Issue 29

A. Castellano et alii, Frattura ed Integrità Strutturale, 29 (2014) 128-138; DOI: 10.3221/IGF-ESIS.29.12 133               2 1 2 1 2 = 1 4 1 I 3 1 J 3 2 1 2 III 1 2 III 1 2 1 2 W                                F B B B B (22) where   and   are the referential shear modulus and referential Poisson’s ratio, respectively, and   0, 1  is a dimensionless material parameter. For the constitutive class (22), the Piola stress takes the form         1 2 1 2 -1 -T -T 4 1 1 1 2 2 III 1 III 1 2                                        B B S F F B F F (23) By following a result contained in [16], it is possible to generate classes of strain energy functions such that the azimuthal shear satisfies both the two equilibrium ODE; in particular, for the present case of a Levinson-Burgess material, deformations of the form (18) are allowed only if the following condition on the material parameters holds: 3 4   (24) Finally, in view of (23), (19)-(21) and (24), after a non-trivial calculation we obtain from (15) a single ODE, whose solution is       2 2 2 2 2 2 2 1 1 2 R R R R R R 1 R,            (25) Periodic twist-like bifurcations We now consider the possibility of deformations which bifurcate from the primary equilibrium pure circular shear (18) as the loading parameter (angle)  increases from 0 (natural state). A condition ( only necessary ) for the occurrence of a local branch of bifurcating solutions at a fixed  is the existence of a non-trivial solution of the homogeneous linearized equilibrium problem which corresponds to an adjacent equilibrium state. Such solutions superposed to the primary azimuthal shear must satisfy the adjacent equilibrium Eq. (26) and the incremental boundary conditions (27)-(28), obtained by linearizing (15)-(17) around the fundamental deformation    x f X  , which is conveniently assumed as the independent variable:         grad r, div in   0  u F   (26) 1 on   u 0  (27)       grad r, z 2 z z = 0, = on     u e e e 0  u F   (28) In particular,   3 :  u x   represents an incremental displacement field, “div” and “grad” are the divergence and gradient operators with respect to x , respectively, and  is the fourth-order instantaneous elasticity tensor, given by:               r, r, r, r, 2 T : D , W Lin                H H H F F F F  (29) For the constitutive class (22), it follows from (29), (20)-(21) and (24) that                       r, T T 1 1 T grad 1 grad grad grad 4 3 2 1 grad grad grad 4 1 2 2                                   u B B u u u B I u I u     u F  (30) where B  is given by (20) 2 . We now restrict our analysis to the class           3 r 1 2 z 3 1 2 : : u v r cos z, u v r cos z, u v r sin z, n , n 0, 1, 2,.., r r H                          u u u  0   (31)