Research Papers

Uniqueness of Neutral Elastic Circular Nano-Inhomogeneities in Antiplane Shear and Plane Deformations

[+] Author and Article Information
Ming Dai

State Key Laboratory of Mechanics and
Control of Mechanical Structures,
Nanjing University of Aeronautics
and Astronautics,
Nanjing 210016, China;
Department of Mechanical Engineering,
University of Alberta,
Edmonton, AB T6G 1H9, Canada
e-mail: mdai1@ualberta.ca

Peter Schiavone

Department of Mechanical Engineering,
University of Alberta,
Edmonton, AB T6G 1H9, Canada
e-mail: P.Schiavone@ualberta.ca

Cun-Fa Gao

State Key Laboratory of Mechanics and
Control of Mechanical Structures,
Nanjing University of Aeronautics
and Astronautics,
Nanjing 210016, China
e-mail: cfgao@nuaa.edu.cn

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received April 12, 2016; final manuscript received July 5, 2016; published online July 25, 2016. Assoc. Editor: Daining Fang.

J. Appl. Mech 83(10), 101001 (Jul 25, 2016) (5 pages) Paper No: JAM-16-1178; doi: 10.1115/1.4034118 History: Received April 12, 2016; Revised July 05, 2016

In elasticity theory, a neutral inhomogeneity is defined as a foreign body which can be introduced into a host solid without disturbing the stress field in the solid. The existence of circular neutral elastic nano-inhomogeneities has been established for both antiplane shear and plane deformations when the interface effect is described by constant interface parameters, and the surrounding matrix is subjected to uniform external loading. It is of interest to determine whether noncircular neutral nano-inhomogeneities can be constructed under the same conditions. In fact, we prove that only the circular elastic nano-inhomogeneity can achieve neutrality under these conditions with the radius of the inhomogeneity determined by the corresponding (constant) interface parameters and bulk elastic constants. In particular, in the case of plane deformations, the (uniform) external loading imposed on the matrix must be hydrostatic in order for the corresponding circular nano-inhomogeneity to achieve neutrality. Moreover, we find that, even when we relax the interface condition to allow for a nonuniform interface effect (described by variable interface parameters), in the case of plane deformations, only the elliptical nano-inhomogeneity can achieve neutrality.

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Milton, G. W. , Briane, M. , and Willis, J. R. , 2006, “ On Cloaking for Elasticity and Physical Equations With a Transformation Invariant Form,” New J. Phys., 8(10), p. 248. [CrossRef]
Farhat, M. , Guenneau, S. , and Enoch, S. , 2009, “ Ultrabroadband Elastic Cloaking in Thin Plates,” Phys. Rev. Lett., 103(2), p. 024301. [CrossRef] [PubMed]
Stenger, N. , Wilhelm, M. , and Wegener, M. , 2012, “ Experiments on Elastic Cloaking in Thin Plates,” Phys. Rev. Lett., 108(1), p. 014301. [CrossRef] [PubMed]
Kadic, M. , Bückmann, T. , Schittny, R. , and Wegener, M. , 2015, “ Experiments on Cloaking in Optics, Thermodynamics and Mechanics,” Phil. Trans. R. Soc. A, 373(2049), p. 20140357. [CrossRef]
Lee, M. K. , and Kim, Y. Y. , 2016, “ Add-On Unidirectional Elastic Metamaterial Plate Cloak,” Sci. Rep., 6, p. 20731. [CrossRef] [PubMed]
Mansfield, E. H. , 1953, “ Neutral Holes in Plane Stress—Reinforced Holes Which Are Elastically Equivalent to the Uncut Sheet,” Q. J. Mech. Appl. Math., 6(3), pp. 370–378. [CrossRef]
Ru, C. Q. , 1998, “ Interface Design of Neutral Elastic Inclusions,” Int. J. Solids Struct., 35, pp. 559–572. [CrossRef]
Benveniste, Y. , and Miloh, T. , 1999, “ Neutral Inhomogeneities in Conduction Phenomena,” J. Mech. Phys. Solids, 47(9), pp. 1873–1892. [CrossRef]
Schiavone, P. , 2003, “ Neutrality of the Elliptic Inhomogeneity in the Case of Non-Uniform Loading,” Int. J. Eng. Sci., 41(18), pp. 2081–2090. [CrossRef]
Vasudevan, M. , and Schiavone, P. , 2005, “ Design of Neutral Elastic Inhomogeneities in Plane Elasticity in the Case of Non-Uniform Loading,” Int. J. Eng. Sci., 43, pp. 1081–1091. [CrossRef]
Hashin, Z. , 1962, “ The Elastic Moduli of Heterogeneous Materials,” ASME J. Appl. Mech., 29(1), pp. 143–150. [CrossRef]
Milton, G. W. , and Serkov, S. K. , 2001, “ Neutral Coated Inclusions in Conductivity and Anti-Plane Elasticity,” Proc. R. Soc. London A, 457(2012), pp. 1973–1997. [CrossRef]
Chen, T. , Benveniste, Y. , and Chuang, P. C. , 2002, “ Exact Solutions in Torsion of Composite Bars: Thickly Coated Neutral Inhomogeneities and Composite Cylinder Assemblages,” Proc. R. Soc. London A, 458(2023), pp. 1719–1759. [CrossRef]
Wang, X. , and Schiavone, P. , 2012, “ Neutrality in the Case of N-Phase Elliptical Inclusions With Internal Uniform Hydrostatic Stresses,” Int. J. Solids Struct., 49(5), pp. 800–807. [CrossRef]
Wang, X. , and Schiavone, P. , 2013, “ A Neutral Multi-Coated Sphere Under Non-Uniform Electric Field in Conductivity,” ZAMM, 64(3), pp. 895–903.
Wang, X. , and Schiavone, P. , 2015, “ Neutrality of Eccentrically Coated Elastic Inclusions,” Math. Mech. Complex Syst., 3(2), pp. 163–175. [CrossRef]
Wang, X. , and Schiavone, P. , 2012, “ Neutral Coated Circular Inclusions in Finite Plane Elasticity of Harmonic Materials,” Eur. J. Mech. A Solids, 33, pp. 75–81. [CrossRef]
Jiménez, S. , Vernescu, B. , and Sanguinet, W. , 2013, “ Nonlinear Neutral Inclusions: Assemblages of Spheres,” Int. J. Solids Struct., 50, pp. 2231–2238. [CrossRef]
Bolaños, S. J. , and Vernescu, B. , 2015, “ Nonlinear Neutral Inclusions: Assemblages of Coated Ellipsoids,” R. Soc. Open Sci., 2(4), p. 140394. [CrossRef] [PubMed]
Gurtin, M. E. , and Murdoch, A. I. , 1975, “ A Continuum Theory of Elastic Material Surfaces,” Arch. Ration. Mech. Anal., 57(4), pp. 291–323. [CrossRef]
Gurtin, M. E. , Weissmüller, J. , and Larche, F. , 1998, “ A General Theory of Curved Deformable Interfaces in Solids at Equilibrium,” Philos. Mag. A, 78(5), pp. 1093–1109. [CrossRef]
Fang, Q. H. , and Liu, Y. W. , 2006, “ Size-Dependent Elastic Interaction of a Screw Dislocation With a Circular Nano-Inhomogeneity Incorporating Interface Stress,” Scr. Mater., 55(1), pp. 99–102. [CrossRef]
Sharma, P. , and Ganti, S. , 2004, “ Size-Dependent Eshelby's Tensor for Embedded Nano-Inclusions Incorporating Surface/Interface Energies,” ASME J. Appl. Mech., 71(5), pp. 663–671. [CrossRef]
Tian, L. , and Rajapakse, R. , 2007, “ Analytical Solution for Size-Dependent Elastic Field of a Nanoscale Circular Inhomogeneity,” ASME J. Appl. Mech., 74(3), pp. 568–574. [CrossRef]
Muskhelishvili, N. I. , 1975, Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, Groningen, The Netherlands.
Dai, M. , Schiavone, P. , and Gao, C. F. , 2016, “ Surface Tension-Induced Stress Concentration Around an Elliptical Hole in an Anisotropic Half-Plane,” Mech. Res. Commun., 73, pp. 58–62. [CrossRef]
Benveniste, Y. , and Miloh, T. , 2007, “ Soft Neutral Elastic Inhomogeneities With Membrane-Type Interface Conditions,” J. Elast., 88(2), pp. 87–111. [CrossRef]


Grahic Jump Location
Fig. 1

Nanosized neutral inhomogeneity in an elastic plane



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