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Large elastic deformation of micromorphic shells. II: Isogeometric analysis. (English) Zbl 07273391

Summary: In Part I of this study [the authors, ibid. 24, No. 12, 3920–3956 (2019; Zbl xxx)], a variational formulation was presented for the large elastic deformation problem of micromorphic shells. Using the novel matrix-vector format presented for the kinematic model, constitutive relations, and energy functions, an isogeometric analysis (IGA)-based solution strategy is developed, which appropriately estimates the macro- and micro-deformation field components. Due to the capability of constructing exact geometries and the powerful mesh refinement tools, IGA can be successfully applied to solve the equilibrium equations with dominant nonlinear terms. It is known that different types of locking phenomena take place in the conventional finite element analysis of thin shells based on low-order elements. Non-standard finite element models with mixed interpolation schemes and additional degrees of freedom (DOFs) or the ones used the high-order Lagrangian shell elements which require high computational costs, are the available solutions to tackle locking issues. The present 16-DOFs IGA is found to be efficient because of possessing a good rate of convergence and providing locking-free stable responses for micromorphic shells. Such a conclusion is found from several comparative studies with available data in the well-known macro-scale benchmark problems based on the classical elasticity as well as the corresponding numerical examples studied in nano-scale beam-, plate-, cylindrical shell- and spherical shell-type structures on the basis of the micromorphic continuum theory.

MSC:

74-XX Mechanics of deformable solids
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[1] Bathe, K-J. Finite element procedures. Klaus-Jurgen Bathe, 2006.
[2] Hughes, TJ. The finite element method: linear static and dynamic finite element analysis. Courier Corporation, 2012.
[3] Brink, U, Stein, E. A posteriori error estimation in large-strain elasticity using equilibrated local Neumann problems. Comput Methods Appl Mech Eng 1998; 161: 77-101. · Zbl 0943.74062 · doi:10.1016/S0045-7825(97)00310-1
[4] Reese, S. On the equivalent of mixed element formulations and the concept of reduced integration in large deformation problems. Int J Nonlinear Sci Numer Simul 2002; 3: 1-34. · Zbl 1079.74057 · doi:10.1515/IJNSNS.2002.3.1.1
[5] Shojaei, MF, Yavari, A. Compatible-strain mixed finite element methods for incompressible nonlinear elasticity. J Comput Phys 2018; 361: 247-279. · Zbl 1390.74177 · doi:10.1016/j.jcp.2018.01.053
[6] Jackson, RL, Green, I. A finite element study of elasto-plastic hemispherical contact against a rigid flat. J Tribol 2005; 127: 343-354. · doi:10.1115/1.1866166
[7] Ghaednia, H, Pope, SA, Jackson, RL, et al. A comprehensive study of the elasto-plastic contact of a sphere and a flat. Tribol Int 2016; 93: 78-90. · doi:10.1016/j.triboint.2015.09.005
[8] Moës, N, Belytschko, T. Extended finite element method for cohesive crack growth. Eng Fract Mech 2002; 69: 813-833. · doi:10.1016/S0013-7944(01)00128-X
[9] Ferté, G, Massin, P, Moës, N. 3D crack propagation with cohesive elements in the extended finite element method. Comput Methods Appl Mech Eng 2016; 300: 347-374. · Zbl 1425.74413 · doi:10.1016/j.cma.2015.11.018
[10] Pezeshki, M, Loehnert, S, Wriggers, P, et al. 3D dynamic crack propagation by the extended finite element method and a gradient-enhanced damage model. In: Multiscale modeling of heterogeneous structures. Springer, 2018, pp.277-299. · doi:10.1007/978-3-319-65463-8_14
[11] Girault, V, Raviart, P-A. Finite element methods for Navier-Stokes equations: theory and algorithms. Springer Science & Business Media, 2012.
[12] de Frutos, J, García-Archilla, B, John, V, et al. Analysis of the grad-div stabilization for the time-dependent Navier-Stokes equations with inf-sup stable finite elements. Adv Comput Math 2018; 44: 195-225. · Zbl 1404.65188 · doi:10.1007/s10444-017-9540-1
[13] Civalek, Ö, Demir, C. A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method. Appl Math Comput 2016; 289: 335-352. · Zbl 1410.74033 · doi:10.1016/j.amc.2016.05.034
[14] Ansari, R, Shojaei, MF, Mohammadi, V, et al. Triangular Mindlin microplate element. Comput Methods Appl Mech Eng 2015; 295: 56-76. · Zbl 1423.74531 · doi:10.1016/j.cma.2015.06.004
[15] Reddy, J, Romanoff, J, Loya, JA. Nonlinear finite element analysis of functionally graded circular plates with modified couple stress theory. Eur J Mech A Solids 2016; 56: 92-104. · Zbl 1406.74448 · doi:10.1016/j.euromechsol.2015.11.001
[16] Liu, K, Melkote, SN. Finite element analysis of the influence of tool edge radius on size effect in orthogonal micro-cutting process. Int J Mech Sc 2007; 49: 650-660. · doi:10.1016/j.ijmecsci.2006.09.012
[17] Norouzzadeh, A, Ansari, R. Finite element analysis of nano-scale Timoshenko beams using the integral model of nonlocal elasticity. Physica E 2017; 88: 194-200. · doi:10.1016/j.physe.2017.01.006
[18] Ansari, R, Torabi, J, Norouzzadeh, A. Bending analysis of embedded nanoplates based on the integral formulation of Eringen’s nonlocal theory using the finite element method. Physica B 2018; 534: 90-97. · doi:10.1016/j.physb.2018.01.025
[19] Neff, P, Forest, S. A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling, existence of minimizers, identification of moduli and computational results. J Elast 2007; 87: 239-276. · Zbl 1206.74019 · doi:10.1007/s10659-007-9106-4
[20] Zervos, A, Papanicolopulos, S-A, Vardoulakis, I. Two finite-element discretizations for gradient elasticity. J Eng Mech 2009; 135: 203-213. · Zbl 1156.74382 · doi:10.1061/(ASCE)0733-9399(2009)135:3(203)
[21] Isbuga, V, Regueiro, RA. Three-dimensional finite element analysis of finite deformation micromorphic linear isotropic elasticity. Int J Eng Sci 2011; 49: 1326-1336. · Zbl 1423.74013 · doi:10.1016/j.ijengsci.2011.04.006
[22] Ansari, R, Bazdid-Vahdati, M, Shakouri, A, et al. Micromorphic prism element. Math Mech Solids 2017; 22: 1438-1461. · Zbl 1371.74255
[23] Ansari, R, Bazdid-Vahdati, M, Shakouri, A, et al. Micromorphic first-order shear deformable plate element. Meccanica 2016; 51: 1797-1809. · Zbl 1388.74078 · doi:10.1007/s11012-015-0325-7
[24] Huang, F-Y, Yan, B-H, Yan, J-L, et al. Bending analysis of micropolar elastic beam using a 3-D finite element method. Int J Eng Sci 2000; 38: 275-286. · Zbl 1210.74166 · doi:10.1016/S0020-7225(99)00041-5
[25] Zhang, H, Wang, H, Wriggers, P, et al. A finite element model for contact analysis of multiple Cosserat bodies. Comput Mech 2005; 36: 444-458. · Zbl 1100.74059 · doi:10.1007/s00466-005-0680-7
[26] Sharbati, E, Naghdabadi, R. Computational aspects of the Cosserat finite element analysis of localization phenomena. Comput Mater Sci 2006; 38: 303-315. · doi:10.1016/j.commatsci.2006.03.003
[27] Ansari, R, Shakouri, A, Bazdid-Vahdati, M, et al. A nonclassical finite element approach for the nonlinear analysis of micropolar plates. J Comput Nonlinear Dyn 2017; 12: 011019. · doi:10.1115/1.4034678
[28] Ansari, R, Norouzzadeh, A, Shakouri, A, et al. Finite element analysis of vibrating micro-beams and-plates using a three-dimensional micropolar element. Thin-Walled Struct 2018; 124: 489-500. · doi:10.1016/j.tws.2017.12.036
[29] Eremeyev, VA, Skrzat, A, Stachowicz, F. On finite element computations of contact problems in micropolar elasticity. Adv Mater Sci Eng 2016: 2016. · doi:10.1155/2016/9675604
[30] Başar, Y, Ding, Y. Finite-rotation shell elements for the analysis of finite-rotation shell problems. Int J Numer Methods Eng 1992; 34: 165-169. · doi:10.1002/nme.1620340109
[31] Bischoff, M, Ramm, E. Shear deformable shell elements for large strains and rotations. Int J Numer Methods Eng 1997; 40: 4427-4449. · Zbl 0892.73054 · doi:10.1002/(SICI)1097-0207(19971215)40:23<4427::AID-NME268>3.0.CO;2-9
[32] Eberlein, R, Wriggers, P. Finite element concepts for finite elastoplastic strains and isotropic stress response in shells: theoretical and computational analysis. Comput Methods Appl Mech Eng 1999; 171: 243-279. · Zbl 0957.74047 · doi:10.1016/S0045-7825(98)00212-6
[33] Başar, Y, Ding, Y, Schultz, R. Refined shear-deformation models for composite laminates with finite rotations. Int J Solids Struct 1993; 30: 2611-2638. · Zbl 0794.73036 · doi:10.1016/0020-7683(93)90102-D
[34] Witkowski, W. 4-Node combined shell element with semi-EAS-ANS strain interpolations in 6-parameter shell theories with drilling degrees of freedom. Comput Mech 2009; 43: 307-319. · Zbl 1162.74479 · doi:10.1007/s00466-008-0307-x
[35] Büchter, N, Ramm, E. 3D-extension of nonlinear shell equations based on the enhanced assumed strain concept. Comput Methods Appl Sci 1992: 55-62.
[36] Braun, M, Bischoff, M, Ramm, E. Nonlinear shell formulations for complete three-dimensional constitutive laws including composites and laminates. Comput Mech 1994; 15: 1-18. · Zbl 0819.73042 · doi:10.1007/BF00350285
[37] Büchter, N, Ramm, E, Roehl, D. Three-dimensional extension of non-linear shell formulation based on the enhanced assumed strain concept. Int J Numer Methods Eng 1994; 37: 2551-2568. · Zbl 0808.73046 · doi:10.1002/nme.1620371504
[38] Sansour, C, Kollmann, F. Families of 4-node and 9-node finite elements for a finite deformation shell theory. An assesment of hybrid stress, hybrid strain and enhanced strain elements. Comput Mech 2000; 24: 435-447. · Zbl 0959.74072 · doi:10.1007/s004660050003
[39] Betsch, P, Menzel, A, Stein, E. On the parametrization of finite rotations in computational mechanics: a classification of concepts with application to smooth shells. Comput Methods Appl Mech Eng 1998; 155: 273-305. · Zbl 0947.74060 · doi:10.1016/S0045-7825(97)00158-8
[40] El-Abbasi, N, Meguid, S. A new shell element accounting for through-thickness deformation. Comput Methods Appl Mech Eng 2000; 189: 841-862. · Zbl 1011.74068 · doi:10.1016/S0045-7825(99)00348-5
[41] Kim, K, Liu, G, Han, S. A resultant 8-node solid-shell element for geometrically nonlinear analysis. Comput Mech 2005; 35: 315-331. · Zbl 1109.74360 · doi:10.1007/s00466-004-0606-9
[42] Belytschko, T, Wong, BL, Stolarski, H. Assumed strain stabilization procedure for the 9-node Lagrange shell element. Int J Numer Methods Eng 1989; 28: 385-414. · Zbl 0674.73054 · doi:10.1002/nme.1620280210
[43] Park, HC, Cho, C, Lee, SW. An efficient assumed strain element model with six DOF per node for geometrically non-linear shells. Int J Numer Methods Eng 1995; 38: 4101-4122. · Zbl 0843.73074 · doi:10.1002/nme.1620382403
[44] Lee, S, Kanok-Nukulchai, W. A nine-node assumed strain finite element for large-deformation analysis of laminated shells. Int J Numer Methods Eng 1998; 42: 777-798. · Zbl 0915.73059 · doi:10.1002/(SICI)1097-0207(19980715)42:5<777::AID-NME365>3.0.CO;2-P
[45] Baumann, M, Schweizerhof, K, Andrussow, S. An efficient mixed hybrid 4-node shell element with assumed stresses for membrane, bending and shear parts. Eng Comput 1994; 11: 69-80. · doi:10.1108/02644409410799164
[46] Sansour, C. A theory and finite element formulation of shells at finite deformations involving thickness change: circumventing the use of a rotation tensor. Arch Appl Mech 1995; 65: 194-216. · Zbl 0827.73044 · doi:10.1007/s004190050012
[47] Sze, K, Zheng, S-J. A hybrid stress nine-node degenerated shell element for geometric nonlinear analysis. Comput Mech 1999; 23: 448-456. · Zbl 0944.74075 · doi:10.1007/s004660050424
[48] Saleeb, A, Chang, T, Graf, W, et al. A hybrid/mixed model for non-linear shell analysis and its applications to large-rotation problems. Int J Numer Methods Eng 1990; 29: 407-446. · Zbl 0724.73227 · doi:10.1002/nme.1620290213
[49] Kollmann, F, Bergmann, V. Numerical analysis of viscoplastic axisymmetric shells based on a hybrid strain finite element. Comput Mech 1990; 7: 89-105. · Zbl 0718.73036 · doi:10.1007/BF00375924
[50] Bischoff, M, Ramm, E. On the physical significance of higher order kinematic and static variables in a three-dimensional shell formulation. Int J Solids Struct 2000; 37: 6933-6960. · Zbl 1003.74045 · doi:10.1016/S0020-7683(99)00321-2
[51] Simo, J-C, Armero, F. Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. Int J Numer Methods Eng 1992; 33: 1413-1449. · Zbl 0768.73082 · doi:10.1002/nme.1620330705
[52] Arciniega, R, Reddy, J. Tensor-based finite element formulation for geometrically nonlinear analysis of shell structures. Comput Methods Appl Mech Eng 2007; 196: 1048-1073. · Zbl 1120.74802 · doi:10.1016/j.cma.2006.08.014
[53] Arciniega, R, Reddy, J. Large deformation analysis of functionally graded shells. Int J Solids Struct 2007; 44: 2036-2052. · Zbl 1108.74038 · doi:10.1016/j.ijsolstr.2006.08.035
[54] Hughes, TJ, Cottrell, JA, Bazilevs, Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 2005; 194: 4135-4195. · Zbl 1151.74419 · doi:10.1016/j.cma.2004.10.008
[55] Cottrell, JA, Hughes, TJ, Bazilevs, Y. Isogeometric analysis: toward integration of CAD and FEA. John Wiley & Sons, 2009. · Zbl 1378.65009 · doi:10.1002/9780470749081
[56] Bazilevs, Y, Calo, VM, Cottrell, JA, et al. Isogeometric analysis using T-splines. Comput Methods Appl Mech Eng 2010; 199: 229-263. · Zbl 1227.74123 · doi:10.1016/j.cma.2009.02.036
[57] Norouzzadeh, A, Ansari, R, Rouhi, H. Pre-buckling responses of Timoshenko nanobeams based on the integral and differential models of nonlocal elasticity: an isogeometric approach, Appl Phys A 2017; 123: 330. · doi:10.1007/s00339-017-0887-4
[58] Thai, S, Thai, H-T, Vo, TP, et al. Size-dependant behaviour of functionally graded microplates based on the modified strain gradient elasticity theory and isogeometric analysis. Comput Struct 2017; 190: 219-241. · doi:10.1016/j.compstruc.2017.05.014
[59] Phung-Van, P, Lieu, QX, Nguyen-Xuan, H, et al. Size-dependent isogeometric analysis of functionally graded carbon nanotube-reinforced composite nanoplates. Compos Struct 2017; 166: 120-135. · doi:10.1016/j.compstruct.2017.01.049
[60] Nguyen, HX, Atroshchenko, E, Nguyen-Xuan, H, et al. Geometrically nonlinear isogeometric analysis of functionally graded microplates with the modified couple stress theory. Comput Struct 2017; 193: 110-127. · doi:10.1016/j.compstruc.2017.07.017
[61] Echter, R, Oesterle, B, Bischoff, M. A hierarchic family of isogeometric shell finite elements. Comput Methods Appl Mech Eng 2013; 254: 170-180. · Zbl 1297.74071 · doi:10.1016/j.cma.2012.10.018
[62] Norouzzadeh, A, Ansari, R. Nonlinear dynamic behavior of small-scale shell-type structures considering surface stress effects: an isogeometric analysis. Int J Non-Linear Mech 2018; 101: 174-186. · doi:10.1016/j.ijnonlinmec.2018.01.008
[63] Keller, HB . Numerical solution of bifurcation and nonlinear eigenvalue problems. In: Rabinowitz, PH , (ed.) Application of bifurcation theory. Academic Press, 1977, pp.359-384. · Zbl 0581.65043
[64] Mohamed, N, Eltaher, M, Mohamed, S, et al. Numerical analysis of nonlinear free and forced vibrations of buckled curved beams resting on nonlinear elastic foundations. Int J Non-Linear Mech 2018; 101: 157-173. · doi:10.1016/j.ijnonlinmec.2018.02.014
[65] Gholami, R, Ansari, R. Nonlinear harmonically excited vibration of third-order shear deformable functionally graded graphene platelet-reinforced composite rectangular plates. Eng Struc 2018; 156: 197-209. · doi:10.1016/j.engstruct.2017.11.019
[66] Smith, A. Inequalities between the constants of a linear micro-elastic solid. Int J Eng Sci 1968; 6: 65-74. · Zbl 0159.56903 · doi:10.1016/0020-7225(68)90020-7
[67] Eringen, AC. Microcontinuum field theories. I. Foundations and solids. Springer Science & Business Media, 2012. · Zbl 0953.74002
[68] Eremeyev, VA, Lebedev, LP, Cloud, MJ. Acceleration waves in the nonlinear micromorphic continuum. Mech Res Commun 2018; 93: 70-74. · doi:10.1016/j.mechrescom.2017.07.004
[69] Sze, K, Liu, X, Lo, S. Popular benchmark problems for geometric nonlinear analysis of shells. Finite Elem Anal Des 2004; 40: 1551-1569. · doi:10.1016/j.finel.2003.11.001
[70] Massin, P, Al Mikdad, M. Nine node and seven node thick shell elements with large displacements and rotations. Comput Struct 2002; 80: 835-847. · doi:10.1016/S0045-7949(02)00050-0
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