Nonlocal elasticity in nanobeams: the stress-driven integral model

Nonlocal elastic models have attracted an increasing amount of attention in the past years, due to the promising feature of providing a viable simulation for scale effects in nano-structures and especially in nano-beams designed for use as actuators or sensors. In adapting ERINGEN's nonlocal model of elasticity to flexure of nano-beams, the bending field is expressed as convolution between the elastic curvature field and an averaging kernel. Basic difficulties are involved in this approach due to conflicting requirements imposed on the bending field by equilibrium on one hand and by constitutive conditions on the other one. In the newly proposed constitutive theory the bending field is placed in the proper position of input variable, giving to the elastic curvature field the role of output of the constitutive law, evaluated by convolution between the bending field and an averaging kernel. Conflicting restrictions on the bending field are thus eliminated and existence and uniqueness of the solution are assured under any data. Equivalence between integral and differential constitutive relations is proven to hold for nonlocal laws with a special kernel, under constitutive boundary conditions stemming naturally from the integral relation. When compared with the local limit, a stiffer elastic response is evaluated by the stress-driven nonlocal model, due to normalisation of the kernel. The theory provides an effective methodology for investigating small scale effects in nanobeams, by well-posed problems.