Solution strategies for 1D elastic continuum with long-range interactions: Smooth and fractional decay

Mechanics Research Communications

Volumn 37, Issue 1, 2010, Pages 13-21

  Failla, G ^a   Santini, A ^a   Zingales, M ^b  

^a UNIVERSITY MEDITERRANEA OF REGGIO CALABRIA   (Italy)

^b UNIVERSITY OF PALERMO   (Italy)

Author keywords

Fractional calculus; Fractional finite differences; Long range interactions; Non local elasticity; Weak formulation of elastic problems

Indexed keywords

ELASTIC PROBLEMS; FINITE DIFFERENCE; FRACTIONAL CALCULUS; LONG RANGE INTERACTIONS; NONLOCAL; WEAK FORMULATION;

CONTINUUM MECHANICS; FINITE DIFFERENCE METHOD;

ELASTICITY;

EID: 75849147541     PISSN: 00936413     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.mechrescom.2009.09.006     Document Type: Article

References (20)

1. Aifantis E.C. Gradient effects at macro micro and nano scales. Journal of the Mechanical Behavior of Materials 5 (1994) 355-375
2. Andrianov I.V., and Awrejcewicz J. Continuous models for 1D discrete media valid for higher-frequency domain. Physics Letters A 345 (2005) 55-62
3. Borino G., Failla B., and Parrinello F. A symmetric non-local damage theory. International Journal of Solids and Structures 40 (2003) 3621-3645
4. Cottone G., Di Paola M., and Zingales M. Fractional mechanical model for the dynamics of non-local continuum. In: Mastorakis N., and Sakellaris J. (Eds). Advances in Numerical Methods. Lecture Notes in Electrical Engineering vol. 11 (2009), Springer-Verlag
5. Di Paola M., and Zingales M. Long-range cohesive interactions of non-local continuum faced by fractional calculus. International Journal of Solids and Structures 45 21 (2008) 5642-5659
6. Di Paola, M., Failla, G., Zingales, M., 2009. Physically-based approach to the mechanics of strong non-local linear elasticity theory. Journal of Elasticity, doi:10.1007/s10659-009-9211-7.
7. Eringen A.C. Linear theory of non-local elasticity and dispersion of plane waves. International Journal of Engineering Science 10 (1972) 425-435
8. Fuschi P., and Pisano A.A. Closed form solution for a nonlocal elastic bar in tension. International Journal of Solids and Structures 40 (2003) 13-23
9. Kroner E. Elasticity theory of materials with long-range cohesive forces. International Journal of Solids and Structures 3 (1967) 731-742
10. Laskin N., and Zaslavsky G. Nonlinear fractional dynamics on a lattice with long range interactions. Physica A: Statistical Mechanics and its Applications 368 (2006) 38-54
11. Marotti De Sciarra F. Variational formulations and a consistent finite-element procedure for a class of non-local elastic continua. International Journal of Solids and Structures 45 (2008) 4184-4202
12. Metrikine A.V., and Askes H. One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure. Part 1: Generic formulation. European Journal of Mechanics A/Solids 21 (2002) 555-572
13. Metrikine A.V., and Askes H. One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure. Part 2: Static and dynamic response. European Journal of Mechanics A/Solids 21 (2002) 573-588
14. Podlubny I. Fractional Differential Equations (1993), Academic Press, New-York
15. Polizzotto C. Nonlocal elasticity and related variational principles. International Journal of Solids and Structures 38 (2001) 7359-7380
16. Polizzotto C. Gradient elasticity and non standard boundary conditions. International Journal of Solids and Structures 40 (2003) 7399-7423
17. Samko S.G., Kilbas A.A., and Marichev O.I. Fractional Integral and Derivatives (1988), Gordon & Breach, Amsterdam
18. Silling S.A. Reformulation of elasticity theory for discontinuities and long-range forces. Journal of Mechanics and Physics of Solids 48 (2000) 175-209
19. Silling S.A., Zimmermann M., and Abeyaratne R. Deformation of a peridynamic bar. Journal of Elasticity 73 (2003) 173-190
20. Shkanukov M.K. On the convergence of difference schemes for differential equations with a fractional derivative. Doklady Akademii Nauk 348 (1996) 746-748