Identification of two-dimensional prestress fields in~inhomogeneous~plates

I. V. Bogachev ; R. D. Nedin

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 23 (2023) no. 4, pp. 456-471.

Résumé

Based on the model of in-plane oscillations of inhomogeneous prestressed plates, the new inverse problems of identifying the components of the prestress tensor via acoustic response probing are considered for the plates with and without holes and inclusions; the prestress components are assumed to be functions of two coordinates. Prestresses were set as a result of solving auxiliary problems of static loading of plates by some initial mechanical load. To solve the main and auxiliary problems of calculating the plates’ displacement functions, a finite element (FE) scheme was developed based on the derived corresponding weak problem statements, implemented in the form of software systems in the FE package FreeFem++. Rectangular plates clamped along one face, both solid and having a hole or a rigid insert, were considered. Inverse problems of identification of three prestress functions depending on two coordinates are formulated on the basis of additional data about the acoustic response on the non-clamped edges of the plates as a result of considering several sets of probing loads at several frequencies. In view of the nonlinearity of the inverse problems under study, an iterative approach was developed to solve them, which combines solving the direct problems for current approximations of the desired functions and the determination of the corresponding corrections from the operator equation built at each iteration. To solve the operator equation, a projection method has been employed that allows one to present the corrections in the form of expansions in terms of some smooth given functions and reduce the problem solution to the study of ill-conditioned SLAEs with respect to sets of the expansion coefficients using the A. N. Tikhonov method. The results of computational experiments on the simultaneous identification of two-dimensional prestress fields corresponding to various types of initial actions on the considered plates are discussed.

Bibliographie

[1] Vatulyan A. O., Dudarev V. V., Nedin R. D., Prestresses: Modeling and Identification, Southern Federal University Publ, Rostov-on-Don, 2015, 206 pp. (in Russian)

[2] Carpinteri A., Pugno N., “Thermal loading in multi-layer and/or functionally graded materials: Residual stress field, delamination, fatigue and related size effects”, International Journal of Solids and Structures, 43:3–4 (2006), 828–841 | DOI | Zbl

[3] Dorodov P. V., Pospelova I. G., “Investigation of the stress state in a plate weakened by a stress concentrator”, Achievements of Science and Technology of the Agro-industrial Complex, 8 (2013), 67–70 (in Russian)

[4] Schajer G. S., Practical Residual Stress Measurement Methods, Wiley, 2013, 560 pp.

[5] Guo J., Fu H., Pan B., Kang R., “Recent progress of residual stress measurement methods: A review”, Chinese Journal of Aeronautics, 34:2 (2021), 54–78 | DOI

[6] Uzun F., Korsunsky A. M., “The use of eigenstrain theory and fuzzy techniques for intelligent modeling of residual stress and creep relaxation in welded superalloys”, Materials Today: Proceedings, 33:4 (2020), 1880–1883 | DOI

[7] Ma W., Zhang H., Zhu W., Xu F., Yang C., “Study on Residual Stress of Welded Hoop Structure”, Applied Sciences, 10:8 (2021), 2838 | DOI

[8] Li N., Zhang M., Ye J.-L., Liu C., “Experimental investigation on residual stress distribution in zirconium/titanium/steel tri-metal explosively welded composite plate after cutting and welding of a cover plate”, Journal of Manufacturing Processes, 64 (2021), 55–63 | DOI

[9] Yi S., Wu Y., Gong H., Peng C., He Y., “Experimental Analysis and Prediction Model of Milling-Induced Residual Stress of Aeronautical Aluminum Alloys”, Applied Sciences, 11:13 (2021), 5881 | DOI

[10] Huang C., Wang L., Wang K., “Residual stress identification in thin plates based on modal data and sensitivity analysis”, International Journal of Solids and Structures, 236–237:4 (2022), 111350 | DOI | MR

[11] Nedin R. D., Vatulyan A. O., Bogachev I. V., “Direct and inverse problems for prestressed functionally graded plates in the framework of the Timoshenko model”, Mathematical Methods in the Applied Sciences, 41:4 (2018), 1600–1618 | DOI | MR | Zbl

[12] Nedin R. D., “Modeling and frequency analysis of prestressed functionally graded plates with holes”, Computational Continuum Mechanics, 12:2 (2019), 192–201 | DOI

[13] Bogachev I. V., “Determination of Prestress in Circular Inhomogeneous Solid and Annular Plates in the Framework of the Timoshenko Hypotheses”, Applied Sciences, 11:21 (2021), 9819 | DOI

[14] Nedin R. D., Vatulyan A. O., “Inverse Problem of Non-homogeneous Residual Stress Identification in Thin Plates”, International Journal of Solids and Structures, 50:13 (2013), 2107–2114 | DOI

[15] Zhukov M. Yu., Shiryaeva E. V., Using the FreeFem++ Finite Element Package for Problems in Fluid Dynamics, Electrophoresis and Biology, Southern Federal University Publ, Rostov-on-Don, 2008, 256 pp. (in Russian)

[16] Vatulyan A. O., Coefficient inverse problems of mechanics, Fizmatlit, M., 2019, 272 pp. (in Russian)

[17] Bogachev I. V., Vatulyan A. O., Dudarev V. V., Lapina P. A., Nedin R. D., “Identification of Properties of Inhomogeneous Plate in the Framework of the Timoshenko Model”, Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 17:4 (2017), 419–430 (in Russian) | DOI | MR

[18] Nedin R. D., Vatulyan A. O., Dudarev V. V., Bogachev I. V., “Detection of nonuniform residual strain in a pipe”, International Journal of Solids and Structures, 139–140 (2018), 121–128 | DOI

[19] Tihonov A. N., Arsenin V. Ya., Methods of ill-posed problems solving, Nauka, M., 1986, 288 pp. (in Russian)

[20] Vatulyan A. O., Bogachev I. V., “The projection method for identification of the characteristics of inhomogeneous solids”, Doklady Physics, 63:2 (2018), 82–85 | DOI | DOI | MR | MR