Nonlocal integral elasticity: 2D finite element based solutions

International Journal of Solids and Structures

Volumn 46, Issue 21, 2009, Pages 3836-3849

  Pisano, A A ^a   Sofi, A ^a   Fuschi, P ^a  

^a UNIVERSITY MEDITERRANEA OF REGGIO CALABRIA   (Italy)

Author keywords

Eringen type model; Nonlocal finite element; Nonlocal integral elasticity; Nonlocal stiffness matrices

Indexed keywords

COMPUTATIONAL ISSUES; ELASTIC BEHAVIOUR; ELASTIC MODELS; ELASTIC PROBLEMS; ERINGEN-TYPE MODEL; FINITE ELEMENT; GLOBAL STIFFNESS MATRIX; MATERIAL BEHAVIOUR; NON-LOCAL FINITE ELEMENT METHOD; NONLOCAL; NONLOCAL FINITE ELEMENT; NONLOCAL INTEGRAL ELASTICITY; NONLOCAL STIFFNESS MATRICES; NONLOCALITIES; STRESS-STRAIN LAW; TOTAL POTENTIAL ENERGY;

ELASTICITY; STIFFNESS; STIFFNESS MATRIX;

FINITE ELEMENT METHOD;

EID: 69249214067     PISSN: 00207683     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.ijsolstr.2009.07.009     Document Type: Article

References (63)

1. Adali S. Variational principles for multi-walled carbon nanotubes undergoing buckling based on nonlocal elasticity theory. Physics Letters A 372 (2008) 5701-5705
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. Microsystem Technologies 15 (2009) 109-115
4. Altan S.B. Uniqueness of the initial-value problems in nonlocal elastic solids. International Journal of Solids and Structures 25 (1989) 1271-1278
5. Altan S.B. Existence in nonlocal elasticity. Archive Mechanics 41 (1989) 25-36
6. Altan S.B. Uniqueness in nonlocal thermoelasticity. Journal of Thermal Stresses 14 (1991) 121-128
7. Bažant Z.P., and Chang T.P. Instability of nonlocal continuum and strain averaging. Journal of Engineering Mechanics 10 (1984) 1441-1450
8. Bažant Z.P., and Jirásek M. Nonlocal integral formulations of plasticity and damage: survey of progress. Journal of Engineering Mechanics 11 (2002) 1119-1149
9. Benvenuti E., Borino G., and Tralli A. A thermodynamically consistent nonlocal formulation for damaging materials. European Journal of Mechanics A/Solids 21 (2002) 535-553
10. Benvenuti E., and Tralli A. The fast Gauss transform for non-local integral FE models. Communications in Numerical Methods in Engineering 22 (2006) 505-533
11. Borino G., Failla B., and Parrinello F. A symmetric nonlocal damage theory. International Journal of Solids and Structures 40 (2003) 3621-3645
12. Ece M.C., and Aydogdu M. Nonlocal elasticity effect on vibration of in-plane loaded double-walled carbon nano-tubes. Acta Mechanica 190 (2007) 185-195
13. Edelen D.G.B., Green A.E., and Laws N. Nonlocal continuum mechanics. Archive for Rationale Mechanics and Analysis 43 (1971) 36-44
14. Edelen D.G.B., and Laws N. On the thermodynamics of systems with nonlocality. Archive for Rationale Mechanics and Analysis 43 (1971) 24-35
15. Eftis J., and Liebowitz H. On surface energy and on the continuum thermodynamics of brittle fracture. Engineering Fracture Mechanics 8 (1976) 459-485
16. Emmrich E., and Weckner O. Analysis and numerical approximation of an integro-differential equation modeling non-local effects in linear elasticity. Mathematics and Mechanics of Solids 12 (2007) 363-384
17. Eringen A.C. Nonlocal polar elastic continua. International Journal Engineering Science 10 (1972) 1-16
18. Eringen A.C. Line crack subjected to shear. International Journal of Fracture 14 (1978) 367-379
19. Eringen A.C. Line crack subjected to anti-plane shear. Engineering Fracture Mechanics 12 (1979) 211-219
20. Eringen A.C. On nonlocal plasticity. International Journal Engineering Science 12 (1981) 1461-1474
21. Eringen A.C. Theories of nonlocal plasticity. International Journal Engineering Science 21 (1983) 741-751
22. Eringen A.C. Theory of nonlocal elasticity and some applications. Res Mechanica 21 (1987) 313-342
23. Eringen A.C., and Edelen D.G.B. On nonlocal elasticity. International Journal of Engineering Science 10 (1972) 233-248
24. Eringen A.C., and Kim B.S. Stress concentration at the tip of a crack. Mechanics Research Communications 1 (1974) 233-237
25. Eringen A.C., Speziale C.G., and Kim B.S. Crack-tip problem in non-local elasticity. Journal of Mechanics Physics and Solids 25 (1977) 339-355
26. Fleck N.A., Muller G.M., Ashby M.F., and Hutchinson J.W. Strain gradient plasticity: theory and experiments. Acta Metallurgica Materialia 42 (1994) 475-487
27. Gao J. An asymmetric theory of nonlocal elasticity. Part 2: continuum field. International Journal of Solids and Structures 36 (1999) 2959-2971
28. Heireche H., Tounsi A., Benzair A., Maachon M., and Adda Beida E.A. Sound wave propagation in single-walled carbon nanotubes using nonlocal elasticity. Physica E 40 (2008) 2791-2799
29. Hinton E., and Owen D.R.J. Finite Element Programming (1989), Academic Press, Harcourt Brace Jovanovich Publishers, London
30. Jirásek M. Nonlocal models for damage and fracture: comparison of approaches. International Journal of Solids and Structures 35 (1998) 4133-4145
31. Kröner E. Elasticity theory of materials with long range cohesive forces. International Journal of Solids and Structures 3 (1967) 731-742
32. Krumhansl J.A. Some considerations on the relations between solid state physics and generalized continuum mechanics. In: Kröner E. (Ed). Mechanics of Generalized Continua (1968), Springer-Verlag, Berlin 298-331
33. Kunin, I.A., 1968. The theory of elastic media with microstructure and the theory of dislocation, in: E. Kröner (Ed.), Mechanics of Generalised Continua, Proceedings of IUTAM Symposium 1967, Springer, New York.
34. Kunin I.A. Elastic Media with Microstructure vol. I (1982), Springer-Verlag, Berlin
35. Liang J. The nonlocal theory solution of a Mode-I crack in functionally graded materials. Science in China Series E: Technological Sciences 52 (2009) 1101-1111
36. Liebe T., Steinmann P., and Benallal A. Theoretical and computational aspects of thermodynamically consistent framework of geometrically linear gradient damage. Computer Methods in Applied Mechanics and Engineering 190 (2001) 6555-6576
37. Murmu T., and Pradhan S.C. Buckling analysis of a single-walled carbon nanotubes embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM. Physica E 41 (2009) 1232-1239
38. Nguyen Q.S. A thermodynamic description of the running crack problem. In: Nemat-Nasser S. (Ed). Three-dimensional Constitutive Relations and Ductile Fracture (1980), North-Holland, Amsterdam 315-330
39. Peddieson J., Buchanan G.R., and McNitt R.P. Application of nonlocal continuum models to nanotechnology. International Journal of Engineering Science 41 (2003) 305-312
40. Pisano A.A., and Fuschi P. Closed form solution for a nonlocal elastic bar in tension. International Journal of Solids and Structures 41 (2003) 13-23
41. Polizzotto C. Nonlocal elasticity and related variational principles. International Journal of Solids and Structures 38 (2001) 7359-7380
42. Polizzotto C. Thermodynamics and continuum fracture mechanics for nonlocal-elastic plastic materials. European Journal of Mechanics A/Solids 21 (2002) 85-103
43. Polizzotto C. Unified thermodynamic framework for nonlocal/gradient continuum theories. European Journal of Mechanics A/Solids 22 (2003) 651-668
44. Polizzotto C. Strain-gradient elastic-plastic material models and assessment of the higher order boundary conditions. European Journal of Mechanics A/Solids 26 (2007) 189-211
45. Polizzotto C. Thermodynamics-based gradient plasticity theories with an application to interface models. International Journal of Solids and Structures 45 (2008) 4820-4834
46. Polizzotto C. Shakedown theorems for elastic-plastic solids in the framework of gradient plasticity. International Journal of Plasticity 24 (2008) 218-241
47. Polizzotto C. A nonlocal strain gradient plasticity theory for finite deformations. International Journal of Plasticity 25 (2009) 1280-1300
48. Polizzotto C., and Borino G. A thermodynamics-based formulation of gradient-dependent plasticity. European Journal of Mechanics A/Solids 17 (1998) 741-761
49. Polizzotto, C., Borino, G., Fuschi, P., 2000. Shakedown of cracked bodies with nonlocal elasticity. In: CD-Rom Proceedings of ECCOMAS-2000, CIMNE, Barcelona.
50. Polizzotto. C., Fuschi, P., Pisano, A.A., 2001. Boundary effects and FEM techniques in nonlocal elasticity. In: CD-Rom Proceedings of AIMETA 2001, Taormina, Italy.
51. Polizzotto C., Fuschi P., and Pisano A.A. A strain-difference-based nonlocal elasticity model. International Journal of Solids and Structures 41 (2004) 2383-2401
52. Polizzotto C., Fuschi P., and Pisano A.A. A nonhomogeneous nonlocal elasticity model. European Journal of Mechanics A/Solids 25 (2006) 308-333
53. Rogula D. Introduction to nonlocal theory of material media. In: Rogula D. (Ed). Nonlocal Theory of Material Media (1982), Springer-Verlag, Berlin 125-222
54. Silling S.A. Reformulation of elasticity theory for discontinuities and long-range forces. Journal of Mechanics and Physics of Solids 48 (2000) 175-209
55. Silling S.A., Zimmermann M., and Abeyaratne R. Deformation of a peridynamic bar. Journal of Elasticity 73 (2003) 173-190
56. Sluys, L.J., 1992. Wave propagation, localization and dispersion in softening solids. Ph.D. Thesis, Civil Engrg. Dept. of Delft University of Technology.
57. Strifors H.C. A generalized force measure of conditions at crack tips. International Journal of Solids and Structures 10 (1974) 1389-1404
58. Wang L. Dynamical behaviors of double-walled carbon nanotubes conveying fluid accounting for the role of small length scale. Computational Materials Science 45 (2009) 584-588
59. Wang J., and Dhaliwal R.S. Uniqueness in generalized nonlocal thermoelasticity. Journal of Thermal Stresses 16 (1993) 71-78
60. Wang Q., and Wang C.M. The constitutive relation and small scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes. Nanotechnology 18 (2007) 075702 (4pp.)
61. Zhang C., and Karihaloo B.L. A Thermodynamic framework of fracture mechanics. Engineering Fracture Mechanics 46 (1993) 1023-1030
62. Zhang Y.Q., Lin G.R., and Xie X.Y. Free transverse vibrations of double-walled carbon nanotubes using a theory of nonlocal elasticity. Physics Review B 781 (2005) L95404
63. Zhou Z.G., and Wang B. Investigation of the interaction of two collinear cracks in anisotropic elasticity materials by means of the nonlocal theory. International Journal of Engineering Science 43 (2005) 1107-1120