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Axisymmetric thermoelastic contact of an FGM-coated half-space under a rotating punch

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Abstract

This paper investigates the axisymmetric thermoelastic contact of a functionally graded material (FGM) coated half-space. A rigid insulated punch rotates on the surface of the FGM coating with a constant angular velocity. The frictional heating is generated within the contact region. The thermoelastic properties of the coating vary exponentially along the thickness direction. By using the Hankel integral transform, the problem is reduced to Cauchy singular integral equations, which are solved numerically to obtain the normal contact stress, radial stress and surface temperature. The effects of the friction coefficient, angular velocity, gradient index, coating thickness and punch geometry on the surface thermoelastic contact characteristics are investigated and discussed in detail.

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Acknowledgements

The work described in this paper was supported by the National Natural Science Foundation of China under Grant Nos.11725207, 12021002 and 12011530056. Jing Liu gratefully acknowledges the China Scholarship Council (CSC) for supporting her visiting scholarship at the Chair of Structural Mechanics, Department of Civil Engineering, School of Science and Technology, University of Siegen, Germany.

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Authors and Affiliations

  1. College of Engineering, Huazhong Agricultural University, Wuhan, 430070, China

    Jing Liu

  2. Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing, 100044, China

    Liaoliang Ke

  3. Department of Civil Engineering, University of Siegen, D-57068, Siegen, Germany

    Jing Liu & Chuanzeng Zhang

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  1. Jing Liu
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  2. Liaoliang Ke
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  3. Chuanzeng Zhang
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Correspondence to Liaoliang Ke.

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Appendix A

Appendix A

1.1 Components of the matrices \(\left[ {M(s,h)} \right]\) and \(\left[ {H(s,h)} \right]\)

The components of the matrices \(\left[ {M(s,h)} \right]\) and \(\left[ {H(s,h)} \right]\) in Eq. (29) are given by the following expressions:

$$ M_{1i} = e^{{ - m_{i} h}} ,\;M_{15} = - e^{ - sh} ,\;M_{16} = he^{ - sh}, $$
(48)
$$ M_{2i} = a_{i} {\text{e}}^{{ - m_{i} h}} ,\;M_{25} = e^{ - sh} ,\;M_{26} = - [{{(3 - 4\nu_{2} )} \mathord{\left/ {\vphantom {{(3 - 4\nu_{2} )} s}} \right. \kern-\nulldelimiterspace} s} + h]e^{ - sh}, $$
(49)
$$ M_{3i} = \frac{{{\text{e}}^{{ - m_{i} h}} }}{{1 - 2\nu_{1} }}[(1 - \nu_{1} )a_{i} m_{i} + \nu_{1} s],\;M_{35} = {{se^{(\beta - s)h} \mu_{2} } \mathord{\left/ {\vphantom {{se^{(\beta - s)h} \mu_{2} } {\mu_{1} }}} \right. \kern-\nulldelimiterspace} {\mu_{1} }}, $$
(50)
$$ M_{36} = {{ - (2 - 2\nu_{2} + hs)e^{(\beta - s)h} \mu_{2} } \mathord{\left/ {\vphantom {{ - (2 - 2\nu_{2} + hs)e^{(\beta - s)h} \mu_{2} } {\mu_{1} }}} \right. \kern-\nulldelimiterspace} {\mu_{1} }}, $$
(51)
$$ M_{4i} = (m_{i} - a_{i} s){\text{e}}^{{ - m_{i} h}} ,\;M_{45} = {{ - 2se^{(\beta - s)h} \mu_{2} } \mathord{\left/ {\vphantom {{ - 2se^{(\beta - s)h} \mu_{2} } {\mu_{1} }}} \right. \kern-\nulldelimiterspace} {\mu_{1} }},\;M_{36} = {{ - 2(2\nu_{2} - 1 - hs)e^{(\beta - s)h} \mu_{2} } \mathord{\left/ {\vphantom {{ - 2(2\nu_{2} - 1 - hs)e^{(\beta - s)h} \mu_{2} } {\mu_{1} }}} \right. \kern-\nulldelimiterspace} {\mu_{1} }}, $$
(52)
$$ M_{6i} = \mu_{1} (m_{i} - a_{i} s),\;i = 1,2,3,4, $$
(53)
$$ M_{55} = M_{56} = M_{65} = M_{66} = 0, $$
(54)
$$ H_{1l} = \frac{{1 + \nu_{1} }}{{1 - \nu_{1} }}\alpha_{1} \frac{{d_{1l} }}{{d_{0l} }}e^{{ - n_{l} h}} ,\;H_{2l} = \frac{{1 + \nu_{1} }}{{1 - \nu_{1} }}\alpha_{1} \frac{{d_{2l} }}{{d_{0l} }}e^{{ - n_{l} h}} ,\;H_{13} = H_{23} = \frac{{ - \alpha_{2} (1 + \nu_{2} )}}{s}e^{ - sh}, $$
(55)
$$ H_{3l} = \frac{{1 + \nu_{1} }}{{1 - \nu_{1} }}\alpha_{1} \frac{{s^{2} (n_{l}^{2} - s^{2} )}}{{d_{0l} }}e^{{ - n_{l} h}} \;H_{4l} = \frac{{1 + \nu_{1} }}{{1 - \nu_{1} }}\alpha_{1} \frac{{2s(n_{l} + \beta )(s^{2} - n_{l}^{2} )}}{{d_{0l} }}e^{{ - n_{l} h}}, $$
(56)
$$ H_{5l} = \frac{{2\mu_{1} (1 + \nu_{1} )}}{{1 - \nu_{1} }}\alpha_{1} \frac{{s^{2} (n_{l}^{2} - s^{2} )}}{{d_{0l} }},\;H_{6l} = \frac{{\mu_{1} (1 + \nu_{1} )}}{{1 - \nu_{1} }}\alpha_{1} \frac{{2s(n_{l} + \beta )(s^{2} - n_{l}^{2} )}}{{d_{0l} }},\;l = 1,2, $$
(57)
$$ H_{33} = H_{43} = H_{53} = H_{63} = 0. $$
(58)

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Liu, J., Ke, L. & Zhang, C. Axisymmetric thermoelastic contact of an FGM-coated half-space under a rotating punch. Acta Mech 232, 2361–2378 (2021). https://doi.org/10.1007/s00707-021-02940-7

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