An efficient model for large deformations of nonlinear viscoelastic elastomeric membranes using predictor-corrector algorithm

Abstract

Many engineering systems incorporate viscoelastic membranes of different geometries and boundary conditions experiencing large deformations. This paper presents a formulation based on the theory of nonlinear viscoelasticity. First, the kinematics of membrane deformation is expressed in three-dimensional space, and then the viscoelastic formulation for membranes is obtained based on the multiplicative decomposition of the tensor of deformation gradient. Also, the right Cauchy−Green viscoelastic tensor is considered as an internal variable. To solve the integration of evolution equation, a predictor-corrector method is used. Finally, due to the nonlinearity of the equations governing the problem, a nonlinear finite element formulation is derived. To check the effectiveness of the obtained formulation, several problems are studied. The comparisons show that the results of this formulation are in good agreement with the analytical and experimental results in the literature. It is shown that the current simplified viscoelastic model can successfully predict the results in the literature with more complicated viscoelastic models. Moreover, it is proven that the present model can predict the experimental data with just four material parameters, while the previous models should employ 12 material parameters. Therefore, the model presented in this paper is capable of predicting the experimental results more accurately with fewer material parameters.

Graphical abstract

Keywords

elastomers / rubber-like materials / non-linear viscoelasticity theory / membranes / finite element analysis

References

[1] Demir Ç, Civalek Ö. On the analysis of microbeams. International Journal of Engineering Science, 2017, 121: 14–33

[2] Abouelregal A E, Ersoy H, Civalek Ö. Solution of Moore–Gibson–Thompson equation of an unbounded medium with a cylindrical hole. Mathematics, 2021, 9(13): 1536

[3] Adiga S P, Jin C, Curtiss L A, Monteiro-Riviere N A, Narayan R J. Nanoporous membranes for medical and biological applications. Wiley Interdisciplinary Reviews. Nanomedicine and Nanobiotechnology, 2009, 1(5): 568–581

[4] Radu E R, Voicu S I, Thakur V K. Polymeric membranes for biomedical applications. Polymers, 2023, 15(3): 619

[5] Firouzi N, Misra A. New insight into large deformation analysis of stretch-based and invariant-based rubber-like hyperelastic elastomers. Thin-walled Structures, 2023, 192: 111162

[6] Cooley C G, Lowe R L. Nonlinear vibration of dielectric elastomer membranes with axial inertia effects. International Journal of Mechanical Sciences, 2023, 248: 108205

[7] Zhao B, Hu J, Chen W, Chen J, Jing Z. A nonlinear uniaxial stress-strain constitutive model for viscoelastic membrane materials. Polymer Testing, 2020, 90: 106633

[8] Li E H, Li Y Z, Xie J Y, Sun Y H, Yang L Z, Ning X W. A fuzzy coordination control of a water membrane evaporator cooling system for aerospace electronics. Applied Thermal Engineering, 2021, 191: 116872

[9] Huang H, Li X, Xue S, Luo Y, Shi D, Hou X, Liu Y, Li N. Performance and measurement devices for membrane buildings in civil Engineering: A review. Applied Sciences, 2022, 12(17): 8648

[10] Tian L, Chen F, Zhu L, Yu T. Large deformation of square plates under pulse loading by combined saturation analysis and membrane factor methods. International Journal of Impact Engineering, 2020, 140: 103546

[11] Hou X, Mao Y, Zhang R, Fang D. Super-flexible polyimide nanofiber cross-linked polyimide aerogel membranes for high efficient flexible thermal protection. Chemical Engineering Journal, 2021, 417: 129341

[12] Khmelinskii I, Makarov V I. On the effects of mechanical stress of biological membranes in modeling of swelling dynamics of biological systems. Scientific Reports, 2020, 10(1): 8395

[13] Yin Y, Pu D, Xiong J. Analysis of the comprehensive tensile relationship in electrospun silk fibroin/polycaprolactone nanofiber membranes. Membranes, 2017, 7(4): 67

[14] Guerra N B, Pegorin G S A, Boratto M H, de Barros N R, de Oliveira Graeff C F, Herculano R D. Biomedical applications of natural rubber latex from the rubber tree Hevea brasiliensis. Materials Science and Engineering C, 2021, 126: 112126

[15] Lima L R, Caetano G F, Soares V O, Santos R J, Malmonge J A, Silva M J, Yarin A L. Evaluation of tensile, thermal, and biological properties of natural rubber-based biocomposite with biosilicate and 45SS-K bioglass. Journal of Applied Polymer Science, 2023, 140(22): e53894

[16] Tamadapu G, Dasgupta A. Finite inflation of a hyperelastic toroidal membrane over a cylindrical rim. International Journal of Solids and Structures, 2014, 51(2): 430–439

[17] Farsad K, de Camilli P. Mechanisms of membrane deformation. Current Opinion in Cell Biology, 2003, 15(4): 372–381

[18] Foster B, Knobloch E. Elastic fingering in a rotating hele-shaw cell. Physical Review E, 2023, 107(6): 065104

[19] Véron A R, Martins A F. Tensorial form of leslie-ericksen equations and applications. Molecular Crystals and Liquid Crystals, 2009, 508(1): 309–336

[20] MarkmanMSimanovskyL. Method of making a parabolic-cylindrical concentrator and device for its implementation. 1982

[21] Gruttmann F, Taylor R. Theory and finite element formulation of rubberlike membrane shells using principal stretches. International Journal for Numerical Methods in Engineering, 1992, 35(5): 1111–1126

[22] Feng X, Huang R Y. Liquid separation by membrane pervaporation: A Review. Industrial & Engineering Chemistry Research, 1997, 36(4): 1048–1066

[23] Cheng T, Dai C, Gan R Z. Viscoelastic properties of human tympanic membrane. Annals of Biomedical Engineering, 2007, 35(2): 305–314

[24] Murayama T, Yoshino M, Hirata T. Three-dimensional lattice boltzmann simulation of two-phase flow containing a deformable body with a viscoelastic membrane. Communications in Computational Physics, 2011, 9(5): 1397–1413

[25] LauberMWeymouthG DLimbertG. Immersed-boundary fluid-structure interaction of membranes and shells. Journal of Physics Conference Series, 2023, 2647, 052002

[26] Mierunalan S, Dassanayake S P, Mallikarachchi H M Y C, Upadhyay S H. Simulation of ultra-thin membranes with creases. International Journal of Mechanics and Materials in Design, 2022, 19(1): 73–94

[27] Zhang X, Ma Z Q, Wu Y, Liu J. Response of mechanical properties of polyvinyl chloride geomembrane to ambient temperature in axial tension. Applied Sciences, 2021, 11(22): 10864

[28] Sarkezi-SelskyPRastedtMLatzAJahnkeT. Lattice-boltzmann simulation of two-phase flows in the GDL and MPL of polymer electrolyte fuel cells. In: EFCF 2019 Proceedings (PEY). Lucerne: EFCF, 2019

[29] van Nguyen C. Numerical investigation of circle defining curve for two-dimensional problem with general boundaries using the scaled boundary finite element method. Frontiers of Structural and Civil Engineering, 2019, 13(1): 92–102

[30] Bhandarkar S D, Betcher J, Smith R, Lairson B M, Ayers T. Constitutive models for the viscoelastic behavior of polyimide membranes at room and deep cryogenic temperatures. Fusion Science and Technology, 2016, 70(2): 332–340

[31] Ghosh A, Both A K, Cheung C L. Characterization of three-dimensional fractional viscoelastic models through complex modulus analysis and polar decomposition. Physics of Fluids, 2022, 34(7): 077115

[32] Geith M A, Eckmann J D, Haspinger D C, Agrafiotis E, Maier D, Szabo P, Sommer G, Schratzenstaller T G, Holzapfel G A. Experimental and mathematical characterization of coronary polyamide-12 balloon catheter membranes. PLoS One, 2020, 15(6): e0234340

[33] Cai R, Jin Y, Rabczuk T, Zhuang X, Djafari-Rouhani B. Propagation and attenuation of rayleigh and pseudo surface waves in viscoelastic metamaterials. Journal of Applied Physics, 2021, 129(12): 124903

[34] Firouzi N. Mechanics of nonlinear visco-hyperelastic-hysteresis membranes. International Journal of Non-linear Mechanics, 2022, 147: 104231

[35] Shafei E, Faroughi S, Rabczuk T. Nonlinear transient vibration of viscoelastic plates: A NURBS-based isogeometric HSDT approach. Computers & Mathematics with Applications, 2021, 84: 1–15

[36] Zhou X, Qiu D, Peng L, Lai X. Modeling the viscoelastic-viscoplastic behavior of glassy polymer membrane with consideration of non-uniform sub-chains in entangled networks. International Journal of Solids and Structures, 2023, 282: 112469

[37] Brighenti R, Rabczuk T, Zhuang X. Phase field approach for simulating failure of viscoelastic elastomers. European Journal of Mechanics. A, Solids, 2021, 85: 104092

[38] Khaniki H B, Ghayesh M H, Chin R, Amabili M. A review on the nonlinear dynamics of hyperelastic structures. Nonlinear Dynamics, 2022, 110(2): 963–994

[39] Khaniki H B, Ghayesh M H, Chin R, Amabili M. Hyperelastic structures: A review on the mechanics and biomechanics. International Journal of Non-linear Mechanics, 2023, 148: 104275

[40] AmabiliM. Nonlinear mechanics of shells and plates: Composite, Soft and Biological Materials. New York, NY: Cambridge University Press, 2018

[41] Amabili M. Derivation of nonlinear damping from viscoelasticity in case of nonlinear vibrations. Nonlinear Dynamics, 2019, 97(3): 1785–1797

[42] Khorasani M, Soleimani-Javid Z, Arshid E, Lampani L, Civalek Ö. Thermo-elastic buckling of honeycomb micro plates integrated with FG-GNPs reinforced Epoxy skins with stretching effect. Composite Structures, 2021, 258: 113430

[43] Shafei E, Faroughi S, Rabczuk T. Multi-patch NURBS formulation for anisotropic variable angle tow composite plates. Composite Structures, 2020, 241: 111964

[44] Sobhani E, Masoodi A R, Civalek Ö, Ahmadi-Pari A R. Agglomerated impact of CNT vs. GNP nanofillers on hybridization of polymer matrix for vibration of coupled hemispherical-conical-conical shells. Aerospace Science and Technology, 2022, 120: 107257

[45] Mercan K, Baltacıoglu A K, Civalek Ö. Free vibration of laminated and FGM/CNT composites annular thick plates with shear deformation by discrete singular convolution method. Composite Structures, 2018, 186: 139–153

[46] Franchini G, Breslavsky I D, Holzapfel G A, Amabili M. Viscoelastic characterization of human descending thoracic aortas uncer cyclic load. Acta Biomaterialia, 2021, 130: 291–307

[47] Le Tallec P, Rahier C, Kaiss A. C, Rahier, A Kaiss, Three-dimensional incompressible viscoelasticity in large strains: Formulation and numerical approximation. Computer Methods in Applied Mechanics and Engineering, 1993, 109(3–4): 233–258

[48] Reese S, Govindjee S. A theory of finite viscoelasticity and numerical aspects. International Journal of Solids and Structures, 1998, 35(26–27): 3455–3482

[49] HolzapfelG A. Nonlinear Solid Mechanics, A Continuum Approach for Engineering. New York: Wiley, 2000

[50] Firouzi N, Żur K K, Amabili M, Rabczuk T. On the time-dependent mechanics of membranes via the nonlinear finite element method. Computer Methods in Applied Mechanics and Engineering, 2023, 407: 115903

[51] ReddyJ N. An Introduction to Nonlinear Finite Element Analysis. Oxford: Oxford University Press, 2015

[52] Kroon M. A constitutive framework for modelling thin incompressible viscoelastic materials under plane stress in the finite strain regime. Mechanics of Time-Dependent Materials, 2011, 15(4): 389–406

[53] Jridi N, Arfaoui M, Hamdi A, Salvia A, Bareille O, Ichchou M, Ben Abdallah J. Separable finite viscoelasticity: Integral-based models vs. experiments. Mechanics of Time-Dependent Materials, 2019, 23(3): 295–325

[54] Bernstein B, Kearsley E, Zapas L. A study of stress relaxation with finite strain. Transactions of the Society of Rheology, 1963, 7(1): 391–410

[55] Christensen R M. A nonlinear theory of viscoelasticity for application to elastomers. Journal of Applied Mechanics, 1980, 47(4): 762–768

[56] Simo J C. On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects. Computer Methods in Applied Mechanics and Engineering, 1987, 60(2): 153–173

[57] Fosdick R, Yu J H. Thermodynamics stability and non-linear oscillations of viscoelastic solids-II history type solids. International Journal of Non-linear Mechanics, 1998, 33(1): 165–188

[58] Patil A, Nordmark A, Eriksson A. Free and constrained inflation of a pre-stretched cylindrical membrane. Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences, 2015, 470: 20140282

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