To consider second gradient continua as constrained microstructured continua à la Germain simplifies numerical analysis of metamaterials

In this work we base our analysis on a seminal paper by Paul Germain (1973) in which the principle of virtual work is used to found microstructured continuum mechanics. Recently, it was shown that a particular considered microstructured continuum can be regarded as a second gradient continuum of the kind studied by Germain. To prove how these theoretical ideas can be useful in applications, we present the paradigmatic case of pantographic metamaterials, in which the deformation energy may depend only on placement second gradient being independent on placement first gradient. In this case, the numerical simulations become more efficient by introducing the Lagrange multipliers which are dual in work of the introduced kinematical constraint, so proving that the viewpoint about stress presented originally by Lagrange is the most fruitful one.