پیاده‌سازی عددی مدل ساختاری هایپرالاستیک-ویسکوپلاستیک برای شبیه‌سازی رفتار مکانیکی پلیمرهای الاستومر گرمانرم با ساختار دو فازی

نوع مقاله : مقاله پژوهشی

نویسندگان

1 دانشکده مهندسی مکانیک، دانشگاه صنعتی اصفهان، اصفهان ، ایران

2 دانشکده مهندسی مکانیک، دانشگاه صنعتی اصفهان ، اصفهان ، ایران

چکیده

کاربرد گسترده‌ی الاستومرهای گرمانرم پلی یورتان (TPU) به‌­ویژه در صنایع خودروسازی و پزشکی، نیاز به توسعه‌ی مدل‌های عددی دقیق جهت پیش‌بینی پاسخ این مواد دو­فازی در مواجهه با بارگذاری‌های مختلف را به‌عنوان یک ضرورت مطرح کرده است. هدف از این مطالعه، شناسایی رفتار تغییر شکل الاستومرهای گرمانرم پلی یورتان­‌ها از طریق انجام آزمایش‌های تجربی و تطبیق یک مدل ماده مناسب با آن و سپس پیاده­‌سازی مدل ساختاری در نرم‌افزار اجزای محدود آباکوس به­روش حل­گر صریح از طریق زیرروال VUMAT است. چارچوب مادی پدیده شناختی شامل یک مؤلفه تعادلی برای توصیف پاسخ غیراتلافی هایپرالاستیک فاز نرم بی­شکل و یک مؤلفه الاستیک-ویسکوپلاستیک برای تغییرات در فاز سخت بلورین TPU است. در این چارچوب، فاز نرم به­‌صورت یک عنصر فنری هایپرالاستیک مدل هشت زنجیره­ای آرودا-بویس و فاز سخت به­‌صورت ترکیبی از یک فنر الاستیک خطی، میراگر ویسکوپلاستیک غیرخطی مدل ری- اِیرینگ اصلاح شده و یک عنصر اصطکاکی مدل­‌سازی شده است. صحت پیاده‌­سازی مدل با استفاده از آزمون‌­های فشار تک‌­محوره به‌­صورت یکنواخت و چرخ‌ه­ای در نرخ‌ کرنش‌­های مختلف در شرایط هم­دما مورد بررسی قرار گرفت. نتایج نشان دادند که با افزایش درصد فاز سخت، TPU رفتار­های پلاستیکی قوی­‌تر، با اتلاف انرژی بیشتر و بازیابی شکل کمتر نشان می­‌دهد و با اعمال کرنش فشاری یک، پس از هر چرخه بارگذاری- باربرداری، کرنش پسماند حاصل، حتی تا چند هفته پس از پایان آزمایش به­‌طور کامل بازیابی نمی‌­شود. تطابق بین مدل و داده‌‌های آزمایشگاهی برای TPU­های نماینده، نه­‌تنها بینش فیزیکی از مکانیزم‌­های تغییر شکل مواد TPU ارائه داد بلکه قابلیت­‌های شبیه­‌‌سازی پدیده­‌های اصلی از جمله رفتار­های الاستیک و ویسکو پلاستیک وابسته به زمان و رژیم نرم‌­شوندگی را که وابسته به درصد فاز سخت TPU هستند، به­‌خوبی توصیف کرد.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

Numerical Implementation of a Hyperelastic-Viscoplastic Constitutive Model to Simulate the Mechanical Behaviour of Two Segmented Thermoplastic Elastomer Polymers

نویسندگان [English]

  • Ehsan Ahmadi 1
  • Mohammad Rea Forouan 1
  • Peiman Mosaddegh 1
  • Manizheh Aghaei 2
1 Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, Iran
2 Department of Mechanical Engineering, Isfahan university of Technology, Isfahan, Iran
چکیده [English]

Thermoplastic polyurethane (TPU) elastomers are widely used in industries such as automotive and medical due to their unique mechanical properties. However, their complex deformation behavior, resulting from the interaction between soft amorphous and hard crystalline phases, necessitates accurate numerical modeling for reliable prediction under various loading conditions. This study aims to characterize the deformation behavior of TPU through experimental testing and the implementation of a suitable constitutive model. A phenomenological material framework was developed and implemented in the ABAQUS/Explicit finite element software via a user-defined VUMAT subroutine. The model consists of an equilibrium hyperelastic component representing the soft phase, based on the Arruda-Boyce eight-chain model, and a elastic-viscoplastic- component for the hard phase. The latter is formulated using a linear elastic spring, a nonlinear viscous damper based on a modified Ree-Eyring model, and a frictional element. The model parameters were calibrated using a series of uniaxial compression tests under monotonic and cyclic loading at various strain rates at room temperature. The results showed that, as the fraction of the hard component increased, TPU exhibits stronger plastic behavior, with more energy dissipation and less shape recovery. Moreover, when subjected to a strain of -1.0 after each loading-unloading cycle, it exhibits a residual strain that is not fully recovered even several weeks after the end of the test. Furthermore, the model demonstrates the capability to simulate TPU response under other loading scenarios such as tension and stress relaxation. This work offers physical insight into the deformation mechanisms of TPU and provides a practical modeling tool for its complex elastomeric-plastic behavior.

کلیدواژه‌ها [English]

  • Thermoplastic polyurethane elastomer
  • Hyperelastic–viscoplastic behavior
  • Arruda–Boyce model
  • Modified Ree–Eyring viscoplastic flow
  • Uniaxial compression and tension tests
  • Stress relaxation test

مراجع

  1.  

    1. Slater, C., Davis, C., and Strangwood, M. Compression set of thermoplastic polyurethane under different thermal-mechanical-moisture conditions, Polymer Degradation and Stability, 2011; 96 (12): 2139-2144. https://doi.org/10.1016/j.polymdegradstab.2011.09.012.
    2. Holden, G., Thermoplastic elastomers (Applied Plastics Engineering Handbook). William Andrew Publishing, 2011. https://doi.org/10.1016/B978-0-323-88667-3.00020-5
    3. Scetta, G., Selles, N., Heuillet, P., Ciccotti, M., and Creton, C. Cyclic fatigue failure of TPU using a crack propagation approach, Polymer Testing, 2021; 97: 107140.

        https://doi.org/10.1016/j.polymertesting.2021.107140.

    1. Petrovic, Z. S. and Ferguson, J. Ployurethane Elastomers, Prog. Polym. Sci, 1991; 16(5): 695-836. https://doi.org/10.1016/0079-6700(91)90011-9
    2. Christensona, E. M., Anderson, J. M., Hiltner, A., and Baer, E. Relationship between nanoscale deformation processes and elastic behavior of polyurethane elastomers, Polymer, 2005; 46(25): 11744–11754. https://doi.org/10.1016/j.polymer.2005.08.083
    3. Qi, H. J., Boyce, M. C. Stress-Strain Behavior of Thermoplastic Polyurethane , Mechanics of Materials, 2005; 37(8): 817-839.

          https://doi.org/10.1016/j.mechmat.2004.08.001.

    1. Ahmadi, E., Forouzan, M. R., and Mosaddegh, P. Permanent Deformation of Thermoplastic Polyurethane in the Solid-State Rolling Process, Advances in Polymer Technology, 2025; 2025(1): 8811192. https://doi.org/10.1155/adv/8811192.
    2. LeMonte, B. Growing Interest In TPUs For Automobile Design, Materials and Design, Vol. 19, pp. 69-72, 1998.

          https://doi.org/10.1016/S0261-3069(98)00005-3.

    1. Yi, J., Boyce, M. C., Lee, G. F., and Balizer, E. Large deformation rate-dependent stress–strain behavior of polyurea and polyurethanes, Polymer, 2006; 47(1): 319–329.

           https://doi.org/10.1016/j.polymer.2005.10.107.

    1. Sarva, S. S., Stephanie, D., Boyce, M. C., and Chen, W. Stress strain behavior of a polyurea and a polyurethane from low to high strain rates, 2007; 48(5): 2208-2213.

          https://doi.org/10.1016/j.polymer.2007.02.058.

    1. Prisacariua, C., Buckley, C. P., and Caraculacu, A. A. Mechanical response of dibenzyl-based polyurethanes with diol chain extension, Polymer, 2005; 46(11): 3884–3894.

          https://doi.org/10.1016/j.polymer.2005.03.046.

    1. Russo, R. and Thomas, E. L. Phase separation in linear and cross-linked polyurethanes, J. Macromol Sci Phy B, 1983; . 22(4): 553-575.

            https://doi.org/10.1080/00222348308224776.

    1. O’sickey, M. J., Lawrey, B. D., and wilkes, G. L. Structure–Property Relationships of Poly (urethane urea)s with Ultra-low Monol Content Poly(propylene glycol) Soft Segments. I. Influence of Soft Segment Molecular Weight and Hard Segment Content, Journal of Applied Polymer Science, 200; 84(2): 229–243. https://doi.org/10.1002/app.10168.
    2. Bartolomé, L., Aurrekoetxea, J., A.Urchegui, M., and Tato, W. The influences of deformation state and experimental conditions on inelastic behaviour of an extruded thermoplastic polyurethane elastomer, Materials and Design, 2013; 49, 974–980.

           https://doi.org/10.1016/j.matdes.2013.02.055.

    1. Miao, Y., He, H., and Li, Z., Strain hardening behaviors and mechanisms of polyurethane under various strain rate loading, Polym Eng Sci., 2020; 60(5): 1083-1092. https://doi.org/10.1002/pen.25364
    2. Scetta, G., Selles, N., Heuillet, P., Ciccotti, M., and Creton, C. Cyclic fatigue failure of TPU using a crack propagation approach, Polymer Testing, 2021; 97: 107140.

         https://doi.org/10.1016/j.polymertesting.2021.107140

    1. Frick, A., Borm, M., Kaoud, N., Kolodziej, J., and Neudeck, J. Microstructure and thermomechanical properties relationship of segmented thermoplastic polyurethane (TPU), AIP Conf. Proc.,2014; 1593(1): 520–525. https://doi.org/10.1063/1.4873835.
    2. Cho, H., Rinaldi, G, R., and Boyce, M. C. Constitutive modeling of the rate-dependent resilient and dissipative large deformation behavior of a segmented copolymer polyurea, Soft Matter, 2013; 9: 6319-6330. https://doi.org/10.1039/c3sm27125k.
    3. Bergstrom, J. and Boyce, M. Constitutive modeling of the large strain timedependent behavior of elastomers, J. Mech. Phys. Solids 1998; 46(5): 931-954. https://doi.org/10.1016/S0022-5096(97)00075-6.
    4. Anand, L., Ames, N. M., Srivastava, V., and Chester, S. A. A thermo-mechanically coupled theory for large deformations of amorphous polymers. Part I: formulation, 2009; 25(8): 1474-1494.

            https://doi.org/10.1016/j.ijplas.2008.11.004.

    1. Boyce, M. C., Parks, D. M., and Argon, A. S. Large inelastic deformation of glassy polymers. Part I: rate dependent constitutive model, Mech. Mater., 1988; 7(1): 15-33, 1988.

            https://doi.org/10.1016/0167-6636(88)90003-8.

    1. Mulliken, A. and Boyce, M. Mechanics of the rate-dependent elasticeplastic deformation of glassy polymers from low to high strain rates, Int. J. Solids Struct., 2006; 43(5): 1331-1356.

           https://doi.org/10.1016/j.ijsolstr.2005.04.016.

    1. Wang, Y. and Arruda, E. M. Constitutive Modeling of a Thermoplastic Olefin Over a Broad Range of Strain Rates, Journal of Engineering Materials and Technology, 2006; 128(4): 551-558.

           https://doi.org/10.1115/1.2349501.

    1. Yu, Z. Q., Maa, Q., Su, X. M., Lai, X. M., and Tibbenham, P. C. Constitutive modeling for large deformation behavior of thermoplastic olefin, Materials and Design, 2010; 31(4):1881–1886, 2010.

            https://doi.org/10.1016/j.matdes.2009.10.059.

    1. Parenteau, T., Bertevas, E., Ausias, G., Stocek, R., Grohens, Y., and Pilvin, P. Characterisation and micromechanical modelling of the elasto-viscoplastic behavior of thermoplastic elastomers, Mechanics of Materials,2014; 71: 114–125, 2014.

           https://doi.org/10.1016/j.mechmat.2013.06.010.

    1. Boyce, C., Kearb, M., Socratea, K., and Shawa, K. Deformation of thermoplastic vulcanizates, Journal of the Mechanics and Physics of Solids, 2001; 49(5): 1073 -1098.

           https://doi.org/10.1016/S0022-5096(00)00066-1.

    1. Somarathna, H. M. C. C., Raman, S. N., Mohotti, D., Mutalib, A. A., and Badri, K. H. Hyper-viscoelastic constitutive models for predicting the material behavior of polyurethane under varying strain rates and uniaxial tensile loading, Construction and Building Materials, 2020; 236(10): 117417.

           https://doi.org/10.1016/j.conbuildmat.2019.117417.

    1. Mohotti, D., Ali, M., Ngo, T., Lu, J., and Mendis, P. Strain rate dependent constitutive model for predicting the material behaviour of polyurea under high strain rate tensile loading, Materials and Design, 2014; 53: 830-837.

           https://doi.org/10.1016/j.matdes.2013.07.020.

    1. Shim, J. and Mohr, D. Rate dependent finite strain constitutive model of polyurea, International Journal of Plasticity, 2011; 27(6): 868–886.

           https://doi.org/10.1016/j.ijplas.2010.10.001.

    1. Nikoukalam, M. T. and Sideris, P. Experimental characterization and constitutive modeling of polyurethanes for structural applications, accounting for damage, hysteresis, loading rate and long term effects, Engineering Structures, 2019; 198: 109462.

           https://doi.org/10.1016/j.engstruct.2019.109462.

    1. Bai, Y., Liu, C., Huang, G., Li, W., and Feng, S. A Hyper-Viscoelastic Constitutive Model for Polyurea under Uniaxial Compressive Loading, Polymer, 2016; 8(4): 133.

           https://doi.org/10.3390/polym8040133.

    1. Cho, H., Mayer, S., Poselt, E., Susoff, M., Veld, P., Rutledge, G. C., Boyce, M. C. Deformation mechanisms of thermoplastic elastomers: Stress-strain behavior and constitutive modeling, Polymer 2017; 128: 87-99.

           https://doi.org/10.1016/j.polymer.2017.08.065.

    1. Idrissa, A. K. M., Wang, K., Ahzi, S., Patlazhan, S., Rémond, Y. A composite approach for modeling deformation behaviors of thermoplastic polyurethane considering soft-hard domains transformation, 2018; 11: 381-388. https://doi.org/10.1007/s12289-017-1369-0.
    2. Wang, Y., Luo, W., Huang, J., Peng, C., Wang, H., Yuan, C., Chen, G., Zeng, B., and Dai, L. Simplification of Hyperelastic Constitutive Model and Finite Element Analysis of Thermoplastic Polyurethane Elastomers, Macromol. Theory Simul, 2020; 29(4): 1-12.

           https://doi.org/10.1002/mats.202000009.

    1. Bartolom´e, L., aginagalde, A., mart´inez, A. B., urchegui, M. A., and tato, W. Experimental characterization and modelling of largestrain viscoelastic behavior ofathermoplastic polyurethane elastomer, Rubber chemistry and technology, 2013; 86(1): 146–164.

           https://doi.org/ 10.5254/rct.13.87998.

    1. Reyes, S. I., Vassiliou, M. F., and Konstantinidis, D. Experimental characterization and constitutive modeling of thermoplastic polyurethane under complex uniaxial loading, J. Mech. Phys. Solids, 2024; 186: 105582.

           https://doi.org/10.1016/j.jmps.2024.105582.

    1. http://www.dgxionglin-tpufilm.com
    2. Drobny, J. G., Handbook of Thermoplastic Elastomers. William Andrew Publishing, 2007.
    3. Indukuri, K. K. Deformation characteristics of thermoplastic elastomers. Ph.D. Thesis, University of Massachusetts Amherst, 2006. Available from: https://ui.adsabs.harvard.edu.
    4. Mullins, L. Softening of rubber by deformation, Rubber Chem. Technol, 1969; 42(1): 339-362.

           https://doi.org/10.5254/1.3539210.

    1. Mullins, L. and Tobin, N. R. Theroretical model for the elastic behavior of filler-reinforced vulcanized rubbers. Rubber Chem. Technol, 1957; 30(2): 555-571. https://doi.org/10.5254/1.3542705.
    2. Marckmann, G., Verron, E., Gornet, L., Chagnon, G., Charrier, P., and Fort, P. A theory of network alteration for the Mullins effect, Journal of the Mechanics and Physics of Solids, 2002; 50(9): 2011–2028.

           https://doi.org/10.1016/S0022-5096(01)00136-3.

    1. Rinaldi, R., Boyce, M., Weigand, S., Londono, D., and Guise, M. Microstructure evolution during tensile loading histories of a polyurea, J. Polym. Sci. Part B Polym. Phys., 2011; 49(23): 1660-1671.

           https://doi.org/10.1002/polb.22352.

    1. Rinaldi, R., Hsieh, A., and Boyce, M. Tunable microstructures and mechanical deformation in transparent poly (urethane urea) s, J. Polym. Sci. Part B Polym. Phys., 2011; 49(2): 123-135.

           https://doi.org/10.1002/polb.22128.

    1. Mullins, L. and Tobin, N. Stress softening in rubber vulcanizates. Part I. Use of a strain amplification factor to describe the elastic behavior of filler-reinforced vulcanized rubber, J. Appl. Polym. Sci., 1965; 9(9): 2993-3009.

           https://doi.org/10.1002/app.1965.070090906.

    1. Castagna, A. M., Pangon, A., Choi, T., Dillon, G. P., and Runt, J. The role of soft segment molecular weight on microphase separation and dynamics of bulk polymerized polyureas, Macromolecules, 2012: 45(20): 8438-8444.

          https://doi.org/ 10.1021/ma3016568.

    1. Bonart, R., Morbitzer, L., and Hentze, G. X-ray investigations concerning the physical structure of cross-linking in urethane elastomers. II. Butanediol as chain extender, Macromol. Sci., Phys., 1969; 3(2): 337-356.

           https://doi.org/10.1080/00222346908205099.

    1. Eyring, H., Viscosity, plasticity, and diffusion as examples of absolute reaction rates, J. Chem. Phys., 1936; 4(4) 283-291.

           https://doi.org/10.1063/1.1749836.

    1. Ree, T. and Eyring, H., Theory of non-newtonian flow. I. Solid plastic system, J. Appl. Phys., 1955; 26(7): 793-800.

           https://doi.org/10.1063/1.1722098.

    1. Lee, E. Elastic–plastic deformation at finite strains J. Appl. Mech., 1969; 36(1): 1-6.

           https://doi.org/10.1115/1.3564580.

    1. Boyce, M. C., Weber, G., and Parks, D. M. On the kinematics of finite strain plasticity J. Mech. Phys. Solids, 1989; 37(5): 647-665.

           https://doi.org/10.1016/0022-5096(89)90033-1.

    1. Gurtin, M. E. and Anand, L. The decomposition F=FeFp, material symmetry, and plastic irrationality for solids that are isotropic-viscoplastic or amorphous, Int. J. Plasticity, 2005; 21(9): 686-1719. https://doi.org/10.1016/j.ijplas.2004.11.007.
    2. Tømmernes, V. Implementation of the Arruda-Boyce Material Model for Polymers in Abaqus. MSc Thesis, Norwegian University of Science and Technology, 2014. Available from:

           https://ntnuopen.ntnu.no.

    1. Tomaš, I., Cisilino, A. P., and Frontini, P. M., Acccurate, efficient and robust explicit and implicit integration schemes for the arruda-boyce viscoplatic model, Mecánica Computacional, Vol. XXVII, 2008. Available from: https://www.semanticscholar.org.
    2. Arif, A. F. M., Pervez, T., and Mughal, M. P. Performance of a finite element procedure for hyperelastic-viscoplastic large deformation problems, Finite Elements in Analysis and Design, 2000; 34(1): 89-112.

           https://doi.org/10.1016/S0168-874X(99)00031-1.

    1. Arruda, E. M. and Boyce, M. C. A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials, J. Mech. Phys. Solids, 1993; 41(2): 389-412.

           https://doi.org/10.1016/0022-5096(93)90013-6.

    1. Cohen, A., A Pade approximant to the inverse Langevin function, Rheol. Acta, 1991; 30(3): 270-273.

         https://link.springer.com/article/10.1007/BF00366640

    1. Miehe, C. and Keck, J. Superimposed finite elastic-viscoelastic-plastoelastic stress response with damage in filled rubbery polymers. Experiments, modelling and algorithmic implementation, J. Mech. Phys. Solids, 2000; 48(2): 323-365.

           https://doi.org/ 10.1016/S0022-5096(99)00017-4.

    1. Qi, H. and Boyce, M. Constitutive model for stretch-induced softening of the stress-stretch behavior of elastomeric materials, J. Mech. Phys. Solids, 2004; 52(10): 2187-2205.

           https://doi.org/10.1016/j.jmps.2004.04.008.

    1. Argon, A. A theory for the low-temperature plastic deformation of glassy polymers, Philos. Mag., 1973; 28(4): 839-865.

           https://doi.org/10.1080/14786437308220987.

    1. Weber, G. and Anand, L. Finite deformation constitutive equations and a time integration procedure for isotropic, hyperelastic-viscoplastic solids, Computer methods in applied mechanics and engineering, 1990; 79(2): 173-202.

           https://doi.org/10.1016/0045-7825(90)90131-5.

    1. Bergstrom, J. S., Mechanics of Solid polymers :Theory and Computational Modeling. William Andrew, 2015.

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