p-Extension of C0 continuous mixed finite elements for plane strain gradient elasticity

Markolefas, S.; Papathanasiou, T. K.; Georgantzinos, S. K.

doi:10.24423/aom.3219

Artykuł - szczegóły

Tytuł artykułu

p-Extension of C0 continuous mixed finite elements for plane strain gradient elasticity

Autorzy

Markolefas S. , Papathanasiou T. K. , Georgantzinos S. K.

Wybrane pełne teksty z tego czasopisma

https://am.ippt.pan.pl/index.php/am

Identyfikatory

DOI

10.24423/aom.3219

Warianty tytułu

Języki publikacji

Abstrakty

A mixed finite element formulation is developed for the general 2D plane strain, linear isotropic gradient elasticity problem. Form II of the dipolar strain gradient theory for micro-structured solids is considered. The main variables are the double stress tensor μ and the displacement field vector u. Standard C0−continuous, high polynomial order hierarchical basis functions are employed for the finite element solution spaces (p-extension). The formulation is numerically validated against the standard axial tension patch test and the Mode I crack problem. The theoretical convergence rates of the uniform h- and p-extensions are confirmed using a benchmark problem where only double stresses appear. Results for the crack problem demonstrate that proper mesh refinement at areas of steep gradients ensures reproduction of the exact solution behaviour at different length scales. More specifically, the asymptotic exponents of the crack face opening displacement and the crack head true stress solutions of the Mode I crack problem are recovered. Finally, the upper bound of the true tensile normal stress near the crack tip is estimated. This upper bound is of major importance since the nature of the exact solution may change radically as we proceed from the macro- to micro-scale.

Słowa kluczowe

dipolar plane strain gradient elasticity mixed finite elements p-version mode I crack problem

Wydawca

Instytut Podstawowych Problemów Techniki PAN

Czasopismo

Archives of Mechanics

Rocznik

2019

Tom

Vol. 71, nr 6

Strony

567--593

Opis fizyczny

Bibliogr. 68 poz., rys.

Twórcy

autor

Markolefas S.

Strength of Materials Laboratory, National and Kapodistrian University of Athens, Psachna Evias 34400, Greece

autor

Papathanasiou T. K.

theodosios.papathanasiou@brunel.ac.uk

Department of Civil & Environmental Engineering, Brunel University London, UK

autor

Georgantzinos S. K.

Metallic Structures Laboratory, National and Kapodistrian University of Athens, Psachna Evias 34400, Greece

Bibliografia

1. M.H. Ghayesh, A. Farajpour, A review on the mechanics of functionally graded nanoscale and microscale structures, International Journal of Engineering Science, 137, 8–36, 2019.
2. A.E. Giannakopoulos, K. Stamoulis, Structural analysis of gradient elastic components, International Journal of Solids and Structures, 44, 3440–3451, 2007.
3. D.C.C. Lam, F. Yang, A.C.M. Chong, J. Wang, P. Tong, Experiments and theory in strain gradient elasticity, Journal of Mechanics and Physics of Solids, 51, 1477–1508, 2003.
4. L. Li, S. Xie, Finite element method for linear micropolar elasticity and numerical study of some scale effects phenomena in MEMS, International Journal of Mechanical Sciences, 46, 1571–1587, 2004.
5. L. Lu, X. Guo, J. Zhao, A unified size-dependent plate model based on nonlocal strain gradient theory including surface effects, Applied Mathematical Modelling, 68, 583–602, 2019.
6. K. Stamoulis, A.E. Giannakopoulos, A second gradient elasto-plastic model for fatigue of small-scale metal components, International Journal of Structural Integrity, 1, 3, 193–208, 2010.
7. A. Charalambopoulos, D. Polyzos, Plane strain gradient elastic rectangle in tension, Archive of Applied Mechanics, 85, 1421–1438, 2015.
8. H. Petryk, S. Stupkiewicz, A minimal gradient-enhancement of the classical continuum theory of crystal plasticity. Part I: The hardening law, Archives of Mechanics, 68, 459–485, 2016.
9. S. Stupkiewicz, H. Petryk, A minimal gradient-enhancement of the classical continuum theory of crystal plasticity. Part II: Size effects, Archives of Mechanics, 68, 487–513, 2016.
10. D. Polyzos, G. Huber, G. Mylonakis, T. Triantafyllidis, S. Papargyri-Beskou, D. E. Beskos, Torsional vibrations of a column of fine-grained material: a gradient elastic approach, Journal of the Mechanics and Physics of Solids, 76, 338–358, 2015.
11. S.N. Iliopoulos, F. Malm, C.U. Grosse, D G. Aggelis, D. Polyzos, Concrete wave dispersion interpretation through Mindlin’s strain gradient elastic theory, Journal of the Acoustics Society of America, 142, 1, 2017.
12. I.D. Gavardinas, A.E. Giannakopoulos, Th. Zisis, A von Karman plate analogue for solving anti-plane problems in couple stress and dipolar gradient elasticity, International Journal of Solids and Structures, 148–149, 169–180, 2018.
13. S. Lurie, Y. Solyaev, Revisiting bending theories of elastic gradient beams, International Journal of Engineering Science, 126, 1–21, 2018.
14. D.M. Manias, T.K. Papathanasiou, S.I. Markolefas, E.E. Theotokoglou, Analysis of a gradient-elastic beam on Winkler foundation and applications to nano-structure modelling, European Journal of Mechanics A/Solids, 56, 45–58, 2016.
15. A. Triantafyllou, A.E. Giannakopoulos, Structural analysis using a dipolar elastic Timoshenko beam, European Journal of Mechanics A/Solids, 39, 218–228, 2013.
16. G.J. Tsamasphyros, C.C. Koutsoumaris, Mixed nonlocal-gradient elastic materials with applications in wave propagation of beams, International Conference of Computational Methods in Sciences and Engineering 2016 (ICCMSE 2016), AIP Conference Proceedings, 1790, 150031-1–150031-4, DOI: 10.1063/1.4968770.
17. P.A. Gourgiotis, Th. Zisis, H.G. Georgiadis, On concentrated surface loads and Green’s functions in the Toupin–Mindlin theory of strain-gradient elasticity, International Journal of Solids and Structures, 130–131, 153–171, 2018.
18. H.G. Georgiadis, P.A. Gourgiotis, D.S. Anagnostou, The Boussinesq problem in dipolar gradient elasticity, Archives of Applied Mechanics, 84, 1373–1391, 2014.
19. A.M. Nikolarakis, E.E. Theotokoglou, Numerical analysis of transient stress field of a functionally graded nickel-zirconia profile under thermal loading, Journal of Thermal Stresses, 38, 10, 1085–1103, 2015.
20. H. G. Georgiadis, The mode III crack problem in microstructured solids governed by dipolar gradient elasticity: static and dynamic anlaysis, Journal of Applied Mechanics 70, 517–530, 2003.
21. P.A. Gourgiotis, H.G. Georgiadis, Plane-strain crack problems in microstructured solids governed by dipolar gradient elasticity, Journal of the Mechanics and Physics of Solids, 57, 1898–1920, 2009.
22. M.X. Shi, Y. Huang, K.C. Hwang, Fracture in a higher-order elastic continuum, Journal of the Mechanics and Physics of Solids, 48, 2513–2538, 2000.
23. K.P. Baxevanakis, P.A. Gourgiotis, H.G. Georgiadis, Interaction of cracks with dislocations in couple-stress elasticity. Part I: Opening mode, International Journal of Solids and Structures, 118–119, 179–191, 2017.
24. P.A. Gourgiotis, A. Piccolroaz, Steady-state propagation of a Mode II crack in couple stress elasticity, International Journal of Fracture, 188, 119–145, 2014.
25. P.A. Gourgiotis, H.G. Georgiadis, Distributed dislocation approach for cracks in couple stress elasticity: shear modes, International Journal of Fracture, 147, 83–102, 2007.
26. L. Zhang, Y. Huang, J.Y. Chen, K.C. Hwang, The mode III full-field solution in elastic materials with strain gradient effects, International Journal of Fracture, 92, 325–348, 1998.
27. S. Khakalo, J. Niiranen, Gradient-elastic stress analysis near cylindrical holes in a plane under bi-axial tension fields, International Journal of Solids and Structures, 110– 111, 351–366, 2017.
28. A. Kordolemis, A.E. Giannakopoulos, N. Aravas, Pretwisted beam subjected to thermal loads: a gradient thermoelastic analogue, Journal of Thermal Stresses, 40, 10, 1231–1253, 2017.
29. G.F. Karlis, A. Charalambopoulos, D. Polyzos, An advanced boundary element method for solving 2D and 3D static problems in Mindlin’s strain-gradient theory of elasticity , International Journal for Numerical Methods in Engineering 83, 1407–1427, 2010.
30. G.F. Karlis, S.V. Tsinopoulos, D. Polyzos, D.E. Beskos, Boundary element analysis of mode I and mixed mode (I and II) crack problems of 2-D gradient elasticity, Computer Methods in Applied Mechanics and Engineering, 196, 5092–5103, 2007.
31. V. Balobanov, J. Kiendl, S. Khakalo, J. Niiranen, Kirchhoff–Love shells within strain gradient elasticity: weak and strong formulations and an H3-conforming isogeometric implementation, Computer Methods in Applied Mechanics and Engineering, 344, 837–857, 2019.
32. J. Niiranen, S. Khakalo, V. Balobanov, A.H. Niemi, Variational formulation and isogeometric analysis for fourth-order boundary value problems of gradient-elastic bar and plane strain/stress problems, Computer Methods in Applied Mechanics and Engineering, 308, 182–211, 2016.
33. J. Niiranen, J. Kiendl, A.H. Niemi, A. Reali, Isogeometric analysis for sixth-order boundary value problems of gradient-elastic Kirchhoff plates, Computer Methods in Applied Mechanics and Engineering, 316, 328–348, 2017.
34. S. Khakalo, J. Niiranen, Isogeometric analysis of higher-order gradient elasticity by user elements of a commercial finite element software, Computer-Aided Design, 82, 154–169, 2017.
35. J. Niiranen, S. Khakalo, V. Balobanov, Isogeometric finite element analysis of mode I cracks within strain gradient elasticity, Journal of Structural Mechanics 50, 337–340, 2017.
36. A. Beheshti, Finite element analysis of plane strain solids in strain-gradient elasticity, Acta Mechanica 228, 3543–3559, 2017.
37. S. Akarapu, H.M. Zbib, Numerical analysis of plane cracks in strain-gradient elastic materials, International Journal of Fracture, 141, 403–430, 2006.
38. P. Fischer, J. Mergheim, P. Steinmann, On the C1 continuous discretization of non-linear gradient elasticity: a comparison of NEM and FEM based on Bernstein–Bézier patches, International Journal for Numerical Methods in Engineering, 82, 1282–1307, 2010.
39. E. Amanatidou, N. Aravas, Mixed finite element formulations of strain-gradient elasticity problems, Computer Methods in Applied Mechanics and Engineering, 191, 1723–1751, 2002.
40. S.I. Markolefas, D.A. Tsouvalas, G.I. Tsamasphyros, Mixed finite element formulation for the general anti-plane shear problem, including mode III crack computations, in 592 S. Markolefas et al. the framework of dipolar linear gradient elasticity, Computational Mechanics, 43, 715–730, 2009.
41. S.-A. Papanicolopulos, F. Gulib, A. Marinelli, A novel efficient mixed formulation for strain-gradient models , International Journal for Numerical Methods in Engineering, 117, 926–937, 2019.
42. G. I. Tsamasphyros, S. Markolefas, D.A. Tsouvalas, Convergence and performance of the h- and p-extensions with mixed finite element C0-continuity formulations, for tension and buckling of a gradient elastic beam, International Journal of Solids and Structures, 44, 5056–5074, 2007.
43. L. Zybell, U. Mühlich, M. Kuna, Z.L. Zhang, A three-dimensional finite element for gradient elasticity based on a mixed-type formulation, Computational Materials Science, 52, 1, 268–273, 2012.
44. V. Phunpeng, P.M. Baiz, Mixed finite element formulations for strain-gradient elasticity problems using the FEniCS environment, Finite Elements in Analysis and Design, 96, 23–40, 2015.
45. Y. Wei, A new finite element method for strain gradient theories and applications to fracture analyses, European Journal of Mechanics A/Solids, 25, 897–913, 2006.
46. A. Zervos, Finite elements for elasticity with microstructure and gradient elasticity, International Journal for Numerical Methods in Engineering, 73, 564–595, 2008.
47. J. Zhao, W. J. Chen, S. H. Lo, A refined nonconforming quadrilateral element for couple stress/strain gradient elasticity, International Journal for Numerical Methods in Engineering, 85, 269–288, 2011.
48. U. Andreaus, F. dell’Isola, I. Giorgio, L. Placidi, T. Lekszycki, N.L. Rizzi, Numerical simulations of classical problems in two-dimensional (non) linear second gradient elasticity, International Journal of Engineering Science, 108, 34–50, 2016.
49. N. Auffray, On the isotropic moduli of 2D strain-gradient elasticity, Continuum Mechanics and Thermodynamics, 27, 5–19, 2015 DOI 10.1007/s00161-013-0325-6.
50. G. Rosi, L. Placidi, N. Auffray, On the validity range of strain gradient elasticity: a mixed static-dynamic identification procedure, European Journal of Mechanics/A Solids, 69, 179–191, 2018.
51. N.M. Cordero, S. Forest, E.P. Busso, Second strain gradient elasticity of nanoobjects, Journal of the Mechanics and Physics of Solids, 97, 92–124, 2016.
52. B.E. Abali, W.H. Müller, F. dell’Isola, Theory and computation of higher gradient elasticity theories based on action principles, Archive of Applied Mechanics, 87, 1495–1510, 2017.
53. R.D. Mindlin, Micro-structure in linear elasticity, Archive for Rational Mechanics and Analysis, 16, 51–78, 1964.
54. H. Askes, E.C. Aifantis, Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results, International Journal of Solids and Structures, 48, 1962–1990, 2011.
55. J.L. Bleustein, A note on the boundary conditions of Toupin’s strain-gradient theory, International Journal of Solids and Structures, 3, 1053–1057, 1967.
56. S. Papargyri-Beskou, S. Tsinopoulos, Lamé’s strain potential method for plane gradient elasticity problems, Archive of Applied Mechanics, 85, 1399– p 1419, 2015.
57. R.D. Mindlin, N.N. Eshel, On first-gradient theories in linear elasticity, International Journal of Solids and Structures, 4, 109–124, 1968.
58. M. Lazar, Irreducible decomposition of strain gradient tensor in isotropic strain gradient elasticity, ZAMM – Journal of Applied Mathematics and Mechanics 96, 11, 1291–1305. 2016. DOI 10.1002/zamm.201500278.
59. S.I. Markolefas, D.A. Tsouvalas, G.I. Tsamasphyros, Theoretical analysis of a class of mixed, C0 continuity formulations for general dipolar Gradient Elasticity boundary value problems, International Journal of Solids and Structures, 44, 546–572, 2007.
60. S. Balasundaram, P.K. Bhattacharyya, A mixed finite element method for fourth order elliptic equations with variable coefficients, Computers and Mathematics with Applications, 10, 245–256, 1984.
61. S. I. Markolefas, D.A. Tsouvalas, G.I. Tsamasphyros, Some C0-continuous mixed formulations for general dipolar linear gradient elasticity boundary value problems and the associated energy theorems, International Journal of Solids and Structures, 45, 3255–3281, 2008.
62. I. Babuška, M. Suri, The p- and h-p versions of the finite element method, an overview, Computer Methods in Applied Mechanics and Engineering, 80, 5–26, 1990.
63. B. Szabo, I. Babuška, Finite Element Analysis, John Wiley & Sons, Inc. New York, 1991.
64. K.-J. Bathe, The inf-sup condition and its evaluation for mixed finite element methods, Computers and Structures, 79, 243–252, 2001.
65. F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer, New York, 1991.
66. F. Brezzi, K.J. Bathe, A discourse on the stability conditions for mixed finite element formulations, Computer Methods in Applied Mechanics and Engineering, 82, 27–57, 1990.
67. S. Khakalo, V. Balobanov, J. Niiranen, Modelling size-dependent bending, buckling and vibrations of 2D triangular lattices by strain gradient elasticity models: applications to sandwich beams and auxetics, International Journal of Engineering Science, 127, 33–52, 2018.
68. S.I. Markolefas, D. Tsouvalas, G. Tsamasphyros, High polynomial order mixed finite element methods for strain gradient elasticity problems: a posteriori error estimation and adaptivity, Proceedings of the 3rd International Conference “From Scientific Computing to Computational Engineering”, Athens, Greece, 9–12 July, 2008.

Typ dokumentu

Bibliografia

Identyfikator YADDA

bwmeta1.element.baztech-242188ec-d122-4863-9a92-af590e2e8aca