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Fully isogeometric modeling and analysis of nonlinear 3D beams with spatially varying geometric and material parameters. (English) Zbl 1440.74191

Summary: We present a fully isogeometric modeling and simulation method for geometrically exact, nonlinear 3D beams with spatially varying geometric and material distributions, both along the beam axis and through its cross-section. The approach is based on the modeling of 3D beams using the Cosserat rod theory and the numerical discretization using B-Spline and NURBS parameterizations in an isogeometric collocation method. Transversally varying material constitutions are represented using non-homogeneous, functionally graded beam cross-section definitions such as laminates and continuously graded cross-sections. Furthermore, to model the axial variation of material and geometry, we introduce the parameterization of cross-section properties as spline curves along the beam centerlines. This fully isogeometric modeling and analysis concept, which is based on spline parameterizations of initial beam centerline curves, kinematic unknowns and axially varying material and geometric parameters, has various practical applications enabled by advances in manufacturing technology, including multi-material 3D printing and advanced manufacturing of composites with automated fiber placement. We verify and demonstrate the modeling and simulation approach using several numerical studies and highlight its practical applicability.

MSC:

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
65D07 Numerical computation using splines
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Software:

UMFPACK
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References:

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