Joint finite size influence and frictional influence on the contact behavior of thermoelectric strip

Abstract

The severe electric and thermal environments may cause localized deterioration of the contact behavior of thermoelectric devices. The contact responses of the thermoelectricity under the joint effect of the finite size and the friction are analyzed. The law of the Coulomb friction is adopted. The obtained Fredholm kernel functions reveal the influence of the thickness and the friction. The known Jacobi polynomials are employed to discretize the obtained singular integral equation. The effects of the thermoelectric loadings (the total electric current and the total energy flux), friction coefficient, the thermoelectric strip thickness, and the elastic and thermoelectric material constants on the distribution of the normal traction and the surface in-plane stress are demonstrated in detail. The smaller thermal expansion coefficient and shear modulus will contribute to the lower stress concentration at both contact edges. The surface in-plane tensile stress behind the trailing edge can be alleviated as the thermoelectric strip becomes thinner.

Appendices

Appendix A

Expressions of $\tilde{u}_{i} \left( {s,y} \right),\;\tilde{w}_{i} \left( {s,y} \right),\left( {i = 1,2} \right)$ in Eq. (10).

The general solutions of Eq. (4) can be expressed as follows:

$$\begin{aligned} \tilde{u}_{1} \left( {s,y} \right) = & e^{ - \left| s \right|y} \left[ {C_{1} \left( s \right) + C_{2} \left( s \right)y} \right] + e^{\left| s \right|y} \left[ {C_{3} \left( s \right) + C_{4} \left( s \right)y} \right] \\ \tilde{w}_{1} \left( {s,y} \right) = & - \frac{is}{{\left| s \right|}}\left\{ {e^{ - \left| s \right|y} \left[ {C_{1} \left( s \right) + C_{2} \left( s \right)\left( {y + \frac{\kappa }{\left| s \right|}} \right)} \right] - e^{\left| s \right|y} \left[ {C_{3} \left( s \right) + C_{4} \left( s \right)\left( {y - \frac{\kappa }{\left| s \right|}} \right)} \right]} \right\} \\ \end{aligned}$$

(A.1)

, where $C_{1} \left( s \right)$, $C_{2} \left( s \right)$, $C_{3} \left( s \right)$, and $C_{4} \left( s \right)$ are to be determined.

The particulars solutions of displacement fields can be given as follows:

$$\begin{aligned} \tilde{u}_{2} \left( {s,y} \right) = & \frac{{i\beta^{ * } }}{{\pi \lambda \gamma \left( {\kappa - 1} \right)}}\int\limits_{ - \infty }^{\infty } \Big[ A_{11} \left( {s - \xi } \right)A_{11} \left( \xi \right)\frac{{{\rm Z}_{1} }}{{\Pi_{1} }}e^{{\eta_{1} y}} + A_{12} \left( {s - \xi } \right)A_{12} \left( \xi \right)\frac{{{\rm Z}_{2} }}{{\Pi_{2} }}e^{{\eta_{2} y}}\\& + A_{11} \left( {s - \xi } \right)A_{12} \left( \xi \right)\frac{{{\rm Z}_{3} }}{{\Pi_{3} }}e^{{\eta_{3} y}} + A_{12} \left( {s - \xi } \right)A_{11} \left( \xi \right)\frac{{{\rm Z}_{4} }}{{\Pi_{4} }}e^{{\eta_{4} y}} \Big] {\text{d}}\xi \\ & \;\; - \frac{{2i\beta^{ * } }}{{\lambda \left( {\kappa + 1} \right)}}y\frac{\left| s \right|}{s}\left[ {A_{21} \left( s \right)e^{\left| s \right|y} - A_{22} \left( s \right)e^{ - \left| s \right|y} } \right] \\ \tilde{w}_{2} \left( {s,y} \right) = & \frac{{ - \beta^{ * } }}{{\pi \lambda \gamma \left( {\kappa - 1} \right)}}\int\limits_{ - \infty }^{\infty } \Big[ A_{11} \left( {s - \xi } \right)A_{11} \left( \xi \right)\frac{{\Psi_{1} }}{{\Pi_{1} }}e^{{\eta_{1} y}} + A_{12} \left( {s - \xi } \right)A_{12} \left( \xi \right)\frac{{\Psi_{2} }}{{\Pi_{2} }}e^{{\eta_{2} y}}\\ & + A_{11} \left( {s - \xi } \right)A_{12} \left( \xi \right)\frac{{\Psi_{3} }}{{\Pi_{3} }}e^{{\eta_{3} y}} + A_{12} \left( {s - \xi } \right)A_{11} \left( \xi \right)\frac{{\Psi_{4} }}{{\Pi_{4} }}e^{{\eta_{4} y}} \Big] {\text{d}}\xi \\ & \;\; + \frac{{2\beta^{ * } }}{{\lambda \left( {\kappa + 1} \right)}}\left[ {A_{21} \left( s \right)\left( {y + \frac{1}{\left| s \right|}} \right)e^{\left| s \right|y} + A_{22} \left( s \right)\left( {y - \frac{1}{\left| s \right|}} \right)e^{ - \left| s \right|y} } \right] \\ \end{aligned}$$

(A.2)

, where ${\rm Z}_{j}$, $\Psi_{j}$ and $\Pi_{j}$ ($j = 1,2,3,4$) have the following forms

$${\rm Z}_{j} = - s^{3} + s\eta_{j}^{2} ,\;\;\Psi_{j} = \eta_{j}^{3} - s^{2} \eta_{j} ,\;\;\Pi_{j} = \frac{\kappa + 1}{{\kappa - 1}}\left( {s^{2} - \eta_{j}^{2} } \right)^{2}$$

(A.3)

Appendix B

Expressions of $b_{ij} (i = 1,2;j = 1,2,3,4,5,6)$ and $a_{ij} (i = 1,2;j = 1,2,3,4)$ appearing in Eqs. (17) and (18).

$$\begin{gathered} b_{11} \left( {\eta_{1} ,s} \right) = \frac{{\beta^{ * } \left[ {2Gs^{2} a_{11} - is\left( {1 + 2Ga_{12} \eta_{1} + a_{13} e^{{h\eta_{1} }} } \right) + a_{14} \eta_{1} e^{{h\eta_{1} }} } \right]}}{{\pi \lambda \gamma \left( {\kappa + 1} \right)\left( {s^{2} - \eta_{1}^{2} } \right)}} \hfill \\ b_{12} \left( {\eta_{2} ,s} \right) = \frac{{\beta^{ * } \left[ {2Gs^{2} a_{11} - is\left( {1 + 2Ga_{12} \eta_{2} + a_{13} e^{{h\eta_{2} }} } \right) + a_{14} \eta_{2} e^{{h\eta_{2} }} } \right]}}{{\pi \lambda \gamma \left( {\kappa + 1} \right)\left( {s^{2} - \eta_{2}^{2} } \right)}} \hfill \\ b_{13} \left( {\eta_{3} ,s} \right) = \frac{{\beta^{ * } \left[ {2Gs^{2} a_{11} - is\left( {1 + 2Ga_{12} \eta_{3} + a_{13} e^{{h\eta_{3} }} } \right) + a_{14} \eta_{3} e^{{h\eta_{3} }} } \right]}}{{\pi \lambda \gamma \left( {\kappa + 1} \right)\left( {s^{2} - \eta_{3}^{2} } \right)}} \hfill \\ b_{14} \left( {\eta_{4} ,s} \right) = \frac{{\beta^{ * } \left[ {2Gs^{2} a_{11} - is\left( {1 + 2Ga_{12} \eta_{4} + a_{13} e^{{h\eta_{4} }} } \right) + a_{14} \eta_{4} e^{{h\eta_{4} }} } \right]}}{{\pi \lambda \gamma \left( {\kappa + 1} \right)\left( {s^{2} - \eta_{4}^{2} } \right)}} \hfill \\ b_{15} \left( s \right) = \frac{{2\beta^{ * } \left[ { - 2iGsa_{12} - isha_{13} e^{h\left| s \right|} + a_{14} e^{h\left| s \right|} + h\left| s \right|a_{14} e^{h\left| s \right|} } \right]}}{{\lambda \left( {\kappa + 1} \right)\left| s \right|}} \hfill \\ b_{16} \left( s \right) = \frac{{2\beta^{ * } \left[ {2iGsa_{12} + isha_{13} e^{ - h\left| s \right|} - a_{14} e^{ - h\left| s \right|} + h\left| s \right|a_{14} e^{ - h\left| s \right|} } \right]}}{{\lambda \left( {\kappa + 1} \right)\left| s \right|}} \hfill \\ \end{gathered}$$

(B.1)

$$\begin{gathered} b_{21} \left( {\eta_{1} ,s} \right) = \frac{{\beta^{ * } \left[ {\eta_{1} + 2Gs^{2} a_{21} - 2iGsa_{22} \eta_{1} - isa_{23} e^{{h\eta_{1} }} + a_{24} \eta_{1} e^{{h\eta_{1} }} } \right]}}{{\pi \lambda \gamma \left( {\kappa + 1} \right)\left( {s^{2} - \eta_{1}^{2} } \right)}} \hfill \\ b_{22} \left( {\eta_{2} ,s} \right) = \frac{{\beta^{ * } \left[ {\eta_{2} + 2Gs^{2} a_{21} - 2iGsa_{22} \eta_{2} - isa_{23} e^{{h\eta_{2} }} + a_{24} \eta_{2} e^{{h\eta_{2} }} } \right]}}{{\pi \lambda \gamma \left( {\kappa + 1} \right)\left( {s^{2} - \eta_{2}^{2} } \right)}} \hfill \\ b_{23} \left( {\eta_{3} ,s} \right) = \frac{{\beta^{ * } \left[ {\eta_{3} + 2Gs^{2} a_{21} - 2iGsa_{22} \eta_{3} - isa_{23} e^{{h\eta_{3} }} + a_{24} \eta_{3} e^{{h\eta_{3} }} } \right]}}{{\pi \lambda \gamma \left( {\kappa + 1} \right)\left( {s^{2} - \eta_{3}^{2} } \right)}} \hfill \\ b_{24} \left( {\eta_{3} ,s} \right) = \frac{{\beta^{ * } \left[ {\eta_{4} + 2Gs^{2} a_{21} - 2iGsa_{22} \eta_{4} - isa_{23} e^{{h\eta_{4} }} + a_{24} \eta_{4} e^{{h\eta_{4} }} } \right]}}{{\pi \lambda \gamma \left( {\kappa + 1} \right)\left( {s^{2} - \eta_{4}^{2} } \right)}} \hfill \\ b_{25} \left( s \right) = \frac{{2\beta^{ * } \left[ {1 - 2iGsa_{22} - isha_{23} e^{h\left| s \right|} + a_{24} e^{h\left| s \right|} + h\left| s \right|a_{24} e^{h\left| s \right|} } \right]}}{{\lambda \left( {\kappa + 1} \right)\left| s \right|}} \hfill \\ b_{26} \left( s \right) = \frac{{2\beta^{ * } \left[ { - 1 + 2iGsa_{22} + isha_{23} e^{ - h\left| s \right|} - a_{24} e^{ - h\left| s \right|} + h\left| s \right|a_{24} e^{ - h\left| s \right|} } \right]}}{{\lambda \left( {\kappa + 1} \right)\left| s \right|}} \hfill \\ \end{gathered}$$

(B.2)

The expressions $a_{ij} (i = 1,2;j = 1,2,3,4)$ in Eqs. (B.1) and (B.2) can be written as

$$\begin{gathered} a_{11} = - i\frac{{\left[ {\left( { - 1 + e^{2\left| s \right|h} } \right)^{2} \left( { - 1 + \kappa } \right)\kappa - 8h^{2} s^{2} e^{2\left| s \right|h} } \right]}}{4Gs\Lambda } \hfill \\ a_{12} = \frac{{\left( {\kappa + 1} \right)\left[ {\kappa \left( {e^{4\left| s \right|h} - 1} \right) + 4h\left| s \right|e^{2\left| s \right|h} } \right]}}{4G\left| s \right|\Lambda } \hfill \\ a_{13} = - \frac{{\left( {\kappa + 1} \right)e^{\left| s \right|h} \left[ {\left( {\kappa + 1} \right)\left( {e^{2\left| s \right|h} + 1} \right) - 2h\left| s \right|\left( {e^{2\left| s \right|h} - 1} \right)} \right]}}{2\Lambda } \hfill \\ a_{14} = i\frac{{\left( {\kappa + 1} \right)\left| s \right|e^{\left| s \right|h} \left[ {\left( { - 1 + \kappa } \right)\left( {e^{2\left| s \right|h} - 1} \right) + 2h\left| s \right|\left( {e^{2\left| s \right|h} + 1} \right)} \right]}}{2s\Lambda } \hfill \\ \end{gathered}$$

(B.3)

$$\begin{gathered} a_{21} = \frac{{\left( {\kappa + 1} \right)\left[ {\kappa \left( {e^{4\left| s \right|h} - 1} \right) - 4h\left| s \right|e^{2\left| s \right|h} } \right]}}{4G\left| s \right|\Lambda } \hfill \\ a_{22} = i\frac{{\left[ {\left( { - 1 + e^{2\left| s \right|h} } \right)^{2} \left( {\kappa^{2} - \kappa } \right) - 8h^{2} s^{2} e^{2\left| s \right|h} } \right]}}{4Gs\Lambda } \hfill \\ a_{23} = i\frac{{\left( {\kappa + 1} \right)\left| s \right|e^{\left| s \right|h} \left[ {\left( {1 - \kappa } \right)\left( {e^{2\left| s \right|h} - 1} \right) + 2h\left| s \right|\left( {e^{2\left| s \right|h} + 1} \right)} \right]}}{2s\Lambda } \hfill \\ a_{24} = - \frac{{\left( {\kappa + 1} \right)e^{\left| s \right|h} \left[ {\left( {\kappa + 1} \right)\left( {e^{2\left| s \right|h} + 1} \right) + 2h\left| s \right|\left( {e^{2\left| s \right|h} - 1} \right)} \right]}}{2\Lambda } \hfill \\ \end{gathered}$$

(B.4)

with $\Lambda = 4h^{2} s^{2} e^{2\left| s \right|h} + e^{2\left| s \right|h} \left( {1 + \kappa^{2} } \right) + \kappa e^{4\left| s \right|h} + \kappa$.