Modulus bulk longitudinal

Longitudinal wave speed Shear wave speed Shear Modulus Poisson s ratio Young s Modulus Bulk Modulus... 

The two basic types of mechanical deformation, from a physical and molecular standpoint, are shear and dilatation. The experimental methods described in the preceding three chapters yield information primarily about shear only in extension measurements on hard solids does a perceptible volume change influence the results. By combining shear and extension measurements, the bulk properties can be calculated by difference, as for example in creep by equation 55 of Chapter 1, but the subtraction is unfavorable for achieving a precise result. Alternatively, bulk properties can be measured directly, or they can be obtained by combining data on shear and bulk longitudinal def ormations (corresponding to the modulus M discussed in Chapter 1), where the subtraction does not involve such a loss of precision. Methods for such measurements will now be described. They have been reviewed in more detail by Marvin and McKinney. ... 

Another commonly used elastic constant is the Poisson s ratio V, which relates the lateral contraction to longitudinal extension in uniaxial tension. Typical Poisson s ratios are also given in Table 1. Other less commonly used elastic moduH include the shear modulus G, which describes the amount of strain induced by a shear stress, and the bulk modulus K, which is a proportionaHty constant between hydrostatic pressure and the negative of the volume... 

It is not necessary to know the bulk modulus to convert E to G. If the transverse strain, , of a specimen is determined during a uniaxial tensile test in addition to the extensional or longitudinal strain e their ratio, called F oisson s ratio, v can be used ... 

A fluid cannot support shear waves over any appreciable distance, so only longitudinal waves can propagate in a fluid. In this case the appropriate modulus is the adiabatic bulk modulus B. Extending to three dimensions with the Laplacian operator... 

The first equation is scalar, and has a wave solution with velocity Vi = -J c /p). This is the longitudinal wave of eqn (6.7). It is sometimes called an irrotational wave, because V x u = 0 and there is no rotation of the medium. The second equation is vector, and has two degenerate orthogonal solutions with velocity v = s/(cu/p)- These are the transverse or shear waves of eqn (6.6) the degenerate solutions correspond to perpendicular polarization. They are sometimes called divergence-free waves, because V u = 0 and there is no dilation of the medium. Waves in fluids may be considered as a special case with C44 = 0, so that the transverse solutions vanish, and C = B, the adiabatic bulk modulus. 

Elasticity of solids determines their strain response to stress. Small elastic changes produce proportional, recoverable strains. The coefficient of proportionality is the modulus of elasticity, which varies with the mode of deformation. In axial tension, E is Young s modulus for changes in shape, G is the shear modulus for changes in volume, B is the bulk modulus. For isotropic solids, the three moduli are interrelated by Poisson s ratio, the ratio of traverse to longitudinal strain under axial load. 

The agreement between fee bulk modulus deduced from Brillouin scattering measurements and fee ADX results is very good. The determination of fee elastic moduli by ultrasonics was made by fee measurement of surface acoustic wave velocities on thin films [22], The second ultrasonics experiment was made on sintered powder, by measuring fee longitudinal and transverse sound velocity at ambient and under uniaxial compression. From feat, fee bulk modulus and its pressure derivative were deduced, but this result seems to be quite imprecise. The ultrasonics experiment on thin films gives rise to a very small difference in fee bulk modulus (5%), but fee ADX or Brillouin determination should be utilised for preference. 

In solids the situation is more complicated than in liquids. Here we have two types of waves, viz. the longitudinal and the shear waves. In contrast with liquids the longitudinal wave in solids is not only determined by the bulk or compression modulus but also by the shear modulus, or alternatively by the Poisson ratio. 

The bulk modulus K is defined as the reciprocal of the isothermal compressibility, and Young s modulus E is defined as the ratio of longitudinal tensile stress and longitudinal tensile strain ... 

As in true strain, the expression above takes into account cross-sectional area changes in a certain cohesive structure (or cake) of powdered material. If the material is isotropic, another possible expression that includes the Poisson s ratio p (the ratio of transverse strain and axial strain resulting from uniformly distributed axial radial stress during static compression of the material in absolute value). The Poisson s ratio, or bulk modulus, permits prediction of the transverse contraction or expansion that occurs when a stress is applied longitudinally. 

The modulus of elasticity, E, and the modulus of rigidity, G, as defined above, apply under longitudinal and shear forces, respectively. When a hydrostatic force is applied, a third elastic modulus, the modulus of compressibility or bulk modulus, K, is used. It is the reciprocal of compressibility, /J, and is defined as the ratio of the hydrostatic pressure, cri,> to the volumetric strain, AV/Vo ... 

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