Long-range cohesive interactions of non-local continuum faced by fractional calculus
By Mario Di Paola, Massimiliano Zingales. 2008
153 citations recorded by the source corpus.
Abstract 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.
Subfields: Materials Chemistry, Mechanics of Materials
View Paper: https://www.semanticscholar.org/paper/0cb906833ac3f267203cab07d99b9c9e6c107a5d
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In Materials Chemistry
In Mechanics of Materials
Citations in the corpus
- A Fractional Model of Continuum Mechanics (2012, methodology): “We remark that the shapes of the displacements u look similar to those obtained in [13, 17] for a finite bar.”
- The mechanically-based approach to 3D non-local linear elasticity theory: Long-range central interactions (2010, methodology): “(8) has been also provided for a 1D bar in some previous works by Di Paola and Zingales (2008) and by Di Paola et al. (2009).”
- Physically-Based Approach to the Mechanics of Strong Non-Local Linear Elasticity Theory (2009, methodology): “Specifically, the resultant Qj of long-range interactions applied to the volume Vj has been provided in the form (see Fig. 1c) [ 36 ]”