Fig. 4

A spurious singular mode observed for Q8G4 element in a, and three spurious singular modes for Q4G1 element in b–d (typical hourglassing modes are shown also in Fig. 1b). a–c correspond to deformed configurations for which the micromorphic field vanishes, i.e. \(v_1(\vec {X}) = 0\), whereas d shows a surface plot of \(v_1(\vec {X})\) for which the mean field vanishes, i.e. \(\vec {v}_0(\vec {X}) = \vec {0}\). Notation QiGj designates a quadrilateral element with i nodes and j integration points. In all cases, bottom left node is fixed in both directions, whereas the right bottom node is fixed in the vertical direction to eliminate rigid body modes. A microstructure with a square stacking of holes used, i.e., \(n = 1\)