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General Interface Conditions in Surface Elasticity of Nanoscaled Solids in General Curvilinear Coordinates

Published online by Cambridge University Press:  05 May 2011

C.-N. Weng  and
T. Y. Chen
C.-N. Weng*
Affiliation:
Department of Civil Engineering, National Cheng Kung University, Tainan, Taiwan 70101, R.O.C.
T. Y. Chen*
Affiliation:
Department of Civil Engineering, National Cheng Kung University, Tainan, Taiwan 70101, R.O.C.
*
*Ph.D. student
**Professor, corresponding author
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Abstract

We consider an arbitrarily curved three-dimensional thin interphase with surface stresses between two anisotropic solids. Letting the interphase be infinitely thin and assuming that the kinematic constraints between the two anisotropic solids remain intact during the deformation, we derive the interface jump conditions along the interface. These conditions are derived analytically in general non-orthogonal curvilinear coordinates in the setting of linear elasticity and steady state conduction. The proof is made directly from a force balance consideration of a small element of the curved interface. Simplified results are also deduced for oblique coordinate systems in which the coordinate axes are straight lines that are not perpendicular to each other. When the axes are orthonormal, we prove that our results agree with the previous known Young-Laplace conditions in solids.

Keywords

Surface/interface stressesGeneralized Young-Laplace equationsElasticity and conduction
Type
Articles
Information
Journal of Mechanics , Volume 26 , Issue 1 , March 2010 , pp. 81 - 86
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2010

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References

1.Gibbs, J. W., The Collected Works of J.W. Gibbs, Longmans, New York, 1, p. 315 (1928).Google Scholar
2.Freund, L. B. and Suresh, S.Thin Film Materials: Stress, Defect Formation and Surface Evolution, Cambridge University Press, Cambridge (2003).Google Scholar
3.Zhou, L. G. and Huang, H. C., “Are Surfaces Elastically Softer or Stiffer,” Applied Physics Letters, 84, p. 1940 (2004).CrossRefGoogle Scholar
4.Duan, H. L.Wang, J., Huang, Z. P. and Karihaloo, B. L.“Size-Dependent Effective Elastic Constants of Solids Containing Nano-Inhomogeneities with Interface 85 Stress,” Journal of the Mechanics and Physics of Solids, 53, pp. 1574–1596(2005).CrossRefGoogle Scholar
5.Yang, F. Q., “Size-Dependent Effective Modulus of Elastic Composite Materials: Spherical Nanocavities at Dilute Concentrations,” Journal of Applied Physics, 95, p. 3516(2004).Google Scholar
6.Gurtin, M. E. and Murdoch, A. I., “A Continuum Theory of Elastic Material Surfaces,” Archive for Rational Mechanics and Analysis, 59, pp. 291–323 (1975).CrossRefGoogle Scholar
7.Song, W. and Yang, F.“Surface Evolution of a Stressed Elastic Layer in an Electric Field,” Journal of Physics D: Applied Physics, 39, pp. 4634–4642 (2006)CrossRefGoogle Scholar
8.Li, X. F. and Peng, X. L., “Theoretical Analysis of Surface Stress for a Micro-Cantilever with Varying Widths,” Journal of Physics D: Applied Physics, 41, pp. 065–301 (2008).Google Scholar
9.Chen, T. and Dvorak, G. J., “Fibrous Nanocomposites with Interface Stress: Hill's and Levin's Connections for Effective Moduli,” Applied Physics Letters, 88, 211912, 3p (2006).CrossRefGoogle Scholar
10.Landau, L. D. and Lifshitz, E. M., Fluid Mechanics, 2nd Ed., Pergamon Press, Oxford (1987).Google Scholar
11.Povstenko, Y. Z., “Theoretical Investigation of Phenomena Caused by Heterogeneous Surface Tension in Solids,” Journal of the Mechanics and Physics of Solids, 41, pp. 1499–1514(1993).CrossRefGoogle Scholar
12.Benveniste, Y. and Miloh, T., “Imperfect Soft and Stiff Interfaces in Two-Dimensional Elasticity,” Mechanics of Materials, 33, p. 309 (2001).CrossRefGoogle Scholar
13.Sharma, P., Ganti, S. and Bhate, N.“Effect of Surfaces on the Size-Dependent Elastic State of Nano-Inhomogeneities,” Applied Physics Letters, 82, pp. 535–537 (2003).CrossRefGoogle Scholar
14.Chen, T., Dvorak, G. J. and Yu, C. C., “Solids Containing Spherical Nano-Inclusions with Interface Stresses: Effective Properties and Thermal-Mechanical Connections,” International Journal of Solids and Structures, 44, pp. 941–955 (2007).CrossRefGoogle Scholar
15.Chen, T., Dvorak, G. J. and Yu, C. C., “Size-Dependent Elastic Properties of Unidirectional Nano-Composites with Interface Stresses,” Acta Mechanica, 188, pp. 39–54 (2007).Google Scholar
16.Chen, T., Chiu, M. S. and Weng, C. N., “Derivation of the Generalized Young-Laplace Equation of Curved Interfaces in Nano-Scaled Solids,” Journal of Applied Physics, 100, 074308, 5 p (2006).Google Scholar
17.Thostenson, E. T., Ren, Z. and Chou, T. W., “Advances in the Science and Technology of Carbon Nanotubes and Their Composites: A Review,” Composites Science and Technology, 61, pp. 1899–1912 (2001).CrossRefGoogle Scholar
18.Qian, H.Xu, K. Y. and Ru, C. Q., “Surface Forces Driven Wrinkling of an Elastic Half-Space Coated with a Thin Stiff Surface Layer,” Journal of Elasticity, 86, pp. 205–219 (2007).Google Scholar
19.Zhao, L.“The Generalized Theory of Perfectly Matched Layers (GT-PML) in Curvilinear Coordinates,” International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 13, pp. 457–469 (2000).Google Scholar
20.Maliska, C. R. and Raithby, G. D., “A Method for Computing Three Dimensional Flows Using Non-Orthogonal Boundary-Fitted Coordinates,” International Journal for Numerical Methods in Fluids, 4, pp. 519–537(1984).CrossRefGoogle Scholar
21.Nix, W. D. and Gao, H.“An Atomistic Interpretation of Interface Stress,” Scripta Materialia, 39, pp. 1653–1661 (1998).CrossRefGoogle Scholar
22.Young, E. C., Vector and Tensor Analysis, M. Dekker, New York (1978).Google Scholar
23.Arfken, G. B. and Weber, H. J., Mathematical Methods for Physicists, Academic Press, San Diago (2001).Google Scholar
24.Benveniste, Y., “A General Interface Model for a Three-Dimensional Curved Thin Anisotropic Interphase Between Two Anisotropic Media,” Journal of the Mechanics and Physics of Solids, 54, pp. 708–734 (2006).CrossRefGoogle Scholar