Analytical solution of axisymmetric indentation of multi-layer coating on elastic substrate body
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O R I G I NA L PA P E R
Kotaro Miura
· Makoto Sakamoto · Yuji Tanabe
Analytical solution of axisymmetric indentation of multi-layer coating on elastic substrate body
Received: 24 December 2019 / Revised: 27 May 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract We consider the axisymmetric contact problem of a multi-elastic layer with various elastic constants bonded to an elastic semi-infinite substrate indented by rigid flat-ended cylindrical and spherical indenters. The transfer matrix method is applied to each elastic layer, and dual integral equations are reduced to an infinite system of simultaneous equations by expressing the normal contact stress at the surface elastic layer as an appropriate series with Chebyshev orthogonal polynomials. Numerical results demonstrate the effects of the elastic constant of each elastic layer and the semi-infinite elastic substrate on the radial distribution of the normal contact stress and normal displacement of the free surface of the elastic layer, stress singularity factor at the edge of the cylindrical indenter, and axial load of a rigid indenter which penetrates the multi-layer material to a constant depth. The results of axial load are in good agreement with previously reported results. The numerical results are given for several combinations of the shear modulus of each elastic layer and the substrate. These results will contribute to the establishment of indentation tests for composite materials and serve as guidelines for the design of appropriate mechanical properties of layered materials.
1 Introduction Improvements in the stiffness of material surfaces are of great interest in the thin film and tribology fields. Indentation tests are widely used to measure the local mechanical properties of materials such as metal coating layers and biological tissues. However, the load–displacement curve obtained from indentation tests includes contributions from the material bulk, not only the surface, making it difficult to estimate the mechanical properties of the surface. Therefore, an understanding of the stress and displacement of layered materials is needed. Indentation tests are based on a contact problem known as Boussinesq’s problem. This problem for a semi-infinite space indented by conical, spherical, and flat-ended cylindrical indenters was solved by Harding and Sneddon [1], Sneddon [2], and Muki [3]. Lebedev and Ufliand [4] solved the contact problem for an elastic layer resting frictionlessly on a rigid foundation indented by a flat-ended cylindrical indenter by reducing the dual integral equations of stress and displacement to a single Fredholm integral equation of the second kind. Hayes et al. [5] solved the problem for an elastic layer bonded to a rigid foundation indented by flat-ended K. Miura (B) Department of Systems Design Engineering, Seikei University, 3-3-1 Kichijojikitamachi, Musashino-shi, Tokyo 180-8633, Japan E-mail: [email protected] M. Sakamoto Department of Health Sciences, Niigata University School of Medicine
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