Does the conventional reciprocal theorem break down in strain gradient elasticity?

The force method and displacement method on the basis of the reciprocal theorem play an important role in the field of structural mechanics and have been successfully applied in structural mechanics. However, it is interestingly found that the unexpected paradox exists when the authors attempt to apply it to problems of deformations of strain gradient beams. The reciprocal relation between higher order stresses and higher order strains within the framework of linear elastic strain gradient elasticity is proposed with a view toward studying the physical nature of this paradoxical phenomenon, and it is then used to prove the updated reciprocal theorem. At the same time, the reciprocal theorem of any gradients of any second-order symmetric stress tensors and their corresponding gradients of displacements are derived according to the proposed reciprocal relation. The results show that the essential reason for the failure of the conventional reciprocal theorem is that the effect of higher order surface forces and surface stresses that are produced by strain gradients contributes to the reciprocal work. When the strain gradients work-conjugating to stress gradients are considered, they satisfy the local reciprocal relation that cannot be degenerated to the conventional reciprocal theorem in the form of body forces and inertial forces. The theory developed in this paper may have an increasingly profound effect on continuum mechanics and is expected to be a helpful tool for the mechanics of cellular structures homogenized by strain gradient elasticity.