Coordinate free nonlinear incremental discrete mechanics

Abstract : In this paper we extend the Kinematic Structural Stability (KISS) issues obtained in the punctual linear case to a full non linear framework. Kinematic Structural Stability (KISS) refers to the stability of conservative or non-conservative elastic systems under kinematic constraints. We revisite and sometimes introduce some new mechanical and geometrically-related concepts, using the systematic language of vector bundles: transversality-stability, tangent stiffness tensor, loading paths and KISS. We bring an intrinsic geometrical meaning to these concepts and we extend the main results to a general nonlinear framework. The lack of connection on the configuration manifold in order to make derivatives of sections is bypassed thanks to the key rule of the nil section of the cotangent bundle. Two mechanical examples illustrate the interest of the new concepts, namely a discrete nonconservative elastic system under non-conservative loading (Ziegler's column) and, in more details, a four-sphere granular system with nonconservative interactions called diamond pattern.
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Jean Lerbet, Nicolas Challamel, François Nicot, Félix Darve. Coordinate free nonlinear incremental discrete mechanics. Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Wiley-VCH Verlag, 2018, 98 (10), pp.1813-1833. ⟨10.1002/zamm.201700133⟩. ⟨hal-01857781⟩

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