A new form of strain gradient elasticity

Following the definition of internal characteristic length vector, the strain energy density is postulated to be dependent on both the strain tensor and the strain gradient tensor. A new form of strain gradient elasticity constitutive relation is derived directly from the strain energy density expansion method at the initial state. When internal characteristic length vector is zero, the expressions can be degenerated to the classical elastic theory. Based on the principle of virtual work, the variational principle, boundary condition and finite element formulation of the new form of strain gradient elasticity are given.