Finite deformation analysis of the rotating cylindrical hollow disk composed of functionally-graded incompressible hyper-elastic material

doi:10.1007/s10483-023-3014-6

Abstract

Abstract: The deformations and stresses of a rotating cylindrical hollow disk made of incompressible functionally-graded hyper-elastic material are theoretically analyzed based on the finite elasticity theory. The hyper-elastic material is described by a new micro-macro transition model. Specially, the material shear modulus and density are assumed to be a function with a power law form through the radial direction, while the material inhomogeneity is thus reflected on the power index m. The integral forms of the stretches and stress components are obtained. With the obtained complicated integral forms, the composite trapezoidal rule is utilized to derive the analytical solutions, and the explicit solutions for both the stretches and the stress components are numerically obtained. By comparing the results with two classic models, the superiority of the model in our work is demonstrated. Then, the distributions of the stretches and normalized stress components are discussed in detail under the effects of m. The results indicate that the material inhomogeneity and the rotating angular velocity have significant effects on the distributions of the normalized radial and hoop stress components and the stretches. We believe that by appropriately choosing the material inhomogeneity and configuration parameters, the functionally-graded material (FGM) hyper-elastic hollow cylindrical disk can be designed to meet some unique requirements in the application fields, e.g., soft robotics, medical devices, and conventional aerospace and mechanical industries.

Key words: functionally-graded material(FGM), incompressible, hyper-elastic model, rotating hollow cylindrical disk

2010 MSC Number:

Libiao XIN, Yang WANG, Zhiqiang LI, Y. B. LI. Finite deformation analysis of the rotating cylindrical hollow disk composed of functionally-graded incompressible hyper-elastic material. Applied Mathematics and Mechanics (English Edition), 2023, 44(8): 1367-1384.

TrendMD

References

[1] MARK, J. E., ERMAN, B., and ROLAND, M. The Science and Technology of Rubber, Academic Press, New York (2013)
[2] BERTOLDI, K., VITELLI, V., CHRISTENSEN, J., and VAN HECKE, M. Flexible mechanical metamaterials. Nature Reviews Materials, 2, 1-11(2017)
[3] WU, L., WANG, Y., CHUANG, K., WU, F., WANG, Q., LIN, W., and JIANG, H. A brief review of dynamic mechanical metamaterials for mechanical energy manipulation. Materials Today, 44, 168-193(2021)
[4] QI, J., CHEN, Z., JIANG, P., HU, W., WANG, Y., ZHAO, Z., CAO, X., ZHANG, S., TAO, R., and LI, Y. Recent progress in active mechanical metamaterials and construction principles. Advanced Science, 9, 2102662(2022)
[5] SHEPHERD, R. F., ILIEVSKI, F., CHOI, W., MORIN, S. A., STOKES, A. A., MAZZEO, A. D., CHEN, X., WANG, M., and WHITESIDES, G. M. Multigait soft robot. Proceedings of the National Academy of Sciences, 108, 20400-20403(2011)
[6] EL-ATAB, N., MISHRA, R. B., AL-MODAF, F., JOHARJI, L., ALSHARIF, A. A., ALAMOUDI, H., DIAZ, M., QAISER, N., and HUSSAIN, M. M. Soft actuators for soft robotic applications:a review. Advanced Intelligent Systems, 2, 2000128(2020)
[7] JI, X., LIU, X., CACUCCIOLO, V., IMBODEN, M., CIVET, Y., EL-HAITAMI, A., CANTIN, S., PERRIARD, Y., and SHEA, H. An autonomous untethered fast soft robotic insect driven by low-voltage dielectric elastomer actuators. Science Robotics, 4, eaaz6451(2019)
[8] RUS, D. and TOLLEY, M. T. Design, fabrication and control of soft robots. nature, 521, 467-475(2015)
[9] ROGERS, J. A., SOMEYA, T., and HUANG, Y. Materials and mechanics for stretchable electronics. Science, 327, 1603-1607(2010)
[10] KIM, D. H., LU, N., MA, R., KIM, Y. S., KIM, R. H., WANG, S., WU, J., WON, S. M., TAO, H., and ISLAM, A. Epidermal electronics. Science, 333, 838-843(2011)
[11] WANG, C., WANG, C., HUANG, Z., and XU, S. Materials and structures toward soft electronics. Advanced Materials, 30, 1801368(2018)
[12] BEEBE, D. J., MOORE, J. S., BAUER, J. M., YU, Q., LIU, R. H., DEVADOSS, C., and JO, B. H. Functional hydrogel structures for autonomous flow control inside microfluidic channels. nature, 404, 588-590(2000)
[13] YANG, C. and SUO, Z. Hydrogel ionotronics. Nature Reviews Materials, 3, 125-142(2018)
[14] LIU, X., LIU, J., LIN, S., and ZHAO, X. Hydrogel machines. Materials Today, 36, 102-124(2020)
[15] ZHAO, X., CHEN, X., YUK, H., LIN, S., LIU, X., and PARADA, G. Soft materials by design:unconventional polymer networks give extreme properties. Chemical Reviews, 121, 4309-4372(2021)
[16] TRELOAR, L. G. The Physics of Rubber Elasticity, Oxford University Press, Oxford (1975)
[17] HUDGINS, R. G. Development of a constitutive relation for elastomers exhibiting selfreinforcement. Polymer Engineering & Science, 46, 919-929(2006)
[18] JAMES, H. M. and GUTH, E. Theory of the elastic properties of rubber. The Journal of Chemical Physics, 11, 455-481(1943)
[19] FLORY, P. J. and REHNER, J., JR. Statistical mechanics of cross-linked polymer networks, I:rubberlike elasticity. The Journal of Chemical Physics, 11, 512-520(1943)
[20] ARRUDA, E. M. and BOYCE, M. C. A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. Journal of the Mechanics and Physics of Solids, 41(2), 389-412(1993)
[21] DAL, H., GUELTEKIN, O., and ACIKGOZ, K. An extended eight-chain model for hyperelastic and finite viscoelastic response of rubberlike materials:theory, experiments and numerical aspects. Journal of the Mechanics and Physics of Solids, 145, 104159(2020)
[22] TRELOAR, L. The photoelastic properties of short-chain molecular networks. Transactions of the Faraday Society, 50, 881-896(1954)
[23] TRELOAR, L. R. G. and RIDING, G. A non-Gaussian theory for rubber in biaxial strain, I:mechanical properties. Proceedings of the Royal Society of London Series A:Mathematical, Physical and Engineering Sciences, 369, 261-280(1979)
[24] WU, P. D. and IESSEN, E. On improved network models for rubber elasticity and their applications to orientation hardening in glassy polymers. Journal of the Mechanics and Physics of Solids, 41, 427-456(1993)
[25] RIVLIN, R. S. Large elastic deformations of isotropic materials, IV:further developments of the general theory. Philosophical Transactions of the Royal Society A:Mathematical, Physical and Engineering Sciences, 241(835), 379-397(1948)
[26] OGDEN, R. W. Large deformation isotropic elasticity-on the correlation of theory and experiment for incompressible rubberlike solids. Proceedings of the Royal Society A:Mathematical, Physical & Engineering Sciences, 326(1567), 565-584(1972)
[27] BECHIR, H., CHEVALIER, L., CHAOUCHE, M., and BOUFALA, K. Hyperelastic constitutive model for rubber-like materials based on the first Seth strain measures invariant. European Journal of Mechanics A/Solids, 25, 110-124(2006)
[28] CARROLL, M. M. A strain energy function for vulcanized rubbers. Journal of Elasticity, 103, 173-187(2011)
[29] DAVIDSON, J. D. and GOULBOURNE, N. C. A nonaffine network model for elastomers undergoing finite deformations. Journal of the Mechanics and Physics of Solids, 61, 1784-1797(2013)
[30] MANGAN, R., DESTRADE, M., and SACCOMANDI, G. Strain energy function for isotropic non-linear elastic incompressible solids with linear finite strain response in shear and torsion. Extreme Mechanics Letters, 9, 204-206(2016)
[31] KHIÊM, V. N. and ITSKOV, M. Analytical network-averaging of the tube model:rubber elasticity. Journal of the Mechanics and Physics of Solids, 95, 254-269(2016)
[32] XIANG, Y., ZHONG, D., WANG, P., MAO, G., YU, H., and QU, S. A general constitutive model of soft elastomers. Journal of the Mechanics and Physics of Solids, 117, 110-122(2018)
[33] MENG, S., IMTIAZ, H., and LIU, B. A simple interpolation-based approach towards the development of an accurate phenomenological constitutive relation for isotropic hyperelastic materials. Extreme Mechanics Letters, 49, 101485(2021)
[34] SHEN, S., ZHONG, D., QU, S., and XIAO, R. A hyperelastic-damage model based on the strain invariants. Extreme Mechanics Letters, 52, 101641(2022)
[35] DAL, H., ACIKGOZ, K., and BADIENIA, Y. On the performance of isotropic hyperelastic constitutive models for rubber-like materials:a state of the art review. Applied Mechanics Reviews, 73, 020802(2021)
[36] OGDEN, R. W., SACCOMANDI, G., and SGURA, I. Fitting hyperelastic models to experimental data. Computational Mechanics, 34, 484-502(2004)
[37] ZHAN, L., WANG, S., QU, S., STEINMANN, P., and XIAO, R. A new micro-macro transition for hyperelastic materials. Journal of the Mechanics and Physics of Solids, 171, 105156(2023)
[38] BOGGARAPU, V., GUJJALA, R., OJHA, S., ACHARYA, S., CHOWDARY, S., and GARA, D. K. State of the art in functionally graded materials. Composite Structures, 262, 113596(2021)
[39] LOH, G. H., PEI, E., HARRISON, D., and MONZÓN, M. D. An overview of functionally graded additive manufacturing. Additive Manufacturing, 23, 34-44(2018)
[40] NAEBE, M. and SHIRVANIMOGHADDAM, K. Functionally graded materials:a review of fabrication and properties. Applied Materials Today, 5, 223-245(2016)
[41] NIKBAKHT, S., KAMARIAN, S., and SHAKERI, M. A review on optimization of composite structures, part II:functionally graded materials. Composite Structures, 214, 83-102(2019)
[42] SALEH, B., JIANG, J., FATHI, R., AL-HABABI, T., XU, Q., WANG, L., SONG, D., and MA, A. 30 years of functionally graded materials:an overview of manufacturing methods, applications and future challenges. Composites Part B:Engineering, 201, 108376(2020)
[43] SURESH, S. and MORTENSEN, A. Fundamentals of Functionally Graded Materials, IOM Communications, London (1998)
[44] IKEDA, Y., KASAI, Y., MURAKAMI, S., and KOHJIYA, S. Preparation and mechanical properties of graded styrene-butadiene rubber vulcanizates. Journal of the Japan Institute of Metals, 62, 1013-1017(1998)
[45] IKEDA, Y. Preparation and properties of graded styrene-butadiene rubber vulcanizates. Journal of Polymer Science Part B:Polymer Physics, 40, 358-364(2002)
[46] BILGILI, E. Controlling the stress-strain inhomogeneities in axially sheared and radially heated hollow rubber tubes via functional grading. Mechanics Research Communications, 30, 257-266(2003)
[47] BILGILI, E. Functional grading of rubber tubes within the context of a molecularly inspired finite thermoelastic model. Acta Mechanica, 169, 79-85(2004)
[48] BATRA, R. and BAHRAMI, A. Inflation and eversion of functionally graded non-linear elastic incompressible circular cylinders. International Journal of Non-Linear Mechanics, 44, 311-323(2009)
[49] NIE, G. and BATRA, R. Stress analysis and material tailoring in isotropic linear thermoelastic incompressible functionally graded rotating disks of variable thickness. Composite Structures, 92, 720-729(2010)
[50] ANANI, Y. and RAHIMI, G. H. Stress analysis of thick pressure vessel composed of functionally graded incompressible hyperelastic materials. International Journal of Mechanical Sciences, 104, 1-7(2015)
[51] ANANI, Y. and RAHIMI, G. H. Stress analysis of rotating cylindrical shell composed of functionally graded incompressible hyperelastic materials. International Journal of Mechanical Sciences, 108, 122-128(2016)
[52] SHARIYAT, M., KHOSRAVI, M., ARIATAPEH, M. Y., and NAJAFIPOUR, M. Nonlinear stress and deformation analysis of pressurized thick-walled hyperelastic cylinders with experimental verifications and material identifications. International Journal of Pressure Vessels and Piping, 188, 104211(2020)
[53] CHEN, W., YAN, Z., and WANG, L. On mechanics of functionally graded hard-magnetic soft beams. International Journal of Engineering Science, 157, 103391(2020)
[54] BARTLETT, N. W., TOLLEY, M. T., OVERVELDE, J. T., WEAVER, J. C., MOSADEGH, B., BERTOLDI, K., WHITESIDES, G. M., and WOOD, R. J. A 3D-printed, functionally graded soft robot powered by combustion. Science, 349, 161-165(2015)
[55] WANG, Z., WANG, Z., ZHENG, Y., HE, Q., and CAI, S. Three-dimensional printing of functionally graded liquid crystal elastomer. Science Advances, 6, eabc0034(2020)
[56] LI, Y., FENG, Z., HAO, L., HUANG, L., XIN, C., WANG, Y., BILOTTI, E., ESSA, K., ZHANG, H., and LI, Z. A review on functionally graded materials and structures via additive manufacturing:from multi-scale design to versatile functional properties. Advanced Materials Technologies, 5, 1900981(2020)
[57] SARKAR, P. R. and RAHMAN, A. S. Effect of magnetic field on the thermo-elastic response of a rotating FGM circular disk with non-uniform thickness. The Journal of Strain Analysis for Engineering Design, 57, 116-131(2022)
[58] OGDEN, R. W. Non-Linear Elastic Deformations, Dover Publications Inc., New York (1984)
[59] TIMOSHENKO, S. and GOODIER, J. N. Theory of Elasticity, Mcgraw-Hill Book Company, New York (1951)
[60] KINCAID, D. R. and CHENEY, E. W. Numerical Analysis:Mathematics of Scientific Computing, Cole Publishing Co., California (2002)