The Boussinesq flat-punch indentation problem within the context of linearized viscoelasticity

Hiromichi Itou; Victor A. Kovtunenko; Kumbakonam R. Rajagopal

doi:10.1016/j.ijengsci.2020.103272

The Boussinesq flat-punch indentation problem within the context of linearized viscoelasticity

Hiromichi Itou, Victor A. Kovtunenko, Kumbakonam R. Rajagopal

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

The Boussinesq problem, namely the indentation of a flat-ended cylindrical punch into a viscoelastic half-space is studied. We assume a linear viscoelastic model wherein the linearized strain is expressed as a function of the stress. However, this expression is not invertible, which makes the problem very interesting. Based on the Papkovich–Neuber representation in potential theory and using the Fourier–Bessel transform for axisymmetric bodies, an analytical solution of the resulting time-dependent integral equation is constructed. Consequently, distribution of the displacement and the stress fields in the half space with respect to time is obtained in the closed form.

Original language	English
Article number	103272
Journal	International Journal of Engineering Science
Volume	151
DOIs	https://doi.org/10.1016/j.ijengsci.2020.103272
Publication status	Published - Jun 2020

Keywords

Axisymmetric body
Boussinesq problem
Closed form solution
Fourier–Bessel transform
Implicit material response
Linear viscoelasticity
Papkovich–Neuber representation
Potential theory
Punch indentation

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@article{147e34a9cbd4477eb0dd683ab13a9bb8,

title = "The Boussinesq flat-punch indentation problem within the context of linearized viscoelasticity",

abstract = "The Boussinesq problem, namely the indentation of a flat-ended cylindrical punch into a viscoelastic half-space is studied. We assume a linear viscoelastic model wherein the linearized strain is expressed as a function of the stress. However, this expression is not invertible, which makes the problem very interesting. Based on the Papkovich–Neuber representation in potential theory and using the Fourier–Bessel transform for axisymmetric bodies, an analytical solution of the resulting time-dependent integral equation is constructed. Consequently, distribution of the displacement and the stress fields in the half space with respect to time is obtained in the closed form.",

keywords = "Axisymmetric body, Boussinesq problem, Closed form solution, Fourier–Bessel transform, Implicit material response, Linear viscoelasticity, Papkovich–Neuber representation, Potential theory, Punch indentation",

author = "Hiromichi Itou and Kovtunenko, {Victor A.} and Rajagopal, {Kumbakonam R.}",

note = "Funding Information: H. Itou is partially supported by Grant-in-Aid for Scientific Research (C)(No. 18K03380 ) and (B)(No. 17H02857 ) of Japan Society for the Promotion of Science (JSPS). V.A. Kovtunenko is supported by the Austrian Science Fund (FWF) project P26147-N26 : PION. H.I. and V.A.K. thank JSPS and Russian Foundation for Basic Research (RFBR) research projects J19-721 and 16-51-50004 for partial support. K.R. Rajagopal thanks the Office of Naval Research and the National Science Foundation for their support. ",

year = "2020",

month = jun,

doi = "10.1016/j.ijengsci.2020.103272",

language = "English",

volume = "151",

journal = "International Journal of Engineering Science",

issn = "0020-7225",

publisher = "Elsevier Limited",

}

TY - JOUR

T1 - The Boussinesq flat-punch indentation problem within the context of linearized viscoelasticity

AU - Itou, Hiromichi

AU - Kovtunenko, Victor A.

AU - Rajagopal, Kumbakonam R.

N1 - Funding Information: H. Itou is partially supported by Grant-in-Aid for Scientific Research (C)(No. 18K03380 ) and (B)(No. 17H02857 ) of Japan Society for the Promotion of Science (JSPS). V.A. Kovtunenko is supported by the Austrian Science Fund (FWF) project P26147-N26 : PION. H.I. and V.A.K. thank JSPS and Russian Foundation for Basic Research (RFBR) research projects J19-721 and 16-51-50004 for partial support. K.R. Rajagopal thanks the Office of Naval Research and the National Science Foundation for their support.

PY - 2020/6

Y1 - 2020/6

N2 - The Boussinesq problem, namely the indentation of a flat-ended cylindrical punch into a viscoelastic half-space is studied. We assume a linear viscoelastic model wherein the linearized strain is expressed as a function of the stress. However, this expression is not invertible, which makes the problem very interesting. Based on the Papkovich–Neuber representation in potential theory and using the Fourier–Bessel transform for axisymmetric bodies, an analytical solution of the resulting time-dependent integral equation is constructed. Consequently, distribution of the displacement and the stress fields in the half space with respect to time is obtained in the closed form.

AB - The Boussinesq problem, namely the indentation of a flat-ended cylindrical punch into a viscoelastic half-space is studied. We assume a linear viscoelastic model wherein the linearized strain is expressed as a function of the stress. However, this expression is not invertible, which makes the problem very interesting. Based on the Papkovich–Neuber representation in potential theory and using the Fourier–Bessel transform for axisymmetric bodies, an analytical solution of the resulting time-dependent integral equation is constructed. Consequently, distribution of the displacement and the stress fields in the half space with respect to time is obtained in the closed form.

KW - Axisymmetric body

KW - Boussinesq problem

KW - Closed form solution

KW - Fourier–Bessel transform

KW - Implicit material response

KW - Linear viscoelasticity

KW - Papkovich–Neuber representation

KW - Potential theory

KW - Punch indentation

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DO - 10.1016/j.ijengsci.2020.103272

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VL - 151

JO - International Journal of Engineering Science

JF - International Journal of Engineering Science

SN - 0020-7225

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