Finite element analysis with staggered gradient elasticity

Computers and Structures

Volumn 86, Issue 11-12, 2008, Pages 1266-1279

  Askes, Harm ^a   Morata, Irene ^b   Aifantis, Elias C ^c,d  

^a UNIVERSITY OF SHEFFIELD   (United Kingdom)

^b UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

^c ARISTOTLE UNIVERSITY OF THESSALONIKI   (Greece)

^d MICHIGAN TECHNOLOGICAL UNIVERSITY   (United States)

Author keywords

Gradient elasticity; Higher order continuum; Length scale; Singularity; Size effect; Staggered solution

Indexed keywords

ALGORITHMS; BOUNDARY CONDITIONS; FINITE ELEMENT METHOD; INTERPOLATION; TENSORS;

FINITE ELEMENT DISCRETISATIONS; HELMHOLTZ EQUATIONS; STAGGERED GRADIENT ELASTICITY; STRAIN TENSORS;

ELASTICITY;

EID: 43049121709     PISSN: 00457949     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.compstruc.2007.11.002     Document Type: Article

References (78)

1. Toupin R.A. Elastic materials with couple-stresses. Arch Ration Mech Anal 11 (1962) 385-414
2. Mindlin R.D. Micro-structure in linear elasticity. Arch Ration Mech Anal 16 (1964) 51-78
3. Altan B.S., and Aifantis E.C. On some aspects in the special theory of gradient elasticity. J Mech Behav Mater 8 (1997) 231-282
4. Aifantis E.C. Update on a class of gradient theories. Mech Mater 35 (2003) 259-280
5. Aifantis E.C. On the microstructural origin of certain inelastic models. ASME J Eng Mater Technol 106 (1984) 326-330
6. Aifantis E.C. The physics of plastic deformation. Int J Plasticity 3 (1987) 211-247
7. Aifantis K.E., and Willis J.R. The role of interfaces in enhancing the yield strength of composites and polycrystals. J Mech Phys Solids 53 (2005) 1047-1070
8. Ru C.Q., and Aifantis E.C. A simple approach to solve boundary-value problems in gradient elasticity. Acta Mech 101 (1993) 59-68
9. Altan B.S., and Aifantis E.C. On the structure of the mode III crack tip in gradient elasticity. Scripta Metall Mater 26 (1992) 319-323
10. Aifantis E.C. Gradient deformation models at nano, micro, and macro scales. ASME J Eng Mater Technol 121 (1999) 189-202
11. Aifantis E.C. Gradient effects at macro, micro, and nano scales. J Mech Behav Mater 5 (1994) 355-375
12. Gutkin M.Y., and Aifantis E.C. Dislocations in the theory of gradient elasticity. Scripta Mater 40 (1999) 559-566
13. Gutkin M.Y. Nanoscopics of dislocations and disclinations in gradient elasticity. Rev Adv Mater Sci 1 (2000) 27-60. http://www.ipme.ru Also available from:
14. Gutkin M.Y., and Aifantis E.C. Dislocations and disclinations in gradient elasticity. Phys Status Solid B 214 (1999) 245-284
15. Rubin M.B., Rosenau P., and Gottlieb O. Continuum model of dispersion caused by an inherent material characteristic length. J Appl Phys 77 (1995) 4054-4063
16. Chen W., and Fish J. A dispersive model for wave propagation in periodic heterogeneous media based on homogenization with multiple spatial and temporal scales. ASME J Appl Mech 68 (2001) 153-161
17. Metrikine A., and Askes H. One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure. Part 1: generic formulation. Eur J Mech A/Solids 21 (2002) 555-572
18. Askes H., and Metrikine A. One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure. Part 2: static and dynamic response. Eur J Mech A/Solids 21 (2002) 573-588
19. Andrianov I.V., Awrejcewicz J., and Barantsev R.G. Asymptotic approaches in mechanics: new parameters and procedures. ASME Appl Mech Rev 56 (2003) 87-110
20. Triantafyllidis N., and Aifantis E.C. A gradient approach to localization of deformation. I. Hyperelastic materials. J Elasticity 16 (1986) 225-237
21. Schreyer H.L., and Chen Z. One-dimensional softening with localization. ASME J Appl Mech 53 (1986) 791-797
22. Aifantis E.C. On the role of gradients in the localization of deformation and fracture. Int J Eng Sci 30 (1992) 1279-1299
23. Aifantis E.C. Pattern formation in plasticity. Int J Eng Sci 33 (1995) 2161-2178
24. Mühlhaus H.-B., and Aifantis E.C. A variational principle for gradient plasticity. Int J Solids Struct 28 (1991) 845-857
25. de Borst R., and Mühlhaus H.-B. Gradient-dependent plasticity: formulation and algorithmic aspects. Int J Numer Methods Eng 35 (1992) 521-539
26. Pamin J. Gradient-dependent plasticity in numerical simulation of localization phenomena. Dissertation, Delft University of Technology; 1994.
27. Peerlings R.H.J., de Borst R., Brekelmans W.A.M., and de Vree J.H.P. Gradient enhanced damage for quasi-brittle materials. Int J Numer Methods Eng 39 (1996) 3391-3403
28. Sluys LJ. Wave propagation, localisation and dispersion in softening solids. Dissertation, Delft University of Technology; 1992.
29. Aifantis E.C. Higher order gradients and size effects. In: Carpinteri A. (Ed). Size-scale effects in the failure mechanisms of materials and structures (1996), E. & F.N. Spon, London 231-242
30. Aifantis E.C. Strain gradient interpretation of size effects. Int J Fract 95 (1999) 299-314
31. Askes H., and Aifantis E.C. Numerical modeling of size effects with gradient elasticity - formulation, meshless discretization and examples. Int J Fract 117 (2002) 347-358
32. Carmeliet J. Optimal estimation of gradient damage parameters from localization phenomena in quasi-brittle materials. Mech Cohes-Frict Mater 4 (1999) 1-16
33. Efremidis G., Carpinteri A., and Aifantis E.C. Griffith's theory versus gradient elasticity in the evaluation of porous materials tensile strength. J Mech Behav Mater 12 (2001) 95-105
34. Efremidis G., Carpinteri A., and Aifantis E.C. Multifractal scaling law versus gradient elasticity in the evaluation of disordered materials compressive strength. J Mech Behav Mater 12 (2001) 107-120
35. Gutiérrez M.A., and de Borst R. Deterministic and stochastic analysis of size effects and damage evolution in quasi-brittle materials. Arch Appl Mech 69 (1999) 655-676
36. Tsagrakis I., and Aifantis E.C. Recent developments in gradient plasticity - part I: formulation and size effects. ASME J Eng Mater Technol 124 (2002) 352-357
37. Tsagrakis I., Konstantinidis A., and Aifantis E.C. Strain gradient and wavelet interpretation of size effects in yield and strength. Mech Mater 35 (2003) 733-745
38. Zhu H.T., Zbib H.M., and Aifantis E.C. Strain gradients and continuum modeling of size effect in metal matrix composites. Acta Mech 121 (1997) 165-176
39. Petera J., and Pittman J.F.T. Isoparametric Hermite elements. Int J Numer Methods Eng 37 (1994) 3489-3519
40. Zervos A., Papanastasiou P., and Vardoulakis I. A finite element displacement formulation for gradient elastoplasticity. Int J Numer Methods Eng 50 (2001) 1369-1388
41. Zervos A., Papanastasiou P., and Vardoulakis I. Modelling of localisation and scale effect in thick-walled cylinders with gradient elastoplasticity. Int J Solids Struct 38 (2001) 5081-5095
42. Askes H., Pamin J., and de Borst R. Dispersion analysis and element-free Galerkin solutions of second and fourth-order gradient-enhanced damage models. Int J Numer Methods Eng 49 (2000) 811-832
43. Pamin J., Askes H., and de Borst R. Two gradient plasticity theories discretized with the element-free Galerkin method. Comput Methods Appl Mech Eng 192 (2003) 2377-2403
44. Tang Z., Shen S., and Atluri S.N. Analysis of materials with strain-gradient effects: a meshless local Petrov-Galerkin (MLPG) approach, with nodal displacements only. Comput Model Eng Sci 4 (2003) 177-196
45. Tsagrakis I., and Aifantis E.C. Element-free Galerkin implementation of gradient plasticity. Part 1: formulation and application to 1D strain localization. J Mech Behav Mater 14 (2003) 199-231
46. Tsagrakis I., and Aifantis E.C. Element-free Galerkin implementation of gradient plasticity. Part 2: applications to 2D strain localization and size effects. J Mech Behav Mater 14 (2003) 233-253
47. Askes H., and Sluys L.J. Explicit and implicit gradient series in damage mechanics. Eur J Mech A/Solids 21 (2002) 379-390
48. Askes H., Suiker A.S.J., and Sluys L.J. A classification of higher-order strain gradient models - linear analysis. Arch Appl Mech 72 (2002) 171-188
49. Askes H., and Sluys L.J. A classification of higher-order strain gradient models in damage mechanics. Arch Appl Mech 73 (2003) 448-465
50. Cirak F., Ortiz M., and Schröder P. Subdivision surfaces: a new paradigm for thin-shell finite-element analysis. Int J Numer Methods Eng 47 (2000) 2039-2072
51. Ioannidou T., Pouget J., and Aifantis E.C. Kink dynamics in a lattice model with long-range interactions. J Phys A: Math Gen 34 (2001) 4269-4280
52. Shu J.Y., King W.E., and Fleck N.A. Finite elements for materials with strain gradient effects. Int J Numer Methods Eng 44 (1999) 373-391
53. Matsushima T., Chambon R., and Caillerie D. Large strain finite element analysis of a local second gradient model: application to localization. Int J Numer Methods Eng 54 (2002) 499-521
54. Amanatidou E., and Aravas N. Mixed finite element formulations of strain-gradient elasticity problems. Comput Meth Appl Mech Eng 191 (2002) 1723-1751
55. Comi C., and Driemeier L. On gradient regularization for numerical analysis in the presence of damage. In: de Borst R., and van der Giessen E. (Eds). Material instabilities in solids (1998), Wiley 425-440 [Chapter 26]
56. Engel G., Garikipati K., Hughes T.J.R., Larson M.G., Mazzei L.M., and Taylor R.L. Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput Methods Appl Mech Eng 191 (2002) 3669-3750
57. Wells G.N., Garikipati K., and Molari L. A discontinuous Galerkin formulation for a strain gradient-dependent damage model. Comput Methods Appl Mech Eng 193 (2004) 3633-3645
58. Tenek L.T., and Aifantis E.C. A two-dimensional finite element implementation of a special form of gradient elasticity. Comput Model Eng Sci 3 (2002) 731-741
59. Peerlings R.H.J., de Borst R., Brekelmans W.A.M., de Vree J.H.P., and Spee I. Some observations on localisation in non-local and gradient damage modelism. Eur J Mech A/Solids 15 (1996) 937-953
60. 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. Int J Solids Struct 38 (2001) 7723-7746
61. Askes H., and Gutiérrez M.A. Implicit gradient elasticity. Int J Numer Methods Eng 67 (2006) 400-416
62. Rodríguez-Ferran A., Morata I., and Huerta A. A new damage model based on non-local displacements. Int J Numer Anal Methods Geomech 29 (2005) 473-493
63. Gitman I.M., Askes H., and Sluys L.J. Representative volume size as a macroscopic length scale parameter. In: Li V.C., Leung C.K.Y., Willam K.J., and Billington S.L. (Eds). Fracture Mechanics of Concrete Structures (2004), Ia-FraMCoS 483-489
64. Gitman I.M., Askes H., and Aifantis E.C. The representative volume size in static and dynamic micro-macro transitions. Int J Fract 135 (2005) L3-L9
65. Gitman I.M., Askes H., and Aifantis E.C. Gradient elasticity with internal length and internal inertia based on the homogenisation of a representative volume element. J Mech Behav Mater 18 (2007) 1-16
66. Vardoulakis I., Exadaktylos G., and Aifantis E.C. Gradient elasticity with surface energy: mode-III crack problem. Int J Solids Struct 33 (1996) 4531-4559
67. Vardoulakis I., and Aifantis E.C. On the role of microstructure in the behavior of soils: effects of higher order gradients and internal inertia. Mech Mater 18 (1994) 151-158
68. Unger D.J., and Aifantis E.C. Strain gradient elasticity theory for antiplane shear cracks. Part I: oscillatory displacements. Theor Appl Fract Mech 34 (2000) 243-252
69. Unger D.J., and Aifantis E.C. Strain gradient elasticity theory for antiplane shear cracks. Part II: monotonic displacements. Theor Appl Fract Mech 34 (2000) 253-265
70. Askes H., and Aifantis E.C. Gradient elasticity theories in statics and dynamics - a unification of approaches. Int J Fract 139 (2006) 297-304
71. de Borst R., Geers M.G.D., Kuhl E., and Peerlings R.H.J. Enhanced damage models for concrete fracture. In: de Borst R., Bićanić N., Mang H., and Meschke G. (Eds). Computational modelling of concrete structures (1998), Balkema 231-248
72. Mazars J., and Pijaudier-Cabot G. Continuum damage theory - application to concrete. ASCE J Eng Mech 115 (1989) 345-365
73. Polizzotto C. Gradient elasticity and nonstandard boundary conditions. Int J Solids Struct 40 (2003) 7399-7423
74. Tsepoura K.G., Papargyri-Beskou S., Polyzos D., and Beskos D.E. Static and dynamic analysis of a gradient-elastic bar in tension. Arch Appl Mech 72 (2002) 483-497
75. Askes H., and Metrikine A. Higher-order continua derived from discrete media: continualisation aspects and boundary conditions. Int J Solids Struct 42 (2005) 187-202
76. Simone A., Askes H., and Sluys L.J. Incorrect initiation and propagation of failure in non-local and gradient-enhanced media. Int J Solids Struct 41 (2004) 351-363
77. In: Carpinteri A. (Ed). Size-scale effects in the failure mechanisms of materials and structures (1996), E. & F.N. Spon, London
78. Bažant Z.P. Scaling of quasibrittle failure: asymptotic analysis. Int J Fract 83 (1997) 19-40