On the development of an approximate theory of direct and inverse problem for an inhomogeneous rectangular region
DOI:
https://doi.org/10.7242/1999-6691/2026.19.1.8Keywords:
plane region, material properties, elastic modulus, compliance, approximate theory, Airy stress function, inverse problem, reconstruction, beam, Timoshenko model, FEMAbstract
The paper is devoted to the development of an approximate theory for solving direct and inverse problems as applied to an inhomogeneous planar domain. The domain is statically deformed under the action of a certain mechanical load. In the direct problem, given the known law of inhomogeneity of material parameters, boundary conditions, and the geometry of the domain, it is required to determine the displacement field. In the inverse problem, it is necessary to reconstruct the law of inhomogeneity of the elastic characteristic using additional data on the measured displacements of a certain unloaded part of the domain’s boundary. Most often, such problems for inhomogeneous bodies are solved numerically – for example, using the finite element method (FEM). However, from the perspective of solution analysis and their application to inverse problems, greater interest lies in: 1) developing simplified models and 2) finding approximate analytical solutions. An approach is proposed for constructing analytical expressions for the components of the two‑dimensional displacement field in the direct problem by introducing a stress function. A comparative analysis is carried out of the solutions obtained using the proposed approximate theory, the Bernoulli–Euler and Timoshenko 1D beam models, and the FEM in a 2D implementation. The comparison demonstrated good accuracy of the obtained analytical representations for various aspect ratios of rectangular domains and inhomogeneity laws. Based on the developed approximate theory and the Timoshenko beam model, an approach is proposed for solving the inverse problem of reconstructing the 1D law of inhomogeneity for the elastic characteristic. Explicit formulas are derived for determining the sought characteristics through the displacement functions specified on one face of a rectangular domain. The results of computational experiments are presented, demonstrating the effectiveness of the proposed approaches for solving direct and inverse problems.
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Lomakin V.A. Teoriya uprugosti neodnorodnykh tel. Moscow: URSS, LENAND, 2014. 367 p.
Vatul’yan A.O. Koeffitsiyentnyye obratnyye zadachi mekhaniki. Moscow: Fizmatlit, 2019. 272 p.
Sobhy M., Zenkour A.M. Porosity and inhomogeneity effects on the buckling and vibration of double-FGM nanoplates via a quasi-3D refined theory. Composite Structures. 2019. Vol. 220. P. 289–303. DOI: 10.1016/j.compstruct.2019.03.096
Shafiei N., Mirjavadi S.S., MohaselAfshari B., Rabby S., Kazemi M. Vibration of two-dimensional imperfect functionally graded (2D-FG) porous nano-/micro-beams. Computer Methods in Applied Mechanics and Engineering. 2017. Vol. 322. P. 615–632. DOI: 10.1016/J.CMA.2017.05.007
Ebrahimi F., Hosseini S.H.S. Resonance analysis on nonlinear vibration of piezoelectric/FG porous nanocomposite subjected to moving load. The European Physical Journal Plus. 2020. Vol. 135, no. 2. P. 1–23. DOI: 10.1140/epjp/s13360-019-00011-4
Constantinides G., Ravi Chandran K.S., Ulm F.-J., Van Vliet K.J. Grid indentation analysis of composite microstructure and mechanics: Principles and validation. Materials Science and Engineering: A. 2006. Vol. 430, no. 1/2. P. 189–202. DOI: 10.1016/J.MSEA.2006.05.125
Randall N.X., Vandamme M., Ulm F.-J. Nanoindentation analysis as a two-dimensional tool for mapping the mechanical properties of complex surfaces. Journal of Materials Research. 2009. Vol. 24, no. 3. P. 679–690. DOI: 10.1557/jmr.2009.0149
Brown L., Allison P.G., Sanchez F. Use of nanoindentation phase characterization and homogenization to estimate the elastic modulus of heterogeneously decalcified cement pastes. Materials & Design. 2018. Vol. 142. P. 308–318. DOI: 10.1016/j.matdes.2018.01.030
Vignesh B., Oliver W., Siva Kumar G., Sudharshan Phani P. Critical assessment of high speed nanoindentation mapping technique and data deconvolution on thermal barrier coatings. Materials & Design. 2019. Vol. 181. P. 108084. DOI: 10.1016/j.matdes.2019.108084
Ariza-Echeverri E.A., Masoumi M., Nishikawa A.S., Mesa D.H., Marquez-Rossy A.E., Tschiptschin A.P. Development of a new generation of quench and partitioning steels: Influence of processing parameters on texture, nanoindentation, and mechanical properties. Materials & Design. 2020. Vol. 186. P. 108329. DOI: 10.1016/j.matdes.2019.108329
Qiu L., Lin J., Wang F., Qin Q.-H., Liu C.-S. A homogenization function method for inverse heat source problems in 3D functionally graded materials. Applied Mathematical Modelling. 2021. Vol. 91. P. 923–933. DOI: 10.1016/J.APM.2020.10.012
Pasha A., Rajaprakash B.M. Fabrication and mechanical properties of functionally graded materials: A review. Materials Today: Proceedings. 2022. Vol. 52. P. 379–387. DOI: 10.1016/j.matpr.2021.09.066
Rodríguez-Castro R., Wetherhold R.C., Kelestemur M.H. Microstructure and mechanical behavior of functionally graded Al A359/SiCp composite. Materials Science and Engineering: A. 2002. Vol. 323. P. 445–456. DOI: 10.1016/S0921-5093(01)01400-9
Khoddami A.M., Sabour A., Hadavi S.M.M. Microstructure formation in thermally-sprayed duplex and functionally graded NiCrAlY/Yttria-Stabilized Zirconia coatings. Surface and Coatings Technology. 2007. Vol. 201. P. 6019–6024. DOI: 10.1016/j.surfcoat.2006.11.020
Yang C., Jin G., Ye X., Liu Z. A modified Fourier–Ritz solution for vibration and damping analysis of sandwich plates with viscoelastic and functionally graded materials. International Journal of Mechanical Sciences. 2016. Vol. 106. P. 1–18. DOI: 10.1016/j.ijmecsci.2015.11.031
Mirjavadi S.S., Afshari B.M., Shafiei N., Hamouda A.M.S., Kazemi M. Thermal vibration of two-dimensional functionally graded (2D-FG) porous Timoshenko nanobeams. Steel Compos. Struct. 2017. Vol. 25, no. 4. P. 415–426. DOI: 10.12989/scs.2017.25.4.415
Ebrahimi-Nejad S., Shaghaghi G.R., Miraskari F., Kheybari M. Size-dependent vibration in two-directional functionally graded porous nanobeams under hygro-thermo-mechanical loading. The European Physical Journal Plus. 2019. Vol. 134, no. 9. P. 465. DOI: 10.1140/epjp/i2019-12795-6
Torabi J., Ansari R. Crack propagation in functionally graded 2D structures: A finite element phase-field study. Thin-Walled Structures. 2020. Vol. 151. P. 106734. DOI: 10.1016/j.tws.2020.106734
Akamatsu M., Nakamura G., Steinberg S. Identification of Lame coefficients from boundary observations. Inverse Problems. 1991. Vol. 7. P. 335–354. DOI: 10.1088/0266-5611/7/3/003
Imanuvilov O.Y., Yamamoto M. On reconstruction of Lame coefficients from partial Cauchy data. Journal of Inverse and Ill-posed Problems. 2011. Vol. 19. P. 881–891.
Imanuvilov O.Y., Uhlmann G., Yamamoto M. On uniqueness of Lamé coefficients from partial Cauchy data in three dimensions. Inverse Problems. 2012. Vol. 28. P. 125002. DOI: 10.1088/0266-5611/28/12/125002
Vatulyan A.O., Dudarev V.V., Mnukhin R.M., Nedin R.D. Identification of the Lamé parameters of an inhomogeneous pipe based on the displacement field data. European Journal of Mechanics - A/Solids. 2020. Vol. 81. P. 103939. DOI: 10.1016/j.euromechsol.2019.103939
Dudarev V.V., Vatulyan A.O., Mnukhin R.M., Nedin R.D. Concerning an approach to identifying the Lamé parameters of an elastic functionally graded cylinder. Mathematical Methods in the Applied Sciences. 2020. Vol. 43, no. 11. P. 6861–6870. DOI: 10.1002/mma.6428
Bogachev I.V., Nedin R.D., Vatulyan A.O., Yavruyan O.V. Identification of inhomogeneous elastic properties of isotropic cylinder. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik. 2017. Vol. 97, no. 3. P. 358–364. DOI: 10.1002/zamm.201600179
Vatulyan A., Nesterov S., Nedin R. Variable properties reconstruction for functionally graded thermoelectroelastic cylinder. Continuum Mechanics and Thermodynamics. 2024. Vol. 36. P. 745–762. DOI: 10.1007/s00161-024-01292-6
Vatulyan A.O. On Some Approaches to the Study of Coefficient Inverse Problems of Mechanics with Variable Characteristics. Mechanics of Solids. 2024. Vol. 59, no. 6. P. 3417–3448. DOI: 10.1134/S0025654424605767
Vatulyan A.O., Yurov V.O. On a new approach to identifying inhomogeneous mechanical properties of elastic bodies. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024. Vol. 24, no. 2. P. 209–221. DOI: 10.18500/1816-9791-2024-24-2-209-221
Demidov S.P. Teoriya uprugosti. Moscow: Vysshaya shkola, 1979. 432 p.
Zubov L.M., Karyakin M.I. Tenzornoye ischisleniye. Osnovy teorii. Moscow: Vuzovskaya kniga, 2006. 120 p.
Timoshenko S.P. Kurs teorii uprugosti. Kyiv: Naukova dumka, 1972. 508 p.
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