Skip to main content
SpringerLink
Log in

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

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. MARK, J. E., ERMAN, B., and ROLAND, M. The Science and Technology of Rubber, Academic Press, New York (2013)

    Google Scholar 

  2. BERTOLDI, K., VITELLI, V., CHRISTENSEN, J., and VAN HECKE, M. Flexible mechanical metamaterials. Nature Reviews Materials, 2, 1–11 (2017)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  8. RUS, D. and TOLLEY, M. T. Design, fabrication and control of soft robots. nature, 521, 467–475 (2015)

    Article  Google Scholar 

  9. ROGERS, J. A., SOMEYA, T., and HUANG, Y. Materials and mechanics for stretchable electronics. Science, 327, 1603–1607 (2010)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  11. WANG, C., WANG, C., HUANG, Z., and XU, S. Materials and structures toward soft electronics. Advanced Materials, 30, 1801368 (2018)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  13. YANG, C. and SUO, Z. Hydrogel ionotronics. Nature Reviews Materials, 3, 125–142 (2018)

    Article  Google Scholar 

  14. LIU, X., LIU, J., LIN, S., and ZHAO, X. Hydrogel machines. Materials Today, 36, 102–124 (2020)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  16. TRELOAR, L. G. The Physics of Rubber Elasticity, Oxford University Press, Oxford (1975)

    Google Scholar 

  17. HUDGINS, R. G. Development of a constitutive relation for elastomers exhibiting self-reinforcement. Polymer Engineering & Science, 46, 919–929 (2006)

    Article  Google Scholar 

  18. JAMES, H. M. and GUTH, E. Theory of the elastic properties of rubber. The Journal of Chemical Physics, 11, 455–481 (1943)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  22. TRELOAR, L. The photoelastic properties of short-chain molecular networks. Transactions of the Faraday Society, 50, 881–896 (1954)

    Article  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  28. CARROLL, M. M. A strain energy function for vulcanized rubbers. Journal of Elasticity, 103, 173–187 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  36. OGDEN, R. W., SACCOMANDI, G., and SGURA, I. Fitting hyperelastic models to experimental data. Computational Mechanics, 34, 484–502 (2004)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  40. NAEBE, M. and SHIRVANIMOGHADDAM, K. Functionally graded materials: a review of fabrication and properties. Applied Materials Today, 5, 223–245 (2016)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  43. SURESH, S. and MORTENSEN, A. Fundamentals of Functionally Graded Materials, IOM Communications, London (1998)

    Google Scholar 

  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)

    Google Scholar 

  45. IKEDA, Y. Preparation and properties of graded styrene-butadiene rubber vulcanizates. Journal of Polymer Science Part B: Polymer Physics, 40, 358–364 (2002)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  47. BILGILI, E. Functional grading of rubber tubes within the context of a molecularly inspired finite thermoelastic model. Acta Mechanica, 169, 79–85 (2004)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  58. OGDEN, R. W. Non-Linear Elastic Deformations, Dover Publications Inc., New York (1984)

    MATH  Google Scholar 

  59. TIMOSHENKO, S. and GOODIER, J. N. Theory of Elasticity, Mcgraw-Hill Book Company, New York (1951)

    MATH  Google Scholar 

  60. KINCAID, D. R. and CHENEY, E. W. Numerical Analysis: Mathematics of Scientific Computing, Cole Publishing Co., California (2002)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mechanical and Vehicle Engineering, Taiyuan University of Technology, Taiyuan, 030024, China

    Libiao Xin & Yang Wang

  2. College of Aeronautics and Astronautics, Taiyuan University of Technology, Taiyuan, 030024, China

    Zhiqiang Li

  3. Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC, 27606, USA

    Y. B. Li

Authors
  1. Libiao Xin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhiqiang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Y. B. Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Libiao Xin.

Additional information

Conflict of interest

The authors declare no conflict of interest.

Project supported by the National Natural Science Foundation of China (No. 11972144), the Shanxi Province Specialized Research and Development Breakthrough in Key Core and Generic Technologies (Key Research and Development Program) (No. 2020XXX017), and the Fundamental Research Program of Shanxi Province of China (No. 202203021211134)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xin, L., Wang, Y., Li, Z. et al. Finite deformation analysis of the rotating cylindrical hollow disk composed of functionally-graded incompressible hyper-elastic material. Appl. Math. Mech.-Engl. Ed. 44, 1367–1384 (2023). https://doi.org/10.1007/s10483-023-3014-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-023-3014-6

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation