Predictive strain-gradient homogenization of a pantographic material with compliant junctions

This paper presents an architectured material featuring significant strain-gradient effects and called pantographic material. It is easy to fabricate, being a plate made of a single and continuous linear elastic material containing voids. The pattern consists of triangles connected by thin junctions and arranged in such a way that two floppy strain modes are present. A homogenization scheme based on the two-scale asymptotic expansion is suggested, keeping only significant strain-gradient contributions in the homogenized energy through an adequate projection. The predictions from the homogenization scheme are validated against a full-scale simulation and yield very good L² error estimates whereas the classical first-gradient homogenization fails. Furthermore, the relative position of the unit cell in the full-scale computation does not have a significant influence on the quality of the prediction. Finally, with an adequate choice of scalings between the scale separation and the junction thinness, it is possible to ensure that strains as well as displacements in compliant junctions remain bounded while preserving macroscopic strain-gradient effects.