Mechanically-based approach to non-local elasticity: Variational principles

International Journal of Solids and Structures

Volumn 47, Issue 5, 2010, Pages 539-548

  Di Paola, M ^a   Pirrotta, A ^a   Zingales, M ^a  

^a UNIVERSITY OF PALERMO   (Italy)

Author keywords

Long range interactions; Non local elastic potential energy; Non local elasticity; Non local state variables; Variational theorems

Indexed keywords

BODY FORCES; CONSTITUTIVE RELATIONS; ELASTIC CONTINUUM; ELASTIC POTENTIAL ENERGY; ELASTIC PROBLEMS; EULER-LAGRANGE EQUATIONS; GLOBAL MINIMA; LONG RANGE INTERACTIONS; MECHANICAL MODEL; NATURAL BOUNDARY CONDITION; NONLOCAL; RELATIVE DISPLACEMENT; STATE VARIABLES; TENSILE SPECIMENS; TOTAL POTENTIAL ENERGY; VARIATIONAL CALCULUS; VARIATIONAL PRINCIPLES; VIRTUAL WORK PRINCIPLE;

DIFFERENTIAL EQUATIONS; ELASTICITY; EULER EQUATIONS; OPTIMIZATION; VARIATIONAL TECHNIQUES;

POTENTIAL ENERGY;

EID: 72449123798     PISSN: 00207683     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.ijsolstr.2009.09.029     Document Type: Article

References (37)

1. Aifantis E.C. Gradient effects at macro, micro and nano scales. Journal of the Mechanical Behaviour of Materials 5 (1994) 355-375
2. Aifantis E.C. Update on a class of gradient theories. Mechanics of Materials 35 (2003) 259-280
3. Aifantis E.C. Exploring the applicability of gradient elasticity to certain micro/nano reliability problems. Microsystems Technology 15 (2009) 109-115
4. Askes H., and Metrikine A.V. 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
5. Bažant Z.P., and Belytschko T.B. Continuum theory for strain-softening. Journal of Engineering Mechanics 110 (1984) 1666-1692
6. Bažant Z.P., and Jirásek M. Non-local integral formulations of plasticity and damage: a survey on recent results. Journal of Engineering Mechanics 128 (2002) 1129-1239
7. Borino G., Failla B., and Parrinello F. A symmetric non-local damage theory. International Journal of Solids and Structures 40 (2003) 3621-3645
8. Cottone G., Di Paola M., and Zingales M. Dynamics of non-local systems handled by fractional calculus. In: Mastorakis N., and Skellaris J. (Eds). Advances in Numerical Methods, Lecture Notes in Electrical Engineering vol. 11 (2009), Springer, -Verlag, Berlin 389-423
9. 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 (2008) 5642-5659
10. Di Paola M., Failla G., and Zingales M. The Physically-based approach to the analysis of non-local continuum. Journal of Elasticity 97 (2009) 103-130
11. Eringen, A.C., 1967. Mechanics of micromorphic continua. In: Kroner (Ed.), Mechanics of Generalized Continua, Proceedings of the IUTAM Symposium, Springer, Berlin.
12. Eringen A.C. Linear theory of nonlocal elasticity and dispersion of plane waves. International Journal of Engineering Science 10 (1972) 425-435
13. Eringen A.C., and Edelen D.G.B. On non-local elasticity. International Journal of Engineering Science 10 (1972) 233-248
14. Failla, G., Santini, A., Zingales, M., 2009. Solution strategies for 1D elastic continuum with long-range central interactions, Mechanics Research Communications, doi:10.1016/j.mechrescom.2009.09.006.
15. Ganghoffer J.F., and De Borst R. A new framework in non-local mechanics. International Journal of Engineering Science 38 (2000) 453-486
16. Gutkin Y., and Aifantis E.C. Screw dislocations in gradient elasticity. Scripta Materialia 35 (1996) 1353-1358
17. Gutkin Y., and Aifantis E.C. Edge dislocations in gradient elasticity. Scripta Materialia 36 (1997) 129-135
18. Kioseoglou J., Konstantopoulos I., Ribarik G., Dimitrakopulos G.P., and Aifantis E.C. Non-singular dislocation and crack fields: implications to small volumes. Microsystems Technology 15 (2009) 117-121
19. Kroner E. Elasticity theory of materials with long-range cohesive forces. International Journal of Solids and Structures 3 (1967) 731-742
20. Krumhanls J.A. Generalized continuum field representation for lattice vibrations. In: Wallis R.F. (Ed). Lattice Dynamics, Proceedings of the International Conference (1963), Pergamon Press, London
21. Krumhanls, J.A., 1967. Some considerations of the relations between solid state physics and generalized continuum mechanics. In: Kroner (Ed.), Mechanics of Generalized Continua, Proceedings of the IUTAM Symposium, Springer, Berlin.
22. Kunin, I.A., 1967, The theory of elastic media with microstructure and the theory of dislocations. In: Kroner (Ed.), Mechanics of Generalized Continua, Proceedings of the IUTAM Symposium, Springer, Berlin.
23. Lax M., 1963, The relation between microscopic and macroscopic the theories of elasticity. In: Wallis, R.F. (Ed.) Lattice Dynamics, Proceedings of the International Conference, Pergamon Press, London.
24. Maraganti R., and Sharma P. A novel atomistic approach to determine strain gradients elasticity constants: tabulation and comparison for various metals, semiconductors, silica, polymers and the (Ir) relevance for nanotechnologies. Journal of the Mechanics and Physics of Solids 55 (2007) 1823-1852
25. 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
26. 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
27. Mindlin R.D., and Eshel N.N. On first strain-gradient theories in linear elasticity. International Journal of Solids and Structures 4 (1968) 109-124
28. Peerlings R.H.J., Geers M.G.D., De Borst R., and Brekelmans W.A.M. A critical comparison of nonlocal and gradient-enhanced softening continua. International Journal of Solids and Structures 38 (2001) 7723-7746
29. Pijadier-Cabot G., and Bažant Z.P. Non-local damage theory. Journal of Engineering Mechanics (ASCE) 113 (1987) 1512-1533
30. Polizzotto C. Non-local elasticity and related variational principles. International Journal of Solids and Structures 38 (2001) 7359-7380
31. Polizzotto C. Gradient elasticity and non-standard boundary conditions. International Journal of Solids and Structures 40 (2003) 7399-7423
32. Polizzotto C., and Borino G. A thermodynamics-based formulation of gradient-dependent plasticity. European Journal of Mechanics A/Solids 17 (1998) 741-761
33. Rogula D. Nonlocal Theory of Material Media, CISM Courses and Lectures (1982), Springer, Udine (IT)
34. Silling S.A. Reformulation of elasticity theory for discontinuities and long-range forces. Journal of Mechanics and Physics of Solids 48 (2000) 175-209
35. Silling S.A., and Lehoucq R.B. Convergence of peridynamics to classical elasticity theory. Journal of Elasticity 93 (2008) 13-37
36. Silling S.A., Zimmermann S.A., and Abeyaratne S.A. Deformation of a peridynamic bar. Journal of Elasticity 73 (2003) 173-190
37. Toupin R.A. Elastic materials with couple-stresses. Archive of Rational Mechanics and Analysis 11 (1963) 385-414