Research Papers

Plane Deformations of an Inhomogeneity–Matrix System Incorporating a Compressible Liquid Inhomogeneity and Complete Gurtin–Murdoch Interface Model

[+] 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;
School of Mechanical Engineering,
Changzhou University,
Changzhou 213164, China
e-mail: m.dai@foxmail.com

Min Li

State Key Laboratory of Mechanics and
Control of Mechanical Structures,
Nanjing University of Aeronautics
and Astronautics,
Nanjing 210016, China
e-mail: mli5@foxmail.com

Peter Schiavone

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

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 8, 2018; final manuscript received September 7, 2018; published online October 1, 2018. Assoc. Editor: Yashashree Kulkarni.

J. Appl. Mech 85(12), 121010 (Oct 01, 2018) (5 pages) Paper No: JAM-18-1464; doi: 10.1115/1.4041469 History: Received August 08, 2018; Revised September 07, 2018

We consider the plane deformations of an infinite elastic solid containing an arbitrarily shaped compressible liquid inhomogeneity in the presence of uniform remote in-plane loading. The effects of residual interface tension and interface elasticity are incorporated into the model of deformation via the complete Gurtin–Murdoch (G–M) interface model. The corresponding boundary value problem is reformulated and analyzed in the complex plane. A concise analytical solution describing the entire stress field in the surrounding solid is found in the particular case involving a circular inhomogeneity. Numerical examples are presented to illustrate the analytic solution when the uniform remote loading takes the form of a uniaxial compression. It is shown that using the simplified G–M interface model instead of the complete version may lead to significant errors in predicting the external loading-induced stress concentration in gel-like soft solids containing submicro- (or smaller) liquid inhomogeneities.

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


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]
Duan, H. L. , Wang, J. , Huang, Z. P. , and Karihaloo, B. L. , 2005, “ Eshelby Formalism for Nano-Inhomogeneities,” Proc. R. Soc. A, 461(2062), pp. 3335–3353. [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. Solids Struct., 43(17), pp. 5055–5065. [CrossRef]
He, L. H. , and Li, Z. R. , 2006, “ Impact of Surface Stress on Stress Concentration,” Int. J. Solids Struct., 43(20), pp. 6208–6219. [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]
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]
Chen, T. , and Chiu, M. S. , 2011, “ Effects of Higher-Order Interface Stresses on the Elastic States of Two-Dimensional Composites,” Mech. Mater., 43(4), pp. 212–221. [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]
Dai, M. , Gharahi, A. , and Schiavone, P. , 2018, “ Analytic Solution for a Circular Nano-Inhomogeneity With Interface Stretching and Bending Resistance in Plane Strain Deformations,” Appl. Math. Model., 55, pp. 160–170. [CrossRef]
Sharma, P. , and Wheeler, L. T. , 2007, “ Size-Dependent Elastic State of Ellipsoidal Nano-Inclusions Incorporating Surface/Interface Tension,” ASME J. Appl. Mech., 74(3), pp. 447–454. [CrossRef]
Wang, G. F. , and Wang, T. J. , 2006, “ Deformation Around a Nanosized Elliptical Hole With Surface Effect,” Appl. Phys. Lett., 89(16), p. 161901. [CrossRef]
Zeng, X. W. , Wang, G. F. , and Wang, T. J. , 2011, “ Erratum: Deformation Around a Nanosized Elliptical Hole With Surface Effect [Appl. Phys. Lett. 89, 161901 (2006)],” Appl. Phys. Lett, 98(15), p. 159901. [CrossRef]
Tian, L. , and Rajapakse, R. , 2007, “ Elastic Field of an Isotropic Matrix With a Nanoscale Elliptical Inhomogeneity,” Int. J. Solids Struct., 44(24), pp. 7988–8005. [CrossRef]
Campàs, O. , Mammoto, T. , Hasso, S. , Sperling, R. A. , O'Connell, D. , Bischof, A. G. , Maas, R. , Weitz, D. A. , Mahadevan, L. , and Ingber, D. E. , 2014, “ Quantifying Cell-Generated Mechanical Forces Within Living Embryonic Tissues,” Nat. Methods, 11(2), p. 183. [CrossRef] [PubMed]
Style, R. W. , Boltyanskiy, R. , Allen, B. , Jensen, K. E. , Foote, H. P. , Wettlaufer, J. S. , and Dufresne, E. R. , 2015, “ Stiffening Solids With Liquid Inclusions,” Nat. Phys., 11(1), p. 82. [CrossRef]
Chen, X. , Li, M. , Yang, M. , Liu, S. , Genin, G. M. , Xu, F. , and Lu, T. J. , 2018, “ The Elastic Fields of a Compressible Liquid Inclusion,” Extreme Mech. Lett., 22, pp. 122–130. [CrossRef]
Wu, J. , Ru, C. Q. , and Zhang, L. , 2018, “ An Elliptical Liquid Inclusion in an Infinite Elastic Plane,” Proc. R. Soc. A, 474(2215), p. 20170813. [CrossRef]
Xu, Q. , Jensen, K. E. , Boltyanskiy, R. , Sarfati, R. , Style, R. W. , and Dufresne, E. R. , 2017, “ Direct Measurement of Strain-Dependent Solid Surface Stress,” Nat. Commun., 8(1), p. 555. [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]
Style, R. W. , Wettlaufer, J. S. , and Dufresne, E. R. , 2015, “ Surface Tension and the Mechanics of Liquid Inclusions in Compliant Solids,” Soft Matter, 11(4), pp. 672–679. [CrossRef] [PubMed]
Dai, M. , Wang, Y. J. , and Schiavone, P. , 2018, “ Integral-Type Stress Boundary Condition in the Complete Gurtin-Murdoch Surface Model With Accompanying Complex Variable Representation,” J. Elast. (epub).
Muskhelishvili, N. I. , 1975, Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, Groningen, The Netherlands.


Grahic Jump Location
Fig. 1

A liquid inhomogeneity in an infinite plane under a uniform remote in-plane loading

Grahic Jump Location
Fig. 2

Uniaxial compression-induced hoop stress around a circular liquid inhomogeneity for complete and simplified G–M models



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