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

Mario Di Paola, Massimiliano Zingales, Mario Di Paola, Massimiliano Zingales

Risultato della ricerca: Article

83 Citazioni (Scopus)

### 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.
Lingua originale English 5642-5659 17 International Journal of Solids and Structures 45 Published - 2008

### Fingerprint

Fractional Calculus
calculus
Continuum
continuums
Continuum Model
Interaction
Range of data
interactions
Fractional Derivative
Unbounded Domain
Fractional
Mathematical Induction
Model
Derivatives
Double integral
Contact Force
Long-range Interactions
Constitutive Law
Fractional Differential Equation

### All Science Journal Classification (ASJC) codes

• Modelling and Simulation
• Materials Science(all)
• Condensed Matter Physics
• Mechanics of Materials
• Mechanical Engineering
• Applied Mathematics

### Cita questo

Long-range cohesive interactions of non-local continuum faced by fractional calculus. / Di Paola, Mario; Zingales, Massimiliano; Paola, Mario Di; Zingales, Massimiliano.

In: International Journal of Solids and Structures, Vol. 45, 2008, pag. 5642-5659.

Risultato della ricerca: Article

title = "Long-range cohesive interactions of non-local continuum faced by fractional calculus",
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.",
keywords = "Fractional calculus, Fractional finite differences, Long-range forces, Non-local models",
author = "{Di Paola}, Mario and Massimiliano Zingales and Paola, {Mario Di} and Massimiliano Zingales",
year = "2008",
language = "English",
volume = "45",
pages = "5642--5659",
journal = "International Journal of Solids and Structures",
issn = "0020-7683",
publisher = "Elsevier Limited",

}

TY - JOUR

T1 - Long-range cohesive interactions of non-local continuum faced by fractional calculus

AU - Di Paola, Mario

AU - Zingales, Massimiliano

AU - Paola, Mario Di

AU - Zingales, Massimiliano

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

KW - Fractional calculus

KW - Fractional finite differences

KW - Long-range forces

KW - Non-local models

UR - http://hdl.handle.net/10447/42037

M3 - Article

VL - 45

SP - 5642

EP - 5659

JO - International Journal of Solids and Structures

JF - International Journal of Solids and Structures

SN - 0020-7683

ER -