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国际多尺度计算工程期刊
影响因子: 1.016 5年影响因子: 1.194 SJR: 0.554 SNIP: 0.68 CiteScore™: 1.18

ISSN 打印: 1543-1649
ISSN 在线: 1940-4352

国际多尺度计算工程期刊

DOI: 10.1615/IntJMultCompEng.2011002416
pages 579-597

FRACTIONAL DIFFERENTIAL CALCULUS FOR 3D MECHANICALLY BASED NON-LOCAL ELASTICITY

Mario Di Paola
Dipartimento di Ingegneria Strutturale, Aerospaziale e Geotecnica, Università degli Studi di Palermo, Viale delle Scienze, I-90128, Palermo, Italy
Massimiliano Zingales
Dipartimento di Ingegneria Strutturale, Aerospaziale e Geotecnica, Università degli Studi di Palermo, Italy

ABSTRACT

This paper aims to formulate the three-dimensional (3D) problem of non-local elasticity in terms of fractional differential operators. The non-local continuum is framed in the context of the mechanically based non-local elasticity established by the authors in a previous study; Non-local interactions are expressed in terms of central body forces depending on the relative displacement between non-adjacent volume elements as well as on the product of interacting volumes. The non-local, long-range interactions are assumed to be proportional to a power-law decaying function of the interaction distance. It is shown that, as far as an unbounded domain is considered, the elastic equilibrium problem is ruled by a vector fractional differential operator that corresponds to a new generalized expression of a fractional operator referred to as the central Marchaud fractional derivative (CMFD). It is also shown that for bounded solids the corresponding integral operators contain only the integral term of the CMFD and no divergent terms on the boundary appear for a one-dimensional solid case. This aspect is crucial since the mechanical boundary conditions may be easily enforced as in classical local elasticity theory.

REFERENCES

  1. Aifantis, E. C., Gradient effects at macro, micro, and nano scales. DOI: 10.1515/JMBM.1994.5.3.355

  2. Aifantis, E. C., Update on a class of gradient theories. DOI: 10.1016/S0167-6636(02)00278-8

  3. Carpinteri, A. and Cornetti, P., A fractional calculus approach to the description of stress and strain localization in fractal media. DOI: 10.1016/S0960-0779(00)00238-1

  4. Del Piero, G. and Truskinovsky, L., Macro- and micro-cracking in one-dimensional elasticity. DOI: 10.1016/S0020-7683(00)00078-0

  5. Dyskin, A., Effective characteristics and stress concentrations in materials with self-similar microstructure. DOI: 10.1016/j.ijsolstr.2004.06.034

  6. Di Paola, M. and Zingales, M., Long-range cohesive interactions of non-local continuum faced by fractional calculus. DOI: 10.1016/j.ijsolstr.2008.06.004

  7. Di Paola, M. and Zingales, M., The multiscale fractional approach to non-local elasticity: Direct connection with fractional operators.

  8. Di Paola, M., Failla, G., and Zingales, M., Physically-based approach to the mechanics of strong non-local elasticity theory. DOI: 10.1007/s10659-009-9211-7

  9. Di Paola, M., Pirrotta, A., and Zingales, M., Mechanically-based approach to non-local elasticity: Variational principles. DOI: 10.1016/j.ijsolstr.2009.09.029

  10. Epstein, M. and Sniatycki, J., Fractal mechanics. DOI: 10.1016/j.physd.2006.06.008

  11. Eringen, A. C., Non-local polar elastic continua. DOI: 10.1016/0020-7225(72)90070-5

  12. Eringen, A. C., Linear theory of nonlocal elasticity and dispersion of plane waves. DOI: 10.1016/0020-7225(72)90050-X

  13. Eringen, A. C. and Edelen, D. G. B., On non-local elasticity. DOI: 10.1016/0020-7225(72)90039-0

  14. Failla, G., Santini, A., and Zingales, M., Solution strategies for 1D elastic continuum with long-range central interactions: Smooth and fractional decay. DOI: 10.1016/j.mechrescom.2009.09.006

  15. Gutkin, M. Y., Nanoscopics of dislocations and disclinations in gradient elasticity.

  16. Kröner, E., Elasticity theory of materials with long-range cohesive forces. DOI: 10.1016/0020-7683(67)90049-2

  17. Krumhansl, J. A., Some considerations of the relations between solid state physics and generalized continuum mechanics.

  18. Kunin, I. A., The theory of elastic media with microstructure and the theory of dislocations.

  19. Lazopoulos, K. A., Non-local continuum mechanics and fractional calculus. DOI: 10.1016/j.mechrescom.2006.05.001

  20. Mindlin, R. D., Micro-structure in linear elasticity.

  21. Polizzotto, C., Non local elasticity and related variational principles. DOI: 10.1016/S0020-7683(01)00039-7

  22. Samko, S., Kilbas, A., and Marichev, O., Fractional Integrals and Derivatives.

  23. Shkanukov, M. K., On the convergence of difference schemes for differential equations with a fractional derivative.

  24. Silling, S. A., Reformulation of elasticity theory for discontinuities and long-range forces. DOI: 10.1016/S0022-5096(99)00029-0

  25. Silling, S. A. and Lehoucq, R. B., Convergence of peridynamics to classical elasticity theory. DOI: 10.1007/s10659-008-9163-3

  26. Silling, S. A., Zimmermann, S. A., and Abeyaratne, S. A., Deformation of a peridynamic bar. DOI: 10.1023/B:ELAS.0000029931.03844.4f

  27. Tarasov, V. E., Continuous medium model for fractal media. DOI: 10.1016/j.physleta.2005.01.024

  28. Tarasov, V. E., Fractional vector calculus and fractional Maxwell’s equations. DOI: 10.1016/j.aop.2008.04.005

  29. Trovalusci, P., Capecchi, D., and Ruta, G. C., Genesis of the multiscale approach for materials with microstructure. DOI: 10.1007/s00419-008-0269-7


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