Modelling of Cylinder Indentation into a Viscoelastic Layer
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lling of Cylinder Indentation into a Viscoelastic Layer I. G. Goryachevaa,* and A. A. Yakovenkoa,** a
Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow, 119526, Russia *e-mail: [email protected] **e-mail: [email protected] Received May 2, 2020; revised May 15, 2020; accepted May 25, 2020
Abstract—The indentation of a rigid cylinder with a flat base into a viscoelastic layer situated without friction on a rigid base is considered. The interaction process consists of two stages: indentation of a cylinder at a constant speed to a certain depth and its further holding in this depth. The mechanical behavior of a viscoelastic layer is described by equations of linear viscoelasticity with relaxation functions of two types: exponential and power. The solution of the problem is based on the application of the Volterra principle and the use of an elastic solution constructed by the asymptotic “methods of large and small λ” that were developed by I. I. Vorovich and his students. On the basis of the solution obtained, the nature of the time variation in contact stresses and loads during the indentation of the cylinder has been studied for both relatively small and relatively large layer thicknesses. A comparison of the time variation of the studied quantities for two different relaxation functions and with the results of calculations based on a simplified one-dimensional model used to describe the behavior of a viscoelastic layer during indentation is performed. Keywords: contact problem, viscoelastic layer, Volterra principle, exponential relaxation function, power relaxation function, method of large and small λ DOI: 10.3103/S0025654420050088
INTRODUCTION The need to build a model for contact interaction of a solid with a relatively soft layer situated on a rigid base arises in many practical areas, for example, when analyzing the results of indentation of soft biological tissues and polymeric materials. In this case, various conditions for fixing the layer on the base, the shape of the contacting surface of the indenter as well as the mechanical properties of the layer itself are considered. The process of indentation can contain several stages: immersion of the indenter into the tissue, holding it in a certain depth, and extracting it from the layer instantly or within a certain time. Since many materials, including biological tissues, exhibit relaxation properties, when constructing a model of contact interaction, it becomes necessary to use models of viscoelastic bodies. There are a lot of studies dealing with the construction of solutions of contact problems for an elastic layer. A significant contribution to this area was made by the Soviet and Russian mechanic and mathematician Iosif Izrailevich Vorovich. He and his students developed an asymptotic method for solving mixed problems of the theory of elasticity [1–4] that was applied, in particular, when solving axisymmetric contact problems for a relatively thick (“method of large λ”, where λ = h a, a is the radi
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