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A receding contact problem between a graded piezoelectric layer and a piezoelectric substrate

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Abstract

The receding contact problem between a graded piezoelectric layer and a homogeneous piezoelectric substrate is considered in this paper. It is assumed that the gradation of the elastic piezoelectric graded layer is of exponential type through its thickness. Using standard Fourier transform, the contact problem is converted to a system of two singular integral equations in which the contact pressures and the electric charge displacement in addition to the contact dimensions are the unknowns. The integral equations are then solved numerically using Gauss–Jacobi integration formula. The primary objective of this paper is to investigate the effect of material gradation on the contact pressure, electric charge distribution and on the length of the receding contact. The main findings of the paper are that the inhomogeneity parameter has a strong effect on the contact pressure and the electric charge distribution at the receding contact interface. It is concluded that a softer FGPM in \(+z\) direction results in lower contact pressure and electric displacement.

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Acknowledgements

The first author is grateful for the funding provided by Texas A&M University at Qatar. The authors are also grateful for the help of Mrs. Hedia Layouni El-Borgi in typesetting the Latex document.

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Authors and Affiliations

  1. Mechanical Engineering Program, Texas A&M University at Qatar, Engineering Building, P.O. Box 23874, Education City, Doha, Qatar

    Sami El-Borgi

  2. Department of Civil Engineering, Karadeniz Technical University, 61080, Ortahisar-Trabzon, Turkey

    Isa Çömez

  3. College of Engineering and Technology, American University of the Middle East, Egaila, Kuwait

    Mehmet Ali Güler

Authors
  1. Sami El-Borgi
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  2. Isa Çömez
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  3. Mehmet Ali Güler
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Correspondence to Sami El-Borgi.

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Appendix A

Appendix A

1.1 Expressions of quantities appearing in Eq. (17)

$$\begin{aligned} \alpha _6&= -c_{{330}}c_{{440}}\epsilon _{{330}}-c_{{440}}e_{330}^{2} \end{aligned}$$
(A.1a)
$$\begin{aligned} \alpha _5&= -3\,\beta \,c_{{330}}c_{{440}}\epsilon _{{330}}-3\,\beta \,c_{{440}}e_{{330}}^{2} \end{aligned}$$
(A.1b)
$$\begin{aligned} \alpha _4&=-3\,{\beta }^{2}c_{{330}}c_{{440}}\epsilon _{{330}}-3\,{\beta }^{2}c_{{ 440}}{e_{{330}}}^{2}+{\lambda }^{2}c_{{110}}c_{{330}}\epsilon _{{330}}+{ \lambda }^{2}c_{{110}}{e_{{330}}}^{2}\nonumber \\&-{\lambda }^{2}{c_{{130}}}^{2} \epsilon _{{330}}-2\,{\lambda }^{2}c_{{130}}c_{{440}}\epsilon _{{330}}-2 \,{\lambda }^{2}c_{{130}}e_{{150}}e_{{330}}-2\,{\lambda }^{2}c_{{130}}e_ {{310}}e_{{330}}+\nonumber \\&{\lambda }^{2}c_{{330}}c_{{440}}\epsilon _{{110}}+{ \lambda }^{2}c_{{330}}{e_{{150}}}^{2}+2\,{\lambda }^{2}c_{{330}}e_{{150} }e_{{310}}+{\lambda }^{2}c_{{330}}{e_{{310}}}^{2}-2\,{\lambda }^{2}c_{{ 440}}e_{{310}}e_{{330}} \end{aligned}$$
(A.1c)
$$\begin{aligned} \alpha _3&= -{\beta }^{3}c_{{330}}c_{{440}}\epsilon _{{330}}-{\beta }^{3}c_{{440}}{e_ {{330}}}^{2}+2\,\beta \,{\lambda }^{2}c_{{110}}c_{{330}}\epsilon _{{330}} +2\,\beta \,{\lambda }^{2}c_{{110}}{e_{{330}}}^{2}-\nonumber \\&2\,\beta \,{\lambda }^{ 2}{c_{{130}}}^{2}\epsilon _{{330}}-4\,\beta \,{\lambda }^{2}c_{{130}}c_{{ 440}}\epsilon _{{330}}-4\,\beta \,{\lambda }^{2}c_{{130}}e_{{150}}e_{{330 }}-4\,\beta \,{\lambda }^{2}c_{{130}}e_{{310}}e_{{330}}+\nonumber \\ {}&2\,\beta \,{ \lambda }^{2}c_{{330}}c_{{440}}\epsilon _{{110}}+2\,\beta \,{\lambda }^{2} c_{{330}}{e_{{150}}}^{2}+4\,\beta \,{\lambda }^{2}c_{{330}}e_{{150}}e_{{ 310}}+2\,\beta \,{\lambda }^{2}c_{{330}}{e_{{310}}}^{2}-\nonumber \\ {}&4\,\beta \,{ \lambda }^{2}c_{{440}}e_{{310}}e_{{330}} \end{aligned}$$
(A.1d)
$$\begin{aligned} \alpha _2&= {\beta }^{2}{\lambda }^{2}c_{{110}}c_{{330}}\epsilon _{{330}}+{\beta }^{2} {\lambda }^{2}c_{{110}}{e_{{330}}}^{2}-{\beta }^{2}{\lambda }^{2}{c_{{130 }}}^{2}\epsilon _{{330}}-3\,{\beta }^{2}{\lambda }^{2}c_{{130}}c_{{440}} \epsilon _{{330}}-\nonumber \\ {}&3\,{\beta }^{2}{\lambda }^{2}c_{{130}}e_{{150}}e_{{330} }-2\,{\beta }^{2}{\lambda }^{2}c_{{130}}e_{{310}}e_{{330}}+{\beta }^{2}{ \lambda }^{2}c_{{330}}c_{{440}}\epsilon _{{110}}+{\beta }^{2}{\lambda }^{2 }c_{{330}}{e_{{150}}}^{2}+\nonumber \\ {}&3\,{\beta }^{2}{\lambda }^{2}c_{{330}}e_{{150} }e_{{310}}+{\beta }^{2}{\lambda }^{2}c_{{330}}{e_{{310}}}^{2}-3\,{\beta } ^{2}{\lambda }^{2}c_{{440}}e_{{310}}e_{{330}}-{\lambda }^{4}c_{{110}}c_{ {330}}\epsilon _{{110}}-\nonumber \\ {}&{\lambda }^{4}c_{{110}}c_{{440}}\epsilon _{{330}} -2\,{\lambda }^{4}c_{{110}}e_{{150}}e_{{330}}+{\lambda }^{4}{c_{{130}}}^ {2}\epsilon _{{110}}+2\,{\lambda }^{4}c_{{130}}c_{{440}}\epsilon _{{110}} +2\,{\lambda }^{4}c_{{130}}{e_{{150}}}^{2}+\nonumber \\ {}&2\,{\lambda }^{4}c_{{130}}e_{ {150}}e_{{310}}-{\lambda }^{4}c_{{440}}{e_{{310}}}^{2} \end{aligned}$$
(A.1e)
$$\begin{aligned} \alpha _1&= -{\beta }^{3}{\lambda }^{2}c_{{130}}c_{{440}}\epsilon _{{330}}-{\beta }^{3 }{\lambda }^{2}c_{{130}}e_{{150}}e_{{330}}+{\beta }^{3}{\lambda }^{2}c_{{ 330}}e_{{150}}e_{{310}}-{\beta }^{3}{\lambda }^{2}c_{{440}}e_{{310}}e_{{ 330}}-\nonumber \\ {}&\beta \,{\lambda }^{4}c_{{110}}c_{{330}}\epsilon _{{110}}-\beta \,{ \lambda }^{4}c_{{110}}c_{{440}}\epsilon _{{330}}-2\,\beta \,{\lambda }^{4} c_{{110}}e_{{150}}e_{{330}}+\beta \,{\lambda }^{4}{c_{{130}}}^{2} \epsilon _{{110}}+\nonumber \\ {}&2\,\beta \,{\lambda }^{4}c_{{130}}c_{{440}}\epsilon _{{ 110}}+2\,\beta \,{\lambda }^{4}c_{{130}}{e_{{150}}}^{2}+2\,\beta \,{ \lambda }^{4}c_{{130}}e_{{150}}e_{{310}}-\beta \,{\lambda }^{4}c_{{440}}{ e_{{310}}}^{2} \end{aligned}$$
(A.1f)
$$\begin{aligned} \alpha _0&= {\lambda }^{4} \left( c_{{440}}\epsilon _{{110}}+{e_{{150}}}^{2} \right) \left( {\beta }^{2}c_{{130}}+{\lambda }^{2}c_{{110}} \right) \end{aligned}$$
(A.1g)

1.2 Expressions of quantities appearing in Eq. (24)

$$\begin{aligned} \gamma _6&= -c_{{330}}c_{{440}}\epsilon _{{330}}-c_{{440}}e_{330}^{2} \end{aligned}$$
(A.2a)
$$\begin{aligned} \gamma _4&= {\lambda }^{2}c_{{110}}c_{{330}}\epsilon _{{330}}+{\lambda }^{2}c_{{110}} {e_{{330}}}^{2}-{\lambda }^{2}{c_{{130}}}^{2}\epsilon _{{330}}-2\,{ \lambda }^{2}c_{{130}}c_{{440}}\epsilon _{{330}}-\nonumber \\ {}&2\,{\lambda }^{2}c_{{130 }}e_{{150}}e_{{330}}- 2\,{\lambda }^{2}c_{{130}}e_{{310}}e_{{330}}+{ \lambda }^{2}c_{{330}}c_{{440}}\epsilon _{{110}}+{\lambda }^{2}c_{{330}}{ e_{{150}}}^{2}+\nonumber \\ {}&2\,{\lambda }^{2}c_{{330}}e_{{150}}e_{{310}}+{\lambda }^{ 2}c_{{330}}{e_{{310}}}^{2}- 2\,{\lambda }^{2}c_{{440}}e_{{310}}e_{{330}} \end{aligned}$$
(A.2b)
$$\begin{aligned} \gamma _2&= -{\lambda }^{4}c_{{110}}c_{{330}}\epsilon _{{110}}-{\lambda }^{4}c_{{110} }c_{{440}}\epsilon _{{330}}-2\,{\lambda }^{4}c_{{110}}e_{{150}}e_{{330}} +{\lambda }^{4}{c_{{130}}}^{2}\epsilon _{{110}}+\nonumber \\ {}&2\,{\lambda }^{4}c_{{130} }c_{{440}}\epsilon _{{110}}+ 2\,{\lambda }^{4}c_{{130}}{e_{{150}}}^{2}+2 \,{\lambda }^{4}c_{{130}}e_{{150}}e_{{310}}-{\lambda }^{4}c_{{440}}{e_{{ 310}}}^{2} \end{aligned}$$
(A.2c)
$$\begin{aligned} \gamma _0&= {\lambda }^{6}c_{{110}} \left( c_{{440}}\epsilon _{{110}}+e_{{150}}^{2 } \right) \end{aligned}$$
(A.2d)

1.3 Expressions of quantities appearing in Eqs. (33) and (34)

$$\begin{aligned}&A(\lambda ,z) = 2\lambda \left( \sum \limits _{k = 1}^6 (- 1)^k{S_k}\frac{{{D_{4k}}}}{D}{\mathrm{e}^{{m_k}z}} + \sum \limits _{k = 1}^3 ( - 1)^k{S_{2k+5}}\frac{{{{\bar{D}}_{1k}}}}{{\bar{D}}}{\mathrm{e}^{{n_k}z}}\right) \end{aligned}$$
(A.3a)
$$\begin{aligned}&G(\lambda ,z) = 2\lambda \left( \sum \limits _{k = 1}^6 (-1)^k{S_k}\frac{{{D_{6k}}}}{D}{\mathrm{e}^{{m_k}z}} + \sum \limits _{k = 1}^3 (-1)^k{S_{2k+5}}\frac{{{{\bar{D}}_{3k}}}}{{\bar{D}}}{\mathrm{e}^{{n_k}z}}\right) \end{aligned}$$
(A.3b)
$$\begin{aligned}&F_{11}(x,t) = \int _{ 0 }^{ + \infty } 2 \lambda \mathrm{e}^{-\beta h} \left( \sum \limits _{k = 1}^6(- 1)^k {S_k}\frac{{{D_{1k}}}}{D}\right) \sin \left( \lambda (t-x)\right) \mathrm{d}\lambda \end{aligned}$$
(A.3c)
$$\begin{aligned}&F_{12}(x,t) = \int _{ 0 }^{ + \infty } 2 \lambda \mathrm{e}^{-\beta h} \left( \sum \limits _{k = 1}^6(- 1)^k {S_k}\frac{{{D_{3k}}}}{D}\right) \sin \left( \lambda (t-x)\right) \mathrm{d}\lambda \end{aligned}$$
(A.3d)

1.4 Expressions of quantities appearing in Eqs. (35) and (36)

$$\begin{aligned}&B(\lambda ,z) = 2\lambda \left( \sum \limits _{k = 1}^6 (- 1)^k{N_k}\frac{{{D_{4k}}}}{D} + \sum \limits _{k = 1}^3 ( - 1)^k{N_{2k+5}}\frac{{{{\bar{D}}_{1k}}}}{{\bar{D}}}\right) \end{aligned}$$
(A.4a)
$$\begin{aligned}&H(\lambda ,z) = 2\lambda \left( \sum \limits _{k = 1}^6 (-1)^k{N_k}\frac{{{D_{6k}}}}{D} + \sum \limits _{k = 1}^3 (-1)^k{N_{2k+5}}\frac{{{{\bar{D}}_{3k}}}}{{\bar{D}}}\right) \end{aligned}$$
(A.4b)
$$\begin{aligned}&F_{21}(x,t) = \int _{ 0 }^{ + \infty } 2 \lambda \mathrm{e}^{-\beta h} \left( \sum \limits _{k = 1}^6(- 1)^k {N_k}\frac{{{D_{1k}}}}{D}\right) \sin \left( \lambda (t-x)\right) \mathrm{d}\lambda \end{aligned}$$
(A.4c)
$$\begin{aligned}&F_{22}(x,t) = \int _{ 0 }^{ + \infty } 2 \lambda \mathrm{e}^{-\beta h} \left( \sum \limits _{k = 1}^6(- 1)^k {N_k}\frac{{{D_{3k}}}}{D}\right) \sin \left( \lambda (t-x)\right) \mathrm{d}\lambda \end{aligned}$$
(A.4d)

1.5 Expressions of quantities appearing in Eq. (39)

$$\begin{aligned}&{\bar{K}}_{11}(x,t) = \int _{ 0 }^{ + \infty } \left[ A(\lambda ,0) - a_0\right] \sin \left( \lambda (t-x)\right) \mathrm{d}\lambda \end{aligned}$$
(A.5a)
$$\begin{aligned}&{\bar{K}}_{12}(x,t) = \int _{ 0 }^{ + \infty } \left[ G(\lambda ,0) - g_0\right] \sin \left( \lambda (t-x)\right) \mathrm{d}\lambda \end{aligned}$$
(A.5b)
$$\begin{aligned}&{\bar{K}}_{21}(x,t) = \int _{ 0 }^{ + \infty } \left[ B(\lambda ,0) - b_0\right] \sin \left( \lambda (t-x)\right) \mathrm{d}\lambda \end{aligned}$$
(A.5c)
$$\begin{aligned}&{\bar{K}}_{22}(x,t) = \int _{ 0 }^{ + \infty } \left[ H(\lambda ,0) - h_0\right] \sin \left( \lambda (t-x)\right) \mathrm{d}\lambda \end{aligned}$$
(A.5d)

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El-Borgi, S., Çömez, I. & Ali Güler, M. A receding contact problem between a graded piezoelectric layer and a piezoelectric substrate. Arch Appl Mech 91, 4835–4854 (2021). https://doi.org/10.1007/s00419-021-02037-6

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