Solids, bulk modulus

Bulk Compressibility and Bulk Modulus is one of the important constants of aa elastic solid Bulk modulus is defined as the tatio of stress to atrsin when the stress is a pressure applied equally on all surfaces of the sample and the strain is the resulting change in volume per suit volume. The reciprocal of bulk modulus Is called bulk compressibility. One apparatus for the direct exnd measurement of the dynamic bulk modulus of a solid was developed at the NOL, White Oak, Md(Ref 1). Some data obtained, on several HE a, using this apparatus are given in Refa 2, Refs l)NAVORD Kept No 1534(1950) 2)NAVORD Rept No 4380(1956) 3)PATR 1740,Rev 1(1958)... 

The expansion coefficient of a solid can be estimated with the aid of an approximate thermodynamic equation of state for solids which equates the thermal expansion coefficient with the quantity where yis the Griineisen dimensionless ratio, C, is the specific heat of the solid, p is the density of the material, and B is the bulk modulus. For fee metals the average value of the Griineisen constant is near 2.3. However, there is a tendency for this constant to increase with atomic number. 

Orowan (1949) suggested a method for estimating the theoretical tensile fracture strength based on a simple model for the intermolecular potential of a solid. These calculations indicate that the theoretical tensile strength of solids is an appreciable fraction of the elastic modulus of the material. Following these ideas, a theoretical spall strength of Bq/ti, where Bq is the bulk modulus of the material, is derived through an application of the Orowan approach based on a sinusoidal representation of the cohesive force (Lawn and Wilshaw, 1975). 

Chemical hardness is an energy parameter that measures the stabilities of molecules—atoms (Pearson, 1997).This is fine for measuring molecular stability, but energy alone is inadequate for solids because they have two types of stability size and shape. The elastic bulk modulus measures the size stability, while the elastic shear modulus measures the shape stability. The less symmetric solids require the full set of elastic tensor coefficients to describe their stabilities. Therefore, solid structures of high symmetry require at least two parameters to describe their stability. 

Stability is sometimes associated with the bulk modulus alone, but this is not valid because the bulk modulus of a liquid, and its corresponding solid, are nearly equal at the melting temperature, while their mechanical stabilities are very different. For example, take the case of aluminum. The bulk modulus of its liquid is about 0.3Mbar, while that of its solid is about 0.7Mbar, both measured near its melting point. On the other hand, the shear modulus of liquid aluminum is zero, while it is about 0.25 Mbar for solid aluminum. 

Equations of state for solids are often cast in terms of the bulk modulus, Kp, which is the inverse of the isothermal compressibility, Kp, and thus defined as... 

One of the consequences of close packing in solids and liquids is much higher densities in comparison with gases for instance, ice and water have densities that are a thousand times higher than water vapor at room pressure. Another consequence is that solids and liquids have much lower compressibility, so that the density is not sensitive to the pressure. The bulk modulus B is defined as B = —AP/ AV/V), which has the units of pressure. This parameter measures the fractional volumetric response of a material, when pressure is applied to all faces of the material at the same time. 

Fig. 2.17 Frequency dependence of dynamic shear moduli computed using a model for the linear viscoelasticity of a cubic phase based on slip planes, introduced by Jones and McLeish (1995). Dashed line G, solid line G".The bulk modulus is chosen to be G — 105 (arb. units). The calculation is for a slip plane density AT1 - 10 5 and a viscosity ratio rh = = 1, where rjs is the slip plane viscosity and t] is the bulk viscosity. The strain...

Thus a measurement of the ultrasonic velocity and density can be used to determine the adiabatic compressibility (or bulk modulus) of the material. For homogeneous solids measurements of the compression and shear velocities can be used to determine the bulk and shear moduli (see section 2.4). The Young s modulus of rod-like materials (e.g. spaghetti) can be determined by measuring the velocity of ultrasound. 

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