Issue 29

A. Castellano et alii, Frattura ed Integrità Strutturale, 29 (2014) 128-138; DOI: 10.3221/IGF-ESIS.29.12 128 Focussed on: Computational Mechanics and Mechanics of Materials in Italy Geometric numerical integrators based on the Magnus expansion in bifurcation problems for non-linear elastic solids A. Castellano, P. Foti, A. Fraddosio, S. Marzano, M. D. Piccioni Politecnico di Bari – Dipartimento di Scienze dell’Ingegneria Civile e dell’Architettura anna.castellano@poliba.it , pilade.foti@poliba.it , a.fraddosio@poliba.it , s.marzano@poliba.it, m.d.piccioni@poliba.it A BSTRACT . 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. K EYWORDS . Nonlinear elasticity; Bifurcation; Geometric numerical integrators; Magnus expansion. I NTRODUCTION n the framework of three-dimensional non-linear elasticity, a number of “small on large” bifurcation problems lead to the analysis of a system of linear second order ODE’s with varying coefficients. It is a common practice to reduce the system of linear second order ODE’s to a simpler non-autonomous first order linear ODE system. Nevertheless, the resulting differential system may be somewhat complex and only numerically tractable; thus, it is crucial to adopt an adequate strategy for obtaining an accurate numerical solution for determining the critical load and the corresponding bifurcation deformation field. For example, approximate expressions of the matricant of non-autonomous linear ODE systems, based either on the multiplicative integral of Volterra or on the truncated Peano expansion, have been recently proposed for bifurcation problems analyzed by means of the Stroh approach within the so-called sextic formalism [1-3]). In Section 2, we discuss an alternative numerical method – based on the Magnus expansion [4] – which furnishes an approximate exponential representations of the matricant of first order linear ODE systems. The Magnus expansion belongs to the so-called geometric numerical integrators, based on Lie-Group methods [5 -7] for appropriate references). To our knowledge, the applications of this method are not widespread in continuum mechanics, but it has been shown in [8] that for problems involving singularities, bifurcations and wave propagation the geometric numerical integrators may be useful and accurate. In particular, the Magnus expansion may be very efficient for a number of bifurcation analyses, since it leads to the determination of approximate solutions that preserve at any order of approximation the same qualitative properties of the exact (but unknown) solution; moreover, this method exhibits an improved accuracy with respect to other frequently used numerical schemes. As an application of the Magnus method, in A n application of the Magnus method in solid mechanics: periodic twist-like deformations bifurcating from the azimuthal shear of a circular tube we investigate the possibility for a compressible hollow cylinder subject to I