Issue 29

A. Castellano et alii, Frattura ed Integrità Strutturale, 29 (2014) 128-138; DOI: 10.3221/IGF-ESIS.29.12 136 Finally, a solution of the primary ODE problem (34) can be written in the following form       2 1 r r r v U v   (45) which clearly satisfies the boundary conditions (34) 2 (see (41) and (43)). The above analysis shows that the possibility of determining non-trivial solutions for the bifurcation differential problem (34) is strictly related to the knowledge of the 3x3 matrix   2 r U , which represents the “north-east” block of the 6x6 matricant (41) of the Cauchy initial-value problem         1 2 1 6x6 r r r for r r r r                    Y A Y I O Y O I (46) Since (46) has the same structure of the Cauchy initial-value problem (1), in the following we illustrate a strategy based on the Magnus method for determining the solution of (46). A strategy for determining the bifurcating load through the Magnus method The theoretical preliminaries in An overview of geometric numerical integrators with special reference to the Magnus expansion combined with the bifurcation analysis of Subsections The fundamental azimuthal shear equilibrium deformation - Periodic twist-like bifurcations allow us to describe a strategy for determining the critical value cr  of the primary angular displacement, that is the lowest value of the angle  for which the tube may support (during a loading process) a non-trivial twist-like bifurcation of the form (31) from the fundamental azimuthal shear deformation. The Magnus method is now the key ingredient for approximately determining the matricant of (46), whereas (44) plays the role of bifurcation condition. Following the steps outlined in the Section 2, we first fix the radii R 1 and R 2 , the height H, the referential shear modulus  and the referential Poisson’s ratio  (recall that 3/ 4   by (24)). Then, we: - fix the number n of twisting cells in the axial direction (by (31) this is equivalent to fixing  ) starting from n =1; - fix the value of the loading parameter  starting from zero; - compute the matrices (35), which now depend only on r, and then compute   r A in (38) by means of (36); - divide the interval   1 2 R , R into N subintervals   i+1 i r , r such that the Magnus series converges in each of them (see the convergence condition (7)); - employ for simplicity a fourth-order method by truncating the Magnus series   i+1 r  after the second term in each subinterval (see (4)) and by approximating the integrals using a Gauss quadrature rule. This allow us to write                     2 1 2 1 2 i+1 i i i i h h 3 r r r r , r 2 12          A A A A  (47) where     1 2 2 1 i i i i R R 1 3 1 3 r r h, r r h , with h = 2 6 2 6 N                    (48) are two Gauss points in correspondence of which   r A and its employed linear approximation agree. - compute the matrix exponential of   i+1 r  using the function MatrixExp in the software Mathematica; - evaluate the matricant of (46) at the external radius r 2 = R 2 by (8) and (9); - compute the “north-east” block of the matricant at r 2 = R 2 , that is the 3x3 matrix   2 2 r U ; - compute the determinant of   2 2 r U and check if the bifurcation condition (44) holds. Following this procedure, we define n cr  as the smallest value of  which satisfies the bifurcation condition (44) corresponding to the fixed value of n. Of course, when one repeats the above procedure by varying the number of possible axial cells, the corresponding critical load n cr  varies. We have performed a large number of computations which show that as n increases, the corresponding n cr  determined by (44) decreases until a critical value of n above which (44)