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Investigation of the solution for discontinuous contact problem between a functionally graded (FG) layer and homogeneous half-space

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Abstract

In this study, the discontinuous contact problem between a functionally graded (FG) layer, which is loaded symmetrically with point load P through a rigid block, and a homogeneous half-space was solved using the theory of elasticity and integral transform techniques. The shear modulus and density of the layer addressed in the problem vary with an exponential function along with its height. The half-space is homogeneous, and no binder exists on the contact surface containing the FG layer. In the solution, the body force of the FG layer was considered, whereas that of the homogeneous half-space was neglected. The Poisson’s ratios of both the FG layer and homogeneous half-space were assumed to remain constant. Additionally, all the surfaces addressed in the problem were assumed to be frictionless. Using the theory of elasticity and integral transform techniques, the discontinuous contact problem was reduced to two integral equations, wherein the contact stress under the rigid block and the slope of the separation, which occurred at the interface of the FG layer and homogeneous half-space, are unknown. These integral equations were solved numerically for the flat condition of the rigid block profile using the Gauss–Chebyshev integration formula. Consequently, the stress distributions, start–end points of the separation region, and separation displacements between the FG layer and homogeneous half-space were obtained for various dimensionless quantities.

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  • 07 October 2020

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Authors and Affiliations

  1. Department of Civil Engineering, Bayburt University, 69000, Bayburt, Turkey

    Erdal Öner

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

    Ahmet Birinci

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  1. Erdal Öner
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  2. Ahmet Birinci
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Correspondence to Erdal Öner.

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The Original article has been corrected: Figs. 5, 6, 8 and 10 has been corrected.

Appendix

Appendix

$$\begin{aligned} A_{1}= & {} [-e^{hn_{1} }\bar{p}\left( -1+\kappa _{1} \right) (-4F\xi \left( \left( -e^{hn_{2}} +e^{hn_{4}} \right) D_{2} D_{4} m_{3} +D_{3} \left( {\begin{array}{l} \left( e^{hn_{3}}-e^{hn_{4}} \right) D_{4} m_{2} \\ +\left( e^{hn_{2}}-e^{hn_{3}} \right) D_{2} m_{4} \\ \end{array}} \right) \right) \nonumber \\&+\left( e^{hn_{2}}-e^{hn_{3}} \right) C_{4} D_{2} D_{3} \left( 1+\kappa _{2} \right) -\left( e^{hn_{2}}-e^{hn_{4}} \right) C_{3} D_{2} D_{4} \left( 1+\kappa _{2} \right) +\left( e^{hn_{3}}-e^{hn_{4} } \right) \nonumber \\&C_{2} D_{3} D_{4} \left( 1+\kappa _{2} \right) )\,-4e^{hn_{1}}F\varphi \left( {\begin{array}{l} e^{hn_{4}}\left( e^{hn_{2}}-e^{hn_{3}} \right) C_{4} D_{2} D_{3} \\ -\left( {\begin{array}{l} e^{hn_{3} }\left( e^{hn_{2} }-e^{hn_{4} } \right) C_{3} D_{2} \\ -e^{hn_{2} }\left( e^{hn_{3} }-e^{hn_{4} } \right) C_{2} D_{3} \\ \end{array}} \right) D_{4} \\ \end{array}} \right) \mu _{0} ]\,/\varDelta \end{aligned}$$
(A.1)
$$\begin{aligned} A_{2}= & {} [\text{ e}^{hn_{2} }(\bar{p} \left( -1+\kappa _{1} \right) (-4F\xi \left( {\begin{array}{l} -\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{4} } \right) D_{1} D_{4} m_{3} \\ +D_{3} \left( \left( \text{ e}^{hn_{3} }-\text{ e}^{hn_{4} } \right) D_{4} m_{1} +\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{3} } \right) D_{1} m_{4} \right) \\ \end{array}} \right) \nonumber \\&+\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{3} } \right) C_{4} D_{1} D_{3} \left( 1+\kappa _{2} \right) -\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{4} } \right) C_{3} D_{1} D_{4} \left( 1+\kappa _{2} \right) +\left( \text{ e}^{hn_{3} }-\text{ e}^{hn_{4} } \right) \nonumber \\&C_{1} D_{3} D_{4} \left( 1+\kappa _{2} \right) )+4F\varphi \left( {\begin{array}{l} \text{ e}^{hn_{4} }\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{3} } \right) C_{4} D_{1} D_{3} \\ -\left( {\begin{array}{l} \text{ e}^{hn_{3} }\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{4} } \right) C_{3} D_{1} \\ -\text{ e}^{hn_{1} }\left( \text{ e}^{hn_{3} }-\text{ e}^{hn_{4} } \right) C_{1} D_{3} \\ \end{array}} \right) D_{4} \\ \end{array}} \right) \mu _{0} )]/\Delta \end{aligned}$$
(A.2)
$$\begin{aligned} A_{3}= & {} [\text{ e}^{hn_{3} }(-\bar{p} \left( -1+\kappa _{1} \right) (-4F\xi \left( {\begin{array}{l} -\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{4} } \right) D_{1} D_{4} m_{2} \\ +D_{2} \left( \left( \text{ e}^{hn_{2} }-\text{ e}^{hn_{4} } \right) D_{4} m_{1} +\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{2} } \right) D_{1} m_{4} \right) \\ \end{array}} \right) \nonumber \\&+\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{2} } \right) C_{4} D_{1} D_{2} \left( 1+\kappa _{2} \right) -\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{4} } \right) C_{2} D_{1} D_{4} \left( 1+\kappa _{2} \right) +\left( \text{ e}^{hn_{2} }-\text{ e}^{hn_{4} } \right) \nonumber \\&C_{1} D_{2} D_{4} \left( 1+\kappa _{2} \right) )-4F\varphi \left( {\begin{array}{l} \text{ e}^{hn_{4} }\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{2} } \right) C_{4} D_{1} D_{2} \\ -\left( {\begin{array}{l} \text{ e}^{hn_{2} }\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{4} } \right) C_{2} D_{1} \\ -\text{ e}^{hn_{1} }\left( \text{ e}^{hn_{2} }-\text{ e}^{hn_{4} } \right) C_{1} D_{2} \\ \end{array}} \right) D_{4} \\ \end{array}} \right) \mu _{0} )]/\Delta \end{aligned}$$
(A.3)
$$\begin{aligned} A_{4}= & {} [\text{ e}^{hn_{4} }(\bar{p} \left( -1+\kappa _{1} \right) (-4F\xi \left( {\begin{array}{l} -\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{3} } \right) D_{1} D_{3} m_{2} \\ +D_{2} \left( \left( \text{ e}^{hn_{2} }-\text{ e}^{hn_{3} } \right) D_{3} m_{1} +\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{2} } \right) D_{1} m_{3} \right) \\ \end{array}} \right) \nonumber \\&+\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{2} } \right) C_{3} D_{1} D_{2} \left( 1+\kappa _{2} \right) -\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{3} } \right) C_{2} D_{1} D_{3} \left( 1+\kappa _{2} \right) +\left( \text{ e}^{hn_{2} }-\text{ e}^{hn_{3} } \right) \nonumber \\&C_{1} D_{2} D_{3} \left( 1+\kappa _{2} \right) )+4F\varphi \left( {\begin{array}{l} \text{ e}^{hn_{3} }\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{2} } \right) C_{3} D_{1} D_{2} \\ -\left( {\begin{array}{l} \text{ e}^{hn_{2} }\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{3} } \right) C_{2} D_{1} \\ -\text{ e}^{hn_{1} }\left( \text{ e}^{hn_{2} }-\text{ e}^{hn_{3} } \right) C_{1} D_{2} \\ \end{array}} \right) D_{3} \\ \end{array}} \right) \mu _{0} )]/\Delta \end{aligned}$$
(A.4)
$$\begin{aligned} B_{1}= & {} [-\frac{1}{\xi }\text{ e}^{h\xi }\left( -1+2h\xi +\kappa _{2} \right) (\xi \bar{p}(C_{4} \left( -\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{3} } \right) D_{1} D_{3} m_{2} +D_{2} \left( {\begin{array}{l} \left( \text{ e}^{hn_{2} }-\text{ e}^{hn_{3} } \right) D_{3} m_{1} \\ +\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{2} } \right) D_{1} m_{3} \\ \end{array}} \right) \right) \nonumber \\&-C_{3} \left( -\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{4} } \right) D_{1} D_{4} m_{2} +D_{2} \left( \left( \text{ e}^{hn_{2} }-\text{ e}^{hn_{4} } \right) D_{4} m_{1} +\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{2} } \right) D_{1} m_{4} \right) \right) \nonumber \\&+C_{2} \left( -\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{4} } \right) D_{1} D_{4} m_{3} +D_{3} \left( \left( \text{ e}^{hn_{3} }-\text{ e}^{hn_{4} } \right) D_{4} m_{1} +\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{3} } \right) D_{1} m_{4} \right) \right) \nonumber \\&-C_{1} (-\left( \text{ e}^{hn_{2} }-\text{ e}^{hn_{4} } \right) D_{2} D_{4} m_{3} +D_{3} \left( \left( \text{ e}^{hn_{3} }-\text{ e}^{hn_{4} } \right) D_{4} m_{2} +\left( \text{ e}^{hn_{2} }-\text{ e}^{hn_{3} } \right) D_{2} m_{4} \right) )) \nonumber \\&\left( -1+\kappa _{1} \right) +\varphi (-C_{3} \left( {\begin{array}{l} \left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{2} } \right) \left( \text{ e}^{hn_{3} }-\text{ e}^{hn_{4} } \right) C_{4} D_{1} D_{2} \\ +\left( \begin{array}{l} \left( \text{ e}^{hn_{2} }-\text{ e}^{hn_{3} } \right) \left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{4} } \right) C_{2} D_{1} \\ -\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{3} } \right) \left( \text{ e}^{hn_{2} }-\text{ e}^{hn_{4} } \right) C_{1} D_{2} \\ \end{array} \right) D_{4} \\ \end{array}} \right) \nonumber \\&+D_{3} \left( {\begin{array}{l} -\left( \text{ e}^{hn_{2} }-\text{ e}^{hn_{3} } \right) \left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{4} } \right) C_{1} C_{4} D_{2} \\ +C_{2} \left( {\begin{array}{l} \left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{3} } \right) \left( \text{ e}^{hn_{2} }-\text{ e}^{hn_{4} } \right) C_{4} D_{1} \\ -\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{2} } \right) \left( \text{ e}^{hn_{3} }-\text{ e}^{hn_{4} } \right) C_{1} D_{4} \\ \end{array}} \right) \\ \end{array}} \right) )\mu _{0} )]/\Delta \end{aligned}$$
(A.5)
$$\begin{aligned} B_{2}= & {} [-2\text{ e}^{h\xi }(\xi \bar{p}(C_{4} \left( -\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{3} } \right) D_{1} D_{3} m_{2} +D_{2} \left( \left( \text{ e}^{hn_{2} }-\text{ e}^{hn_{3} } \right) D_{3} m_{1} +\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{2} } \right) D_{1} m_{3} \right) \right) \nonumber \\&-C_{3} (-\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{4} } \right) D_{1} D_{4} m_{2} +D_{2} \left( \left( \text{ e}^{hn_{2} }-\text{ e}^{hn_{4} } \right) D_{4} m_{1} +\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{2} } \right) D_{1} m_{4} \right) ) \nonumber \\&+C_{2} \left( -\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{4} } \right) D_{1} D_{4} m_{3} +D_{3} \left( \left( \text{ e}^{hn_{3} }-\text{ e}^{hn_{4} } \right) D_{4} m_{1} +\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{3} } \right) D_{1} m_{4} \right) \right) \nonumber \\&-C_{1} \left( -\left( \text{ e}^{hn_{2} }-\text{ e}^{hn_{4} } \right) D_{2} D_{4} m_{3} +D_{3} \left( \left( \text{ e}^{hn_{3} }-\text{ e}^{hn_{4} } \right) D_{4} m_{2} +\left( \text{ e}^{hn_{2} }-\text{ e}^{hn_{3} } \right) D_{2} m_{4} \right) \right) )\left( -1+\kappa _{1} \right) \nonumber \\&+\varphi (-C_{3} \left( \left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{2} } \right) \left( \text{ e}^{hn_{3} }-\text{ e}^{hn_{4} } \right) C_{4} D_{1} D_{2} +\left( {\begin{array}{l} \left( \text{ e}^{hn_{2} }-\text{ e}^{hn_{3} } \right) \left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{4} } \right) C_{2} D_{1} \\ -\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{3} } \right) \left( \text{ e}^{hn_{2} }-\text{ e}^{hn_{4} } \right) C_{1} D_{2} \\ \end{array}} \right) D_{4} \right) \nonumber \\&+D_{3} \left( {\begin{array}{l} -\left( \text{ e}^{hn_{2} }-\text{ e}^{hn_{3} } \right) \left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{4} } \right) C_{1} C_{4} D_{2} \\ +C_{2} \left( {\begin{array}{l} \left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{3} } \right) \left( \text{ e}^{hn_{2} }-\text{ e}^{hn_{4} } \right) C_{4} D_{1} \\ -\left( \text{ e}^{hn_{1} }-\text{ e}^{hn_{2} } \right) \left( \text{ e}^{hn_{3} }-\text{ e}^{hn_{4} } \right) C_{1} D_{4} \\ \end{array}} \right) \\ \end{array}} \right) )\mu _{0} )]/\Delta \end{aligned}$$
(A.6)
$$\begin{aligned} \bar{p}= & {} \int \limits _0^a {p(t)\cos (\xi t)dt} \end{aligned}$$
(A.7)
$$\begin{aligned} \varphi= & {} \int \limits _b^c {\varphi (t)\sin (\xi t)dt} \end{aligned}$$
(A.8)
$$\begin{aligned} \Delta= & {} (4\text{ e}^{hn_{4} }F\xi C_{4} \left( {-\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{3} }} \right) D_{1} D_{3} m_{2} +D_{2} \left( {\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) D_{3} m_{1} +\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) D_{1} m_{3} } \right) } \right) \nonumber \\&+C_{1} (-4\text{ e}^{hn_{1} }F\xi \left( {-\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) D_{2} D_{4} m_{3} +D_{3} \left( {\left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) D_{4} m_{2} +\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) D_{2} m_{4} } \right) } \right) \nonumber \\&+\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) \left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) C_{4} D_{2} D_{3} \left( {1+\kappa _{2} } \right) )C_{3} (-4\text{ e}^{hn_{3} }F\xi \left( {\begin{array}{l} -\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) D_{1} D_{4} m_{2} \\ +D_{2} \left( {\begin{array}{l} \left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) D_{4} m_{1} \\ +\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) D_{1} m_{4} \\ \end{array}} \right) \\ \end{array}} \right) \,\, \nonumber \\&+\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) \left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) C_{4} D_{1} D_{2} \left( {1+\kappa _{2} } \right) \,+\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) \left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) C_{2} D_{1} D_{4} \left( {1+\kappa _{2} } \right) \, \nonumber \\&-\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{3} }} \right) \left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) C_{1} D_{2} D_{4} \left( {1+\kappa _{2} } \right) )\,+C_{2} (4\text{ e}^{hn_{2} }F\xi \left( {\begin{array}{l} -\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) D_{1} D_{4} m_{3} \\ +D_{3} \left( {\begin{array}{l} \left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) D_{4} m_{1} \\ +\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{3} }} \right) D_{1} m_{4} \\ \end{array}} \right) \\ \end{array}} \right) \, \nonumber \\&-\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{3} }} \right) \left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) C_{4} D_{1} D_{3} \left( {1+\kappa _{2} } \right) +\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) \left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) \,C_{1} D_{3} D_{4} \left( {1+\kappa _{2} } \right) ))\mu _{0} \,\,\,\, \end{aligned}$$
(A.9)
$$\begin{aligned} N_{1} (x,t)= & {} \int \limits _0^\infty [ (\{-16F\frac{z}{h^{2}}^{2}(D_{3} \left( {\begin{array}{l} \left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) \left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) D_{4} m_{1} m_{2} \\ +\left( {\begin{array}{l} \left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) \left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) D_{2} m_{1} \\ -\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{3} }} \right) \left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) D_{1} m_{2} \\ \end{array}} \right) m_{4} \\ \end{array}} \right) \nonumber \\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-m_{3} \left( {\begin{array}{l} -\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) \left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) D_{1} D_{4} m_{2} \\ +D_{2} \left( {\begin{array}{l} \left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{3} }} \right) \left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) D_{4} m_{1} \\ -\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) \left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) D_{1} m_{4} \\ \end{array}} \right) \\ \end{array}} \right) )\left( {\kappa _{1} -1} \right) \nonumber \\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\frac{z}{h}(C_{4} \left( {-\text{ e}^{hn_{2} }\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{3} }} \right) D_{1} D_{3} m_{2} +D_{2} \left( {\begin{array}{l} \text{ e}^{hn_{1} }\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) D_{3} m_{1} \\ +\text{ e}^{hn_{3} }\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) D_{1} m_{3} \\ \end{array}} \right) } \right) \nonumber \\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-C_{3} \left( {-\text{ e}^{hn_{2} }\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) D_{1} D_{4} m_{2} +D_{2} \left( {\begin{array}{l} \text{ e}^{hn_{1} }\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) D_{4} m_{1} \\ +\text{ e}^{hn_{4} }\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) D_{1} m_{4} \\ \end{array}} \right) } \right) \nonumber \\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+C_{2} \left( {-\text{ e}^{hn_{3} }\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) D_{1} D_{4} m_{3} +D_{3} \left( {\begin{array}{l} \text{ e}^{hn_{1} }\left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) D_{4} m_{1} \\ +\text{ e}^{hn_{4} }\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{3} }} \right) D_{1} m_{4} \\ \end{array}} \right) } \right) \nonumber \\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-C_{1} \left( {-\text{ e}^{hn_{3} }\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) D_{2} D_{4} m_{3} +D_{3} \left( {\begin{array}{l} \text{ e}^{hn_{2} }\left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) D_{4} m_{2} \\ +\text{ e}^{hn_{4} }\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) D_{2} m_{4} \\ \end{array}} \right) } \right) ) \nonumber \\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\kappa _{1} -1} \right) \left( {1+\kappa _{2} } \right) \}/\Lambda ^{*})-1]\,[\sin \frac{z}{h}(t-x)]dz\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \end{aligned}$$
(A.10)
$$\begin{aligned} N_{2} (x,t)= & {} \int \limits _0^\infty [ \{4F\frac{z}{h}(C_{4} \left( {-\text{ e}^{h\left( {n_{2} +n_{4} } \right) }\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{3} }} \right) D_{1} D_{3} m_{2} +D_{2} \left( {\begin{array}{l} \text{ e}^{h\left( {n_{1} +n_{4} } \right) }\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) D_{3} m_{1} \\ +\text{ e}^{h\left( {n_{3} +n_{4} } \right) }\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) D_{1} m_{3} \\ \end{array}} \right) } \right) \nonumber \\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-C_{3} \left( {-\text{ e}^{h\left( {n_{2} +n_{3} } \right) }\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) D_{1} D_{4} m_{2} +D_{2} \left( {\begin{array}{l} \text{ e}^{h\left( {n_{1} +n_{3} } \right) }\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) D_{4} m_{1} \\ +\text{ e}^{h\left( {n_{3} +n_{4} } \right) }\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) D_{1} m_{4} \\ \end{array}} \right) } \right) \nonumber \\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+C_{2} \left( {-\text{ e}^{h\left( {n_{2} +n_{3} } \right) }\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) D_{1} D_{4} m_{3} +D_{3} \left( {\begin{array}{l} \text{ e}^{h\left( {n_{1} +n_{2} } \right) }\left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) D_{4} m_{1} \\ +\text{ e}^{h\left( {n_{2} +n_{4} } \right) }\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{3} }} \right) D_{1} m_{4} \\ \end{array}} \right) } \right) \nonumber \\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-C_{1} \left( {-\text{ e}^{h\left( {n_{1} +n_{3} } \right) }\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) D_{2} D_{4} m_{3} +D_{3} \left( {\begin{array}{l} \text{ e}^{h\left( {n_{1} +n_{2} } \right) }\left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) D_{4} m_{2} \\ +\text{ e}^{h\left( {n_{1} +n_{4} } \right) }\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) D_{2} m_{4} \\ \end{array}} \right) } \right) )\mu _{0} \}/\Lambda ] \nonumber \\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,[\cos \frac{z}{h}(t+x)-\cos \frac{z}{h}(t-x)]dz\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \end{aligned}$$
(A.11)
$$\begin{aligned} \Lambda= & {} 4\text{ e}^{hn_{4} }F\xi C_{4} \left( {-\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{3} }} \right) D_{1} D_{3} m_{2} +D_{2} \left( {\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) D_{3} m_{1} +\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) D_{1} m_{3} } \right) } \right) \nonumber \\&\,\,\,\,\,\,\,\,\,-4\text{ e}^{hn_{3} }F\xi C_{3} \left( {-\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) D_{1} D_{4} m_{2} +D_{2} \left( {\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) D_{4} m_{1} +\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) D_{1} m_{4} } \right) } \right) \nonumber \\&\,\,\,\,\,\,\,\,\,+4\text{ e}^{hn_{2} }F\xi C_{2} \left( {-\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) D_{1} D_{4} m_{3} +D_{3} \left( {\left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) D_{4} m_{1} +\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{3} }} \right) D_{1} m_{4} } \right) } \right) \nonumber \\&\,\,\,\,\,\,\,\,\,-4\text{ e}^{hn_{1} }F\xi C_{1} \left( {-\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) D_{2} D_{4} m_{3} +D_{3} \left( {\left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) D_{4} m_{2} +\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) D_{2} m_{4} } \right) } \right) \nonumber \\&\,\,\,\,\,\,\,\,\,+(\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) \left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) C_{3} C_{4} D_{1} D_{2} +\left( {-\text{ e}^{hn_{1} }+\text{ e}^{hn_{3} }} \right) \left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) C_{2} C_{4} D_{1} D_{3} \nonumber \\&\,\,\,\,\,\,\,\,\,+\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) \left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) C_{1} C_{4} D_{2} D_{3} +\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) \left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) C_{2} C_{3} D_{1} D_{4} \nonumber \\&\,\,\,\,\,\,\,\,\,+\left( {-\text{ e}^{hn_{1} }+\text{ e}^{hn_{3} }} \right) \left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) C_{1} C_{3} D_{2} D_{4} +\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) \left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) C_{1} C_{2} D_{3} D_{4} )\left( {1+\kappa _{2} } \right) .\,\,\,\,\,\,\,\, \end{aligned}$$
(A.12)
$$\begin{aligned} \Lambda ^{*}= & {} (1+\kappa _{1} )\Lambda \end{aligned}$$
(A.13)
$$\begin{aligned} N_{3} (x,t)= & {} \int \limits _0^\infty {[\{[} +\text{ e}^{-h\beta }F(C_{3} (\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) \left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) C_{4} D_{1} D_{2} +(\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) \left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) C_{2} D_{1} \nonumber \\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{3} }} \right) \,\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) C_{1} D_{2} )D_{4} )-D_{3} (-\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) \left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) C_{1} C_{4} D_{2} \nonumber \\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+C_{2} (\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{3} }} \right) \left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) C_{4} D_{1} -\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) \left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) C_{1} D_{4} )))\mu _{0} \nonumber \\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\left( {1+\kappa _{2} } \right) \mu _{0} +\text{ e}^{h\beta }\left( {1+\kappa _{1} } \right) \mu _{2} } \right) ]/\mathrm{Z}\}-1]\left[ {\sin \frac{z}{h}(t+x)+\sin \frac{z}{h}(t-x)} \right] dz\,\,\,\,\,\,\,\,\, \end{aligned}$$
(A.14)
$$\begin{aligned} N_{4} (x,t)= & {} \int \limits _0^\infty {[\{[} -4\text{ e}^{-h\beta }F\frac{z}{h}(C_{4} \left( {\begin{array}{l} -\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{3} }} \right) D_{1} D_{3} m_{2} \\ +D_{2} \left( {\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) D_{3} m_{1} +\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) D_{1} m_{3} } \right) \\ \end{array}} \right) \nonumber \\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-C_{3} \left( {-\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) D_{1} D_{4} m_{2} +D_{2} \left( {\begin{array}{l} \left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) D_{4} m_{1} \\ +\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) D_{1} m_{4} \\ \end{array}} \right) } \right) \nonumber \\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+C_{2} \left( {-\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) D_{1} D_{4} m_{3} +D_{3} \left( {\left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) D_{4} m_{1} +\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{3} }} \right) D_{1} m_{4} } \right) } \right) \nonumber \\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-C_{1} \left( {-\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) D_{2} D_{4} m_{3} +D_{3} \left( {\begin{array}{l} \left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) D_{4} m_{2} \\ +\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) D_{2} m_{4} \\ \end{array}} \right) } \right) )\left( {\kappa _{1} -1} \right) ]/\mathrm{T}\}] \nonumber \\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left[ {\cos \frac{z}{h}(t-x)} \right] dz\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \end{aligned}$$
(A.15)
$$\begin{aligned} \varpi= & {} \frac{(1+\kappa _{2} )\mu _{0} e^{-\beta h}}{(1+\kappa _{1} )\mu _{2} } \end{aligned}$$
(A.16)
$$\begin{aligned} \mathrm{Z}= & {} (4\text{ e}^{hn_{4} }F\frac{z}{h}C_{4} \left( {-\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{3} }} \right) D_{1} D_{3} m_{2} +D_{2} \left( {\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) D_{3} m_{1} +\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) D_{1} m_{3} } \right) } \right) \nonumber \\&\,\,\,\,\,-4\text{ e}^{hn_{3} }F\frac{z}{h}C_{3} \left( {-\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) D_{1} D_{4} m_{2} +D_{2} \left( {\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) D_{4} m_{1} +\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) D_{1} m_{4} } \right) } \right) \nonumber \\&\,\,\,\,\,+4\text{ e}^{hn_{2} }F\frac{z}{h}C_{2} \left( {-\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) D_{1} D_{4} m_{3} +D_{3} \left( {\left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) D_{4} m_{1} +\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{3} }} \right) D_{1} m_{4} } \right) } \right) \nonumber \\&\,\,\,\,\,-4\text{ e}^{hn_{1} }F\frac{z}{h}C_{1} \left( {-\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) D_{2} D_{4} m_{3} +D_{3} \left( {\left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) D_{4} m_{2} +\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) D_{2} m_{4} } \right) } \right) ) \nonumber \\&\,\,\,\,\,\left( {-1+\kappa _{1} } \right) \mu _{2} \,+(\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) \left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) C_{3} C_{4} D_{1} D_{2} +\left( {-\text{ e}^{hn_{1} }+\text{ e}^{hn_{3} }} \right) \left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) C_{2} C_{4} D_{1} D_{3} \nonumber \\&\,\,\,\,\,+\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) \left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) C_{1} C_{4} D_{2} D_{3} +\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) \left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) C_{2} C_{3} D_{1} D_{4} +\left( {-\text{ e}^{hn_{1} }+\text{ e}^{hn_{3} }} \right) \nonumber \\&\,\,\,\,\,\,\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) C_{1} C_{3} D_{2} D_{4} +\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) \left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) C_{1} C_{2} D_{3} D_{4} )\left( {\kappa _{1} -1} \right) \left( {\kappa _{2} +1} \right) \mu _{2} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \end{aligned}$$
(A.17)
$$\begin{aligned} \mathrm{T}= & {} (4\text{ e}^{hn_{4} }F\frac{z}{h}C_{4} \left( {-\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{3} }} \right) D_{1} D_{3} m_{2} +D_{2} \left( {\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) D_{3} m_{1} +\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) D_{1} m_{3} } \right) } \right) \nonumber \\&\,\,\,\,-4\text{ e}^{hn_{3} }F\frac{z}{h}C_{3} \left( {-\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) D_{1} D_{4} m_{2} +D_{2} \left( {\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) D_{4} m_{1} +\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) D_{1} m_{4} } \right) } \right) \nonumber \\&\,\,\,+4\text{ e}^{hn_{2} }F\frac{z}{h}C_{2} \left( {-\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) D_{1} D_{4} m_{3} +D_{3} \left( {\left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) D_{4} m_{1} +\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{3} }} \right) D_{1} m_{4} } \right) } \right) \nonumber \\&\,\,\,-4\text{ e}^{hn_{1} }F\frac{z}{h}C_{1} \left( {-\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) D_{2} D_{4} m_{3} +D_{3} \left( {\left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) D_{4} m_{2} +\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) D_{2} m_{4} } \right) } \right) ) \nonumber \\&\,\,\,\left( {\kappa _{1} -1} \right) +(\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) \left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) C_{3} C_{4} D_{1} D_{2} +\left( {-\text{ e}^{hn_{1} }+\text{ e}^{hn_{3} }} \right) \,\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) C_{2} C_{4} D_{1} D_{3} \nonumber \\&\,\,\,+\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) \left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) C_{1} C_{4} D_{2} D_{3} +\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{3} }} \right) \left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{4} }} \right) C_{2} C_{3} D_{1} D_{4} +\left( {-\text{ e}^{hn_{1} }+\text{ e}^{hn_{3} }} \right) \nonumber \\&\,\,\,\,\left( {\text{ e}^{hn_{2} }-\text{ e}^{hn_{4} }} \right) C_{1} C_{3} D_{2} D_{4} +\left( {\text{ e}^{hn_{1} }-\text{ e}^{hn_{2} }} \right) \left( {\text{ e}^{hn_{3} }-\text{ e}^{hn_{4} }} \right) C_{1} C_{2} D_{3} D_{4} )\,\left( {\kappa _{1} -1} \right) \left( {\kappa _{2} +1} \right) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \end{aligned}$$
(A.18)

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Öner, E., Birinci, A. Investigation of the solution for discontinuous contact problem between a functionally graded (FG) layer and homogeneous half-space. Arch Appl Mech 90, 2799–2819 (2020). https://doi.org/10.1007/s00419-020-01750-y

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