The extension of the method of dimensionality reduction to layered elastic media

Abstract

The method of dimensionality reduction (MDR) has been extended to the axisymmetric unilateral contact problem for a layered elastic medium so that the case of continuously inhomogeneous elastic foundation is covered as well. The corresponding MDR formalism has been developed for a circular contact area. Both the non-adhesive contact and the JKR-type adhesive contact are considered. The developed theory is verified by means of two special cases, and new results, in particular, have been derived for the case of a functionally graded solid with an exponential law of inhomogeneity. IntroductionContact problems for elastic layered media [1,2] and functionally graded materials [3][4][5][6] are encountered in a number of applications, including, geotechnical engineering [7], non-destructive indentation testing [8,9], and tribological contacts [10,11]. The effect of material inhomogeneity complicates the solution by many analytical techniques [12,13] (see also a review in [14]), and therefore the advancement of analytical solution methods is urgently needed.In the last decade, Popov and Heß [15] developed the so-called Method of Dimensionality Reduction (MDR), whose main idea is to reduce an axisymmetric contact problem to a certain one-dimensional contact problem which is much more amendable to analysis. So far, the MDR formalism has been worked out for adhesive contact [16,17], viscoelastic contact [18], and non-axisymmetric contact [19,20]. Very recently, the MDR was extended by Heß [21] to a special case of elastically inhomogeneous, power-law graded half-space.In the present paper, we further extend the MDR to a general axially symmetric elastic medium whose elastic properties arbitrarily vary with depth. The developed MDR formalism utilizes the explicit solution p 0 (r; a ) of the frictionless contact problem for a cylindrical flat-ended indenter of radius a and provides a straightforward way to solve the axisymmetric problem of normal contact for an arbitrary indenter, which produces a circular contact area, both in the unilateral setting and the JKR-type adhesive contact framework.The rest of the paper is organized as follows. In Sect. 2, we formulate the axisymmetric unilateral frictionless contact problem with a circular area of contact for a layered media. In Sect. 3, we outline the general solution to the nonadhesive problem by means of the Mossakovskii's incremental method [22]. The new MDR formalism is presented in Sect. 4. Generally speaking, the extension of the MDR presented below is also valid for a semi-infinite inhomogeneous elastic medium with continuously varying properties. The specific feature of the material inhomogeneity exhibits itself in the case of adhesive contact, which is considered in Sect. 5. Three different examples of applications of the developed extension of the MDR are given in Sects. 6-8. Finally, in Sect. 9, we discuss the obtained results and outline possible generalizations. Axisymmetric unilateral contact problem with a circular contact areaWe consider the unilateral frictionle...

Stress Tensor and Gradient of Hydrostatic Pressure in the Contact Plane of Axisymmetric Bodies Under Normal and Tangential Loading

“…Moreover, note that the validity of the superposition is only bound to the linearity of the problem. It is, for example, also valid for contact problems of layered [21] or functionally graded media [22].…”

Stress Tensor and Gradient of Hydrostatic Pressure in the Half-Space Beneath Axisymmetric Bodies in Normal and Tangential Contact

“…The examples show that the absolute values of the pressure gradient are higher under small radii of curvature of the indenting body and singular at the surface, where discontinuities such as contact radius, stick radius, or sudden topography changes of the indenting body are located. It should be noted that the superposition idea is not bound to homogenous half-spaces, but can, for example, also be used for layered, graded media like cartilage (Argatov et al, 2018).…”

From Winkler’s Foundation to Popov’s Foundation

“…But it is more than that. The MDR (and correspondingly the Popov foundation) has been extended to cope with tangential [11] and torsional [38] contacts and to account for viscoelastic material's constitutive relationship [39] as well as for material grading [40,41].…”