  Mechanical couplings such as axial-shear and axial-bending have great
potential in the design of active mechanical metamaterials with directional
control of input and output loads in sensors and actuators. However, the
current ad hoc design of mechanical coupling without theoretical support of
elasticity cannot provide design guidelines for mechanical coupling with
lattice geometries. Moreover, the correlation between mechanical coupling
effects and geometric symmetry is not yet clearly understood. In this work, we
systematically search for all possible mechanical couplings in 2D lattice
structures by determining the non-zero diagonal terms in the decomposed
micropolar elasticity tensor. We also correlate the mechanical couplings with
the point-group symmetry of 2D lattices by applying the symmetry operation to
the decomposed micropolar elasticity tensor. The decoupled micropolar
constitutive equation uncovers eight coupling effects for 2D lattice
structures. The symmetry operation of the decoupled micropolar elasticity
tensor reveals the correlation of the mechanical coupling with the point
groups. Our findings can strengthen the design of mechanical metamaterials with
potential applications in areas including sensors, actuators, soft robots, and
active metamaterials for elastic/acoustic wave guidance and thermal management.
