Deformation bulk modulus

The bulk modulus of an ideal SWNT crystal in the plane perpendicular to the axis of the tubes can also be calculated as shown by Tersoff and Ruoff and is proportional to for tubes of less than 1.0 nm diameter[17]. For larger diameters, where tube deformation is important, the bulk modulus becomes independent of D and is quite low. Since modulus is independent of D, close-packed large D tubes will provide a very low density material without change of the bulk modulus. However, since the modulus is highly nonlinear, the modulus rapidly increases with increasing pressure. These quantities need to be measured in the near future. 

Poison s ratio is used by engineer s in place of the more fundamental quality desired, the bulk modulus. The latter is in fact determined by r for linearly elastic systems—h ncc the widespread use of v engineering equation for large deformations, however, where the Strain is not proportional to the stress, a single value of the hulk modulus may still suffice even when the value of y is not- constant,... 

Amorphous HIPS resin pellets compact very little in the temperature range of 25 °C to about 75 °C, a temperature that is about 25 °C below Tg, as shown in Fig. 4.5. The HIPS resin compacts to a much higher degree for temperatures of 25 °C below Tg up to Tg. Like the LDPE resin, the bulk density at 25 °C and zero pressure was measured using the cell shown in Fig. 4.2 at 0.62 g/cmL At temperatures 25 °C below the Tg (100 °C), the bulk modulus of the resin is relatively high, and thus the pellets do not deform easily under pressure. At higher temperatures, how... 

Note 3 At small deformations, the bulk modulus is related to Young s modulus (E) by... 

Knowledge of the sample pressure is essential in all high-pressure experiments. It is vital for determinations of equations of state, for comparisons with other experimental studies and for comparisons with theoretical calculations. Unfortunately, one cannot determine the sample pressure directly from the applied force on the anvils and their cross-sectional area, as losses due to friction and elastic deformation cannot be accurately accounted for. While an absolute pressure scale can be obtained from the volume and compressibility, by integration of the bulk modulus [109], the most commonly-employed methods to determine pressures in crystallographic experiments are to use a luminescent pressure sensor, or the known equation of state of a calibrant placed into the sample chamber with the sample. W.B. Holzapfel has recently reviewed both fluorescence and calibrant data with the aim of realising a practical pressure scale to 300 GPa [138]. 

Rather surprisingly, all these kinds of deformation can be described in terms of a single modulus. This is a result of the assumption that rubber is virtually incompressible (i.e. bulk modulus much greater than shear modulus). Young s modulus E = 3G (for fdled rubbers the numerical factor may be in fact as high as 4). Indeed, these relationships by no means fully describe the complete stress strain behaviour of real rubbers but may be taken as first approximations. The shear stress relationship is usually good up to strains of 0.4 and the tension relationship approximately true up to 50% extension. 

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. 

Sections A and B of this chapter dealt with purely elastic deformations, i.e. deformations in which the strain was assumed to be a time-independent function of the stress and vice versa. In reality, materials are never purely elastic under certain circumstances they have non-elastic properties. This is especially true of polymers, which may show non-elastic deformation under circumstances in which metals may be regarded as purely elastic. For a better understanding three phenomena may be distinguished, the combination of which is called viscoelasticity. This is elucidated in Table 13.8. The only modulus which is time-independent is the bulk modulus hence its advantage as a basis for additivity. 

Models of viscoelastic behaviours, 412 Modes of deformation, 526 Modulus, 396 bulk, 395,405, 447,514 complex, 410, 418 dynamic, 451 loss, 408... 

Tensile and shear forces are not the only types of loads that can result in deformation. Compressive forces may as well. For example, if a body is subjected to hydrostatic pressure, which exists at any place in a body of fluid (e.g. air, water) owing to the weight of the fluid above, the elastic response of the body would be a change in volume, but not shape. This behavior is quantified by the bulk modulus, B, which is the resistance to volume change, or the specific incompressibihty, of a material. A related, but not identical property, is hardness, H, which is defined as the resistance offered by a material to external mechanical action (plastic deformation). A material may have a high bulk modulus but low hardness (tungsten carbide, B = 439 GPa, hardness = 30 GPa). 

It should be noted that for a polycrystal composed of cubic crystalhtes, the Voigt and Reuss approximations for the bulk modulus are equal to each other, as they should be since the bulk modulus represents a volume change but not shape change. Therefore, in a cube the deformation along the principal strain directions are the same. Hence, Eqs. 10.39 and 10.40 are equal and these equations also hold for an isotropic body. The... 

The mechanical properties of a linear, isotropic material can be specified by a bulk modulus, K, and a shear modulus, G. For an ideal elastic solid, these moduli are real-valued. For real solids undergoing sinusoidal deformation, these are best represented as complex quantities [49] K = K jA and G = G -I- jG". The real parts of K and G represent the component of stress in-phase with strain, giving rise to energy storage in the film (consequently K and G are referred to as storage moduli) the imaginary parts represent the component of stress 90° out of phase with strain, giving rise to power dissipation in the film (thus, K" and G" are called loss moduli). 

Another simple model that can explain how the excess volume in deformations can lead to a reduction of the enthalpy of formation in non-metallic materials is the Birch-Mumaghan (BM) equations of state that give the molar energy as a function of the equilibrium molar energy Eg, the equilibrium molar volume Vo, the actual molar volume V, and the bulk modulus at equilibrium Bo [12]. At OK, the enthalpy of formation coincides with the molar energy of formation between the two deformed materials. 

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