A different approach to Eringen's nonlocal integral stress model with applications for beams

The nonlocal continuum theory, either integral or differential form, has been formulated, evolved and widely used in our era to explain size effect phenomena in micro– and nano– structures. In the case of Euler Bernoulli beam theory (EBBT), the nonlocal integral form produces energy consistent formulas, unlike the nonlocal differential form revealing inconsistencies and giving rise to paradoxes. In this work, our overall research objective mainly focuses on nonlocal integral elasticity analysis of beams by employing a normalized symmetric kernel. This kernel corresponds to a finite domain (and will be called modified kernel from now on), that satisfies all the properties of a probability density function as well as successfully handles the physical inconsistencies of classic types of kernel. Our concern is to investigate the static response of a beam with various types of loading and boundary conditions (BCs) by making use the modified kernel and the kernel corresponding to the two phase nonlocal integral (TPNI) model. Carrying out numerical methods to our problems, the deducing results appear more flexible behavior than those of classic-local and the common nonlocal differential, respectively. What is more, a comparison is made between the results of the aforementioned kernels to demonstrate the advantages of the modified kernel. Unlike the common nonlocal differential form, the nonlocal integral forms do not give rise to paradoxes. Moreover, it is critical that the TPNI model does not raise paradoxes as has been observed in other publications presented in the literature.