On admissible external loadings within the first and second strain gradient elasticity

Recent extended applications of higher-order models of continuum mechanics as strain gradient elasticity and plasticity result in a certain extension of possible external loadings. In addition to known forces and couples, we face double forces and hyper-stresses acting in the volume and on the external surface of an elastic solid body. Despite the fact that such higher-order loadings as the double force and bi-moment are already known in structural mechanics, we still cannot say that these quantities are so obvious in continuum mechanics. Here, we discuss the admissible external loadings including those acting in a volume, on a surface, along a curve, or even at certain points. To establish this possibility, we use the Virtual Work Principle (VWP) generalized to first and second strain gradient elasticity. The principle is a foundation of weak solutions of the boundary-value problems under consideration. From the mathematical standpoint, external loadings form a linear functional in a corresponding functional energy space where a weak solution exists. By admissible loadings, we mean those for which this functional is bounded, i.e., such that the work of external forces is meaningful and the total energy is finite. Then, using Sobolev’s imbedding theorems, we characterize a class of admissible external loadings in the VWP form and discuss their physical meaning.