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Frictional moving contact problem between a functionally graded monoclinic layer and a rigid punch of an arbitrary profile

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Abstract

In this study, the moving contact problem of a rigid flat and cylindrical punch on a graded monoclinic layer is considered based on the linear elasticity theory. The punch is subjected to concentrated normal and tangential forces and moves steadily with a constant velocity on the boundary. The material properties of the graded layer are assumed to vary exponentially through the thickness direction. The friction between the layer and the punch is considered. Using the integral transform technique and boundary conditions of the problem, a second kind singular integral equation is obtained and solved numerically using the Gauss–Jacobi integration formulas. The effect of the moving velocity, the fiber angle, the material inhomogeneity, the friction coefficient, the punch radius and length, and the external load on the contact stress and in-plane stress is investigated.

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

    İsa Çömez

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Appendices

Appendix A

The material constants \(\overline{C}_{ij}^{0}\) can be expressed as follows [73]

$$ \overline{C}_{11}^{0} = C_{11} \cos^{4} \theta + 2(C_{12} + 2C_{66} )\cos^{2} \theta \sin^{2} \theta + C_{22} \sin^{4} \theta, $$
(29)
$$ \overline{C}_{12}^{0} = (C_{11} + C_{22} - 4C_{66} )\cos^{2} \theta \sin^{2} \theta + C_{12} (\cos^{4} \theta + \sin^{4} \theta ), $$
(30)
$$ \overline{C}_{13}^{0} = C_{13} \cos^{2} \theta + C_{23} \sin^{2} \theta, $$
(31)
$$ \overline{C}_{16}^{0} = \cos \theta \sin \theta (\cos^{2} \theta (C_{11} - C_{12} - 2C_{66} ) + \sin^{2} \theta (C_{12} - C_{22} + 2C_{66} )), $$
(32)
$$ \overline{C}_{22}^{0} = C_{11} \sin^{4} \theta + 2(C_{12} + 2C_{66} )\cos^{2} \theta \sin^{2} \theta + C_{22} \cos^{4} \theta, $$
(33)
$$ \overline{C}_{23}^{0} = C_{23} \cos^{2} \theta + C_{13} \sin^{2} \theta, $$
(34)
$$ \overline{C}_{33}^{0} = C_{33}, $$
(35)
$$ \overline{C}_{36}^{0} = (C_{13} - C_{23} )\cos \theta \sin \theta, $$
(36)
$$ \overline{C}_{44}^{0} = C_{44} \cos^{2} \theta + C_{55} \sin^{2} \theta, $$
(37)
$$ \overline{C}_{45}^{0} = (C_{55} - C_{44} )\cos \theta \sin \theta, $$
(38)
$$ \overline{C}_{55}^{0} = C_{55} \cos^{2} \theta + C_{44} \sin^{2} \theta, $$
(39)
$$ \overline{C}_{66}^{0} = (C_{11} - 2C_{12} - 2C_{22} )\cos^{2} \theta \sin^{2} \theta + C_{66} (\sin^{2} \theta - \cos^{2} \theta )^{2}, $$
(40)

where \(C_{ij}\) are the stiffness coefficient of the layer in the directions parallel and perpendicular to the fiber.

Appendix B: Cylindrical punch figures

See Figs. 4, 5, 6, 7, 8 and 9.

Appendix C: Flat punch figures

See Figs. 10, 11, 12, 13, 14 and 15.

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Çömez, İ. Frictional moving contact problem between a functionally graded monoclinic layer and a rigid punch of an arbitrary profile. Acta Mech 233, 1435–1453 (2022). https://doi.org/10.1007/s00707-022-03178-7

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