On Λ-Fractional peridynamic mechanics

K.A. Lazopoulos; E. Sideridis; A.K. Lazopoulos; K.A. Lazopoulos; E. Sideridis; A.K. Lazopoulos

doi:10.3934/matersci.2022042

AIMS Materials Science

2022, Volume 9, Issue 5: 684-701. doi: 10.3934/matersci.2022042

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On Λ-Fractional peridynamic mechanics

1.
Independent researcher
2.
SEMFE, National Technical University of Athens, Greece
3.
Mathematical Sciences Department, Hellenic Army Academy, Vari, Greece

Received: 04 July 2022 Revised: 07 August 2022 Accepted: 12 August 2022 Published: 01 September 2022

Λ-Fractional Mechanics has already been introduced since it combines non-locality with mathematical analysis. It is well established, that conventional mechanics is not a proper theory for describing various phenomena in micro or nanomechanics. Further, various experiments in viscoelasticity, fatigue, fracture, etc., suggest the introduction of non-local mathematical analysis in their description. Fractional calculus has been used in describing those phenomena. Nevertheless, the well-known fractional derivatives with their calculus fail to generate differential geometry, since the established fractional derivatives do not fulfill the prerequisites of differential topology. A Λ-fractional analysis can generate geometry conforming to the prerequisites of differential topology. Hence Λ-fractional mechanics deals with non-local mechanics, describing the various inhomogeneities in various materials with more realistic rules.
- Fractional-order,
- Fractional Integral,
- Fractional Derivative,
- Riemann-Liouville Fractional derivative,
- Λ-fractional space,
- Λ-fractional derivative,
- left and right Λ-spaces,
- fractional peridynamics,
- fractional horizon
Citation: K.A. Lazopoulos, E. Sideridis, A.K. Lazopoulos. On Λ-Fractional peridynamic mechanics[J]. AIMS Materials Science, 2022, 9(5): 684-701. doi: 10.3934/matersci.2022042

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Abstract

Λ-Fractional Mechanics has already been introduced since it combines non-locality with mathematical analysis. It is well established, that conventional mechanics is not a proper theory for describing various phenomena in micro or nanomechanics. Further, various experiments in viscoelasticity, fatigue, fracture, etc., suggest the introduction of non-local mathematical analysis in their description. Fractional calculus has been used in describing those phenomena. Nevertheless, the well-known fractional derivatives with their calculus fail to generate differential geometry, since the established fractional derivatives do not fulfill the prerequisites of differential topology. A Λ-fractional analysis can generate geometry conforming to the prerequisites of differential topology. Hence Λ-fractional mechanics deals with non-local mechanics, describing the various inhomogeneities in various materials with more realistic rules.

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