Evaluation of the integrals of Green’s function of Lamb’s model used in contact problems

Abstract

The dynamic analysis of contact problems is related to great mathematical difficulties and, thus, unsurprisingly has to date not been solved completely. In this work, a semi-analytical method is proposed to evaluate some integrals of Green’s function used in the dynamic analysis of a rectangular plate resting on the surface of an elastic foundation of inertial properties (Lamb’s model). The great challenge herein is overcoming the singularity present in the study of Green’s function related to this problem. The proposed solution involves the discretization of the studied system (a rectangular plate resting on the surface of an elastic foundation of inertial properties), which leads to a numerical solution in matrix form. All the terms of the matrix are doubly indexed, and the singularity is present in the terms having the same indices. Therefore, special efforts are made to calculate the terms of the matrix having the same indices, in order to eliminate the singularity. This requires solving the integrals of the terms of the matrix with the same indices analytically and the integrals of the terms of the matrix of different indices by numerical methods. Finally, this study of Green’s function is used in the dynamic analysis of the above-defined system and was successfully accomplished with a semi-analytical method leading to determinate values of the Eigen-frequencies and the Eigen-shapes of the plate.

Appendices

Appendix A

1.1 Proof of the solution’s precision by transforming the functions of the integrals by a series with only five terms of approximation

Let us bring the graphic comparison of the hypergeometric function 1F2 with its approximation to justify the precision of the results with only five terms of approximation with the following values of the considered elastic foundation with inertial properties:

$$\begin{aligned} E_{0} =250\times 10^{6}\,{\hbox {N/m}}^{2};\quad \nu _{0} =0.3;\quad \rho =1500\,{\hbox {Kg/m}}^{3};\quad t=\frac{1}{\cos \left( \varphi \right) };\quad \varphi _{\max } =\frac{\pi }{4};\;\forall m\in \left[ {1;11} \right] . \end{aligned}$$

The distance c2 (defining the half width of the square element) varies according to the size of the discretization area. For our case, we chose a square contact area with 2m long sides, so the maximum value of c2 will not exceed 0.1 m (Fig. 8).

Fig. 8

(a) Comparison of the hypergeometric function 1F2 with its approximation according to the variation in the excitation frequencies omega when c2 = 0.1 m. (b) Comparison of the hypergeometric function 1F2 with its approximation according to the variation in the length c2 when $\omega = 250$ Hz

Appendix B

With the help of Mathematica, we get the final formula of the Green’s function defining the vertical displacements of the surface of a half-space with inertial properties (Lamb’s model):

$$\begin{aligned} v= & {} \left( {\frac{-k\,\cos \left( {\omega \,t} \right) }{2bc\pi \,G_{0} }} \right) \times \Delta ,\\ \Delta= & {} 8\int \limits _0^{\arctan \left( {\frac{b}{c}} \right) } {\int \limits _0^{\frac{c}{2\cos \phi }} {\left[ {I_{21} \left( {r,\varphi } \right) +I_{3} \left( {r,\varphi } \right) } \right] } \,} rdrd\phi \\= & {} \mathop \sum \limits _{m=1}^{11} b_{2\times m-1} \times \left( {\frac{1}{2m}\left( {\left( {\frac{b2c2\left( {24-\left( {-24+\left( {b2^{2}+3c2^{2}} \right) k^{2}} \right) m} \right) }{6\left( {1+m} \right) }} \right) } \right. } \right. \\&\left. {\left. {-4^{-m}\left( {\frac{b2c2\left( {96-\left( {-96+\left( {b2^{2}+3c2^{2}} \right) k^{2}} \right) m} \right) }{24\left( {1+m} \right) }} \right) } \right) } \right) \\&+d_{0} \times \left( {\frac{1}{k}\times \left( {\frac{16\sqrt{b2^{2}+c2^{2}} \hbox {arctanh}\left[ {\tan \left[ {\frac{1}{2}\hbox {arctan}\left[ {\frac{b2}{c2}} \right] } \right] } \right] }{\sqrt{1+\frac{b2^{2}}{c2^{2}}} }} \right) +\left( {\frac{b2c2}{18}\left( {-72+\left( {b2^{2}+3c2^{2}} \right) k^{2}} \right) } \right) } \right) \\&+(d_{1} )\times \left\{ {\frac{\pi }{2}\times \left( \left( {-\frac{\left( {2+c2k\pi \chi } \right) }{36c2k^{3}\pi ^{2}\chi ^{3}}\left( {24\left( {24-12c2^{2}k^{2}\chi ^{2}+c2^{4}k^{4}\chi ^{4}} \right) \arctan \left[ {\frac{b2}{c2}} \right] } \right. +b2c2k^{2}\chi ^{2}\times } \right. \right. } \right. \\&\times \left[ 12\gamma \left( {-24+\left( {b2^{2}+3c2^{2}} \right) k^{2}\chi ^{2}} \right) +144\left( {3+\log \left[ 4 \right] } \right) -k^{2}\chi ^{2}\times \left( {b2^{2}\left( {19+\log \left[ {4096} \right] } \right) } \right. + \right. \\&\left. \left. { +3c2^{2}\left( {23+\log \left[ {4096} \right] } \right) } \right) +6\left( {-24+\left( {b2^{2}+3c2^{2}} \right) k^{2}\chi ^{2}} \right) \log \left[ {1+\frac{b2^{2}}{c2^{2}}} \right] \right. \\&\left. \left. \left. +12\left( {-24+\left( {b2^{2}+3c2^{2}} \right) k^{2}\chi ^{2}} \right) \log \left[ {c2k\chi } \right] \right] \right) \right) \\&+\left( {\frac{2c2^{2}k\chi }{45\pi }\left[ {b2\sqrt{1+\frac{b2^{2}}{c2^{2}}} \left( {60-\left( {2b2^{2}+5c2^{2}} \right) k^{2}\chi ^{2}} \right) +3c2\left( {-20+c2^{2}k^{2}\chi ^{2}} \right) } \right. } \right. \\&\times \left( {\log \left[ {\cos \left[ {\frac{1}{2}\arctan \left[ {\frac{b2}{c2}} \right] } \right] -\sin \left[ {\frac{1}{2}\arctan \left[ {\frac{b2}{c2}} \right] } \right] } \right] } \right. \\&\left. {\left. {\left. {\left. {-\log \left[ {\cos \left[ {\frac{1}{2}\arctan \left[ {\frac{b2}{c2}} \right] } \right] +\sin \left[ {\frac{1}{2}\arctan \left[ {\frac{b2}{c2}} \right] } \right] } \right] } \right) } \right] } \right) } \right) \\&-\left. {\mathop \sum \limits _{n=0}^{5} \frac{\left( {-\frac{b2c2\left( {\left( {b2^{2}+3c2^{2}} \right) k^{2}\left( {1+n} \right) -24\left( {3+n} \right) } \right) }{6\left( {3+n} \right) }} \right) }{\left( {1+n} \right) \times \chi ^{1+n}}} \right\} \\&+(d_{2} )\times 128^{-1}\times \left\{ \left( \frac{b2c2k^{4}}{2721600}\left[ -241920\left( {3b2^{4}+10b2^{2}c2^{2}+15c2^{4}} \right) \right. \right. \right. \\&+1080\left( {5b2^{6}+21b2^{4}c2^{2}+35b2^{2}c2^{4}+35c2^{6}} \right) k^{2}\\&\left. \left. {-(35b2^{8}+180b2^{6}c2^{2}+378b2^{4}c2^{4}+420b2^{2}c2^{6}+315c2^{8})k^{4}} \right] \right) \\&+16\left[ {4-(\frac{2c2}{3}(b2^{3}+3b2c2^{2})k^{2})(2-2\gamma +Log[4])} \right. \\&\left. {\left. {+\left( {\frac{1}{9}k^{2}\left( {24c2^{4}\arctan \left[ {\frac{b2}{c2}} \right] +b2c2\left( {-7b2^{2}-33c2^{2}+6\left( {b2^{2}+3c2^{2}} \right) \log \left[ {\left( {b2^{2}+c2^{2}} \right) k^{2}} \right] } \right) } \right) } \right) } \right] } \right\} \end{aligned}$$

where [19] $\chi =1.0723562676808107$; $d_{0} =-0.638753325488385$; $d_{1} =0.14255586708614507$, and $d_{2} =-0.45897252064027794.$