Elastic potential energy density

Following the notation of Tucker and Rampton (1972) the elastic potential energy density in the crystal is then... 

This last equation demonstrates that the elastic constant tensor can be obtained by calculating the variations to second order in the potential energy density with respect to small strains. Using this expression, we can write the strain energy density as... 

The density of dislocations is usually stated in terms of the number of dislocation lines intersecting unit area in the crystal it ranges from 10 cm for good crystals to 10 cm" in cold-worked metals. Thus, dislocations are separated by 10 -10 A, or every crystal grain larger than about 100 A will have dislocations on its surface one surface atom in a thousand is apt to be near a dislocation. By elastic theory, the increased potential energy of the lattice near... 

The stiffness matrix, Cy, has 36 constants in Equation (2.1). However, less than 36 of the constants can be shown to actually be independent for elastic materials when important characteristics of the strain energy are considered. Elastic materials for which an elastic potential or strain energy density function exists have incremental work per unit volume of... 

Equations (10.23) and (10.24) hold for the /3-phase as well and could be inserted into Eqn. (10.22). The additivity of pt with respect to the elastic and electric potential is based on 1) the assumption of linear elastic theory (which is an approximation) and 2) the low energy density of the electric field (resulting from the low value of the absolute permittivity e0 = 8.8x10 12 C/Vm). In equilibrium, V/i, = 0 and A V, = df-pf = 0. Therefore, in an ionic system with uniform hydrostatic pressure, the explicit equilibrium condition reads Aa/fi=A)... 

Elastic constants of solids can be related to the fundamental interaction energy terms. In fact it can be shown easily that the bulk modulus, K scales as the energy density, Ulr for an ionic material interacting through Bom-Mayer potential (see later in this section). Thus the molar volume which increases with the presence of larger ions in glasses can be expected to cause a decrease in Young s modulus. 

On a more formal basis we may consider a homogeneous polarization field P, energy density interaction with an elastic field with potential... 

I tested the GAP models on a range of simple materials, based on data obtained from Density Functional Theory. I built interatomic potentials for the diamond lattices of the group IV semiconductors and I performed rigorous tests to evaluate the accuracy of the potential energy surface. These tests showed that the GAP models reproduce the quantum mechanical results in the harmonic regime, i.e. phonon spectra, elastic properties very well. In the case of diamond, I calculated properties which are determined by the anharmonic nature of the PES, such as the temperature dependence of the optical phonon frequency at the F point and the temperature dependence of the thermal expansion coefficient. Our GAP potential reproduced the values given by Density Functional Theory and experiments. 

The K-BKZ Theory Model. The K-BKZ model was developed in the early 1960s by two independent groups. Bernstein, Kearsley, and Zapas (70) of the National Bureau of Standards (now the National Institute of Standards and Technology) first presented the model in 1962 and published it in 1963. Kaye (71), in Cranfield, U.K., published the model in 1962, without the extensive derivations and background thermodynamics associated with the BKZ papers (82,107). Regardless of this, only the final form of the constitutive equation is of concern here. Similar to the idea of finite elasticity theory, the K-BKZ model postulates the existence of a strain potential function U Ii, I2, t). This is similar to the strain energy density function, but it depends on time and, now, the invariants are those of the relative left Cauchy-Green deformation tensor The relevant constitutive equation is... 

The description of deformation in which the development in Section 2.1 is based is retained. However, if the thickness of the film is to be taken into account, then the strain energy of the film material must be included in the calculation of total potential energy. This is accomplished by adopting the strain expression (2.2) for the film as well as the substrate, but augmenting it by the elastic mismatch strain Cm in the former case. The strain energy density throughout the system is then... 

The in-plane tensile stress in the direction normal to the hole and the inplane shear stress on the surface of the hole are zero because of the condition of zero applied traction. The strain energy density as a function of position along the surface of the hole is [/(ly) = (t /2E where E = E/ X — v ) is the plane strain elastic modulus. The chemical potential field over the surface of the hole is then... 

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