Asymptotic homogenization models for pantographic lattices with variable order rotational resistance at pivots

Abstract

In this paper, different asymptotic orders of torsional resistance at the pivots for linear pantographic lattices are introduced within an asymptotic homogenization of the discrete micro-beam model associated with the summation of internal forces and moments at the pivots. The small parameter is defined by the ratio of the square root of a unit cell size for the lattice and the problem domain size. The objective is to systematically identify leading order continuum field equations and to define homogenized elasticity tensors and their properties for increasing rotational torque stiffness. To measure the effect of the torsional spring stiffness on the deformation of the lattice, the torsional resistance of a beam element between pivots which compares the torque stiffness to its bending moment stiffness is introduced as a power of the small parameter. Consistent with previous models for pantographic sheets with zero (perfect) or light torsional resistance, an asymptotic expansion with the small scaling parameter shows that the leading order continuum field equations are characterized by second gradient elasticity theory. The homogenized elasticity coefficients are determined in terms of the fiber characteristics and the torque stiffness. A shear modulus is calculated proportional to the torque stiffness. By increasing the torsional resistance to order one, the resulting field equations move to first-gradient classical elasticity with a shear modulus varying as a homographic function in the torsional resistance. By increasing torsional stiffness to higher orders, the shear modulus is obtained for the limiting case of rigid connections. Numerical results are presented for standard elongation and shear bias tests. Strain energies from the first gradient and second gradient terms present for the models with different orders of torsional resistance are computed to demonstrate the zero and infinite limiting cases, and the need to move from a second-gradient continuum model with low-order torsional resistance as scaled with the small parameter, to a first-gradient model for high-orders of torsional resistance.