Higher-order nonlocal gradient elasticity: A consistent variational theory

A consistent variational theory of the higher-order nonlocal gradient elasticity is conceived to appropriately introduce the nonlocality to the higher-order strain gradient theory. The abstract variational approach, based on appropriate functional spaces of test fields, is applied to establish the higher-order nonlocal gradient mechanics of elastic beams in flexure. Two nonlocal and two gradient characteristic lengths are exploited to describe the size-dependent response of continua with nano-structures. Integral convolutions of the higher-order constitutive law are restored to the equivalent differential problem endowed with non-standard boundary conditions of constitutive-type. The higher-order strain gradient theory, higher-order nonlocal elasticity and modified nonlocal strain gradient theory, extensively adopted in the community of Engineering Science, are demonstrated to be particular cases of the introduced higher-order nonlocal gradient theory. The well-posedness of the developed higher-order nonlocal gradient problem is revealed by studying the flexural response of structures with wide-ranging applications in nano-engineering. Exact analytical solution for elastostatic deflections of nano-beams is derived and new benchmark examples of nano-mechanics interest are detected. The higher-order nonlocal gradient elasticity theory can effectively characterize nanoscopic phenomena in advanced nano-composites and nano-structures. (C) 2020 Elsevier Ltd. All rights reserved.