Long-range cohesive interactions of non-local continuum faced by fractional calculus

A non-local continuum model including long-range forces between non-adjacent volume elements has been studied in this paper. The proposed continuum model has been obtained as limit case of two fully equivalent mechanical models: (i) A volume element model including contact forces between adjacent volumes as well as long-range interactions, distance decaying, between non-adjacent elements. (ii) A discrete point-spring model with local springs between adjacent points and non-local springs with distance-decaying stiffness connecting non-adjacent points. Under the assumption of fractional distance-decaying interactions between non-adjacent elements a fractional differential equation involving Marchaud-type fractional derivatives has been obtained for unbounded domains. It is shown that for unbounded domains the two mechanical models revert to Lazopoulos and Eringen model with fractional distance-decaying functions. It has also been shown that for a confined bar, the stress-strain relation is substantially different from that obtained simply using the truncated Marchaud derivatives since a double integral instead of convolution integral appears. Moreover, in the analysis of bounded domains, the governing equations turn out to an integro-differential equation including only the integral part of Marchaud fractional derivatives on finite interval. The mechanical boundary condition for the proposed model has been introduced consistently on the basis of mechanical considerations, and the constitutive law of the proposed continuum model has been reported by mathematical induction. Several numerical applications have been reported to show, verify and assess the concepts listed in this paper.