Geometric numerical integrators based on the Magnus expansion in bifurcation problems for non-linear elastic solids

• Published in: Frattura ed integritá strutturale Vol. 8; no. 29; pp. 128 - 138
• Main Authors: A. Castellano, P. Foti, A. Fraddosio, S. Marzano, M. D. Piccioni
• Format: Journal Article
• Language: English
• Published: Gruppo Italiano Frattura 01.07.2014
• Subjects: Nonlinear elasticity; Bifurcation; Geometric numerical integrators; Magnus expansion
• ISSN: 1971-8993; 1971-8993
• DOI: 10.3221/IGF-ESIS.29.12

We illustrate a procedure based on the Magnus expansion for studying mechanical problems which lead to non-autonomous systems of linear ODE’s. The effectiveness of the Magnus method is enlighten by the analysis of a bifurcation problem in the framework of three-dimensional non-linear elasticity. In particular, for an isotropic compressible elastic tube subject to an azimuthal shear primary deformation we study the possibility of axially periodic twist-like bifurcations. The approximate matricant of the resulting differential problem and the first singular value of the bifurcating load corresponding to a non-trivial bifurcation are determined by employing a simplified version of the Magnus method, characterized by a truncation of the Magnus series after the second term.