On the strain gradient elasticity theory for isotropic materials

Abstract Classical elasticity mechanics fails to explain size-dependent phenomenon, the strain gradient elasticity theories including the couple stress theory, the simplified strain gradient elasticity theory and the general strain gradient elasticity theory are used to describe size effects. In this paper, the relations among the couple stress theory, the simplified strain gradient elasticity theory and the general strain gradient elasticity theory are derived. The general strain gradient elasticity theory contains the effects of all strain gradients while the couple stress theory and the simplified strain gradient elasticity theory include effects of only part of the strain gradients, thus, the general strain gradient elasticity theory can reduce to the couple stress theory, the simplified strain gradient elasticity theory when some strain gradients are neglected. Subsequently, a reasoning comparison with examples is performed to discuss the differences between the general strain gradient elasticity theory and the reduced theories in describing the size effects: one is the static and dynamic responses of a microbeam and the other is the tensile of a microbar. In the beam bending and vibration example, the material length-scale parameters and the deflections predicted by the reduced theories are larger while the natural frequency is smaller than that of the general strain gradient elasticity theory. The reduced theories underestimate the size-dependent response of a bending beam. In the bar tensile example, the couple stress theory fails to predict the size-dependent tensile behaviour. This illustrates clearly that compared with the reduced theories, the general strain gradient elasticity theory is capable to describe the size effects more appropriately.