Issue 29

A. Castellano et alii, Frattura ed Integrità Strutturale, 29 (2014) 128-138; DOI: 10.3221/IGF-ESIS.29.12 135 which well-known existence theorems and procedures for constructing the solutions are available in the literature. With this aim, once introduced the 3x3 matrices                   1 1 T : r, r, , r, , : r, r, r, , r, ,                          T P R Q K P P R R (36) we rewrite (34) in the form     1 2 1 2 , r r r r r             v Tv Kv v v 0 (37) Then, by setting 6x1 6x6 6x6 6x6 : , : : , :                                         v O I y A v T K I O O O M N O O I O (38) we easily see that (37) is equivalent to the linear homogeneous first order ODE boundary-value problem       1 2 1 2 6x1 r for r r r r r                     y A y My Ny 0 0 (39) Observe that in the notation above we have dropped the dependence of the basic parameters (the inner and outer radii, the height of the tube, the material moduli and the angular displacement  ), usually assumed as prescribed. We only maintain explicit the dependence on r, in order to emphasize that the linear ODE system is non-autonomous. Now, by following elementary results on systems of linear ODE's, we may write a solution of (39) in the form (see (38) 1 and (37) 2 )           1 1 6x1 r r r r r             y Y y Y v 0 (40) where   1 r y is an unknown constant vector and               1 2 1 2 1 1 2 6x6 6x6 r r r , with det r 0, for r < r < r and r r r                      I O U U Y Y Y O I U U (41) is a particular fundamental matrix solution for (39) 1 , usually called matricant . Then, substitution of (40) into (39) 2 yields       1 2 1 r r r               0 MY NY y 0 (42) which, in view of (38) 3-4 , (41) and (40), is equivalent to the homogeneous system of three algebraic equations     2 2 1 r r   U v 0 (43) for which non-trivial solution   1 r v 0   are possible if and only if   2 2 det r 0  U (44)