Axisymmetric indentation of a periodically layered, viscoelastic half-space
type: Item
instance of: scholarly article
title: Axisymmetric indentation of a periodically layered, viscoelastic half-space
zbMATH Open document ID: 1504.74063
author: Deepak Sachan
author: Ishan Sharma
author: T. Muthukumar
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 2022-10-19T00:00:00Z
review text: The authors analyze an indentation problem for a viscoelastic half-space by an axisymmetric indenter. The displacement formulation of the indentation problem is written as: \(\frac{\partial }{\partial x_{j}}\{ \int_{0}^{t}G_{ijkh}(x,t-\tau )\frac{\partial }{\partial \tau }(\frac{ \partial u_{k}}{\partial x_{h}})(x,\tau )d\tau \}=0\), \(i=1,2,3\), in \(\Omega \times \lbrack 0,\infty )\), \(\frac{\partial u_{k}}{\partial x_{i}}\cdot \widehat{e}_{3}=\frac{\partial \phi }{\partial x_{i}}\), \(i=1,2\), on \(\Gamma _{1}(t)\times \lbrack 0,\infty )\), \(\int_{0}^{t}G_{i3kh}(x,t-\tau )\frac{ \partial }{\partial \tau }(\frac{\partial u_{k}}{\partial x_{h}})(x,\tau )d\tau =0\), \(i=1,2\), on \(\partial \Omega \times \lbrack 0,\infty )\), \( \int_{0}^{t}G_{33kh}(x,t-\tau )\frac{\partial }{\partial \tau }(\frac{ \partial u_{k}}{\partial x_{h}})(x,\tau )d\tau =0\) on \(\Gamma _{2}(t)\times \lbrack 0,\infty )\). Here \(\Omega =\mathbb{R}\times \mathbb{R}\times (0,\infty )\), the indenter's surface, is expressed as \(\{x_{3}=\phi (r)\}\), where \(r=\sqrt{x_{1}^{2}+x_{2}^{2}}\) is the radial distance of the surface from the \(x_{3}\)-axis, the indenter is in contact with the boundary surface \( \partial \Omega =\mathbb{R}\times \mathbb{R}\times \{0\}\) of the half-space over a time-varying contact region \(\Gamma _{1}(t)\) in the \(x_{1}-x_{2}\) plane, \(\Gamma _{2}(t)\) is the boundary surface of the half-space exterior to the contact region, and \(G_{ijkh}(x,t)\) are the components of a fourth-order tensor that characterizes the material linear viscoelastic response. The authors then consider a periodic, heterogeneous and viscoelastic medium with a reference cell \(Y=(0,l_{1})\times (0,l_{2})\times (0,l_{3})\) for some \(l_{1},l_{2},l_{3}>0\) and they assume that the relaxation moduli take have different values in the two components of the medium. They set \(G_{ijkh}^{\varepsilon }(x,t)=G_{ijkh}(x/\varepsilon ,t)\). They also introduce a standard two-scale asymptotic expansion of…
reviewed by: Alain Brillard
Mathematics Subject Classification ID: 74M15
Mathematics Subject Classification ID: 74D05
Mathematics Subject Classification ID: 74Q10
Mathematics Subject Classification ID: 74H10
Mathematics Subject Classification ID: 74E05
Mathematics Subject Classification ID: 74E30
Mathematics Subject Classification ID: 74S05
zbMATH DE Number: 7604068
zbMATH Keywords: axisymmetric indenter
zbMATH Keywords: homogenization
zbMATH Keywords: anisotropy
zbMATH Keywords: linear viscoelasticity
zbMATH Keywords: two-scale asymptotic expansion
zbMATH Keywords: Laplace transform
zbMATH Keywords: finite element method
describes a project that uses: ABAQUS
MaRDI profile type: Publication
full work available at URL: s00033-022-01859-w
OpenAlex ID: W4300817576
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DOI: 10.1007/S00033-022-01859-W
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description: scientific article; zbMATH DE number 7604068