Extension of non-linear beam models with deformable cross sections

Соколов, И. А.; Krylov, Slava; Harari, Isaac

doi:10.1007/s00466-015-1215-5

Cited by 9 publications

(3 citation statements)

References 56 publications

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“…These papers considered linearly elastic material and addressed both static and dynamic cases, but they presented different approaches to time-stepping schemes and updating rotation vector: Eulerian [35,36], updated Lagrangian [40], and total Lagrangian [41,39]. Since these papers got published, research tackling the theoretical and computational techniques gained momentum and the advanced research in this field continues to occur till date, for examples: problems related to discretization and interpolation approaches [42,43,44,45,46,47,48], mixed formulation [49], non-linear materials and constitutive law [50,51,52,53], space and time-integration schemes [54,55,56], initially curved configuration [57,32], higher-order Kirchhoff-love beam [58,59,60], and enhanced kinematics [35,61,62,63]. Noteworthy contributions to computational formulation of geometrically-exact beam including shear deformation and their applications (e.g., multi-body dynamics of earth orbiting satellites) was made by Vu-Quoc in collaboration with Simo [64,65,66].…”

Section: Introductionmentioning

confidence: 99%

The mathematical theory of a higher-order geometrically-exact beam with a deforming cross-section

Chadha

Todd

2020

International Journal of Solids and Structures

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This paper investigates the variational formulation and numerical solution of a higher-order, geometrically exact Cosserat type beam with a deforming cross-section, instigated from generalized kinematics presented in earlier works. The generalizations include the effects of a fully-coupled Poisson's and warping deformations in addition to other deformation modes in Simo-Reissner beam kinematics.The kinematics at hand renders the deformation map to be a function of not only the configuration of the beam but also on the elements of the tangent space of the beam's configuration (axial strain vector, curvature, warping amplitude, and their derivatives). This complicates the process of deriving the balance laws and exploring the variational formulation of the beam, at the same time, make it worthwhile. The weak and strong form is derived for the dynamic case considering a general boundary.We restrict ourselves to linear small-strain elastic constitutive law and the static case for numerical implementation. The finite element modeling of this beam has higher regularity requirements. The matrix (discretized) form of the equation of motion is derived. Finally, numerical simulations comparing various beam models are presented.

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Section: Introductionmentioning

confidence: 99%

The mathematical theory of a higher-order geometrically-exact beam with a deforming cross-section

Chadha

Todd

2020

International Journal of Solids and Structures

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“…Sokolov, Krylov and Harari (2015) [62] extend the work of Dasambiagio, Campello and Pimenta [20] to allow distortion of the cross section.…”

Section: Literature Reviewmentioning

confidence: 94%

“…The geometrically exact formulations attracted a lot of researchers since their beginning. Lots of research was done in geometrically exact theories, Simo and Vu-Quoc (1986) [60], Simo and Vu-Quoc (1991) [57], Simo (1992) [54], Pimenta and Yojo (1993) [41], Pimenta (1996) [42], Gruttmann, Sauer and Wagner (1997 [29,30,31] , Pimenta and Campello (2003) [46], Sokolov, Krylov and Harari (2015) [62] to name just a few.…”

Section: Literature Reviewmentioning

confidence: 99%

Modelo geometricamente exato rígido ao cisalhamento de cascas e barras.

Silva¹

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This work presents geometrically exact shear-rigid rod and shell formulations. Displacements and rotations are finite. Linear elastic constitutive equations for small strains are considered in the numerical examples for the rods. A Neo-Hookean material is considered for the shell. Energetically conjugated cross-sectional stresses and strains are defined. A straight reference configuration is assumed for the rod, and a flat reference configuration the shell. Consequently, the use of convective non-Cartesian coordinate systems is not necessary, and only components on orthogonal frames are employed. The parameterization of the rotation field is done by the rotation tensor with the Rodrigues formula, which makes the updating of the rotational variables very simple. The usual Finite Element Method was used and C1 continuity is achieved within the element. This method is used to discretize the potentials on a computational domain in terms of the nodal degrees of freedom. Bearing in mind that the potential is nonlinear a Newton-Raphson iteration scheme is chosen to solve this problem. A set of numerical benchmark examples illustrates the usefulness of the formulation and its numerical implementation. These problems were performed and presented satisfying results. Hence, it can be concluded that this formulation shows great promises to be extensively used for general 3D problems for slender structures. Shear-rigid theories can be widely applied on engineering problems. They can be used in oil drilling rods, robot arms and for rib-reinforced shells that are common in aerospace and automobile industry.

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