A Green operator-based elastic modeling for two-phase pantographic-inspired bi-continuous materials

Abstract We propose an estimate for the effective elastic properties of a new imagined two-phase bi-continuous composite material type with a so-called ``pantographic-inspired” (P-I) architecture in the sense of a matrix reinforcement which is a 3D fiber network capable of large, pantographic-like, deformations, owing to particular properties carried by the fiber interconnections. Such fiber networks co-continuous with the matrix are described from an assemblage of inter-penetrated planar alignments of parallel infinite, identical and equally distant, rods or beams that we call fiber layers or FPAs for ``fiber planar alignments”. Piece-wise linear deformation of such a structure is constrained, as 2D pantographs are, by the specific relations that link the evolutions of the FPA misorientations, of the fiber inter distance and concentration within each FPA, and of the given characteristics to the through-layer interconnections. We derive effective properties within a well defined first gradient elastic homogenization scheme for macro-homogeneous composite structures, based on explicating the involved mean Green operator (mGO) for the representative pattern of such a P-I networked reinforcement in an isotropic matrix. This mGO for P-I composites is built in making use of previously derived and presented mGOs for rod and beam FPAs. Comparisons with Young and shear modulus variations during numerical homogeneous extension simulations show that the proposed modeling provides relevant effective property evolutions with the advantage of being analytical. Comparing so estimated force-displacement curves with numerical ones for 2D pantographs enables to identify ways to further account in the modeling for the specific strengthening effects of the FPA interconnections in pantographs. This tends to prove such P-I composite structures, still to be manufactured, to possibly also behave similarly to pantograph ones.