Axisymmetric contact problems of the theory of elasticity for inhomogeneous layers

Abstract

An approach for constructing semi‐analytical solutions in contact problems of the theory of elasticity for inhomogeneous layers is developed. The approach is efficient for the layer of arbitrary thickness which is either continuously inhomogeneous (functionally graded) or piecewise homogeneous (i.e. presented as a set of homogeneous layers with different elastic properties). The foundation is also assumed to be elastic, but much stiffer than the layer. The loads considered address the case of axisymmetric contact problems under torsion and indentation of a rigid circular punch with the flat base. The technique based on integral transforms is used to reduce the problems to the integral equations. Special approximations for the kernel transforms are used to obtain analytical solutions of the integral equations. The main results include computations of the profiles of contact stresses under the punch and the dependences of displacements with depth for different types of variation of elastic properties in the layer. The results are also compared with those obtained by other methods.

Workshop on Encounter of the third kind on Generalized continua and microstructures in Arpino, 3–7 April 2018: A review of presentations and discussions

“…Andrey Vasiliev, in his talk (''Bilateral asymptotic method for solution of a system of dual integral equations and its application for modeling of nanoindentation experiments''), discussed a method of solution of a system of dual integral equations that arises in the modelling of contact problems in the framework of electroelastic materials. The presented method allows approximated analytical solutions to be computed for a large class of axisymmetric and plane contact problems [76].…”

Simplified Analytical Solution of the Contact Problem on Indentation of a Coated Half-Space by a Spherical Punch

“…Let us denote p hom (r), P hom , δ hom , S hom as the contact stresses, indentation force, punch displacement and indentation stiffness, respectively, in the Hertzian contact (for a homogeneous half-space with elastic properties that coincide with the substrate). We introduce dimensionless variables and functions: (19) In order to analyze the difference in the distribution of contact stresses for a half-space with and without coating, we also consider the relative contact stresses: Logarithmic scale is used. For λ→0 the values of P 0 , δ 0 , S 0 tend to unit, which means that the force, depth and stiffness almost equal to the values corresponding to the Hertzian contact.…”

“…In the papers by Aizikovich [21], Vasiliev et al [17]- [19] and Volkov et al [20] solution of axisymmetric and plane contact problems for elastic solids with a functionally graded coating were constructed using the bilateral asymptotic method. This approach has a few advantages.…”

Contact Problems for an Elastic Inhomogeneous Body With a Cylindrical Cavity

“…При помощи интегрального преобразования Фурье, предполагая, что задача симметрична по z, учитывая убывание перемещений на бесконечности, общее решение уравнений (5) получим в виде 1 1 1 2 2 0 00 ( ) cos( ) , ( ) sin( ) ,…”