Abstract
We present a general finite deformation higher-order gradient elasticity theory. The governing equations of the higher-order gradient solid along with boundary conditions of various orders are derived from a variational principle using integration by parts on the surface. The objectivity of the energy functional is achieved by carefully selecting the invariants under rigid-body transformation. The third-order gradient solid theory includes more than 10.000 material parameters. However, under certain simplifications, the material parameters can be greatly reduced; down to 3. With this simplified formulation, we develop a nonlocal operator method and apply it to several numerical examples. The numerical analysis shows that the high gradient solid theory exhibits a stiffer response compared to a ’conventional’ hyperelastic solid. The numerical tests also demonstrate the capability of the nonlocal operator method in solving higher-order physical problems.
•A general finite deformation higher-order gradient elasticity theory is proposed.•The governing equations of the second-order gradient solid along with boundary conditions of various orders are derived from a variational principle.•The nonlocal operator method and Newton Raphson iteration method are used to solve 2D fifth-order gradient elasticity and 3D third-order gradient elasticity.•The physical properties of the high gradient solid theory are studied.