×

zbMATH — the first resource for mathematics

24-DOF quadrilateral hybrid stress element for couple stress theory. (English) Zbl 1398.74365
Summary: A 24-DOF quadrilateral hybrid stress element for couple stress theory is proposed in this study. In order to satisfy the equilibrium equation in the domain of the element, the \(21\beta\) Airy stress functions are chosen a assumed stress interpolation functions, and beam functions are adopted as the displacement interpolation functions on the boundary. This element can satisfy weak \(\mathrm C^0\) continuity with second-order accuracy and weak \(\mathrm C^1\) continuity simultaneously. So the element can pass the enhanced patch test of a convergence condition. Moreover, the reduced integration and a stresses smooth technique are introduced to improve the element accuracy. Numerical examples presented show that the proposed model can pass the \(\mathrm C^{0-1}\) enhanced patch test and indeed possesses higher accuracy. Besides, it does not exhibit extra zero energy modes and can capture the scale effects of microstructure.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74B99 Elastic materials
74C99 Plastic materials, materials of stress-rate and internal-variable type
74A30 Nonsimple materials
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Fleck, NA; Muller, GM; Ashby, MF; Hutchinson, JW, Stain gradient plasticity: theory and experiment, Acta Metall Mater, 42, 475-487, (1994)
[2] Nix, WD; Gao, H, Indentation size effects in crystalline materials: a law for strain gradient plasticity, J Mech Phys Solids, 46, 411-425, (1998) · Zbl 0977.74557
[3] Wei YG, Wang XZ, Wu XL (2000) Theoretical and experimental study on micro-indentation size effects (in Chinese). Sci China (A) 30:1025-1032 · Zbl 1157.74014
[4] Cosserat E, Cosserat F (1909) Theorie des corps deformables. Herrman, Paris · JFM 40.0862.02
[5] Toupin, RA, Elastic materials with couple stresses, Arch Ration Mech Anal, 11, 385-414, (1962) · Zbl 0112.16805
[6] Koiter, WT, Couple stresses in the theory of elasticity I & II, Proc Koninklijke Nederlandse Akademie van Wetenschappen, 67, 17-44, (1964) · Zbl 0119.39504
[7] Eringen, AC, Nonlocal polar elastic continua, Int J Eng Sci, 10, 1-16, (1972) · Zbl 0229.73006
[8] Eringen, AC, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J Appl Phys, 54, 4703-4710, (1983)
[9] Eringen AC (2002) Nonlocal continuum field theories. Springer, New York · Zbl 1023.74003
[10] Mindlin, RD, Microstructure in linear elasticity, Arch Ration Mech Anal, 16, 51-78, (1964) · Zbl 0119.40302
[11] Fleck, NA; Hutchinson, JW, A phenomenological theory for strain gradient effects in plasticity, J Mech Phys Solids, 41, 1825-1857, (1993) · Zbl 0791.73029
[12] Yang, F; Chong, AM; Lam, DCC; Tong, P, Couple stress based strain gradient theory for elasticity, Int J Solids Struct, 39, 2731-2743, (2002) · Zbl 1037.74006
[13] Argyris, J; Fried, I; Scharpf, DW, The TUBA family of plate elements for the matrix displacement method, Aeronaut J R Aeronaut Soc, 72, 701-709, (1968)
[14] Bell, K, A refined triangular plate bending element, Int J Numer Methods Eng, 1, 101-122, (1969)
[15] Zervos, A; Papanastasiou, P; Vardoulakis, I, Modelling of localisation and scale in thick-walled cylinders with gradient elastoplasticity, Int J Numer Methods Eng, 38, 5081-5095, (2001) · Zbl 0997.74054
[16] Papanicolopulos, S-A; Zervos, A; Vardoulakis, I, A three dimensional C\(^{1}\) finite element for gradient elasticity, Int J Numer Methods Eng, 77, 1396-1415, (2009) · Zbl 1156.74382
[17] Zervos A, Papanicolopulos S-A, Vardoulakis I (2009) Two finite-element discretizations for gradient elasticity. J Eng Mech ASCE 135:203-213 · Zbl 1156.74382
[18] Soh AK, Chen WJ (2004) Finite element formulations of strain gradient theory for microstructures and the C\(^{0-1}\) patch test. Int J Numer Methods Eng 61:433-454 · Zbl 1075.74678
[19] Zhao, J; Chen, WJ; Ji, B, A weak continuity condition of FEM for axisymmetric couple stress theory and 18-DOF triangular axisymmetric element, Finite Elem Anal Des, 46, 632-644, (2010)
[20] Zhao, J; Chen, WJ; Lo, SH, A refined nonconforming quadrilateral element for couple stress/strain gradient elasticity, Int J Numer Methods Eng, 85, 269-288, (2011) · Zbl 1217.74140
[21] Chen, WJ, Enhanced patch test of finite element methods, Sci China, 49, 213-227, (2006) · Zbl 1147.74391
[22] Pian THH (1964) Derivation of element stiffness matrices by assumed stress distributions. AIAA J 2:1333-1336
[23] Pian, THH, Recent advances in hybrid stress finite element method, Adv Mech, 31, 344-349, (2001)
[24] Pian, THH; Sumihara, K, Rational approach for assumed stress finite elements, Int J Numer Methods Eng, 20, 1685-1695, (1984) · Zbl 0544.73095
[25] Tong, P, New displacement hybrid finite-element models for solid continua, Int J Numer Methods Eng, 2, 43-64, (1970) · Zbl 0247.73085
[26] Chen WJ (1981) The generalized hybrid element. Acta Mech Sin 17:582-591
[27] Chen, WJ, Refined hybrid element method and refined quadrilateral plane element, J Dalian Univ Technol, 32, 510-519, (1992)
[28] Sze, KY, Efficient formulation of robust mixed element using orthogonal stress/strain interpolants and admissible matrix formulation, Int J Numer Methods Eng, 35, 1-20, (1992)
[29] Zienkiewicz, OC; Taylor, RL; Too, JM, Reduced integration technique in general analysis of plates and shells, Int J Numer Methods Eng, 3, 275-290, (1971) · Zbl 0253.73048
[30] Zienkiewicz, OC; Zhu, JZ, Superconvergence and superconvergent patch recovery, Finite Elem Anal Des, 19, 11-23, (1995) · Zbl 0875.73292
[31] Boroomand, B; Zienkiewicz, OC, An improved REP recovery and the effectivity robustness test, Int J Numer Methods Eng, 40, 3247-3277, (1997) · Zbl 0895.73065
[32] Chen, WJ; Wang, JZ; Zhao, J, The functions for patch test in finite element analysis of Mindlin plate and thin cylindrical shell, Sci China (G), 52, 762-767, (2009)
[33] Stolken, JS; Evans, AG, A microbend test method for measuring the plasticity length scale, Acta Mater, 46, 5109-5115, (1998)
[34] Bin, Ji; Chen, WJ, A new analytical solution of pure bending beam in couple stress elasto-plasticity: theory and applications, Int J Solids Struct, 47, 779-785, (2010) · Zbl 1193.74074
[35] Shu, JY; King, WE; Fleck, NA, Finite elements for materials with strain gradient effects, Int J Numer Methods Eng, 44, 373-391, (1999) · Zbl 0943.74072
[36] Mindlin, RD, Influence of couple-stresses on stress concentrations, Exp Mech, 3, 1-7, (1963)
[37] Park, SK; Gao, X-L, Variational formulation of a modified couple stress theory and its application to a simple shear problem, Zeitsch Angew Math Phys (ZAMP), 59, 904-917, (2008) · Zbl 1157.74014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.