Displacement-Driven Approach to Nonlocal Elasticity

Abstract: This study presents a physically consistent displacement-driven reformulation of the concept of action-at-a-distance, which is at the foundation of nonlocal elasticity. In contrast to existing approaches that adopts an integral stress-strain constitutive relation, the displacement-driven approach is predicated on an integral strain-displacement relation. The most remarkable consequence of this reformulation is that the (total) strain energy is guaranteed to be convex and positive-definite without imposing any constraint on the symmetry of the kernels. This feature is critical to enable the application of nonlocal formulations to general continua exhibiting asymmetric interactions; ultimately a manifestation of material heterogeneity. Remarkably, the proposed approach also enables a strong satisfaction of the locality recovery condition and of the laws of thermodynamics, which are not foregone conclusions in most classical nonlocal elasticity theories. Additionally, the formulation is frame-invariant and the nonlocal operator remains physically consistent at boundaries. The study is complemented by a detailed analysis of the dynamic response of the nonlocal continuum and of its intrinsic dispersion leading to the consideration that the choice of nonlocal kernels should depend on the specific material. Examples of exponential or power-law kernels are presented in order to demonstrate the applicability of the method to different classes of nonlocal media. The ability to admit generalized kernels reinforces the generalized nature of the displacement-driven approach over existing integral methodologies, which typically lead to simplified differential models based on exponential kernels. The theoretical formulation is also leveraged to simulate the static response of nonlocal beams and plates illustrating the intrinsic consistency of the approach, which is free from unwanted boundary effects.