GRADIENT TYPE OF NONLOCAL HIGHER-ORDER BEAM THEORY AND NEW SOLUTION METHODOLOGY OF NONLOCAL BENDING DEFLECTION

Di erent predictions were found in di erent literatures about the trend of nonlocal e ects on nanostructural sti ness. By employing an iterative method and the Taylor expansion method, an infinite series for nonlocal high-order stress is achieved based on the gradient-type of nonlocal di erential constitutive model. The nonlocal stress consists of the classical bending stress and each order gradient of nonlocal deflection. Consequently, the di erential equation of the bending deflection curve for nonlocal high-order beam is derived. The nonlocal deflection is determined via the regular perturbation method. Some numerical examples are provided to reveal and quantize the e ects of nonlocal scale factor on bending deflection. It is shown that compared with the corresponding classical results, the nonlocal bending deflections of nanostructure may increase, decrease or remain unchanged. The sti ness of nanostructures can be reduced or enhanced or the same as classical structures by considering the gradient-type of nonlocal high-order stress e ect, and the trend depends on external loads and boundary constraints which are found to play significant roles independently in nonlocal bending of nanostructures. Moreover, it is observed for the first time that the position of maximum nonlocal deflection may be influenced by nonlocal scale factor. The present studies are expected to solve the problems in the application of nonlocal elasticity theory to nanostructures, and further provide supports for the development and optimization of such theory.