an elastic medium with exponentially varying shear modulus and density in the radial direction is asymptotically reduced to a Wiener-Hopf integral equation whose solution is obtained by Carleman's method. Uniform asymptotic results are obtained. Explicit results are given for the displacement outside the rigid disc, the moment of the applied forces necessary to oscillate the disc, and the amplitude of oscillation of the disc.

An Effective Stress Model for Liquefaction

The important factors affecting the dynamic response of saturated sand layers to earthquake motions are: (1)The initial shear modulus in situ; (2)the variation of shear modulus with shear strain; (3)contemporaneous generation and dissipation of pore-water pressures; (4)changes in effective mean normal stress; (5)damping; and (6)hardening. Constitutive relations are formulated that take all these factors into account and these are incorporated into a nonlinear method for the dynamic effective stress analysis of saturated sands. The method predicts the phenomenological features of the dynamic response of saturated sand layers that commonly occur as the pore-water pressure rises in the sand during earthquake shaking. It allows the distribution of pore-water pressure and the effects that drainage and internal flow have on the location and time of liquefaction to be determined quantitatively.

Mechanical properties of polymer–paper laminates

Tensile mechanical properties of synthetic polymer–paper laminates were measured. The laminates were constructed by hot pressing a sandwich made of a sheet of paper between polymer films. There is complete penetration of the polymer inside the paper; no voids are left. Two different polymer matrices were used: poly(methyl methacrylate) and polyethylene. Several paper samples were utilized: an unoriented holocellulose paper (a strong paper), a highly oriented holocellulose paper, and an unoriented Whatman filter paper (a weak paper). The laminates contain from 0% to 50% of paper. Young's moduli and breaking strengths of the unoriented holocellulose paper laminates can be theoretically predicted from the properties of their constituents using laws of mixtures. The mechanical properties of the Whatman paper laminates are significantly higher than those predicted from the laws of mixtures. This indicates that the polymer increases the strength of the fiber-to-fiber bonds of the weaker sheets, although it does not change the bond strength of a stronger paper such as the holocellulose paper. For the oriented paper laminates, changes in Young's modulus with angle of measurement are explained by the composite materials theories if the angular variations in shear modulus are taken in to account. Changes in breaking strength with angle for the oriented laminates can be analyzed by Tsai and Azzi's theory for composite materials.

Thermoelasticity and Chain Conformation in Alternating Copolymers of Butadiene

Abstract The internal energy contribution (fe/f) and the temperature coefficient of the unperturbed chain dimensions can be determined by thermoelastic measurements. A particularly convenient technique for such measurements is the determination of the variation of shear moduli as a function of temperature. In previous works, it was found that values of fe/f for random copolymers bear no simple relation to those of respective parent homopolymers. In addition, the signs of the energy contributions are often the reverse of the homopolymers. In this work, thermoelastic measurements of several alternating copolymers were carried out. Specifically, these elastomers are alternating copolymers of butadiene with propylene, methylene, and tetramethylene units. Again no simple relation can be found between copolymers and their parent homopolymers. Implications of these observations will be discussed.