# Inverse Thermoelastic Analysis of a Cylindrical Tribo-Couple

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Problem

## 3. Solution Technique

## 4. Numerical Example and Discussion

^{2}/s], $\alpha {\text{}}_{T}^{(1)}=14\times {10}^{-6}$ [1/K], ${E}_{1}=190$ [GPa], and ${\nu}_{1}=0.3$) and the outer one 2 is made of copper (${\lambda}_{2}=381$ [Wt/(m × K)], $a{\text{}}_{2}=101.9\times {10}^{-6}$ [m

^{2}/s], $\alpha {\text{}}_{T}^{(2)}=17\times {10}^{-6}$ [1/K], ${E}_{2}=121$ [GPa], and ${\nu}_{2}=0.33$).

^{2}× K/Wt], $R{\text{}}_{\ast}=1.1\times {10}^{-3}$ [m

^{2}× K/Wt]; ${\sigma}_{\ast}={10}^{2}$ [MPa], ${T}_{0}=20$ [K], $B\text{}=200$ [K], $C={10}^{2}$, $f=0.25,$ $\omega =1.22$ [rad/s], and $\tau {\text{}}_{m}=2.5$.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Tokovyy, Y.V.; Ma, C.-C. The Direct Integration Method for Elastic Analysis of Nonhomogeneous Solids; Cambridge Scholars Pub.: Cambridge, UK, 2021; 342p. [Google Scholar]
- Dwivedi, S.D.; Vishwakarma, M.; Soni, A. Advances and researches on non destructive testing: A review. Mater. Today Proc.
**2018**, 53690–53698. [Google Scholar] [CrossRef] - Wang, B.; Zhong, S.; Lee, T.-L.; Fancey, K.S.; Mi, J. Non-destructive testing and evaluation of composite materials/structures: A state-of-the-art review. Adv. Mech. Eng.
**2000**, 12, 1–28. [Google Scholar] [CrossRef] [Green Version] - Tokovyy, Y.V.; Ma, C.-C. Elastic analysis of inhomogeneous solids: History and development in brief. J. Mech.
**2019**, 35, 613–626. [Google Scholar] [CrossRef] - Alifanov, O.M. Inverse Heat Transfer Problems; Springer: Berlin, Germany, 1994; 348p. [Google Scholar]
- Beck, J.V.; Blackwell, B.; Clair, C.R. Inverse Heat Conduction: Ill-Posed Problems; Wiley: New York, NY, USA, 1985; 308p. [Google Scholar]
- Ootao, Y. Inverse problem of thermal deformation in a cylinder. In Encyclopedia of Thermal Stresses; Hetnarski, R.B., Ed.; Springer: Dordrecht, The Netherland, 2014; Volume 5, pp. 2578–2585. [Google Scholar] [CrossRef]
- Nowacki, W. Thermoelasticity; Pergamon Press: New York, NY, USA, 1962; 628p. [Google Scholar]
- Hetnarski, R.B.; Eslami, M.R. Thermal Stresses—Advanced Theory and Applications; Springer: Dordrecht, The Netherland, 2009; 560p. [Google Scholar]
- Kalynyak, B.M.; Tokovyy, Y.V.; Yasinskyy, A.V. Direct and inverse problems of thermomechanics concerning the optimization and identification of the thermal stressed state of deformed solids. J. Math. Sci.
**2019**, 236, 21–34. [Google Scholar] [CrossRef] - Chang, W.J.; Lee, H.L.; Yang, Y.C. Estimation of heat flux and thermal stresses in functionally graded hollow circular cylinders. J. Stresses
**2011**, 34, 740–755. [Google Scholar] [CrossRef] - Golbahar Haghighi, M.R.; Malekzadeh, P.; Afshari, M. Inverse estimation of heat flux and pressure in functionally graded cylinders with finite length. Compos. Struct.
**2015**, 121, 1–15. [Google Scholar] [CrossRef] - Grysa, K.; Maciag, A. Solving direct and inverse thermoelasticity problems by means of Trefftz base functions for finite element method. J. Stresses
**2011**, 34, 378–393. [Google Scholar] [CrossRef] - Kushnir, R.M.; Yasinskyy, A.V.; Tokovyy, Y.V. Reconstruction of thermal loading of a functionally-graded hollow sphere by the surface displacements. Math. Methods Phys. Mech. Fields
**2020**, 63, 149–160. (In Ukrainian) [Google Scholar] - Segall, A.E.; Engels, D.; Drapaca, C. Inverse determination of thermal boundary conditions from transient surface temperatures and strains in slabs and tubes. Mater. Manuf. Process.
**2012**, 27, 860–868. [Google Scholar] [CrossRef] - Yasinskii, A.V. Identification of thermal and thermostressed states of a two-layer cylinder from surface displacements. Int. Appl. Mech.
**2008**, 44, 34–40. [Google Scholar] [CrossRef] - Grylitskiy, D.V.; Pyryev, Y.A.; Mandzyk, Y.I. Quasistatic thermoelastic contact problem for infinite two layer circular cylinder under friction heating. J. Therm. Stresses
**1997**, 20, 47–65. [Google Scholar] [CrossRef] - Yevtushenko, A.; Kuciej, M.; Och, E. Theoretical nonlinear model of frictional heat generation in braking. Heat Transf. Res.
**2019**, 50, 1007–1022. [Google Scholar] [CrossRef] - Yevtushenko, A.; Kuciej, M.; Topczewska, K. Frictional heating during braking of the C/C composite disc. Materials
**2020**, 13, 2691. [Google Scholar] [CrossRef] [PubMed] - Yasinskyy, A.; Tokova, L. Inverse problem on the identification of temperature and thermal stresses in an FGM hollow cylinder by the surface displacements. J. Therm. Stresses
**2017**, 40, 1471–1483. [Google Scholar] [CrossRef] - Yasins’kyi, A.V. Reconstruction of temperature fields and thermal stresses for given displacements in the case of friction contact of layers. Mater. Sci.
**2002**, 38, 814–823. [Google Scholar] [CrossRef] - Yasins’kyi, A.V. Inverse problem of evaluation on the coefficient of friction of layers according to the data of measurements of the surface displacements. Mater. Sci.
**2003**, 39, 704–711. [Google Scholar] [CrossRef] - Eslami, M.R.; Hetnarski, R.B.; Ignaczak, J.; Noda, N.; Sumi, N.; Tanigawa, Y. Theory of Thermal Stresses. Explanations, Problems and Solutions; Springer: Dordrecht, The Netherlands, 2013; 789p. [Google Scholar]
- Farlow, S.J. Partial Differential Equations for Scientists and Engineers; Dover Publ.: New York, NY, USA, 1993; 414p. [Google Scholar]
- Corduneanu, C. Integral Equations and Applications; Cambridge Univ. Press: Cambridge, UK, 2008; 366p. [Google Scholar]
- Hudramovich, V.S.; Hart, E.L.; Marchenko, O.A. Reinforcing inclusion effect on the stress concentration within the spherical shell having an elliptical opening under uniform internal pressure. Strength Mater.
**2020**, 52, 832–842. [Google Scholar] [CrossRef] - Delves, L.M.; Mohamed, J.L. Computational Methods for Integral Equations; Cambridge Univ. Press: Cambridge, UK, 2008; 388p. [Google Scholar]
- Blau, P.J. Tribosystem Analysis: A Practical Apprach to the Diagnosis of Wear Problems; CRC Press: Boca Raton, FL, USA, 2016; 194p. [Google Scholar]
- Dykha, A.; Padgurskas, J.; Musial, J.; Matiukh, S. Wear Models and Diagnostics of Cylindrical Sliding Tribosystem; Foundation of Mechatronics Development: Bydgoszcz, Poland, 2020; 198p. [Google Scholar]

**Figure 1.**The scheme of the considered tribo-couple, where the inner and outer cylindrical layers are denoted by “1” and “2”, respectively; the thermal and force loadings are imposed on the inner and outer surfaces ${R}_{1}$ and ${R}_{2}$ and the frictional heating occurs on the interface ${R}_{0}$.

**Figure 2.**The circumferential elastic strain $\epsilon (\tau )={\epsilon}_{\phi \phi}^{(2)}({k}_{2},\tau )\times {10}^{4}$ (

**a**) computed by the temperature ${t}_{1}(\tau )$ given in (19) by solving the direct problem versus the dimensionless time $\tau $; the dimensionless temperature $t{\text{}}_{1}^{(i)}$ on the inner circumference (

**b**) as given by formula (19) (solid lines) and computed by solving the inverse problem (open circles).

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**MDPI and ACS Style**

Kushnir, R.; Yasinskyy, A.; Tokovyy, Y.; Hart, E.
Inverse Thermoelastic Analysis of a Cylindrical Tribo-Couple. *Materials* **2021**, *14*, 2657.
https://doi.org/10.3390/ma14102657

**AMA Style**

Kushnir R, Yasinskyy A, Tokovyy Y, Hart E.
Inverse Thermoelastic Analysis of a Cylindrical Tribo-Couple. *Materials*. 2021; 14(10):2657.
https://doi.org/10.3390/ma14102657

**Chicago/Turabian Style**

Kushnir, Roman, Anatoliy Yasinskyy, Yuriy Tokovyy, and Eteri Hart.
2021. "Inverse Thermoelastic Analysis of a Cylindrical Tribo-Couple" *Materials* 14, no. 10: 2657.
https://doi.org/10.3390/ma14102657