Abstract
In this research, we present an approach to identify variable characteristics of an inhomogeneous thermoelectroelastic radially polarized elongated hollow cylinder. The cylinder’s thermomechanical characteristics depend on the radial coordinate. We consider two loading types for the cylinder—the mechanical and the thermal ones. The radial displacement is considered as the additional data collected on the outer cylinder’s surface under the first type load, while the temperature measured over a certain time interval is considered for the second type load. The direct problem after non-dimensioning and applying the Laplace transform is solved by jointly applying the shooting method and the transform inversion based on expanding the actual space in terms of the shifted Legendre polynomials. The effect of the laws of change in variable characteristics on the input data values taken in the experiment is analyzed. A nonlinear inverse problem on the reconstruction of the cylinder’s variable properties is formulated and solved on the basis of an iterative technique. The initial approximation is set in the class of positive bounded linear functions whose coefficients are determined from the condition of minimizing the residual functional. To find the corrections at each stage of the iterative process, the Fredholm integral equations of the first kind are solved by means of the Tikhonov method. A series of computational experiments on recovering one and two variable characteristics is conducted. The effect of coupling parameters and input noise on the reconstruction results is revealed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- \({\bar{a}}\) :
-
General designation of dimensionless thermomechanical characteristics
- \({\bar{a}}^{0} \) :
-
Initial approximation for the dimensionless thermomechanical characteristic
- \([a_{1},a_{2} ], [b_{1},b_{2} ]\) :
-
Dimensionless time intervals of the additional data measured under thermal and mechanical loads, respectively
- \(c_{\epsilon } \) :
-
Specific volumetric heat capacity at a constant strain tensor
- \({\bar{c}}\) :
-
Dimensionless specific volumetric heat capacity
- \(c_{ijkl} \) :
-
Elasticity tensor components
- \({\bar{c}}_{11} \), \({\bar{c}}_{13} \), \({\bar{c}}_{33} \) :
-
Dimensionless elasticity tensor components
- \(D_{i} =e_{ijkl} u_{k,l} - \epsilon _{ik} \varphi _{,k} +g_{i} \theta \) :
-
Components of the electric displacement vector
- \({\bar{D}}_{r} \) :
-
Dimensionless radial component of the electric displacement vector
- \(\sigma _{ij} =c_{ijkl} u_{k,l} +e_{kij}\varphi _{,k} -\gamma _{ij} \theta \) :
-
Stress tensor components
- \(e_{kij} \) :
-
Piezoelectric tensor components,
- \({\bar{e}}_{31} \), \({\bar{e}}_{33} \) :
-
Dimensionless piezoelectric tensor components
- \(g_{i} \) :
-
Pyroelectric vector
- \({\bar{g}}_{3} \) :
-
Dimensionless component of the pyroelectric vector
- \(H\left( \tau _{1} \right) \) :
-
The Heaviside function
- J :
-
Dimensionless residual functional in the reconstruction of 2 characteristics
- \(J_{1} \), \(J_{2} \) :
-
Dimensionless residual functionals under thermal and mechanical loads, respectively
- \(k_{ij} \) :
-
Thermal conductivity coefficients
- \({\bar{k}}\) :
-
Dimensionless component of the thermal conductivity tensor
- m :
-
Inhomogeneity parameter
- n :
-
Number of iterations
- p :
-
The dimensionless Laplace transform parameter
- \(p_{i} \) :
-
Mechanical load vector components
- \(P_{s}^{*} \) :
-
The shifted Legendre polynomials
- q :
-
Heat flux density
- Q :
-
Dimensionless heat flux density
- \(U_{r} \) :
-
Dimensionless radial displacement
- W :
-
Dimensionless temperature
- \(\beta \) :
-
Noise amplitude
- \(\beta _{1} \), \(\beta _{2} \) :
-
Dimensionless amplitudes of thermal and mechanical loads, respectively
- \(\gamma _{0} \) :
-
Random variable with a uniform distribution law on the interval \([-1,1]\)
- \(\gamma _{ij} \) :
-
Thermal stress tensor components
- \({\bar{\gamma }}_{1} \), \({\bar{\gamma }}_{3} \) :
-
Dimensionless components of the thermal stress vector
- \(\epsilon _{ij} \) :
-
Permittivity tensor components
- \({\bar{\epsilon }}_{3} \) :
-
Dimensionless component of the permittivity vector
- \(\varepsilon _{0} \) :
-
Ratio of characteristic times of sound (\(t_2\)) and thermal (\(t_1\)) perturbations
- \(\chi _{1} \) :
-
Dimensionless electromechanical coupling parameter
- \(\chi _{2} \) :
-
Dimensionless thermomechanical coupling parameter
- \(\chi _{3} \) :
-
Dimensionless thermoelectric coupling parameter
- \(\varphi \) :
-
Electric potential
- \(\varphi \left( \tau _{1} \right) \), \(\phi \left( \tau _{2} \right) \) :
-
dimensionless laws of change for thermal and mechanical loads, respectively
- \(\Phi \) :
-
Dimensionless electric potential,
- \(\theta \) :
-
Increment of body’s temperature from its natural state with the temperature \(T_{0} \)
- \(\rho \) :
-
Density
- \({\bar{\rho }}\) :
-
Dimensionless density
- \(\Omega _{rr} \), \(\Omega _{\varphi \varphi } \) :
-
dimensionless radial and circumferential components of the stress tensor
- \(\tau _{1} \), \(\tau _{2} \) :
-
Dimensionless time under thermal and mechanical load, respectively,
- \(\xi \) :
-
Dimensionless radial coordinate
- \(\xi _{0} \) :
-
Dimensionless value of cylinder’s inner radius
References
Tauchert, T.R., Ashida, F., Noda, N., Adali, S., Verijenko, V.E.: Developments in thermopiezoelasticity with relevance to smart composite structures. Compos. Struct. 48, 31–38 (2000)
Rao, S.S., Sunar, M.: Analysis of distributed thermopiezoelectric sensors and actuators in advanced intelligent structures. AIAA J. 7, 1280–1286 (1993)
Mindlin, R.D.: On the equations of motion of piezoelectric crystals. Problems in Continuum Mechanics, SIAM, pp. 282–290 (1961)
Mindlin, R.D.: Equations of high frequency, vibrations of thermopiezoelectric crystal plates. Int. J. Solids Struct. 10, 625–637 (1974)
Paul, H.S., Raman, G.V.: Wave propagation in a hollow pyroelectric circular cylinder of crystal class 6. Acta Mech. 87, 37–46 (1991)
Ding, H.J., Wang, H.M., Ling, D.S.: Analytical solution of a pyroelectric hollow cylinder for piezothermoelastic axisymmetric dynamic problems. J. Therm. Stress 26, 261–276 (2003)
Shlyakhin, D.A., Kalmova, M.A.: The coupled non-stationary thermo-electro-elasticity problem for a long hollow cylinder. Vestnik Samarskogo Gosudarstvennogo Tekhnichescogo Universiteta-Seriya-Fiziko-Matematicheskiye Nauki 24(4), 677–691 (2020). (in Russian)
Wang, B.L., Noda, N.: Design of a smart functionally graded thermopiezoelectric composite structure. Smart. Mater. Struct. 10, 189–193 (2001)
Marin, M., Öchsner, A.: The effect of a dipolar structure on the Hölder stability in Green–Naghdi thermoelasticity. Contin. Mech. Thermodyn. 29(6), 1365–1374 (2017)
Chirilă, A., Marin, M., Montanaro, A.: On adaptive thermo-electro-elasticity within a Green–Naghdi type II or III theory. Contin. Mech. Thermodyn. 31(5), 1453–1475 (2019)
Vatulyan, A., Nesterov, S., Nedin, R.: Regarding some thermoelastic models of “coating-substrate’’ system deformation. Contin. Mech. Thermodyn. 32(4), 1173–1186 (2020)
Abo-Dahab, S.M., Abouelregal, A.E., Marin, M.: Generalized thermoelastic functionally graded on a thin slim strip non-gaussian laser beam. Symmetry 12(7), 1–16 (2020)
Ootao, Y., Akai, T., Tanigawa, Y.: Transient piezothermoelastic analysis for a functionally graded thermopiezoelectric hollow cylinder. J. Therm. Stress 31(10), 935–955 (2008)
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. Lond. Ser. A 466, 1077–1091 (2010)
Shahani, A.R., Sharifi Torki, H.: Determination of the thermal stress wave propagation in orthotropic hollow cylinder based on classical theory of thermoelasticity. Contin. Mech. Thermodyn. 30(3), 509–527 (2018)
Huang, H., Rao, D.: Thermal buckling of functionally graded cylindrical shells with temperature-dependent elastoplastic properties. Contin. Mech. Thermodyn. 32(5), 1403–1415 (2018)
Ashida, F., Tauchert, T.R., Sakata, S.I., Igi, Y.: Inverse problem of a piezothermoelastic cylinder subject to transient heating. J. Therm. Stress 33(7), 706–718 (2010)
Ying, C., Zhifei, S.: Exact solutions of functionally gradient piezothermoelastic cantilevers and parameter identification. J. Intell. Mater. Syst. Struct., 1–10 (2005)
Chen, Y., Shi, Z.-F.: Analysis of a functionally graded piezothermoelastic hollow cylinder. J. Zhejiang Univ. Sci. A 6, 956–961 (2005)
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 Transf. 128, 1157–1167 (2019)
Vatulyan, A.O., Nesterov, S.A.: On determination of inhomogeneous thermomechanical characteristics of a pipe. J. Eng. Phys. Thermophys. 88(4), 984–993 (2015)
Nedin, R.D., Nesterov, S.A., Vatulyan, A.O.: Concerning identification of two thermomechanical character. advanced structured materials. Advanced Structured Materials. Solid Mechanics, Theory of Elasticity and Creep 185, 247–264 (2023)
Vatulyan, A.O., Nesterov, S.A.: On determination of the thermomechanical characteristics of a functionally graded finite cylinder. Mech. Solids 56(7), 1429–1438 (2021)
Krylov, V.I., Skoblya, N.S.: Methods for the Approximate Fourier Transform and the Laplace Transform Inversion. Nauka, Moscow (1974). ([in Russian])
Tikhonov, A.N., Goncharsky, A.V., Stepanov, V.V., Yagola, A.G.: Numerical Methods for the Solution of Ill-posed Problems. Kluwer, Dordrecht (1995)
Acknowledgements
The study was supported by the Russian Science Foundation, Grant No. 22-11-00265, https://rscf.ru/project/22-11-00265/, at the Southern Federal University.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Vatulyan, A., Nesterov, S. & Nedin, R. Variable properties reconstruction for functionally graded thermoelectroelastic cylinder. Continuum Mech. Thermodyn. (2024). https://doi.org/10.1007/s00161-024-01292-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00161-024-01292-6