Harmonic circular inclusions for non-uniform fields through the use of multi-coating
Authors:
Xu Wang and Peter Schiavone
Journal:
Quart. Appl. Math. 72 (2014), 267-280
MSC (2010):
Primary 74B05; Secondary 30E25
DOI:
https://doi.org/10.1090/S0033-569X-2014-01332-1
Published electronically:
February 4, 2014
MathSciNet review:
3186236
Full-text PDF Free Access
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Abstract: We propose a novel method for rendering a circular inclusion harmonic even in the presence of non-uniform loading. Most significantly, the condition that the inclusion be harmonic is shown to be independent of the specific form of the loading. In addition, we demonstrate that the harmonicity condition obtained actually leads to the stronger property of neutrality when the loading takes a particular form. Our method is based on the idea of ‘multi-coating’ (surrounding the inclusion with a specified number of coatings each with its own separate elastic properties) used in the design of cloaking structures for the conductivity problem. Consequently, the harmonic inclusions designed here can also be thought of as special kinds of ‘near-cloaking’ structures in plane elasticity (in the sense that they are invisible to any changes in mean stress in the structure).
References
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- Dundurs, J., 1969. Elastic interaction of dislocations with inhomogeneities, in Mathematical Theory of Dislocations, T. Mura Ed., ASME, New York, pp. 70–115.
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- Ru, C.Q., 1998. Interface design of neutral elastic inclusions. Int. J. Solids Struct. 35, 559–572.
- Ru, C.Q., 1999. A new method for an inhomogeneity with stepwise graded interphase layer under thermomechanical loadings. J. Elasticity 56, 107$-$127.
- G. F. Wang, P. Schiavone, and C.-Q. Ru, Harmonic shapes in finite elasticity under nonuniform loading, Trans. ASME J. Appl. Mech. 72 (2005), no. 5, 691–694. MR 2168196, DOI https://doi.org/10.1115/1.1979514
- Weinans, H., Huiskes, R., Grootenboer, H.J., 1994. Effect of fit and bonding characteristics of femoral stems on adaptive bone remodeling. J. Biomech. Engrg. 116, 393$-$400.
References
- Ammari, H., Kang, H., Lee, H., and Lim, M., 2011. Enhancement of near cloaking using generalized polarization tensors vanishing structure. Part I: the conductivity problem. arXiv:1104.3936v1.
- Bjorkman, G.S., Richards, R., 1976. Harmonic holes—an inverse problem in elasticity. ASME J. Appl. Mech. 43, 414$-$418.
- Bjorkman, G.S., Richards, R., 1979. Harmonic holes for nonconstant fields. ASME J. Appl. Mech. 46, 573$-$576.
- Dundurs, J., 1969. Elastic interaction of dislocations with inhomogeneities, in Mathematical Theory of Dislocations, T. Mura Ed., ASME, New York, pp. 70–115.
- Firoozbakhsh, K., Aleyaasin, M., 1989. The effect of stress concentration on the bone remodeling: Theoretical predictions. J. Biomech. Engrg. 111, 355–360.
- N. I. Muskhelishvili, Some basic problems of the mathematical theory of elasticity. Fundamental equations, plane theory of elasticity, torsion and bending, P. Noordhoff Ltd., Groningen, 1953. Translated by J. R. M. Radok. MR 0058417 (15,370d)
- Ru, C.Q., 1998. Interface design of neutral elastic inclusions. Int. J. Solids Struct. 35, 559–572.
- Ru, C.Q., 1999. A new method for an inhomogeneity with stepwise graded interphase layer under thermomechanical loadings. J. Elasticity 56, 107$-$127.
- G. F. Wang, P. Schiavone, and C.-Q. Ru, Harmonic shapes in finite elasticity under nonuniform loading, Trans. ASME J. Appl. Mech. 72 (2005), no. 5, 691–694. MR 2168196 (2006c:74039), DOI https://doi.org/10.1115/1.1979514
- Weinans, H., Huiskes, R., Grootenboer, H.J., 1994. Effect of fit and bonding characteristics of femoral stems on adaptive bone remodeling. J. Biomech. Engrg. 116, 393$-$400.
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Additional Information
Xu Wang
Affiliation:
School of Mechanical and Power Engineering, East China University of Science and Technology, 130 Meilong Road, Shanghai 200237, China
Email:
xuwang_sun@hotmail.com
Peter Schiavone
Affiliation:
Department of Mechanical Engineering, University of Alberta, 4-9 Mechanical Engineering Building, Edmonton, Alberta, Canada T6G 2G8
Email:
p.schiavone@ualberta.ca
Keywords:
Harmonic inclusion,
neutral inclusion,
cloaking structure,
inverse problem,
non-uniform loading,
plane elasticity
Received by editor(s):
May 29, 2012
Published electronically:
February 4, 2014
Additional Notes:
The first author was supported by the Innovation Program of Shanghai Municipal Education Commission (No. 12ZZ058)
The second author acknowledges the support of the Natural Sciences and Engineering Research Council of Canada
Article copyright:
© Copyright 2014
Brown University
The copyright for this article reverts to public domain 28 years after publication.
