Large in-plane elastic deformations of bi-pantographic fabrics: asymptotic homogenization and experimental validation

Abstract

Bi-pantographic fabrics are composed of two families of pantographic beams and correspond to a class of architectured materials that are described in plane as second-gradient 2D-continua. On a discrete level, a pantographic beam is a periodic arrangement of cells and looks like an expanding barrier. The materialisation of a bi-pantographic fabrics made by Polyamide was achieved by additive manufacturing techniques. Starting from a discrete spring system, the deformation energy of the corresponding continuum is derived for large strains by asymptotic homogenisation. The obtained energy depends on the second gradient of the deformation through the rate of change in orientation and stretch of material lines directed along the pantographic beams. Displacementcontrolled bias extension tests were performed on rectangular prototypes for total elastic extension up to 25%.Force-displacement measurements complemented by local digital image correlation analyses were used to t the continuum model achieving excellent agreement.

Size effects of mechanical metamaterials: a computational study based on a second-order asymptotic homogenization method

“…-The extension to 3D cases will be conducted in order to be able to predict out-of-plane deformations [11,23,33] and to study 3D metamaterials. -The application of the homogenization method to the analysis of pantographic (bi-pantographic) structures will be explored [9,10,26,27]. -A comparative study will be undertaken between the presented computational homogenization method and that shown in [23,31].…”

“…Many efforts have been made to develop strain gradient continua. Dell'Isola and colleagues developed second gradient homogenization techniques for the so-called pantographic structures which has been extensively studied, for example, in [3,9,11,[23][24][25][26][27]55]. Boutin et al [18] obtained high-order continuum model by means of a micro/macro-identification procedure based on the asymptotic homogenization method of discrete media.…”

Lattice shells composed of two families of curved Kirchhoff rods: an archetypal example, topology optimization of a cycloidal metamaterial

“…Specifically, the first eight vibration modes have been determined. Figures 16,17,18,19,20,21,22 and 23 show a comparison of the modal shapes associated with the natural frequencies for both models. Table 2 lists their natural frequencies with relative errors.…”

“…This is a light material with a huge range of deformation in the elastic regime. Some alternatives of this type of material have been proposed recently for the 1D [18,42,76] and 2D case [16,17].…”

Effective strain gradient continuum model of metamaterials and size effects analysis

“…On the global scale, the material is treated as an equivalent homogeneous continuum with the aforementioned effective properties. There are a number of RVE-based methods, such as computational approaches [75], asymptotic homogenization methods [6,8,15,20,66], and variational-asymptotic methods [17,88]. However, these methods depend on the ratio, , of the unit size l to the size of the global region of interest L. Note that a unit cell can be different from an RVE.…”