Universal Displacements in Linear Strain-Gradient Elasticity

TL;DR

This paper characterizes universal displacement fields in three-dimensional linear strain-gradient elasticity across all symmetry classes, revealing when these displacements coincide with or differ from classical elasticity solutions.

Contribution

It provides a complete classification of universal displacements in strain-gradient elasticity for all 48 symmetry classes, extending previous work and clarifying the impact of higher-order PDEs.

Findings

Universal displacements coincide with classical elasticity in high-symmetry classes.

Lower symmetry classes impose additional differential restrictions on displacements.

Explicit characterization of all symmetry classes' universal displacement fields.

Abstract

We study universal displacement fields in three-dimensional linear strain-gradient elasticity within the Toupin-Mindlin first strain-gradient theory. Building on the approach of Yavari (2020), we derive, for each material symmetry class, the universality PDEs obtained by requiring the equilibrium equations (in the absence of body forces) to hold for any material in that class, and we determine the complete set of universal displacements. Using the full symmetry classification together with compact matrix representations of the elasticity tensors, we provide explicit characterizations for all 48 strain-gradient symmetry classes, including centrosymmetric and chiral classes. For several high-symmetry classes, the strain-gradient universality PDEs impose no additional restrictions beyond the classical ones, so the universal displacement families coincide with those of classical linear elasticity (for example, the isotropic classes SO(3) and O(3)). For lower symmetry classes, the strain-gradient universality PDEs can be stricter than their classical counterparts, so the universal displacements form proper subsets of the classical universal displacement families due to additional higher-order differential conditions.