Research Papers

On Spherical Inhomogeneity With Steigmann–Ogden Interface

[+] Author and Article Information
Anna Y. Zemlyanova

Department of Mathematics,
Kansas State University,
138 Cardwell Hall,
Manhattan, KS 66506
e-mail: azem@ksu.edu

Sofia G. Mogilevskaya

Department of Civil, Environmental,
and Geo-Engineering,
University of Minnesota,
500 Pillsbury Drive S.E.,
Minneapolis, MN 55455
e-mail: mogil003@umn.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received July 25, 2018; final manuscript received September 13, 2018; published online October 1, 2018. Assoc. Editor: Shengping Shen.

J. Appl. Mech 85(12), 121009 (Oct 01, 2018) (10 pages) Paper No: JAM-18-1444; doi: 10.1115/1.4041499 History: Received July 25, 2018; Revised September 13, 2018

The problem of an infinite isotropic elastic space subjected to uniform far-field load and containing an isotropic elastic spherical inhomogeneity with Steigmann–Ogden interface is considered. The interface is treated as a shell of vanishing thickness possessing surface tension as well as membrane and bending stiffnesses. The constitutive and equilibrium equations of the Steigmann–Ogden theory for a spherical surface are written in explicit forms. Closed-form analytical solutions are derived for two cases of loading conditions—the hydrostatic loading and deviatoric loading with vanishing surface tension. The single inhomogeneity-based estimates of the effective properties of macroscopically isotropic materials containing spherical inhomogeneities with Steigmann–Ogden interfaces are presented. It is demonstrated that, in the case of vanishing surface tension, the Steigmann–Ogden model describes a special case of thin and stiff uniform interphase layer.

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.


Judelewicz, M. , Künzi, H. U. , Merk, N. , and Ilschner, B. , 1994, “ Microstructural Development During Fatigue of Copper Foils 20-100 μm Thick,” Mater. Sci. Eng. A, 186(1–2), pp. 135–142. [CrossRef]
Ma, Q. , and Clarke, D. R. , 1995, “ Size Dependent Hardness of Silver Single Crystals,” J. Mater. Res., 10(4), pp. 853–863. [CrossRef]
Hong, S. , and Weil, R. , 1996, “ Low Cycle Fatigue of Thin Copper Foils,” Thin Solid Films, 283(1–2), pp. 175–181. [CrossRef]
Read, D. T. , 1998, “ Tension-Tension Fatigue of Copper Films,” Int. J. Fatigue, 20(3), pp. 203–209. [CrossRef]
Miller, R. E. , and Shenoy, V. B. , 2000, “ Size-Dependent Elastic Properties of Nanosized Structural Elements,” Nanotechnol., 11(3), pp. 139–147. [CrossRef]
Diao, J. , Gall, K. , and Dunn, M. L. , 2003, “ Surface-Stress-Induced Phase Transformation in Metal Nanowires,” Nat. Mater., 2(10), pp. 656–660. [CrossRef] [PubMed]
Gao, X.-L. , 2006, “ An Expanding Cavity Model Incorporating Strain-Hardening and Indentation Size Effects,” Int. J. Solids Struct., 43(21), pp. 6615–6629. [CrossRef]
Gao, X.-L. , 2006, “ New Expanding Cavity Model for Indentation Hardness Including Strain-Hardening and Indentation Size Effects,” J. Mater. Res., 21(5), pp. 1317–1326. [CrossRef]
He, J. , and Lilley, C. M. , 2008, “ Surface Effect on the Elastic Behavior of Static Bending Nanowires,” Nano Lett., 8(7), pp. 1798–1802. [CrossRef] [PubMed]
Cammarata, R. C. , 1994, “ Surface and Interface Stress Effects in Thin Films,” Progr. Surf. Sci., 46(1), pp. 1–38. [CrossRef]
Chhapadia, P. , Mohammadi, P. , and Sharma, P. , 2011, “ Curvature-Dependent Surface Energy and Implications for Nanostructures,” J. Mech. Phys. Solids, 59(10), pp. 2103–2115. [CrossRef]
Chhapadia, P. , Mohammadi, P. , and Sharma, P. , 2012, “ Erratum to: Curvature-Dependent Surface Energy and Implications for Nanostructures,” J. Mech. Phys. Solids, 60(6), pp. 1241–1242. [CrossRef]
Gibbs, J. W. , 1906, The Scientific Papers of J. Willard Gibbs, Vol. 1, Longmans-Green, London.
Shuttleworth, R. , 1950, “ The Surface Tension of Solids,” Proc. Phys. Soc. Lond., A, 63(5), pp. 444–457. [CrossRef]
Nicholson, M. M. , 1955, “ Surface Tension in Ionic Crystals,” Proc. Roy. Soc. Lond., A, 228(1175), pp. 490–510. [CrossRef]
Orowan, E. , 1970, “ Surface Energy and Surface Tension in Solids and Liquids,” Proc. R. Soc. London, A, 316(1527), pp. 473–491. [CrossRef]
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. , and Murdoch, A. I. , 1978, “ Surface Stress in Solids,” Int. J. Solid. Struct., 14(6), pp. 431–440. [CrossRef]
Duan, H. L. , Wang, J. , and Karihaloo, B. L. , 2009, “ Theory of Elasticity at the Nanoscale,” Adv. Appl. Mech., 42, pp. 1–68. [CrossRef]
Javili, A. , Dell'Isola, F. , and Steinmann, P. , 2013, “ Geometrically Nonlinear Higher-Gradient Elasticity With Energetic Boundaries,” J. Mech. Phys. Solids, 61(12), pp. 2381–2401. [CrossRef]
Javili, A. , McBride, A. , Steinmann, P. , and Reddy, B. D. , 2014, “ A Unified Computational Framework for Bulk and Surface Elasticity Theory: A Curvilinear-Coordinate-Based Finite Element Methodology,” Comput. Mech., 54(3), pp. 745–762. [CrossRef]
Javili, A. , Ottosen, N. S. , Ristinmaa, M. , and Mosler, J. , 2018, “ Aspects of Interface Elasticity Theory,” Math. Mech. Solids, 23(7), pp. 1004–1024.
Javili, A. , 2018, “ Variational Formulation of Generalized Interfaces for Finite Deformation Elasticity,” Math. Mech. Solids, 23(9), pp. 1303–1322.
Chatzigeorgiou, G. , Meraghni, F. , and Javili, A. , 2017, “ Generalized Interfacial Energy and Size Effects in Composites,” J. Mech. Phys. Solids, 106, pp. 257–282. [CrossRef]
Sharma, P. , and Ganti, S. , 2002, “ Interfacial Elasticity Corrections to Size-Dependent Strain-State of Embedded Quantum Dots,” Phys. Status Solidi, 234(3), pp. R10–R12. [CrossRef]
Sharma, P. , Ganti, S. , and Bhate, N. , 2003, “ Effect of Surfaces on the Size-Dependent Elastic State of Nano-Inhomogeneities,” Appl. Phys. Lett., 82(4), pp. 535–537. [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]
Duan, H. L. , Wang, J. , Huang, Z. P. , and Luo, Z. Y. , 2005, “ Stress Concentration Tensors of Inhomogeneities With Interface Effects,” Mech. Mater., 37(7), pp. 723–736. [CrossRef]
Duan, H. L. , Wang, J. , Huang, Z. P. , and Karihaloo, B. L. , 2005, “ Eshelby Formalism for Nano-Inhomogeneities,” Proc. R. Soc. London, A, 461(2062), pp. 3335–3353. [CrossRef]
Duan, H. L. , Wang, J. , Huang, Z. P. , and Karihaloo, B. L. , 2005, “ Size-Dependent Effective Elastic Constants of Solids Containing Nano-Inhomogeneoties With Interface Stress,” J. Mech. Phys. Solids, 53(7), pp. 1574–1596. [CrossRef]
Duan, H. L. , Wang, J. , Karihaloo, B. L. , and Huang, Z. P. , 2006, “ Nanoporous Materials Can Be Made Stiffer That Non-Porous Counterparts by Surface Modification,” Acta Mater., 54(11), pp. 2983–2990. [CrossRef]
Lim, C. W. , Li, Z. R. , and He, L. H. , 2006, “ Size Dependent, Non-Uniform Elastic Field Inside a Nano-Scale Spherical Inclusion Due to Interface Stress,” Int. J. Solid. Struct., 43(17), pp. 5055–5065. [CrossRef]
He, L. H. , and Li, Z. R. , 2006, “ Impact of Surface Stress on Stress Concentration,” Int. J. Solid. Struct., 43(20), pp. 6208–6219. [CrossRef]
Mi, C. , and Kouris, D. A. , 2006, “ Nanoparticles Under the Influence of Surface/Interface Elasticity,” J. Mech. Mater. Struct., 1(4), pp. 763–791. [CrossRef]
Duan, H. L. , Yi, X. , Huang, Z. P. , and Wang, J. , 2007, “ A United Scheme for Prediction of Effective Moduli of Multiphase Composites With Interface Effects—Part I: Theoretical Framework,” Mech. Mater., 39(1), pp. 81–93. [CrossRef]
Chen, T. , Dvorak, G. J. , and Yu, C. C. , 2007, “ Size-Dependent Elastic Properties of Unidirectional Nano-Composites With Interface Stresses,” Acta Mech., 188(1–2), pp. 39–54. [CrossRef]
Tian, L. , and Rajapakse, R. K. N. D. , 2007, “ Elastic Field of an Isotropic Matrix With a Nanoscale Elliptical Inhomogeneity,” Int. J. Solids Struct., 44 (24), pp. 7988–8005. [CrossRef]
Mogilevskaya, S. G. , Crouch, S. L. , and Stolarski, H. K. , 2008, “ Multiple Interacting Circular Nano-Inhomogeneities With Surface/Interface Effects,” J. Mech. Phys. Solids, 56(6), pp. 2298–2327. [CrossRef]
Jammes, M. , Mogilevskaya, S. G. , and Crouch, S. L. , 2009, “ Multiple Circular Nano- Inhomogeneities and/or Nano-Pores in One of Two Joined Isotropic Elastic Half-Planes,” Eng. Anal. Bound. Elem., 33(2), pp. 233–248. [CrossRef]
Kushch, V. I. , Mogilevskaya, S. G. , Stolarski, H. K. , and Crouch, S. L. , 2011, “ Elastic Interaction of Spherical Nanoinhomogeneities With Gurtin–Murdoch Type Interfaces,” J. Mech. Phys. Solids, 59(9), pp. 1702–1716. [CrossRef]
Kushch, V. I. , Mogilevskaya, S. G. , and Stolarski, H. K. , 2013, “ Elastic Fields and Effective Moduli of Particulate Nanocomposites With the Gurtin-Murdoch Model of Interfaces,” Int. J. Solid. Struct., 50(7–8), pp. 1141–1153. [CrossRef]
Mi, C. , and Kouris, D. A. , 2013, “ Stress Concentration Around a Nanovoid Near the Surface of an Elastic Half-Space,” Int. J. Solid. Struct., 50(18), pp. 2737–2748. [CrossRef]
Mi, C. , and Kouris, D. A. , 2017, “ Surface Mechanics Implications for a Nanovoided Metallic Thin-Plate Under Uniform Boundary Loading,” Math. Mech. Solids, 22(3), pp. 401–419. [CrossRef]
Gurtin, M. E. , Weissmüller, J. , and Larché, F. , 1998, “ A General Theory of Curved Deformable Interfaces in Solids at Equilibrium,” Philos. Mag., A, 78(5), pp. 1093–1109. [CrossRef]
Gao, X. , Huang, Z. , Qu, J. , and Fang, D. , 2014, “ A Curvature-Dependent Interfacial Energy-Based Interface Stress Theory and Its Applications to Nano-Structured Materials: (I) general Theory,” J. Mech. Phys. Solids, 66, pp. 59–77. [CrossRef]
Gao, X. , Huang, Z. , and Fang, D. , 2017, “ Curvature-Dependent Interfacial Energy and Its Effects on the Elastic Properties of Nanomaterials,” Int. J. Solid. Struct, 113–114, pp. 100–107. [CrossRef]
Steigmann, D. J. , and Ogden, R. W. , 1997, “ Plain Deformations of Elastic Solids With Intrinsic Boundary Elasticity,” Proc. R. Soc. London, A, 453(1959), pp. 853–877. [CrossRef]
Steigmann, D. J. , and Ogden, R. W. , 1999, “ Elastic Surface-Substrate Interactions,” Proc. R. Soc. London, A, 455(1982), pp. 437–474. [CrossRef]
Gao, X.-L. , Park, S. K. , and Ma, H. M. , 2009, “ Analytical Solution for a Pressurized Thick-Walled Spherical Shell Based on a Simplified Strain Gradient Elasticity Theory,” Math. Mech. Solids, 14(8), pp. 747–758. [CrossRef]
Gao, X.-L. , and Ma, H. M. , 2010, “ Solution of Eshelby's Inclusion Problem With a Bounded Domain and Eshelby's Tensor for a Spherical Inclusion in a Finite Spherical Matrix Based on a Simplified Strain Gradient Elasticity Theory,” J. Mech. Phys. Mater., 58(5), pp. 779–797. [CrossRef]
Wu, B. , Chen, W. , and Zhang, C. , 2018, “ On Free Vibration of Piezoelectric Nanospheres With Surface Effect,” Mech. Adv. Mater. Struct., 25(13), pp. 1101–1114. [CrossRef]
Eremeyev, V. , and Lebedev, L. , 2016, “ Mathematical Study of Boundary-Value Problems Within the Framework of Steigmann-Ogden Model of Surface Elasticity,” Continuum Mech. Therm., 28(1–2), pp. 407–422. [CrossRef]
Zemlyanova, A. Y. , 2017, “ A Straight Mixed Mode Fracture With the Steigmann-Ogden Boundary Condition,” Quart. J. Mech. Appl. Math., 70(1), pp. 65–86. [CrossRef]
Zemlyanova, A. Y. , 2018, “ Frictionless Contact of a Rigid Stamp With a Semi-Plane in the Presence of Surface Elasticity in the Steigmann-Ogden Form,” Math. Mech. Solids, 23(8), pp. 1140–1155. [CrossRef]
Zemlyanova, A. Y. , and Mogilevskaya, S. G. , 2018, “ Circular Inhomogeneity With Steigmann–Ogden Interface: Local Fields, Neutrality, and Maxwell's Type Approximation Formula,” Int. J. Solids Struct., 135, pp. 85–98. [CrossRef]
Mi, C. , 2018, “ Elastic Behavior of a Half-Space With a Steigmann–Ogden Boundary Under Nanoscale Frictionless Patch Loads,” Int. J. Eng. Sci., 129, pp. 129–144. [CrossRef]
Han, Z. , Mogilevskaya, S. G. , and Schillinger, D. , 2018, “ Local Fields and Overall Transverse Properties of Unidirectional Composite Materials With Multiple Nanofibers and Steigmann-Ogden Interfaces,” Int. J. Solid. Struct., 147, pp. 166–182. [CrossRef]
Benveniste, Y. , and Miloh, T. , 2001, “ Imperfect Soft and Stiff Interfaces in Two-Dimensional Elasticity,” Mech. Mater., 33(6), pp. 309–323. [CrossRef]
Ogden, R. W. , 1997, Non-Linear Elastic Deformations, Dover, Mineola, NY.
Itskov, M. , 2007, Tensor Algebra and Tensor Analysis for Engineers, Springer International Publishing, Cham, Switzerland.
Christensen, R. M. , and Lo, K. H. , 1979, “ Solutions for Effective Shear Properties in Three Phase Sphere and Cylinder Models,” J. Mech. Phys. Solids, 27(4), pp. 315–330. [CrossRef]
Christensen, R. M. , 2012, Mechanics of Composite Materials, Dover, Mineola, NY.
Mogilevskaya, S. G. , and Crouch, S. L. , 2007, “ On the Use of Somigliana's Formulae and Series of Surface Spherical Harmonics for Elasticity Problems With Spherical Boundaries,” Eng. Anal. Boundary Elem, 31(2), pp. 116–132. [CrossRef]
Hobson, E. W. , 1965, The Theory of Spherical and Ellipsoidal Harmonics, Chelsea, New York.
Wang, J. , Duan, H. L. , Zhang, Z. , and Huang, Z. P. , 2005, “ An Anti-Interpenetration Model and Connections Between Interphase and Interface Models in Particle-Reinforced Composites,” Int. J. Mech. Sci., 47(4–5), pp. 701–718. [CrossRef]
McCartney, L. N. , and Kelly, A. , 2008, “ Maxwell's Far-Field Methodology Applied to the Prediction of Properties of Multi-Phase Isotropic Particulate Composites,” Proc. R. Soc. London, A, 464(2090), pp. 423–446. [CrossRef]
McCartney, L. N. , 2010, “Maxwell's Far-Field Methodology Predicting Elastic Properties of Multi-Phase Composites Reinforced With Aligned Transversely Isotropic Spheroids,” Philos. Mag., 90(31–32), pp. 4175–4207. [CrossRef]
Mogilevskaya, S. G. , Stolarski, H. K. , and Crouch, S. L. , 2012, “ On Maxwell's Concept of Equivalent Inhomogeneity: When Do the Interactions Matter?,” J. Mech. Phys. Solids, 60(3), pp. 391–417. [CrossRef]
Mogilevskaya, S. G. , Zemlyanova, A. Y. , and Zammarchi, M. , 2018, “ On the Elastic Far-Field Response of a Two-Dimensional Coated Circular Inhomogeneity: Analysis and Applications,” Int. J. Solid. Struct., 130–131, pp. 199–210. [CrossRef]
Mogilevskaya, S. G. , Crouch, S. L. , LaGrotta, A. , and Stolarski, H. K. , 2010, “ The Effects of Surface Elasticity and Surface Tension on the Transverse Overall Elastic Behavior of Unidirectional Nano-Composites,” Compos. Sci. Technol., 70(3), pp. 427–434. [CrossRef]


Grahic Jump Location
Fig. 1

Notations related to a spherical material surface

Grahic Jump Location
Fig. 2

A spherical inhomogeneity with Steigmann–Ogden interface in an infinite matrix

Grahic Jump Location
Fig. 3

Normalized jumps (a) △σrr/μ and (b) △σ/μ for φ = 0

Grahic Jump Location
Fig. 4

Normalized jumps (a) △σrr/μ and (b) △σ/μ for φ = 90

Grahic Jump Location
Fig. 5

Normalized jumps (a) △σrr/μ and (b) △σ/μ as for θ = 90

Grahic Jump Location
Fig. 6

Normalized effective shear modulus for the composite material with overall isotropy



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In