Inverse problem of thermoelectricity for a functionally graded layer

A. O. Vatulyan   ; S. A. Nesterov

Vladikavkazskij matematičeskij žurnal, Tome 26 (2024) no. 1, pp. 68-84

Résumé

The formulation of the inverse problem of identification the variable material characteristics of a transversely inhomogeneous thermoelectroelastic layer, the lower face of which is rigidly pinched, shorted and maintained at zero temperature, and an unsteady load is applied on the upper non-electrodated face. Using the Fourier transform, the two-dimensional inverse problem is reduced to a number of one-dimensional problems similar to those for an elastic and thermoelastic rod with modified characteristics. A step-by-step approach is proposed to identify the material characteristics of the layer. Dimensionless direct problems after applying the Laplace transform are solved on the basis of the apparatus of Fredholm integral equations of the 2nd kind and the conversion of transformants based on the theory of residues. Using the linearization method, operator equations of the 1st kind are obtained to solve the inverse problems at each stage. Computational experiments have been carried out to reconstruct the material characteristics of a thermoelectroelastic layer, both in the absence of noise input information and at 1% noise. Effective time intervals for the identification of additional information have been identified. The analysis of the results of the identification of the thermomechanical characteristics of the layer is carried out.

Bibliographie

[1] “Fine-Film Captive Pressure and Temperature”, Sensors and Systems, 58:3(201) (2016), 50–56 (in Russian)

[2] Tauchert T. R., Ashida F., Noda N., “Developments in thermopiezoelasticity with relevance to smart composite structures”, Compos. Struct., 48:1–3 (2000), 31–38 | DOI

[3] Rao S. S., Sunar M., “Analysis of distributed thermopiezoelectric sensors and actuators in advanced intelligent structures”, AIAA Journal, 93:7 (1993), 1280–1286 | DOI

[4] Mindlin R. D., “On the equations of motion of piezoelectric crystals”, Probl. Contin. Mech. SIAM, 1961, 282–290 | DOI | MR

[5] Mindlin R. D., “Equations of high frequency, vibrations of thermopiezoelectric crystal plates”, Int. J. Solid. Struct., 10:6 (1974), 625–637 | DOI | Zbl

[6] “Plane Waves and Fundamental Solutions in Linear Thermoelectroelasticity”, Journal of Applied Mechanics and Technical Physics, 37 (1996), 727–733 | DOI | MR | Zbl

[7] Vatulyan, A. O., “Thermal Shock on the Thermoelectroelastic Layer”, Vestnik Donskogo Gosudarstvennogo Tekhnichescogo Universiteta, 1(7):1 (2001), 82–89 (in Russian)

[8] “Applied Theory of Flexural Vibrations of a Piezoactive Bimorph in the Framework of an Uncoupled Boundary-Value Problem of Thermoelectroelasticity”, Contemporary Mathematics Fundamental Directions, 69, no. 2, 2023, 364–374 (in Russian) | DOI

[9] Zhao X., Iegaink F. J. N., Zhu W. D., Li Y .H., “Coupled thermo-electro-elastic forced vibrations of piezoelectric laminated beams by means of Green's functions”, Int. J. Solid. Struct., 156:9 (2019), 355–369 | DOI

[10] Shen S., Kuang Z. B., “An active control model of laminated piezothermoelastic plate”, Int. J. Solids and Struct., 36:13 (1999), 1925–1947 | DOI | Zbl

[11] Belokon, A. V. and Nasedkin, A. V., “Calculation of Some Types of Thermoelectroelastic Problems Using the ANSYS and ASELAN Packages”, Izvestiya vuzov, Severo-Kavkazskiy region, 2004, spetsial'nyy vypusk, 52–55 (in Russian) | Zbl

[12] Wang B. L., Noda N., “Design of a smart functionally graded thermopiezoelectric composite structure”, Smart Mater. Struct., 10 (2001), 189–193 | DOI

[13] Ying C., Zhifei S., “Exact Solutions of Functionally Gradient Piezothermoelastic Cantilevers and Parameter Identification”, J. Intell. Mater. Syst. Struct., 16:6 (2005), 531–539 | DOI

[14] Soloviev A. N., Chebanenko V. A., Oganesyan P. A., Chao S. F., Liu Y. M., “Applied theory for electroelastic plates with non-homogeneous polarization”, Mater. Phys. Mech., 42:2 (2019), 242–255 | DOI

[15] Vatulyan, A. O. and Nesterov, S. A., “Dynamic Problem of Thermoelectroelasticity for Functionally Graded Layer”, Computational Continuum Mechanics, 10:2 (2017), 117–126 (in Russian) | DOI

[16] Belyankova, T. I. and Kalinchuk, V. V., “On the Modeling of a Prestressed Thermoelastic Half-Space with a Coating”, Mechanics of Solids, 52:1 (2017), 95–110 | DOI

[17] Vatulyan, A. O. and Nesterov, S. A., “Identification of Inhomogeneous Characteristics of Prestressed Pyromaterials”, Chebyshevskii Sbornik, 19:2 (2018), 183–198 (in Russian) | DOI | MR | Zbl

[18] Vatulyan A., Nesterov S., Nedin R., “Some features of solving an inverse problem on identification of material properties of functionally graded pyroelectrics”, Int. J. Heat Mass Transfer., 128 (2019), 1157–1167 | DOI

[19] Vatulyan, A. O. and Nesterov, S. A., Coefficient Inverse Problems of Thermomechanics, 2nd ed., Southern Federal University Press, Rostov-on-Don–Taganrog, 2022, 178 pp. (in Russian) | MR

[20] Vatul'yan, A. O. and Nesterov, S. A., “Study of the Inverse Problems of Thermoelasticity for Inhomogeneous Materials”, Siberian Mathematical Journal, 64:3 (2023), 699–706 | DOI | DOI | MR | MR | Zbl | Zbl

[21] Vatulyan, A. O. and Nesterov, S. A., “Solution of the Inverse Problem of Two Thermomechanical Characteristics Identification of a Functionally Graded Rod”, Izvestiya of SaratovUniversity. Mathematics. Mechanics. Informatics, 22:2 (2022), 180–195 (in Russian) | DOI | MR | Zbl

[22] Vatul'yan, A. O. and Uglich, P. S., “Reconstruction of Inhomogeneous Characteristics of a Transverse Inhomogeneous Layer in Antiplane Vibrations”, Journal of Applied Mechanics and Technical Physics, 55:3 (2014), 499–505 | DOI | MR | Zbl

[23] Vatul'yan, A. O., Bogachev, I. V. and Yavruyan, O. V., “Identifying the Inhomogeneous Properties of an Orthotropic Elastic Layer”, Acoustical Physics, 59:6 (2013), 702–708 | DOI | DOI

[24] “Numerical-Analytical Solution of the Inverse Coefficient Problem of Thermoelasticity for a Plate”, Vestnik Moskovskogo Aviatsionnogo Instituta, 16:6 (2009), 244–249 (in Russian) | MR

[25] Lukasievicz S. A., Babaei R., Qian R. E., “Detection of material properties in a layered body by means of thermal effects”, J. Thermal Stresses, 26:1 (2003), 13–23 | DOI | MR

[26] Vatulyan, A. O. and Nesterov, S. A., “On One Approach to Identifying the Thermomechanical Characteristics of a Layered Biological Tissue”, Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2016, no. 2, 29–36 (in Russian)

[27] Vatulyan, A. O. and Nesterov, S. A., “Numerical Implementation of an Iterative Scheme for Solving Inverse Problems of Thermoelasticity for Inhomogeneous Bodies with Coatings”, Computational Technologies, 22:5 (2017), 14–26 (in Russian) | Zbl

[28] Inverse Coefficient Problems for Differential Equations of Elasticity, Nauka, Novosibirsk, 1990, 304 pp. (in Russian) | MR

[29] Tikhonov, A. N., Goncharskiy, A. V., Stepanov, V. V. and Yagola, A. G., Numerical Methods for Solving Ill-Posed Problems, Nauka, M., 1990, 230 pp. (in Russian) | MR

[30] Raddy J. N., Chin C. D., “Thermoelastic analysis of functionally graded cylinders and plates”, J. Thermal Stresses, 21 (1998), 593–626 | DOI

[31] Babaei M. H., Chen Z. T., “The transient coupled thermo-piezoelectric hollow cylinder to dynamic loadings response of a functionally graded piezoelectric”, Proc. R. Soc. A, 466:2116 (2010), 1077–1091 | DOI | Zbl