Symmetry breaking and emergent shear-normal coupling in two-point periodic lattices

We determine the in-plane effective compliance of Y-type hexagonal lattices and identify the structural origin of shear-normal coupling. The lattice walls are modelled as Euler-Bernoulli beams with homogeneous material properties and cross-sections assigned consistently to symmetry-related members, and the effective compliance is obtained by strain-energy homogenization of the periodic unit cell under uniform macroscopic stress. The planar response is described by six independent compliances; the two shear-normal terms S13 and S23, which couple extensional and shear deformation, are the focus of this work. We show that these terms vanish if and only if the unit cell is invariant under a mirror aligned with the coordinate axes, and that this invariance requires two independent geometric ingredients: the two branches meeting at a junction must be images of one another under that mirror, a property we call branch balance, and the junctions of the repeating unit must all carry the same geometric data, a uniform decoration. Neither ingredient follows from periodicity. The closure identity of a hexagonal cell, whose boundary contains two stems and four branches, holds for every choice of branch projections and therefore constrains nothing, so a uniformly decorated lattice with unbalanced branches is admissible: it is the balanced honeycomb with successive rows slipped horizontally, it retains a single junction type, and it is generically coupled. The construction developed here instead assigns the two branch families to opposite sides at alternating junctions, giving a two-point basis, and for it we obtain closed-form expressions for all six compliances as functions of the dimensionless geometric parameters. The shear-normal compliances are then nonzero for all but a measure-zero set of geometries, vanishing only on isolated surfaces that include the symmetric one-point limit, which is recovered exactly. Because the basis size is a positive integer, two is the smallest basis supporting the alternating arrangement used here; this minimality refers to the arrangement and not to coupling in general, which the slipped lattice already reaches with a single junction type. Using the closed-form solution we map the magnitude and sign of S13 and S23 over the design space, give the loci on which each coefficient changes sign, and show that the coupling grows linearly with the geometric asymmetry near the symmetric limit, so that small departures from a symmetric cell produce first-order coupling. The four first-order sensitivities are obtained in closed form, and one of them vanishes on a curve of the geometry plane through the regular honeycomb, where S13 becomes second order in the angular asymmetry and first order in the length asymmetry alone. The results separate two roles in architected lattice design: the mirror content of the unit cell, set jointly by branch balance and by the decoration of the periodic basis, fixes which couplings are attainable, and the cell geometry then tunes their magnitude and sign.