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Computational finite element model for surface wrinkling of shells on soft substrates. (English) Zbl 07264491
Summary: We provide a robust finite element formulation for quantitative prediction of surface wrinkling of pressurized elastic shells on soft substrates. Our theory is build on three basic assumptions which involve thin shell kinematics, the approximation of the substrate response by a Winkler foundation and a model order reduction of the displacement field. Our element keeps all the nonlinear terms of the reduced model. The proposed formulation does not require any perturbations, either in the initial geometry or in the load, to incite the transition from fundamental to secondary equilibrium path for the considered set of shells, due to inherent asymmetric imperfections in the mesh. Numerical simulations using the derived element and an advanced path-following method on full spheres, hemispheres and spheroids show a very good quantitative agreement with theoretical predictions and experiments on the characteristic wavelength of the pattern as well as the qualitative depiction of the pattern evolution.
MSC:
68W Algorithms in computer science
74 Mechanics of deformable solids
74S Numerical and other methods in solid mechanics
Software:
AceFEM
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[1] Wieghardt, K., Über den balken auf nachgiebiger unterlage, Z angew Math Mech, 2, 3, 165-184 (1922) · JFM 48.0930.04
[2] Biot, M. A., Bending of an infinite beam on an elastic foundation, J Appl Mech, 4, 1-7 (1937)
[3] Reissner, M. E., On the theory of beams resting on a yielding foundation, Proc Natl Acad Sci USA, 23, 6, 328-333 (1937) · JFM 63.0748.04
[4] Gough, G. S.; Elam, C. F.; Tipper, G. H.; De Bruyne, N. A., The stabilisation of a thin sheet by a continuous supporting medium, Aeronaut J, 44, 349, 12-43 (1940)
[5] Allen, H. G., Analysis and design of structural sandwich panels (1969), Pergamon Press: Pergamon Press New York
[6] Stafford, C. M.; Harrison, C.; Beers, K. L.; Karim, A.; Amis, E. J.; VanLandingham, M. R.; Kim, H. C.; Volksen, W.; Miller, R. D.; Simonyi, E. E., A buckling-based metrology for measuring the elastic moduli of polymeric thin films, Nat Mater, 3, 545-550 (2004)
[8] Chan, E. P.; Crosby, A. J., Fabricating microlens arrays by surface wrinkling, Adv Mater, 18, 24, 3238-3242 (2006)
[9] Chan, E. P.; Smith, E. J.; Hayward, R. C.; Crosby, A. J., Surface wrinkles for smart adhesion, Adv Mater, 20, 4, 711-716 (2008)
[10] Yang, S.; Krishnacharya, K.; Lin, P. C., Harnessing surface wrinkle patterns in soft matter, Adv Funct Mater, 20, 16, 2550-2564 (2010)
[11] David, C. S.; Crosby, A. J., Mechanics of wrinkled surface adhesion, Soft Matter, 7, 11, 5373-5381 (2011)
[12] Chung, J. Y.; Youngblood, J. P.; Stafford, C. M., Anisotropic wetting on tunable micro-wrinkled surfaces, Soft Matter, 3, 9, 1163-1169 (2007)
[13] Terwagne, D.; Brojan, M.; Reis, P. M., Smart morphable surfaces for aerodynamic drag control, Adv Mater, 26, 38, 6608-6611 (2014)
[14] Huang, Z.; Hong, W.; Suo, Z., Nonlinear analyses of wrinkles in a film bonded to a compliant substrate, J Mech Phys Solids, 53, 9, 2101-2118 (2005) · Zbl 1176.74116
[15] Audoly, B.; Boudaoud, A., Buckling of a stiff film bound to a compliant substrate-part i: formulation, linear stability of cylindrical patterns, secondary bifurcations, J Mech Phys Solids, 56, 7, 2401-2421 (2008) · Zbl 1171.74349
[16] Cai, S.; Breid, D.; Crosby, A. J.; Suo, Z.; Hutchinson, J. W., Periodic patterns and energy states of buckled films on compliant substrates, J Mech Phys Solids, 59, 5, 1094-1114 (2011) · Zbl 1270.74126
[17] Cao, G.; Chen, X.; Li, C.; Ji, A.; Cao, Z., Self-assembled triangular and labyrinth buckling patterns of thin films on spherical substrates, Phys Rev Lett, 100, 3, 036102(4) (2008)
[19] Breid, D.; Crosby, A. J., Curvature-controlled wrinkle morphologies, Soft Matter, 9, 13, 3624-3630 (2013)
[20] Yin, J.; Han, X.; Cao, Y.; Lu, C., Surface wrinkling on polydimethylsiloxane microspheres via wet surface chemical oxidation, Sci Rep UK, 4, 5710(8) (2014)
[21] Li, B.; Fei, J.; Cao, J. P.; Feng, X. Q.; Gao, H., Surface wrinkling patterns on a core-shell soft sphere, Phys Rev Lett, 106, 23, 234301(4) (2011)
[22] Stoop, N.; Lagrange, R.; Terwagne, D.; Reis, P. M.; Dunkel, J., Curvature-induced symmetry breaking determines elastic surface patterns, Nat Mater, 14, 337-342 (2015)
[23] Ciarlet, P. G., Mathematical elasticity, 3 (2000)
[24] Xu, F.; Potier-Ferry, M., On axisymmetric/diamond-like mode transitions in axially compressed core-shell cylinders, J Mech Phys Solids, 94, 68-87 (2016)
[25] Damil, N.; Potier-Ferry, M., A new method to compute perturbed bifurcations: application to the buckling of imperfect elastic structures, Int J Eng Sci, 28, 9, 943-957 (1990) · Zbl 0721.73018
[26] Zhano, Y.; Cao, Y.; Feng, X. Q.; Ma, K., Axial compression-induced wrinkles on a core-shell soft cylinder: theoretical analysis, simulations and experiments, J Mech Phys Solids, 73, 212-227 (2014) · Zbl 1349.74255
[27] Lagrange, R.; Jimenez, F. L.; Terwagne, D.; Brojan, M.; Reis, P. M., From wrinkling to global buckling of a ring on a curved substrate, J Mech Phys Solids, 89, 77-95 (2016)
[28] Bohinc, U.; Brank, B.; Ibrahimbegovic, A., Discretization error for the discrete kirchoff plate finite element approximation, Comput Method Appl M, 269, 415-436 (2014) · Zbl 1296.65154
[29] Bohinc, U.; Ibrahimbegovic, A.; Brank, B., Model adaptivity for finite element analysis of thin or thick plates based on equilibrated boundary stress resultants, Eng Comput (Swansea), 26, 1/2, 69-99 (2009) · Zbl 1257.74147
[30] Yin, J.; Cao, Z.; Li, C.; Sheinman, I.; Chen, X., Stress-driven buckling patterns in spheroidal core/shell structures, P Natl Acad Sci USA, 105, 49, 19132-19135 (2008)
[31] Yin, J.; Chen, X.; I. Sheinman, I., Anisotropic buckling patterns in spheroidal film/substrate systems and their implications in some natural and biological systems, J Mech Phys Solids, 57, 1470-1484 (2009) · Zbl 1371.74205
[32] Brojan, M.; Terwagne, D.; Lagrange, R.; Reis, P. M., Wrinkling crystallography on spherical surfaces, P Natl Acad Sci USA, 112, 1, 14-19 (2015)
[33] Erber, T.; Hockney, G. M., Equilibrium configurations of n equal charges on a sphere, J Phys A Math Gen, 24, 23, 1369-1377 (1991)
[34] Betsch, P.; Gruttmann, F.; Stein, E., A 4-node finite shell element for the implementation of general hyperelastic 3D-elasticity at finite strains, Comput Meth Appl Mech Eng, 130, 57-79 (1996) · Zbl 0861.73068
[35] Brank, B.; Korelc, J.; Ibrahimbegović, A., Nonlinear shell problem formulation accounting for through-the-thickness stretching and its finite element implementation, Comp Struct, 80, 699-717 (2002)
[36] Brank, B., Nonlinear shell models with seven kinematic parameters, Comput Meth Appl Mech Eng, 194, 2336-2362 (2005) · Zbl 1082.74050
[37] Simo, J. C.; Fox, D. D., On a stress resultant geometrically exact shell model. part i: formulation and optimal parametrization, Comput Meth Appl Mech Eng, 72, 267-304 (1989) · Zbl 0692.73062
[38] Brank, B.; Peri, D.; Damjanić, F. B., On large deformations of thin Elasto-plastic shells: implementation of a finite rotation model for quadrilateral shell element, Int J Numer Methods Eng, 40, 689-726 (1997) · Zbl 0892.73055
[39] Stanić, A.; Brank, B.; Korelc, J., On path-following methods for structural failure problems, Comput Mech, 58, 281-306 (2016) · Zbl 1398.74407
[40] Naghdi, P. M., The theory of shell and plates, mechanics of solids, vol. II, linear theories of elasticity and thermoelasticity, (Truesdell, C., Linear and nonlinear theories of rods, plates and shells (1973), Springer-Verlag: Springer-Verlag Berlin, Heidelberg), 426-640
[41] Xu, F.; Potier-Ferry, M., Quantitative predictions of diverse wrinkling patterns in film/substrate systems, Sci Rep, 7, 18081 (2017)
[42] Jia, F.; Li, B.; Feng, X. Q., Wrinkling pattern evolution of cylindrical biological tissues with differential Growts, Phys Rev, E91, 012403 (2015)
[43] Wu, J.; Chen, X., Buckling patterns of conical thin film/substrate systems, J Phys D Appl Phys, 46, 155306 (2013)
[44] Simo, J. C.; Hughes, T. J.R., On the variational foundations of assumed strain methods, J Appl Mech, 53, 1, 51-54 (1986) · Zbl 0592.73019
[45] Korelc, J.; Wriggers, P., Automation of finite element methods (2016), Springer Inter-national Publishing · Zbl 1367.74001
[46] Mathematica, Version 11.3 (2018), Champaign: Champaign IL
[47] Brank, B.; Damjanić, F. B.; Peri, D., On implementation of a nonlinear four node shell finite element for thin multilayered elastic shells, Comput Mech, 16, 5, 341-359 (1995) · Zbl 0848.73060
[48] Riks, E., Buckling analysis of elastic structures: a computational approach, Adv Appl Mech, 24, 1-76 (1997) · Zbl 0889.73032
[49] Kegl, M.; Brank, B.; Harl, B.; Oblak, M. M., Efficient handling of stability problems in shell optimization by asymmetric worst-case shape imperfection, Int J Numer Meth Eng, 73, 1197-1216 (2008) · Zbl 1159.74028
[50] Dujc, J.; Brank, B., Stress resultant plasticity for shells revisited, Comput Meth Appl Mech Eng, 247/248, 146-165 (2012) · Zbl 1352.74172
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