On the synthesis of complete two-dimensional second-gradient continua: Tri-pantographic fabrics

Abstract

We introduce a notion of completeness for two-dimensional second-gradient elastic continua and propose a microstructural route toward its synthesis. A continuum is said to be complete if the Hessian of the stored energy with respect to the second-gradient variable is locally positive definite, so that every nonzero admissible increment of the placement second gradient is quadratically controlled in the highest-order part of the energy about each configuration. Starting from the Principle of Virtual Work, we derive the constitutive relations, equilibrium equations, and admissible boundary interactions for a broad class of fibrous second-gradient continua whose stored energies depend on fiber stretch, stretch gradients, and curvature. The theory is then applied to several continua motivated by pantographic microstructures. Classical pantographic sheets are shown to be incomplete, while bi-pantographic fabrics enlarge the class of components of the second gradient detected by the energy but remain incomplete. Finally, we formulate a tri-pantographic continuum associated with a proposed three-family architecture and prove that it is complete. The examples illustrate how microstructural architecture influences both the completeness properties of an effective continuum and the pointwise form of its higher-order boundary interactions.