Issue 29

A. Castellano et alii, Frattura ed Integrità Strutturale, 29 (2014) 128-138; DOI: 10.3221/IGF-ESIS.29.12 137 definitively does not hold for any  . In correspondence, a load cr  corresponding to cr n is determined, and this is the critical value of the circular angle of shear which allows for the occurrence of a non-trivial axial periodic deformation of the form (31). Fig. 1 reports the bifurcation curve for a referential shear modulus 1   MPa, referential Poisson’s ratio 0.25   , 3/ 4   and H/R 1 =1. It shows the critical load cr  as a function of the ratio 2 1 R /R between the outer and the inner radius of the tube. We may easily infer, as expected, that twist-like bifurcation may be more easily supported in thin tubes. Figure 1 :  cr vs. R 2 /R 1 , for  = 1,  = 3/4,  = 0.25, H/R 1 =1. C ONCLUSIONS e have studied a bifurcation problem concerning the possibility of periodic twits-like deformations superposed to a primary azimuthal shear state for a Levinson-Burgess isotropic elastic tube. The procedure proposed for determining approximate solutions of the resulting non-autonomous systems of linear ODE’s is based on the Magnus method, which represents an innovative numerical scheme belonging to the more general class of geometric numerical integrators. We have the feeling that these new methods, already employed in many scientific areas, may find interesting applications also in the field of continuum mechanics. In the future we are going to further explore the applicability and the advantages of the Magnus method by studying other bifurcation problems yielding non-autonomous ODE systems like, for example, a Taylor-Couette type bifurcation for a sheared incompressible annular tube modeled by an extended version of the Gent constitutive equation. A CKNOWLEDGMENTS he authors gratefully acknowledge the research projects MIUR-PRIN 2010-2011: “ Dinamica, stabilità e controllo di strutture flessibili ” and MIUR PON-REC: “MASSIME” – Sistemi di sicurezza meccatronici innovativi (cablati e wireless) per applicazioni ferroviarie, aerospaziali e robotiche . R EFERENCES [1] Goriely, A., Vandiver, R., Destrade, M., Nonlinear Euler buckling, Proc. R. Soc. A, 464 (2008) 3003 – 3019. [2] Shuvalov, A. L., A sextic formalism for the three-dimensional elastodynamics of cylindrically anisotropic radially inhomogeneous materials, Proc. R. Soc. Lond., 459 (2003) 1611–1639. [3] Shuvalov, A. L., Poncelet, O., Deschamps, M, General formalism for plane guided waves in transversely inhomogeneous anisotropic plates, Wave Motion, 40 (2004) 413 – 426. W T