High-order stress theory for solids: a more generalized strain gradient theory

This paper seeks to address the size dependence of microstructure by defining a characteristic scale vector to characterize the size effect in materials. It introduces high-order stress moments and high-order momentum moment as foundational concepts. Based on these ideas, we propose a high-order stress theory for solids that integrates a complete second-order displacement gradient, as opposed to solely incorporating a rotation gradient or a strain gradient. This methodology enhances the high-order stress theory, rendering it a more comprehensive and generalized framework. Under certain conditions, this theory can be degenerated into other models that elucidate the size effect, indicating that the high-order stress theory has a wider applicability and is not limited by its own ideal assumptions or prerequisites, unlike other existing theories. The high-order stress theory presented in this paper is applicable not only in the field of micromechanics but also in multi-field analyses. To exemplify its utility, we investigate the flexoelectric effect in dielectric materials using the proposed high-order theory. We compute parameters such as electric field intensity and structural response under various deformation conditions, including tension, bending, shearing, and torsion. Furthermore, we conduct electromechanical coupling experiments on PZT plates within these deformation scenarios. The analysis of the experimental results substantiates the efficacy of the high-order theory.