A Nonlocal Strain Gradient Elastic Model for Beams in Bending

According to laboratory experiments, Wheel et al. (Int J Solids Struct 67–68:84–92, 2015), in small scale structures, size effects can manifest themselves either by an increase or by a decrease of stiffness features due to the presence of internal length scale parameters, which depends on certain states of microstructure. The ever increasing use of metamaterials makes even more interesting the development of theoretical models that are able to predict, through the calibration of specific parameters, such peculiar behaviors. In this context a nonlocal strain gradient elasticity theory, applied to beams in static bending, is discussed. The beam model grounds on a suitable combination of Eringen’s nonlocal strain integral elasticity theory, known to generally predict stiffness softening, Eringen (Nonlocal continuum field theories. Springer-Verlag, New York), and the strain gradient elasticity theory, which in general predicts stiffness hardening, Askes and Aifantis (Int J Solids Struct 48(13):1962–1990, 2011). The proposed beam theory admits integral and differential equivalent approaches and it is exempt from known paradoxical outcomes. Benchmarks are presented considering few cases of beams in static bending.