Universal displacements in linear strain-gradient elasticity

We study universal displacement fields in three-dimensional linear strain-gradient elasticity. For each material symmetry class, by requiring the equilibrium equations to hold for any material in that class in the absence of body forces, we derive the corresponding universality constraints on the displacement field and obtain the complete set of universal displacements. Using the full symmetry classification and compact matrix representations of the elasticity tensors, we give explicit characterizations for all 48 strain-gradient symmetry classes, including centrosymmetric and chiral classes. For several high-symmetry classes, the strain-gradient universality constraints do not impose restrictions beyond the classical ones, so the universal displacement families coincide with those of classical linear elasticity (e.g., the isotropic classes S O ( 3 ) and O ( 3 ) ). For lower symmetry classes, the strain-gradient universality constraints can be stricter than their classical counterpart: the universal displacement fields form a proper subset of the classical universal displacements, obtained by imposing additional higher-order differential constraints that eliminate some of the classical families.