Large in-plane elastic deformations of bi-pantographic fabrics: Discrete modelling, asymptotic homogenization and experimental validation
At LMSSC, Paris, July 2nd 2019, 1 p.m.
Doctorate, Department of Structural and Geotechnical Engineering, University La Sapienza, Rome, Italy
There is a kind of 2D-continua in-plane which can be described exclusively by their placement functions. We derive for large strains the deformation energy of one such continuum by applying a heuristic homogenisation procedure to a family of discrete elastic bi-pantographic structures.
Such structures generalize previously introduced pantographic ones, since they can be regarded as these last being obtained interconnecting discrete pantographic beams constituted by extensional and rotational springs, rather than interconnecting discrete Elasticae. Within the homogenisation process, the overall dimension of the system is kept fixed, while the number of the periodically appearing subsystems is increased, and the stiffnesses are appropriately scaled.
The deformation energy of the continuum limit of discrete pantographic fabrics is non-classic. Indeed, it depends on the second gradient of the placement through the material curvature of said material lines. The deformation energy of the continuum model derived herein does further depend on their stretch gradient. Therefore, in addition to classical essential placement line-boundary conditions and non-classical placement vertex-boundary conditions, further non-classical boundary conditions can be derived by variational deduction for such 2D-continuum, i.e. normal placement gradient line-boundary conditions.
In the spirit of metamaterials, and following the rationale underlying that exploited for pantographic fabrics, we successively propose a possible design of bi-pantographic prototype aimed at obeying the discrete model. Displacement-controlled bias extension tests have been performed on such specimens for large total quasi-elastic deformations up to ca. 25%. Experimental force-displacement measurements -- together with deformation ones obtained by local digital image correlation (DIC) -- are used to fit the discrete and the continuum model, achieving excellent agreement within the measurements' tolerances.