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Finite strain expansion/contraction of a hollow sphere made of strain- and rate- hardening material

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Abstract

This paper presents a semi-analytic rigid/plastic solution for the expansion/contraction of a hollow sphere at large strains. The yield stress depends on the equivalent strain rate and the equivalent strain. No restriction is imposed on this dependence. The solution reduces to a single ordinary differential equation for determining the radial stress. The independent variable in this equation is the equivalent strain. Moreover, the equivalent strain rate is expressed in terms of elementary functions of the equivalent strain, which allows for representing the yield stress as a function of the equivalent strain and a time-like independent variable. In the course of deriving the equations above, the transformation between Eulerian and Lagrangian coordinates is used. A numerical example illustrates the solution for a material model available in the literature. The motivation of this research is that solutions for the expansion/contraction of a hollow sphere are widely used at the micro-level to calculate some material properties at the macro-level. To this end, it is necessary to specify constitutive equations for micromechanical modeling. The accuracy of these equations is questionable. An advantage of the solution found is that it is practically analytic for quite a general material model that accounts for both strain- and rate-hardening. Therefore, it is straightforward to generate a large amount of theoretical data for comparing with measurable quantities at the macro-level.

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References

  1. Hill, R.: The Mathematical Theory of Plasticity. Oxford University Press, Oxford (1950)

    MATH  Google Scholar 

  2. Durban, D., Baruch, M.: Analysis of an elasto-plastic thick walled sphere loaded by internal and external pressure. Int. J. Non-Linear Mech. 12, 9–21 (1977)

    Article  ADS  Google Scholar 

  3. Carroll, M.M., Kim, K.T.: Pressure-density equations for porous metals and metal powders. Powder Metall. 27, 153–159 (1984)

    Article  Google Scholar 

  4. Wilkinson, D.S., Ashby, M.F.: Pressure sintering by power law creep. Acta Metall. 23, 1277–1285 (1975)

    Article  Google Scholar 

  5. Haghi, M., Anand, L.: Analysis of strain-hardening viscoplastic thick-walled sphere and cylinder under external pressure. Int. J. Plast. 7, 123–140 (1991)

    Article  Google Scholar 

  6. Thore, P., Pastor, F., Pastor, J., Kondo, D.: Closed-form solutions for the hollow sphere model with Coulomb and Drucker–Prager materials under isotropic loadings. C. R. Mec. 337, 260–267 (2009)

    Article  ADS  Google Scholar 

  7. Johnson, G.R., Cook, W.Y.: Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Eng. Fract. Mech. 21(1), 31–48 (1985)

    Article  Google Scholar 

  8. Meyer, H.W., Jr., Kleponis, D.S.: Modeling the high strain rate behavior of titanium undergoing ballistic impact and penetration. Int. J. Impact Eng. 26, 509–521 (2001)

    Article  Google Scholar 

  9. Siegel, A., Laporte, S., Sauter-Starace, F.: Johnson–Cook parameter identification for commercially pure titanium at room temperature under quasi-static strain rates. Materials 14, Article 3887 (2021)

  10. Ashrafian, M.M., Kordkheili, S.A.H.: A novel phenomenological constitutive model for Ti–6Al–4V at high temperature conditions and quasi-static strain rates. Proc. IMechE Part G J. Aerosp. Eng. 235(13), 1831–1842 (2021)

    Article  Google Scholar 

  11. Alister, F., Celentano, D., Signorelli, J., Bouchard, P.-O., Munoz, D.P., Cruchaga, M.: Viscoplastic and temperature behavior of Zn–Cu–Ti alloy sheets: experiments, characterization, and modeling. J. Mater. Res. Technol. 15, 3759–3772 (2021)

    Article  Google Scholar 

  12. Rusinek, A., Zaera, R., Klepaczko, J.R.: Constitutive relations in 3-D for a wide range of strain rates and temperatures—application to mild steels. Int. J. Solids Struct. 44, 5611–5634 (2007)

    Article  Google Scholar 

  13. Jia, B., Rusinek, A., Pesci, R., Bahi, S., Bernier, R.: Thermo-viscoplastic behavior of 304 austenitic stainless steel at various strain rates and temperatures: testing, modeling and validation. Int. J. Mech. Sci. 170, 105356 (2020)

    Article  Google Scholar 

  14. Cheng, W., Outeiro, J., Costes, J.-P., M’Saoubi, R., Karaouni, H., Denguir, L., Astakhov, V., Auzenat, F.: Constitutive model incorporating the strain-rate and state of stress effects for machining simulation of titanium alloy Ti6Al4V. Proc. CIRP 77, 344–347 (2018)

    Article  Google Scholar 

  15. Cheng, W., Outeiro, J., Costes, J.-P., M’Saoubi, R., Karaouni, H., Astakhov, V.: A constitutive model for Ti6Al4V considering the state of stress and strain rate effects. Mech. Mater. 137, 103103 (2019)

    Article  Google Scholar 

  16. Dos Santos, T., Outeiro, J.C., Rossi, R., Rosa, P.: A new methodology for evaluation of mechanical properties of materials at very high rates of loading. Proc. CIRP 58, 481–486 (2017)

    Article  Google Scholar 

  17. Kim, H., Yoon, J.W., Chung, K., Lee, M.-G.: A multiplicative plastic hardening model in consideration of strain softening and strain rate: theoretical derivation and characterization of model parameters with simple tension and creep test. Int. J. Mech. Sci. 187, 105913 (2020)

    Article  Google Scholar 

  18. Attar, H.R., Li, N., Foster, A.: A method for determining equivalent hardening responses to approximate sheet metal viscoplasticity. MethodsX 8, 101554 (2021)

    Article  Google Scholar 

  19. Bodner, S.R., Partom, Y.: Constitutive equations for elastic-viscoplastic strain-hardening materials. Trans. ASME J. Appl. Mech. 42, 385–389 (1975)

    Article  ADS  Google Scholar 

  20. Leu, S.-Y.: Analytical and numerical investigation of strain-hardening viscoplastic thick-walled cylinders under internal pressure by using sequential limit analysis. Comput. Methods Appl. Mech. Eng. 196, 2713–2722 (2007)

    Article  ADS  Google Scholar 

  21. Leu, S.-Y.: Limit analysis of strain-hardening viscoplastic cylinders under internal pressure by using the velocity control: analytical and numerical investigation. Int. J. Mech. Sci. 50, 1578–1585 (2008)

    Article  Google Scholar 

  22. Leu, S.-Y.: Investigation of rotating hollow cylinders of strain-hardening viscoplastic materials by sequential limit analysis. Comput. Methods Appl. Mech. Eng. 197, 4858–4865 (2008)

    Article  ADS  Google Scholar 

  23. Alexandrov, S., Hwang, Y.-M.: Plane strain bending with isotropic strain hardening at large strains. Trans. ASME J. Appl. Mech. 77, 064502 (2010)

    Article  ADS  Google Scholar 

  24. Alexandrov, S., Pirumov, A., Jeng, Y.-R.: Expansion/contraction of a spherical elastic/plastic shell revisited. Contin. Mech. Thermodyn. 27, 483–494 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  25. Alexandrov, S., Jeng, Y.-R.: An elastic/plastic solution for a hollow sphere subject to thermo-mechanical loading considering temperature dependent material properties. Int. J. Solids Struct. 200–201, 23–33 (2020)

    Article  Google Scholar 

  26. Collins, I.F., Meguid, S.A.: On the influence of hardening and anisotropy on the plane-strain compression of thin metal strip. ASME J. Appl. Mech. 44, 271–278 (1977)

    Article  ADS  Google Scholar 

  27. Adams, M.J., Briscoe, B.J., Corfield, G.M., Lawrence, C.J., Papathanasiou, T.D.: An analysis of the plane-strain compression of viscoplastic materials. ASME J. Appl. Mech. 64, 420–424 (1997)

    Article  ADS  Google Scholar 

  28. Alexandrov, S., Jeng, Y.-R.: Compression of viscoplastic material between rotating plates. ASME J. Appl. Mech. 76, 031017 (2009)

    Article  ADS  Google Scholar 

  29. Roberts, S.M., Hall, F., Van Bael, A., Hartley, P., Pillinger, I., Sturgess, E.N., Van Houtte, P., Aernoudt, E.: Benchmark tests for 3-D, elasto-plastic, finite-element codes for the modelling of metal forming processes. J. Mater. Process. Technol. 34, 61–68 (1992)

    Article  Google Scholar 

  30. Abali, B.E., Reich, F.A.: Verification of deforming polarized structure computation by using a closed-form solution. Contin. Mech. Thermodyn. 32, 693–708 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  31. Lee, Y., Dawson, P.R.: Obtaining residual stresses in metal forming after neglecting elasticity on loading. ASME J. Appl. Mech. 56, 318–327 (1989)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This work was financially supported by the Ministry of Science and Technology of Taiwan (MOST 106-2923-E-194-002-MY3, 108- 2221-E-006-228-MY3 and 108-2119-M-006-010) and Air Force Office of Science Research (AFOSR) under contract no. FA4869- 06-1-0056 AOARD 064053. Professor Yeau-Ren Jeng would like to acknowledge Medical Device Innovation Center (MDIC) and Intelligent Manufacturing Research Center (iMRC) from The Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE) in Taiwan and AC2T research GmbH (AC2T) in Austria (COMET InTribology, FFG-No.872176).

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Authors and Affiliations

  1. Federal State Autonomous Educational Institution of Higher Education “South Ural State University” (National Research University), 76 Lenin prospekt, Chelyabinsk, Russia, 454080

    Sergei Alexandrov

  2. Department of Biomedical Engineering, National Cheng Kung University (NCKU), Tainan, 70101, Taiwan

    Yeau-Ren Jeng

  3. Center of Smart Material and Manufacturing, National Cheng Kung University, Tainan, 70101, Taiwan

    Yeau-Ren Jeng

  4. Academy of Innovative Semiconductor and Sustainable Manufacturing, National Cheng Kung University, Tainan, 70101, Taiwan

    Yeau-Ren Jeng

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  1. Sergei Alexandrov
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  2. Yeau-Ren Jeng
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Correspondence to Yeau-Ren Jeng.

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Communicated by Andreas Öchsner.

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Alexandrov, S., Jeng, YR. Finite strain expansion/contraction of a hollow sphere made of strain- and rate- hardening material. Continuum Mech. Thermodyn. 34, 1113–1124 (2022). https://doi.org/10.1007/s00161-022-01103-w

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