## Three-dimensional finite elements for large deformation micropolar elasticity.(English)Zbl 1231.74399

Summary: In this paper, we propose a three-dimensional finite element formulation for micropolar elasticity dealing with large displacements and small strains (or equivalently small strains and finite rotations). A comprehensive outline of the theory’s characteristical features is given and we try to elucidate the set-up of a possible non-linear finite element implementation. One focus of the present study is on a sound verification process, featuring the construction of an enhanced Patch Test and the assessment of quadratic asymptotic rates of convergence. Aspects of performance and validity are discussed at a set of numerical examples. We show that, the proposed model is able to reproduce the transition between micropolar and classical continua highly accurate. Finally, we present results underlining the implementation’s applicability in the realm of finite deformation with arbitrarily large rotations.

### MSC:

 74S05 Finite element methods applied to problems in solid mechanics 74B20 Nonlinear elasticity 74A35 Polar materials
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### References:

 [1] Amanatidou, E.; Aravas, N., Mixed finite element formulations of strain-gradient elasticity problems, Comput. methods appl. mech. engrg., 191, 1723-1751, (2002) · Zbl 1098.74678 [2] Begley, M.R.; Hutchinson, J.W., The mechanics of size-dependent indentation, J. mech. phys. solids, 46, 2049-2068, (1998) · Zbl 0967.74043 [3] Belytschko, T.; Liu, W.K.; Moran, B., Non-linear finite elements for continua and structures, (2000), John Wiley & Sons Chichester [4] E. Cosserat, F. Cosserat, Théorie de Corps Déformables, A. Herman et fils, Paris, 1909. · JFM 40.0862.02 [5] Cowin, S.C., An incorrect inequality in micropolar elasticity theory, Z. angew. math. phys. (ZAMP), 21, 494-497, (1970) · Zbl 0198.58503 [6] Cowin, S.C., The stresses around a hole in an linear elastic material with voids, Q. J. mech. appl. math., 37, 441-465, (1984) · Zbl 0543.73016 [7] Diegele, E.; Elsässer, R.; Tsakmakis, Ch., Linear micropolar elastic crack-tip fields under mixed mode loading conditions, Int. J. fract., 129, 309-339, (2004) · Zbl 1187.74171 [8] Ehlers, W.; Diebels, S.; Volk, W., Deformation and compatibility for elastoplastic micropolar materials with application to geomechanical problems, J. phys. IV France, 8, 127-134, (1998) [9] Eringen, A.C., Linear theory of micropolar elasticity, J. math. mech., 15, 909-923, (1966) · Zbl 0145.21302 [10] Eringen, A.C., Theory of micropolar elasticity, () · Zbl 0145.21302 [11] Eringen, A.C.; Kafadar, C.B., Polar field theories, (), 2-75 [12] R. Elsässer, Bruchmechanische Untersuchungen für elastische mikropolare Kontinua, Wissenschaftliche Berichte FZKA 6709, Forschungszentrum Karlsruhe, Technik und Umwelt, 2002. [13] Fleck, N.A.; Müeller, G.M.; Ashby, M.F.; Hutchinson, J.W., Strain gradient plasticity – theory and experiment, Comput. methods appl. mech. engrg., 42, 475-487, (1994) [14] Gauthier, R.D.; Jashman, W.E., A quest for micropolar elastic constants, J. appl. mech., 42, 369-374, (1975) [15] Grammenoudis, P.; Tsakmakis, Ch., Hardening rules for finite deformation plasticity: restrictions imposed by the second law of thermodynamics and the postulate of il’iushin, Continuum mech. thermodyn., 13, 325-363, (2001) · Zbl 1012.74009 [16] Grammenoudis, P.; Tsakmakis, Ch., Prediction of microtorsional experiments by micropolar plasticity, Proc. R. soc., 461, 189-205, (2005) · Zbl 1121.74474 [17] Grammenoudis, P.; Tsakmakis, Ch., Finite element implementation of large deformation micropolar plasticity exhibiting isotropic and kinematic hardening effects, Int. J. numer. methods engrg., 62, 1691-1720, (2005) · Zbl 1121.74474 [18] Huang, F.-Y.; Liang, K.-Z., Torsional analysis of micropolar elasticity using the finite element method, Int. J. engrg. sci., 32, 347-358, (1994) · Zbl 0788.73067 [19] Huang, F.-Y.; Yan, B.-H.; Yan, J.-L.; Yang, D.-U., Bending analysis of micropolar elastic beam using a 3-D finite element method, Int. J. engrg. sci., 38, 275-286, (2000) · Zbl 1210.74166 [20] Hughes, T.J.R., The finite element method – linear static and dynamic finite element analysis, (1987), Prentice-Hall New-Jersey [21] Kaloni, P.N.; Ariman, T., Stress concentration effects in micropolar elasticity, Z. angew. math. phys. (ZAMP), 18, 136-141, (1967) [22] Kozar, I.; Ibrahimbegovic, A., Finite element formulation of the finite rotation solid element, Finite elem. anal. des., 20, 101-126, (1995) · Zbl 0875.73297 [23] Li, L.; Xie, S., Finite element method for linear micropolar elasticity and numerical study of some scale effects phenomena in MEMS, Int. J. mech. sci., 46, 1571-1587, (2004) · Zbl 1098.74054 [24] Macneal, R.H.; Harder, R.L., A proposed standard set of problems to test finite element accuracy, Finite elem. anal. des., 1, 3-20, (1985) [25] Miehe, C., Numerical computation of algorithmic (consistent) tangent moduli in large-strain computational inelasticity, Comput. methods appl. mech. engrg., 134, 223-240, (1996) · Zbl 0892.73012 [26] Mindlin, R.D., Influence of couple-stresses on stress concentrations, Exp. mech., 3, 1-7, (1963) [27] Mindlin, R.D.; Tiersten, H.F., Effects of couple-stresses in linear elasticity, Arch. ration. mech. anal., 11, 415-448, (1963) · Zbl 0112.38906 [28] Mora, R.J.; Waas, A.M., Evaluation of the micropolar elasticity constants for honeycombs, Acta mechanica, 192, 1-16, (2007) · Zbl 1120.74046 [29] Nakamura, S.; Benedict, R.; Lakes, R., Finite element method for orthotropic micropolar elasticity, Int. J. engrg. sci., 22, 319-330, (1984) · Zbl 0536.73007 [30] Neff, P.; Chelminski, K.; Müller, W.; Wieners, C., A numerical solution method for an infinitesimal elasto-plastic Cosserat model, Math. models methods appl. sci., 17, 1211-1239, (2007) · Zbl 1137.74012 [31] Pérez-Foguet, A.; Rodríguez-Ferran, A.; Huerta, A., Numerical differentiation for local and global tangent operators in computational plasticity, Comput. methods appl. mech. engrg., 189, 277-296, (2000) · Zbl 0961.74078 [32] Perić, D.; Yu, J.; Owen, D.R.J., On error estimates and adaptivity in elastoplastic solids: applications to the numerical simulation of strain localization in classical and Cosserat continua, Int. J. numer. methods engrg., 37, 1351-1379, (1994) · Zbl 0805.73066 [33] Providas, E.; Kattis, M.A., Finite element method in plane Cosserat elasticity, Comput. struct., 80, 2059-2069, (2002) [34] Ramezani, S.; Naghdabadi, R.; Sohrabpour, S., Non-linear finite element implementation of micropolar hypo-elastic materials, Comput. methods appl. mech. engrg., 197, 4149-4159, (2008) · Zbl 1194.74468 [35] Stölken, J.S.; Evans, A.G., A microbend test method for measuring the plasticity length scale, Acta materialia, 46, 5109-5115, (1998) [36] Taylor, R.L.; Simo, J.C., The patch test – a condition for assessing FEM convergence, Int. J. numer. methods engrg., 22, 39-62, (1986) · Zbl 0593.73072 [37] Timoshenko, S.; Goodier, J.N., Theory of elasticity, (1951), McGraw-Hill New York · Zbl 0045.26402 [38] Wriggers, P., Nichtlineare finite-elemente-methoden, (2001), Springer Berlin [39] Zienkiewicz, O.C.; Taylor, R.L., The finite element patch test revisited: a computer test for convergence, validation and error estimates, Comput. methods appl. mech. engrg., 149, 223-254, (1997) · Zbl 0918.73134 [40] Zienkiewicz, O.C.; Taylor, R.L., The finite element method volume i, Basic formulations and linear problems, (1988), McGraw-Hill Book Company London [41] Zienkiewicz, O.C.; Taylor, R.L., The finite element method volume II, Solid and fluid mechanics dynamics and non-linearity, (1988), McGraw-Hill Book Company London
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