Homogenization of helical beam-like structures: application to single-walled carbon nanotubes

Abstract

International audienceThis work is devoted to the computation of axial stiffness of helical beam-like structures. Starting from the homogenization theory of periodic slender domains and taking benefit of the property of helical symmetry, the overall elastic behavior can be obtained from the solution of three-dimensional problems posed on a reduced basic cell. The mechanical analysis of this reduced basic cell performed using a concise FE model allows therefore to compute easily the anisotropic beam homogenized stiffness coefficients. The accuracy and usefulness of this approach is demonstrated by comparisons with reference solutions and large FE model results for two numerical volume structure examples: a wire spring and a stranded “6 + 1” rope. The homogenization procedure is then applied to single-walled carbon nanotubes and it is shown from the two helical symmetries that their basic cell can be reduced to three beam elements

Derivation of the Young's and Shear Moduli Ofsingle-Walled Carbon Nanotubes Through a Computational Homogenization Approach

“…The numerical implementation of the basic cell problems using FE software appears to be very easy. The data corresponding to a macroscopic strain are imposed through a set of linear relationships relating displacement components of the opposite nodes (situated on the+ and-sides) (Cartraud and Messager, 2006;Messager and Cartraud, 2008):…”

“…then expressed in the local cylindrical coordinates (i = r, 9, 3) (Messager and Cartraud, 2008). Thus, one can define a problem posed on this helical basic cell oflength "A/ e to compute the axial properties of an SWCNT.…”

“…The results for Samceflarge FE models are also given. The homogenized stiffness in torsion can also be validated using the same approach [see Messager and Cartraud (2008)].…”

Two dimensional modeling of helical structures, an application to simple strands

“…Nawrocki and Labrosse [15] emphasized on the effect that the helix kinematic considerations have on the stiffness terms of single layer helical strands. Cartraud and Messager [16] applied homogenization theory to beams with periodic micro-structures to simulate the response of helical bodies, applying their modeling approach to single layer engineering strands, while Messager and Cartraud [17] presented a modeling technique based on volumetric finite elements. In more recent works, Usabiaga and Pagalday [18] introduced a modeling approach for the simulation of axial and torsional helix loading, while Ghoreishi et al [19] pointed out limitations of several analytical schemes.…”

Analytical closed-form expressions for the structural response of helical constructions to thermal loads

“…The measurements have provided useful insights into the role of the strand's structural arrangement on the deformation profile, as well as on the forces and moments generated. On the numerical modeling side, different finite element models have been developed to simulate the helix axial and torsional loading mechanical response [18,19,20], their applicability being primarily illustrated in the context of engineering helical strands [21].…”