Asymptotic Behavior of Stable Structures Made of Beams

Abstract

In this paper, we study the asymptotic behavior of an $\varepsilon $ ε -periodic 3D stable structure made of beams of circular cross-section of radius $r$ r when the periodicity parameter $\varepsilon $ ε and the ratio ${r/\varepsilon }$ r / ε simultaneously tend to 0. The analysis is performed within the frame of linear elasticity theory and it is based on the known decomposition of the beam displacements into a beam centerline displacement, a small rotation of the cross-sections and a warping (the deformation of the cross-sections). This decomposition allows to obtain Korn type inequalities. We introduce two unfolding operators, one for the homogenization of the set of beam centerlines and another for the dimension reduction of the beams. The limit homogenized problem is still a linear elastic, second order PDE.

Symbolic homogenization and structure optimization for a periodically perforated cylindrical shell

“…The goal is to parametrize the periodicity cell Y with symbolic variables and combine it with the classical numerical computation. First, we start with considering a fixed periodicity cell and follow the procedure presented in [14], [28], and analysis in [2]. There a reduction to 1D beam finite elements is presented.…”

“…Homogenization and dimension reduction for perforated cylindrical shell, passing to a homogenized 2d orthotropic shell and influence of different boundary conditions (fixation on the curves and strait boundaries) is discussed in [1]. The dimension reduction for periodic frame of beams to the elasticity problem on a one‐dimensional fiber network is recently proven in [2, 38]. The obtained limiting homogeneous two‐dimensional cylindrical orthotropic shell is then used to describe the effects of having a point load on the shell.…”

“…In [1] we started with the full 3D linear elasticity problem for a heterogeneous shell and reduced it to an equivalent homogeneous 2D formulation, where the effective properties were obtained by six auxiliary cell experiments. After that, in [2], each cell experiment was reduced to the 1d elasticity problems on a two‐periodic frame. Our goal in this section is to derive an analytic solution to the homogenized problem for a point load, which we call in the following a pinching load, acting on the shell.…”

Simulation of Leather Visco-Elastic Behavior Based on Collagen Fiber-Bundle Properties and a Meso-Structure Network Model

“…The thickness of fibers is small compared to their length. Thus, the dimension of the auxiliary problems in the representative cell can be reduced further by an asymptotic approach with respect to the fiber thickness [19,[22][23][24]. Finally, it is numerically solved by the finite element method with frictional contact [20,21], and extended to visco-elastic relaxation [25,26].…”

Periodic unfolding for anisotropically bounded sequences

“…Such method has largely found application (see e.g. [5,7,8,9,10,11,12]) and also for thin periodic structures like periodically perforated shells (see [18]), textiles made of long curved beams (see [15,16]) and stable lattice structures made of beams (see [14,17]).…”