We postulate a new gradient elasticity formulation with three higher-order terms and, thus, three independent length scales. The formulation has one higher-order stiffness term and two higher-order inertia terms. It is shown that this formulation is better able to capture the wave disperion characteristics as seen in nano-scale experiments or layered composites.

We derive a C0 finite element implementation based on a newly developed operator split, which allows the governing fourth-order partial differential equations to be rewritten in a set of fully coupled second-order partial differential equations. The fundamental unknowns of these coupled equations are the micro-scale and macro-scale displacement; thus, this is intrisically a fully coupled multi-scale formulation. With the reduction of the order of the equations, finite element implementation becomes straightforward.

Finally, we address the identification of the three length scales. We show that for layered composites it is possible to derive the three length scales as closed-form expressions of the laminate thickness, thereby greatly simplifying the quantification of the additional constitutive parameters.