Evolution equation of Lie-type for finite deformations, time-discrete integration, and incremental methods

0428756 - ÚTAM 2018 RIV AT eng J - Journal Article
Fiala, Zdeněk
Evolution equation of Lie-type for finite deformations, time-discrete integration, and incremental methods.
Acta Mechanica. Roč. 226, č. 1 (2015), s. 17-35. ISSN 0001-5970. E-ISSN 1619-6937
R&D Projects: GA ČR(CZ) GA103/09/2101
Institutional support: RVO:68378297
Keywords : solid mechanics * finite deformations * evolution equation of Lie-type * time-discrete integration
OECD category: Statistics and probability
Impact factor: 1.694, year: 2015
http://link.springer.com/article/10.1007%2Fs00707-014-1162-9#page-1

While the position and shape of a deformed body take place in the usual three-dimensional Euclidean space, a corresponding progress of the deformation tensor makes up a trajectory in the space of all symmetric positive-definite matrices - a negatively curved Riemannian symmetric manifold. In this context, we prove that a well-known relation between deformation rate and symmetric velocity gradient, via deformation gradient, can be actually interpreted as an equation of Lie-type describing evolution of the right Cauchy-Green deformation tensor on the configuration space .As a consequence, this interpretation leads to geometrically consistent time-discrete integration schemes for finite deformation processes, such as the Runge-Kutta-Munthe-Kaas method. The need to solve such equation arises from an incremental numerical modelling of deformations of nonlinear materials.
Permanent Link: http://hdl.handle.net/11104/0235626