The purpose of this paper is to demonstrate the applicability of Particle Swarm Optimization algorithm to determine material parameters in incompressible isotropic elastic strain-energy functions using combined tension and torsion loading. Simulation of rubber behavior was conducted from the governing equations of the deformation of a cylinder composed of isotropic hyperelastic incompressible materials. Four different forms of strain-energy function were considered based respectively on polynomial, exponential and logarithmic terms to reproduce load force (N) and torque (M) trends using natural rubber experimental data. After highlighting the minimization of the objective function generated in the fitting process, the study revealed that a particle swarm optimization algorithm could be successfully used to identify the best material parameters and characterize the behavior of rubber-like hyperelastic materials.
References
[1]
Martins, P.A.L.S., Natal Jorge, R.M. and Ferreira, A.J.M. (2006) A Comparative Study of Several Material Models for Prediction of Hyperelastic Properties: Application to Silicone-Rubber and Soft Tissues. Strain, 42, 135-147. https://doi.org/10.1111/j.1475-1305.2006.00257.x
[2]
Wu, Y., Wang, H. and Li, A. (2016) Parameter Identification Methods for Hyperelastic and Hyper-Viscoelastic Models. Applied Sciences, 6, 386. https://doi.org/10.3390/app6120386
[3]
Nunes, L.C.S. (2011) Mechanical Characterization of Hyperelastic Polydimethylsiloxane by Simple Shear Test. Materials Science and Engineering, 528, 1799-1804. https://doi.org/10.1016/j.msea.2010.11.025
[4]
Rachik, M., Schmidtt, F., Reuge, N., Le Maoult, Y. and Abbeé, F. (2001) Elastomer Biaxial Characterization Using Bubble Inflation Technique. II: Numerical Investigation of Some Constitutive Models. Polymer Engineering and Science, 41, 532-541. https://doi.org/10.1002/pen.10750
[5]
Beda, T. (2005) Optimizing the Ogden Strain Energy Expression of Rubber Materials. Journal of Engineering Materials and Technology, 127, 351-353. https://doi.org/10.1115/1.1925282
[6]
Beda, T. (2006) Combining Approach in Stages with Least Squares for Fits of Data in Hyperelasticity. Comptes Rendus Mecanique, 334, 628-633. https://doi.org/10.1016/j.crme.2006.06.004
[7]
Fernández, J.R., López-Campos, J.A., Segade, A. and Vilán, J.A. (2018) A Genetic Algorithm for the Characterization of Hyperelastic Materials. Applied Mathematics and Computation, 329, 239-250. https://doi.org/10.1016/j.amc.2018.02.008
[8]
Blaise, B.B., Betchewe, G. and Beda, T. (2019) Optimization of the Model of Ogden Energy by the Genetic Algorithm Method. Applied Rheology, 29, 21-29. https://doi.org/10.1515/arh-2019-0003
[9]
López-Campos, J.A., Segade, A., Casarejos, E., Fernández, J.R. and Días, G.R. (2019) Hyperelastic Characterization Oriented to Finite Element Applications Using Genetic Algorithms. Advances in Engineering Software, 133, 52-59. https://doi.org/10.1016/j.advengsoft.2019.04.001
[10]
Alam, M.N. (2016) Particle Swarm Optimization: Algorithm and Its Codes in MATLAB.
[11]
Ramzanpour, M., Hosseini-Farid, M., Ziejewski, M. and Karami, G. (2019) Particle Swarm Optimization Method for Hyperelastic Characterization of Soft Tissues. International Mechanical Engineering Congress and Exposition, Salt Lake City, 11-14 November 2019. https://doi.org/10.1115/IMECE2019-11829
[12]
Duan, Y., Harley, R.G. and Habetler, T.G. (2009) Comparison of Particle Swarm Optimization and Genetic Algorithm in the Design of Permanent Magnet Motors. IEEE 6th International Power Electronics and Motion Control Conference, Wuhan, 17-20 May 2009, 822-825. https://doi.org/10.1109/IPEMC.2009.5157497
[13]
Rivlin, R.S. and Saunders, D.W. (1951) Large Elastic Deformations of Isotropic Materials VII. Experiments on the Deformation of Rubber. Philosophical Transactions of the Royal Society A, 243, 251-288. https://doi.org/10.1098/rsta.1951.0004
[14]
Haupt, P. and Sedlan, K. (2001) Viscoplasticity of Elastomeric Materials. Experimental Facts and Constitutive Modeling. Archive of Applied Mechanics, 71, 89-109. https://doi.org/10.1007/s004190000102
[15]
Mars, W.V. and Fatemi, A. (2004) A Novel Specimen for Investigating the Mechanical Behavior of Elastomers under Multiaxial Loading Conditions. Experimental Mechanics, 44, 136-146. https://doi.org/10.1007/BF02428173
[16]
Suphadon, N., Thomas, A.G. and Busfield, J.J.C. (2009) Viscoelastic Behavior of Rubber under a Complex Loading. Polymer, 113, 693-699. https://doi.org/10.1002/app.30102
[17]
Lectez, A.S., Verron, E., Huneau, B., Béranger, A.S. and Le Brazidec, F. (2013) Characterization of Elastomers under Simultaneous Tension and Torsion for Application to Engine Mounts. In: Constitutive Models for Rubber VIII, CRC Press, Boca Raton, 585-590.
[18]
Humphrey, J.D. (2002) Cardiovascular Solid Mechanics. Springer-Verlag, New York, 758-766. https://doi.org/10.1007/978-0-387-21576-1
[19]
Rivlin, R.S. (1949) Large Elastic Deformations of Isotropic Materials VI. Further Results in the Theory of Torsion, Shear and Flexure. Philosophical Transactions of the Royal Society A, 242, 173-195. https://doi.org/10.1007/978-1-4612-2416-7_10
[20]
Rivlin, R.S. and Saunders, D.W. (1951) Large Elastic Deformations of Isotropic Materials VII. Experiments on the Deformation of Rubber. Philosophical Transactions of the Royal Society A, 243, 251-288. https://doi.org/10.1007/978-1-4612-2416-7_12
[21]
Sigaeva, T. and Czekanski, A. (2016) Coupling of Surface Effect and Hyperelasticity in Combined Tension and Torsion Deformations of a Circular Cylinder. International Journal of Solids and Structures, 85-86, 172-179. https://doi.org/10.1016/j.ijsolstr.2016.02.019
[22]
Suphadon, N. and Busfield, J.J.C. (2009) Elastic Behaviour of Rubber Cylinders under Combined Torsion and Tension Loading. Plastics Rubber and Composites, 38, 337. https://doi.org/10.1179/146580109X12473409436788
[23]
Kashaev, R.M. (2008) On Tension-Torsion Testing of Solid Cylindrical Specimens. Letters on Materials, 8, 346-352. https://doi.org/10.22226/2410-3535-2018-3-346-352
[24]
Kirkinis, E. and Ogden, R.W. (2002) On Extension and Torsion of a Compressible Elastic Circular Cylinder. Mathematics and Mechanics of Solids, 7, 373-392. https://doi.org/10.1177/108128028476
[25]
Michelle, S., Fatt, H. and Ouyang, X. (2007) Integral-Based Constitutive Equation for Rubber at High Strain Rates. International Journal of Solids and Structures, 44, 6491-6506. https://doi.org/10.1016/j.ijsolstr.2007.02.038
[26]
Lectez, A.S., Verron, E. and Huneau, B. (2014) How to Identify a Hyperelastic Constitutive Equation for Rubber-Like Materials with Multiaxial Tension-Torsion Experiments. International Journal of Non-Linear Mechanics, 65, 260-270. https://doi.org/10.1016/j.ijnonlinmec.2014.06.007
[27]
Treloar, L.R.G., Hopkins, H.G., Rivlin, R.S. and Ball, J.M. (1976) The Mechanics of Rubber Elasticity [and Discussions]. Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences, 351, 301-330. https://doi.org/10.1098/rspa.1976.0144
[28]
Rivlin, R.S. (1948) Large Elastic Deformations of Isotropic Materials, I, II, III, Fundamental Concepts. Philosophical Transactions of the Royal Society A, 240, 459. https://doi.org/10.1098/rsta.1948.0002
[29]
Yeoh, O.H. (1990) Characterization of Elastic Properties of Carbon-Black-Filled Rubber Vulcanizates. Rubber Chemistry and Technology, 63, 792-805. https://doi.org/10.5254/1.3538289
[30]
Gent, A.N. and Thomas, A.G. (1958) Forms for the Stored (Strain) Energy Function for Vulcanized Rubber. Journal of Polymer Science, 28, 625-628. https://doi.org/10.1002/pol.1958.1202811814
[31]
Fung, Y.C.B. (1967) Elasticity of Soft Tissues in Sample Elongation. American Journal of Physiology, 213, 1532-1544. https://doi.org/10.1152/ajplegacy.1967.213.6.1532
[32]
Chatelin, S., Constantinesco, A. and Willinger, R. (2010) Fifty Years of Brain Tissue Mechanical Testing: From in Vitro to in Vivo Investigations. Biorheology, 47, 255-276. https://doi.org/10.3233/BIR-2010-0576
[33]
Evans, S.L. (2017) How Can We Measure the Mechanical Properties of Soft Tissues? In: Avril, S. and Evans, S.L., Eds., Material Parameter Identification and Inverse Problems in Soft Tissue Biomechanics, Springer, London, 67-83. https://doi.org/10.1007/978-3-319-45071-1_3
[34]
Staber, B. and Guilleminot, J. (2017) Stochastic Hyperelastic Constitutive Laws and Identification Procedure for Soft Biological Tissues with Intrinsic Variability. Journal of the Mechanical Behavior of Biomedical Materials, 65, 743-752. https://doi.org/10.1016/j.jmbbm.2016.09.022
[35]
Wex, C., Arndt, S., Stoll, A., Bruns, C. and Kupriyanova, Y. (2015) Isotropic Incompressible Hyperelastic Models for Modelling the Mechanical Behaviour of Biological Tissues: A Review. Biomedical Engineering/Biomedizinische Technik, 60, 577-592. https://doi.org/10.1515/bmt-2014-0146
[36]
Hartmann, S. (2001) Numerical Studies on the Identification of the Material Parameters of Rivlin’s Hyperelasticity Using Tension-Torsion Tests. Acta Mechanica, 148, 129-155. https://doi.org/10.1007/BF01183674
[37]
Lectez, A.S. (2014) Comportement multiaxial de pièces élastomères précontraintes: Application aux suspensions moteur. Mécanique des matériaux [physics.class-ph] Ecole Centrale de Nantes, 151-170.
[38]
Kennedy, J. and Eberhart, R.C. (1995) Particle Swarm Optimization. Proceedings IEEE Conference on Neural Networks, Perth, 27 November-1 December 1995, 1942-1948.
[39]
Boehner, K., DePaula, R., Dourish, P. and Sengers, P. (2007) How Emotion Is Made and Measured. International Journal of Human-Computer Studies, 65, 275-291. https://doi.org/10.1016/j.ijhcs.2006.11.016
[40]
Shi, Y. and Eberhart, R. (1998) A Modified Particle Swarm Optimizer. IEEE International Conference on Evolutionary Computation, Anchorage, 4-9 May 1998, 69-73.
[41]
Rivlin, R.S. (1948) Large Elastic Deformations of Isotropic Materials III. Some Simple Problems in Cylindrical Polar Coordinates. Philosophical Transactions of the Royal Society A, 240, 509-525. https://doi.org/10.1007/978-1-4612-2416-7_7