A reformulation of constitutive relations in the strain gradient elasticity theory for isotropic materials

Abstract

• The Abstract and the Conclusion parts have been rewritten and polished. • The use of the English language has been improved with the help of the language editing service. • The research objectives and procedures are presented more clearly in the revised manuscript. The general isotropic strain gradient elasticity theory with five higher-order elastic constants is reformulated by introducing two different orthogonal decompositions of the strain gradient tensor. Just applying the mathematical reformulations, no extra conditions needed, the constitutive relations, equilibrium equation and boundary conditions are reformulated. In the reformulated theory, the number of independent higher-order elastic constants is proved to be three for isotropic materials, which indicates that the five higher-order elastic constants in the general isotropic strain gradient elasticity theory are dependent with each other. Therefore, the general strain gradient elasticity theory contains only three independent material length-scale parameters for isotropic materials in addition to the Lame constants. The new theory is different from the existed strain gradient elasticity theory with one or three material length-scale parameters, which introduces extra conditions during deriving process. Moreover, the reformulated theory can be directly reduced to that of incompressible materials by assuming the terms associated with hydrostatic strains to be zero. Some examples, such as torsion of cylindrical bars, shearing of fixed-end layers, and pure bending of thin beams, are performed to reveal the necessity of including multi-length-scale parameters in the strain gradient elasticity theory to predict size effects at micron scale.