A simplified deformation gradient theory and its experimental verification

In this paper, materials with nonlocal properties are considered as a continuum model composed of micro-elements with certain volumes. Based on this hypothesis, the deformation and corresponding energy of micro-structure system are studied in detail, and the equivalent governing equations in simplified form are given. In the framework of micro-structure system, a simplified deformation gradient theory (SDG) with two length-scale parameters is obtained by defining the micro-strain and micro-rotation of elements specifically from the perspective of deformation, which has definite physical significance. The generalized strain energy is introduced in the SDG, which gives a new explanation of elastic moduli, and the nonlocal effect parameter is defined to capture nonlocal properties of materials quantitatively. Under certain micro-deformation assumptions, the SDG can be degenerated into couple stress theory, strain gradient theory and classical continuum theory. The nonlocal deformation consists of two branches: one is the macro-deformation of non-uniform materials and the other is the micro-deformation of materials with micro-structures. For the macro-tension of particle reinforced composites, the SDG successfully verifies and predicts the approximatively linear relationship between elastic moduli and particle sizes on the micron scale. Moreover, the theoretical solution to nonlocal micro-torsion based on the SDG agrees well with the experiment results and also predicts the torsion stiffness of cylinder at smaller diameters.