Issue 29

A. Castellano et alii, Frattura ed Integrità Strutturale, 29 (2014) 128-138; DOI: 10.3221/IGF-ESIS.29.12 132 A deformation         : = r, , z R, , Z       f X x f X f    of  is assumed to be a smooth function with gradient     :   F X f X . Since the tube is elastic and isotropic, the general form of the strain energy function is given by     W W I , II , III  F B B B , where           2 2 2 2 2 T 1 1 I tr , II tr tr tr , III det det 2 2                 B F F B B F F FF B F B B B (12) are the orthogonal principal invariants of T :  B F F ; thus, the Piola stress takes the form       T 1 2 3 DW W W I W III 2 2 2       S F F F F BF F B B (13) with 1 2 3 W W W W : , W : , W : I II III          B B B (14) We assume that the inner cylinder is kept fixed, whereas the outer is subject to a uniform angular displacement 0   around its axis. On the bases of  only tangential displacements are admitted. This leads to the following mixed boundary-value problem:   Div = in S 0 F  (15)               1 R 2 2 1 Z 1 2 2 1 2 , 0 at R = R = R cos 1 at R = R 0 at R = R = on R sin at R = R = 0 at R = R R                  f X X e f X X e f X X e  (16)         2 = 0, = on        f X X e S e e 0 F  (17) For the equilibrium problem (15)-(17), we consider the possibility of an azimuthal shear deformation f  defined by   r R, R , z Z        (18) where  , the angular displacement field, is assumed to be a smooth function satisfying     1 2 R 0, R 0       1 on   (19) Consequently, in view of (18) we have     2 2 R r r R R , R R                      I e I   F e B e e e e e e (20) where the prime denotes differentiation with respect to R and   r z , ,  e e e is the deformed cylindrical orthonormal basis at   r, , z  . Notice that the azimuthal shear is an isochoric deformation whose principal invariants (12) are given by 2 2 , I II 3 R II 1 I         B B B (21) Moreover, (18) trivially satisfies the displacement boundary conditions (16)-(17) 1 and, in view of (13) and (20), it is easily seen that also the traction boundary condition (17) 2 holds. It remains to study the equilibrium field Eq. (15). The explicitly evaluation of (15) by means of (13) and (20) yields an over-determined system of two ODE for the single unknown function   R  . In particular, here we consider the class C 2 Levinson-Burgess strain energy function