Case of a shell

In this section we consider the model of a shallow shell analysed in the previous section and prove the convergence of solutions provided that the length of the boundary crack tends to zero. 

Let f2 C be a bounded domain with a smooth boundary L, and G r, G r, x (see Fig.4.3). Consider the smooth curve F with the ends x, x by assuming F c fl. Let x G F, and the length of the curve F with the ends a °, x be equal to 5 0. The points x, x and x, x are assumed not to belong to F, F, respectively. The curve F divides fl into two subdomain fli, 0,2- In the domain fig = fI F5 we consider the equilibrium problem for the shallow shell. Our aim is to prove that the solutions of this problem converge to the solution of the equilibrium problem found in the smooth domain 0 as (5 — 0. 

Here n is a unit exterior vector to F. As it was indicated in the previous section, in the domain fig the following problem can be solved. 

Choosing a subsequence still denoted by x we assume that as 5 — 0 

This allows us to define the function % = (IF, w), such that 

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The importance of the off-diagonal QA/i terms is most easily demonstrated in the case of a shell of N equivalent orbitals. Then Ax may be reduced to... 

Non-relativistic and relativistic cases of a shell of equivalent electrons... 

Matrix elements of the energy operator in the case of a shell lN... 

Damping Characteristics. Damping causes vibrations to decay in an elastic structure and depends on the vibration frequency, the material of the elastic structure, the geometry, and the physical properties of the surrounding fluid (in the case of a shell-and-tube heat exchanger, the surrounding fluid is the shell fluid). 

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