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Analytical solution for contact and crack problem ın homogeneous half-plane

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Abstract

In this study, the frictionless contact and crack problem of an elastic homogeneous semi-infinite plane has been investigated according to the elasticity theory. The problem has been solved as a superposition of the separate solutions of the contact and crack problem. The aim of this study is to find sub-punch stress distributions and stress intensity factors due to opening mode and shear mode for different loading conditions and geometric sizes. There are two rigid punches on the semi-infinite plane and P and Q loads are transferred to the semi-infinite plane by these punches. Problem has been considered as plain strain because of the geometry of the problem. The effect of the mass forces has not been included, the stress and displacement expressions to be used for the contact problem have been obtained by using Navier equations and Fourier integral transformation technique, and the boundary conditions determined for the problem has been applied. The equations to be used for the crack problem have been specified and the boundary conditions for the crack problem have been applied to these equations. The problem has been reduced to an integral equation system consisting of four singular integral equations where contact stresses and crack displacements are unknown. Numerical solution of the integral equation system has been realized by using Jacobi polynomials. Numerical results on sub-punch stress distributions and stress intensity factors have been obtained for different loading conditions, geometric sizes and presented by graphics.

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Authors and Affiliations

  1. Karadeniz Technical University, Civil Engineering, 61080, Ortahisar, Trabzon, Turkey

    Ayhan Üstün, Gökhan Adıyaman & Talat Şükrü Özşah¡n

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  1. Ayhan Üstün
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Appendix

Appendix

$$K_{11} (t,y) = K_{22} (t,y) = - \frac{1}{t + y} + \frac{6y}{{t + y}} - \frac{{4y^{2} }}{{(t + y)^{3} }}$$
(34a)
$$K_{13} (t,y) = K_{14} (t,y) = - \frac{2yt}{{(y + t)^{2} }}$$
(34b)
$$K_{23} (t,y) = K_{24} (t,y) = - \frac{{2y^{2} t}}{{(y^{2} + t^{2} )^{2} }}$$
(34c)
$$K_{31} (t,x) = K_{41} (t,x) = - \frac{{4xt^{2} }}{{(x^{2} + t^{2} )^{2} }}$$
(34d)
$$K_{32} (t,x) = K_{42} (t,x) = \frac{{4x^{3} }}{{(x^{2} + t^{2} )^{2} }}$$
(34e)
$$M_{11} (r_{1} ,s_{1} ) = - \frac{1}{\pi }\frac{e}{2}K_{11} \left( { - \frac{e}{2}r_{1} + \frac{e}{2}, - \frac{e}{2}s_{1} + \frac{e}{2}} \right)$$
(35a)
$$M_{13} (r_{3} ,s_{1} ) = - \frac{1}{\pi }\frac{a - b}{2}K_{13} \left( {\frac{a - b}{2}r_{3} + \frac{b + a}{2}, - \frac{e}{2}s_{1} + \frac{e}{2}} \right)$$
(35b)
$$M_{14} (r_{4} ,s_{1} ) = \frac{1}{\pi }\frac{d - c}{2}K_{14} \left( {\frac{d - c}{2}r_{4} + \frac{d + c}{2}, - \frac{e}{2}s_{1} + \frac{e}{2}} \right)$$
(35c)
$$M_{22} (r_{2} ,s_{2} ) = - \frac{1}{\pi }\frac{e}{2}K_{22} \left( { - \frac{e}{2}r_{2} + \frac{e}{2}, - \frac{e}{2}s_{2} + \frac{e}{2}} \right)$$
(35d)
$$M_{23} (r_{3} ,s_{2} ) = - \frac{1}{\pi }\frac{a - b}{2}K_{23} \left( {\frac{a - b}{2}r_{3} + \frac{b + a}{2}, - \frac{e}{2}s_{2} + \frac{e}{2}} \right)$$
(35e)
$$M_{24} (r_{4} ,s_{2} ) = \frac{1}{\pi }\frac{d - c}{2}K_{24} \left( {\frac{d - c}{2}r_{4} + \frac{d + c}{2}, - \frac{e}{2}s_{2} + \frac{e}{2}} \right)$$
(35f)
$$M_{31} (r_{1} ,s_{3} ) = - \frac{1}{\pi }\frac{e}{2}K_{31} \left[ { - \frac{e}{2}r_{1} + \frac{e}{2},\frac{a - b}{2}s_{3} + \frac{b + a}{2}} \right]$$
(35g)
$$M_{32} (r_{2} ,s_{3} ) = - \frac{1}{\pi }\frac{e}{2}K_{32} \left[ { - \frac{e}{2}r_{2} + \frac{e}{2},\frac{a - b}{2}s_{3} + \frac{b + a}{2}} \right]$$
(35h)
$$M_{41} (r_{1} ,s_{4} ) = - \frac{1}{\pi }\frac{e}{2}K_{41} \left[ { - \frac{e}{2}r_{1} + \frac{e}{2},\frac{d - c}{2}s_{4} + \frac{d + c}{2}} \right]$$
(35i)
$$M_{42} (r_{2} ,s_{4} ) = - \frac{1}{\pi }\frac{e}{2}K_{42} \left[ { - \frac{e}{2}r_{2} + \frac{e}{2},\frac{d - c}{2}s_{4} + \frac{d + c}{2}} \right]$$
(35j)

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Üstün, A., Adıyaman, G. & Özşah¡n, T.Ş. Analytical solution for contact and crack problem ın homogeneous half-plane. Arch Appl Mech 93, 4399–4423 (2023). https://doi.org/10.1007/s00419-023-02500-6

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