Shell displacements

We consider an equilibrium problem for a shell with a crack. The faces of the crack are assumed to satisfy a nonpenetration condition, which is an inequality imposed on the horizontal shell displacements. The properties of the solution are analysed - in particular, the smoothness of the stress field in the vicinity of the crack. The character of the contact between the crack faces is described in terms of a suitable nonnegative measure. The stability of the solution is investigated for small perturbations to the crack geometry. The results presented were obtained in (Khludnev, 1996b). 

This section is concerned with an extreme crack shape problem for a shallow shell (see Khludnev, 1997a). The shell is assumed to have a vertical crack the shape of which may change. From all admissible crack shapes with fixed tips we have to find an extreme one. This means that the shell displacements should be as close to the given functions as possible. To be more precise, we consider a functional defined on the set describing crack shapes, which, in particular, depends on the solution of the equilibrium problem for the shell. The purpose is to minimize this functional. We assume that the... 

When the charges are treated adiabatically, a self-consistent method must be used to solve for the shell displacements, d, (just as with the dipoles fi, in the previous section). Combining Eqs. [26], [28], and [29], we can write the total energy of the shell model system as... 

As mentioned earlier, the shell model is closely related to those based on polarizable point dipoles in the limit of vanishingly small shell displacements, they are electrostatically equivalent. Important differences appear, however, when these electrostatic models are coupled to the nonelectrostatic components of a potential function. In particular, these interactions are the nonelectrostatic repulsion and van der Waals interactions—short-range interactions that are modeled collectively with a variety of functional forms. Point dipole-and EE-based models of molecular systems often use the Lennard-Jones potential. On the other hand, shell-based models frequently use the Buckingham or Born-Mayer potentials, especially when ionic systems are being modeled. 

Static simulations of perfect lattices give the lattice energy and crystal structure of the garnets at 0 K. In the static limit, the lattice stmcture is determined by the condition 9 //9A = 0, where U is the internal energy, and the variables A define the structure (i.e., the lattice vectors, the atomic positions in the garnet unit cell, and the oxygen shell displacements). 

Differentiating with respect to core and shell displacements gives... 

The shell displacements are removed from the equation relating to the cores by rearranging the equation relating to the shells to give the shell displacements ... 

The treatment of the defective lattice follows the customary two-region approach (Catlow and Mackrodt, 1982 this volume Chapter 7) in which the total energy of the defective system is minimized by variation of the nuclear positions (and shell displacements) around the defect. The crystal is partitioned into an inner region, immediately surrounding the defect where the relaxation is assumed to be greatest, and a less perturbed outer region. In the inner region the... 

The above potential is based on a rigid-ion-model (RIM), as no effect of atomic polarization is taken into account. A shell model (SM) was also developed, which considers a split core-shell structure for polarizable O atoms. As usual [23], core and shell are coupled by an elastic spring of force constant k, and are characterized by different electric charges zqc and ZQs- In addition to k, and zqs, also the core-shell displacement, is to be optimized, and contributes three positional parameters (unless reduced by symmetry) for eaeh O atom in the asymmetric unit. When an O atom is involved in the two-body interaction, the repulsive and, possibly, dispersive energy is eomputed by reference to the 0 shell position. All other atoms and interactions are treated as for the RIM case. 

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