Bulk modulus

The bulk modulus K is a measure of the pressure Ap needed to change a 

The minus sign accounts for the fact that positive pressure usually causes a 

The faces in low density LDPE foams are partly buckled or wrinkled, as a result of processing (a.l7). This affects both the bulk modulus and the Young s modulus. The foam bulk modulus Kp is predicted, using the Kelvin closed cell foam model, to be  

This is not true for single crystals because they are generally anisotropic, i.e., the velocity of propagation is not the same in the [100] as in the [110] directions. However, polycrystalline materials and glasses are isotropic because of the random nature (or absence) of their grain structure. Therefore, the above relationship holds for bulk polycrystalline solids. This allows a number of relationships between B, E, G, and v to be derived. Using the relation between the Poisson ratio and Cn and C12 to eliminate C12, the bulk modulus may be related to Young s modulus by 

Using Equation 7.12 to relate C44 to Cn and C12 for isotropic materials, we can write 

A number of simple empirical potential functions were presented in Chapter 3. We will now attempt to use some of these functions in order to derive relationships between the various elastic properties of materials. 

Let us consider the pressure of electron gas within the framework of the independent electron approximation. 

The average energy of free-electron gas containing N electrons is given by 

Because the electrons do not interact with one another the energy N Ep is a pure kinetic. The pressure of this gas can be found as p = — dE/dV)i.i. Bulk modulus can be calculated taking the derivative of the pressure, B = —V (9p/9V)j,. Note that the number of electrons is constant. 

Substitution of the values to the end expression of (5.32) and calculation leads to a design formula 

The theory predicts results close to the experiment for potassium, rabidium and cesium and fails in the case of lithium and sodium (Table 5.1). Note that the model of independent electrons fails for elements with a high electron density (Vs = 3.23 and 3.93). 

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Alexander S, Chaikin P M, Grant P, Morales G J, Pincus P and Hone D 1984 Charge renormalisation, osmotic pressure, and bulk modulus of colloidal crystals theory J. Chem. Phys. 80 5776-81... 

There has not been as much progress computing the properties of crystals as for molecular calculations. One property that is often computed is the bulk modulus. It is an indication of the hardness of the material. 

Material properties can be further classified into fundamental properties and derived properties. Fundamental properties are a direct consequence of the molecular structure, such as van der Waals volume, cohesive energy, and heat capacity. Derived properties are not readily identified with a certain aspect of molecular structure. Glass transition temperature, density, solubility, and bulk modulus would be considered derived properties. The way in which fundamental properties are obtained from a simulation is often readily apparent. The way in which derived properties are computed is often an empirically determined combination of fundamental properties. Such empirical methods can give more erratic results, reliable for one class of compounds but not for another. 

For crystalline polymers, the bulk modulus can be obtained from band-structure calculations. Molecular mechanics calculations can also be used, provided that the crystal structure was optimized with the same method. 

The basic wave velocity is a function of the bulk modulus and density of the fluid, where K = bulk modulus, Pa. 

Material Young s modulus, E, GPa Proportionality limit d, % extension Shear modulus, G, GPa Poisson s ratio, P Bulk modulus, B, GPa "... 

The are the elastic constants the bulk modulus of the material is computed as -B = + 2c 2 )/3- Values in parentheses are estimates. 

Mechanical Properties. Measuremeat of the mechanical properties of diamoad is compHcated, and references should be consulted for the vahous qualifications (7,34). Table 1 compares the theoretical and experimental bulk modulus of diamond to that for cubic BN and for SiC (29) and compares the compressive strength of diamond to that for cemented WC, and the values for the modulus of elasticity E to those for cemented WC and cubic BN. 

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... 

P = hquid bulk modulus of elasticity E = elastic modulus of pipe waU D = pipe inside diameter b = pipe wall thickness... 

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. 

A guide to tire stabilities of inter-metallic compounds can be obtained from the semi-empirical model of Miedema et al. (loc. cit.), in which the heat of interaction between two elements is determined by a contribution arising from the difference in work functions, A0, of tire elements, which leads to an exothermic contribution, and tire difference in the electron concentration at tire periphery of the atoms, A w, which leads to an endothermic contribution. The latter term is referred to in metal physics as the concentration of electrons at the periphery of the Wigner-Seitz cell which contains the nucleus and elecUonic structure of each metal atom within the atomic volume in the metallic state. This term is also closely related to tire bulk modulus of each element. The work function difference is very similar to the electronegativity difference. The equation which is used in tire Miedema treatment to... 

Ps = PoCiiAri + Bri + Crj + Drj + ), which upon differentiation yields the isentropic bulk modulus... 

Because the Griineisen ratio relates the isentropic pressure, P, and bulk modulus, K, to the Hugoniot pressure, P , and Hugoniot bulk modulus, K , it is a key equation of state parameter. 

Ruoff (1967) first showed how the coefficients of the shock-wave equation of state are related to the zero pressure isentropic bulk modulus, and its first and second pressure derivatives, K q and Kq, via... 

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). 

A proportionality between the theoretical spall strength and the bulk modulus is obtained when a two-parameter model is chosen to represent the intermolecular potential. Other two-parameter representations of the intermolecular potential, such as the Lennard-Jones 6-12 potential, will yield a similar proportionality although the numerical coefficients will differ slightly. 

With a three-parameter model of the intermolecular potential, the theoretical spall strength is not simply a constant times the bulk modulus. Although the slightly greater accuracy obtained is not critical to the present investigation, an energy balance is revealed in the analysis which is not immediately transparent in the Orowan approach. 

In (8.1), is the specific cohesive energy, v = 1/p is the specific volume and the reciprocal of the density p. Vq is the specific volume at zero pressure as shown in Fig. 8.1. The final parameter a is constrained by the relation for the bulk modulus... 

Note that the theoretical spall strength now depends upon the cohesive energy as well as the bulk modulus. Representative values for selected metals are shown in Table 8.1. These can be compared with experimental spall strengths in later sections. 

F(FG = normal (shear) component of force A = area u(w) = normal (shear) component of displacement o-(e ) = true tensile stress (nominal tensile strain) t(7) = true shear stress (true engineering shear strain) p(A) = external pressure (dilatation) v = Poisson s ratio = Young s modulus G = shear modulus K = bulk modulus. 

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