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Polarization-Optical Parameters of the Axisymmetric Residual Stresses in an Isotropic Dielectric Cylinder

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We consider the direct problem of determination of axisymmetric residual stresses in a finite isotropic dielectric cylindrical body caused by the diagonal tensor of inconsistent eigenstrains whose components depend only on the radial coordinate. The parameters of the stress field are investigated by using the obtained solution of the problem. These parameters can be empirically found by the polarization-optical method as a result of sounding of the body with rays of polarized light in different directions and then used as input data for the inverse problems of nondestructive determination of the residual stress-strain state. We consider three sets of directions in the planes of space symmetry of the stress-strain state. For each set, we establish the integral relations connecting the values of the corresponding polarizationoptical parameters with the distributions of the components of stresses in a body in the direction of raysounding. We perform quantitative investigations of the polarization-optical parameters depending on the level of residual stresses and their gradients. The results of these investigations enable us to estimate the informativity of the empirical data that can be obtained as a result of ray-sounding of the cylinder along these three sets of directions.

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References

  1. Kh. K. Aben, Integral Photoelasticity [in Russian], Valgus, Tallinn (1975).

    Google Scholar 

  2. V. F. Chekurin, “A variational method for solving direct and inverse problems of the theory of elasticity for semiinfinite strip,” Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, No. 2, 58–70 (1999); English translation: Mech. Solids, 34, No. 2, 49–59 (1999).

  3. V. F. Chekurin, “An approach to solving the stress state tomography problems of elastic solids with inconsistent strains,” Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, No. 6, 38–48 (2000); English translation: Mech. Solids, 35, No. 6, 29–37 (2000).

  4. V. F. Chekurin, “Inverse problem of nondestructive control of the level of hardening of sheet glass,” Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, No. 3, 86–97 (1998); English translation: Mech. Solids, 33, No. 3, 68–77 (1998).

  5. V. F. Chekurin, “Variational method for the solution of the problems of tomography of the stressed state of solids,” Fiz.-Khim. Mekh. Mater., 35, No. 5, 23–32 (1999); English translation: Mater. Sci., 35, No. 5, 623–633 (1999; 10.1007/BF02359348.

  6. V. F. Chekurin and L. I. Postolaki, “Application of the variational method of homogeneous solutions for the optimal control of the axisymmetric thermoelastic state of a cylinder,” Mat. Met. Fiz.-Mekh. Polya, 60, No. 2, 105–116 (2017); English translation: J. Math. Sci., 243, No. 1, 128–144 (2019); 10.1007/s10958-019-04531-3.

  7. M. Achintha and D. Nowell, “Eigenstrain modelling of residual stresses generated by laser shock peening,” J. Mater. Proc. Technol., 211, No. 6, 1091–1101 (2011); https://doi.org/10.1016/j.jmatprotec.2011.01.011.

    Article  Google Scholar 

  8. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, North-Holland, Amsterdam (1977).

    Google Scholar 

  9. B. A. Balan and M. Achintha, “Assessment of stresses in float and tempered glass using eigenstrains, Exp. Mech., 55, No. 7, 1301–1315 (2015); https://doi.org/10.1007/s11340-015-0036-y.

  10. V. F. Chekurin, “Integral photoelasticity relations for inhomogeneously strained dielectrics,” Math. Model. Comput., 1, No. 2, 144–155 (2014).

    Article  MathSciNet  Google Scholar 

  11. V. Chekurin and L. Postolaki, “A variational method of homogeneous solutions for axisymmetric elasticity problems for cylinder,” Math. Model. Comput., 2, No. 2, 128–132 (2015); 10.23939/mmc2015.02.128.

  12. V. Chekurin and L. Postolaki, “Application of the least squares method in axisymmetric biharmonic problems,” Math. Probl. Eng., 2016, Article ID 3457649 (2016); https://doi.org/10.1155/2016/3457649.

  13. V. Chekurin and L. Postolaki, “Residual stresses in a finite cylinder. Direct and inverse problems and their solving using the variational method of homogeneous solutions,” Math. Model. Comput., 5, No. 2, 119–133 (2018); 10.23939/mmc2018.02.119.

  14. G. S. Glaesemann, Optical Fiber Mechanical Reliability. Review of Research at Corning’s Optical Fiber Strength Laboratory, White Paper WP8002. ISO 9001 Registered, Geneva, ISO (2017); https://www.corning.com/media/worldwide/coc/documents/Fiber/RC-%20White%20Papers/WP-General/WP8002_7.17.pdf.

  15. Handbook of Residual Stress and Deformation of Steel, G. Totten, M. Howes, and T. Inoue (editors), ASM International, Novelty, OH (2002).

  16. M. Inoue, S. Nakano, H. Harada, Y. Miyamura, B. Gao, Y. Kangawa, and K. Kakimoto, “Numerical analysis of the dislocation density in multicrystalline silicon for solar cells by the vertical Bridgman process,” Int. J. Photoenergy, 2013, Article ID 706923 (2013); https://doi.org/10.1155/2013/706923.

  17. T. Iwaki and N. Kobayashi, “Thermal and residual stresses of Czochralski-grown semiconducting material,” Int. J. Solids Struct., 22, No. 3, 307–314 (1986); https://doi.org/10.1016/0020-7683(86)90094-6.

    Article  MATH  Google Scholar 

  18. I. P. Kaminow, “Polarization-maintaining fibers,” Appl. Sci. Res., 41, 257–270 (1984); 10.1007/BF00382456.

  19. A. Kumar, R. G. R. Prasath, V. Pogue, K. Skenes, C. Yang, S. N. Melkote, and S. Danyluk, “Effect of growth rate and wafering on residual stress of diamond wire sawn silicon wafers,” Procedia Manuf., 5, 1382–1393 (2016); 10.1016/j.promfg.2016.08.108.

  20. . X. Liang, L. Cheng, Q. Chen, Q. Yang, and A. C. To, “A modified method for estimating inherent strains from detailed process simulation for fast residual distortion prediction of single-walled structures fabricated by directed energy deposition,” Addit. Manuf., 23, 471-486 (2018); 10.1016/j.addma.2018.08.029.

  21. V. Luzin, “Use of the eigenstrain concept for residual stress analysis,” Mater. Sci. Forum, 768–769, 193–200 (2013); https://doi.org/10.4028/www.scientific.net/MSF.768-769.193.

  22. R. Montanari, A. Fava, and G. Barbieri, “Experimental techniques to investigate residual stress in joints,” in: Residual Stress Analysis on Welded Joints by means of Numerical Simulation and Experiments, P. Ferro and F. Berto (editors), IntechOpen, London (2017), pp. 1–28; https://doi.org/10.5772/intechopen.71564.

    Chapter  Google Scholar 

  23. T. Mura, Micromechanics of Defects in Solids, Martinus Nijhoff, Dordrecht (1987); DOI: https://doi.org/10.1007/978-94-009-3489-4.

  24. J. H. Nielsen, “Remaining stress-state and strain-energy in tempered glass fragments,” Glass Struct. Eng., 2, No. 1, 45–56 (2017); https://doi.org/10.1007/s40940-016-0036-z.

    Article  Google Scholar 

  25. L. Pagnotta and A. Poggialini, “Measurement of residual internal stresses in optical fiber preforms,” Exp. Mech., 43, No. 1, 69–76 (2003); https://doi.org/10.1007/BF02410486.

    Article  Google Scholar 

  26. M. H. Sadd, Elasticity: Theory, Applications, and Numerics, Elsevier, Amsterdam (2005).

    Google Scholar 

  27. E. Salvati, T. Sui, A. J. G. Lunt, and A. M. Korsunsky, “The effect of eigenstrain induced by ion beam damage on the apparent strain relief in FIB-DIC residual stress evaluation,” Mater. Design., 92, 649–658 (2016); https://doi.org/10.1016/j.matdes.2015.12.015.

  28. G. S. Schajer (ed.), Practical Residual Stress Measurement Methods, Wiley, New York (2013).

    Google Scholar 

  29. S. M. Sze and M.-K. Lee, Semiconductor Devices: Physics and Technology, Wiley, New York (2012).

    Google Scholar 

  30. P. J. Withers, “Residual stress and its role in failure,” Rep. Prog. Phys., 70, No. 12, 2211–2264 (2007); https://doi.org/10.1088/0034-4885/70/12/R04.

    Article  Google Scholar 

  31. M.-X. Yang, R.-G. Li, P Jiang, F.-P. Yuan, Y.-D. Wang, Y.-T. Zhu, and X.-L. Wu, “Residual stress provides significant strengthening and ductility in gradient structured materials,” Mater. Res. Lett., 7, No. 11, 433-438 (2019); https://doi.org/10.1080/21663831.2019.1635537.

    Article  Google Scholar 

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Authors and Affiliations

  1. Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine

    V. F. Chekurin, L. I. Postolaki & V. V. Dyakiv

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  1. V. F. Chekurin
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  2. L. I. Postolaki
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  3. V. V. Dyakiv
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Correspondence to V. F. Chekurin.

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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 62, No. 4, pp. 95–111, October–December, 2019.

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Chekurin, V.F., Postolaki, L.I. & Dyakiv, V.V. Polarization-Optical Parameters of the Axisymmetric Residual Stresses in an Isotropic Dielectric Cylinder. J Math Sci 265, 454–473 (2022). https://doi.org/10.1007/s10958-022-06064-8

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