A general interface model for a three-dimensional curved thin anisotropic interphase between two anisotropic media. (English) Zbl 1120.74323

Summary: An arbitrarily curved three-dimensional anisotropic thin interphase between two anisotropic solids is considered. The purpose of this study is to model this interphase as a surface between its two neighbouring media by means of appropriately devised interface conditions on it. The analysis is carried out in the setting of unsteady heat conduction and dynamic elasticity, and makes use of the simple idea of a Taylor expansion of the relevant fields in thin regions. It consists of a generalization of a previous study by Bövik [1994. On the modelling of thin interface layers in elastic and acoustic scattering problems. Q. J. Mech. Appl. Math. 47, 17-42] which was confined to the isotropic setting. The remarkable feature of the presently derived anisotropic interface model is that formally it has a more compact form than that of Bövik’s isotropic version. This is achieved by a judicious choice of surface differential operators which have been used in the derivation, and makes possible to show that several previously known classical interface models are recovered as special cases of the one obtained in this study, once suitable assumptions are made on the magnitude of the conductivity and elasticity tensors of the interphase.


74A50 Structured surfaces and interfaces, coexistent phases
74E10 Anisotropy in solid mechanics
74A40 Random materials and composite materials
Full Text: DOI


[1] Achenbach, J.D.; Zhu, H., Effect of interfacial zone on mechanical behaviour and failure of fiber reinforced composites, J. mech. phys. solids, 37, 381-393, (1989)
[2] Benveniste, Y.; Miloh, T., The effective conductivity of composites with imperfect contact at constituent interfaces, Int. J. eng. sci., 24, 1537-1552, (1986) · Zbl 0594.73116
[3] Benveniste, Y.; Miloh, T., Imperfect soft and stiff interphases in two-dimensional elasticity, Mech. mater., 33, 309-324, (2001)
[4] Bövik, P., On the modelling of thin interface layers in elastic and acoustic scattering problems, Q. J. mech. appl. math., 47, 17-42, (1994) · Zbl 0803.73025
[5] Bövik, P.; Olsson, P., Effective boundary conditions for the scattering of two-dimensional SH waves from a curved thin elastic layer, Proc. R. soc. London A, 439, 257-269, (1992) · Zbl 0778.73019
[6] Cheng, H.; Torquato, S., Effective conductivity of periodic arrays of spheres with interfacial resistance, Proc. R. soc. London A, 453, 145-161, (1997)
[7] Cheng, H.; Torquato, S., Effective conductivity of dispersion of spheres with superconducting interface, Proc. R. soc. London A, 453, 1331-1344, (1997)
[8] Datta, S.K.; Ledbetter, H.M.; Shindo, Y.; Shah, A.H., Phase velocity and attenuation of plane-elastic waves in a particle-reinforced composite medium, Wave motion, 10, 171-182, (1988) · Zbl 0632.73006
[9] Dingreville, R.; Qu, J.; Cherkaoui, M., Surface free energy and its effect on the elastic behavior of nano-sized particles, wires and films, J. mech. phys. solids, 53, 1829-1854, (2005) · Zbl 1120.74683
[10] Duan, H.L.; Wang, J.; Huang, Z.P.; Karihaloo, B.L., Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress, J. mech. phys. solids, 53, 1574-1596, (2005) · Zbl 1120.74718
[11] Gol’denveizer, A.L., The theory of elastic thin shells, (1961), Pergamon Press New York
[12] Goland, M.; Reissner, E., The stresses in cemented joints, J. appl. mech., A17-A27, (1944)
[13] Gurtin, M.E.; Murdoch, A.I., A continuum theory of elastic surfaces, Arch. ration. mech. anal., 57, 291-323, (1975) · Zbl 0326.73001
[14] Gurtin, M.E.; Weissmuller, J.; Larché, F., A general theory of curved deformable interfaces in solids at equilibrium, Philos. mag. A, 78, 1093-1109, (1998)
[15] Hashin, Z., The spherical inclusion with imperfect interface, J. appl. mech., 58, 444-449, (1991)
[16] Hashin, Z., Thin interphase/imperfect interface in conduction, J. appl. phys., 89, 2261-2267, (2001)
[17] Hashin, Z., Thin interphase/imperfect interface in elasticity with application to coated fiber composites, J. mech. phys. solids, 50, 2509-2537, (2002) · Zbl 1080.74006
[18] Hasselman, D.P.H.; Johnson, L.F., Effective thermal conductivity of composites with interfacial barrier resistance, J. compos. mater., 21, 508-515, (1987)
[19] He, Q.C.; Benveniste, Y., Exactly solvable anisotropic thermoelastic microstructures, J. mech. phys. solids, 52, 2661-2682, (2004) · Zbl 1087.74022
[20] Herring, C., The use of classical microscopic concepts in surface energy problems, (), 5-81
[21] Jones, J.P.; Whittier, J.S., Waves at a flexible bonded interface, J. appl. mech., 34, 905-909, (1967)
[22] Kapitza, P.L., 1941. The study of heat transfer in helium II. Zh. Eksp. Teor. Fiz. 11, 1 [J. Phys. (USSR) 4, 181 (1941)]; In: ter Haar, D. (Ed.), Collected Papers of P. L. Kapitza, vol. 2. Pergamon, Oxford, 1965, p. 581.
[23] Klarbring, A.; Movchan, A.B., Asymptotic modeling of adhesive joints, Mech. mater., 28, 137-145, (1998)
[24] Lipton, R., Variational methods, bounds, and size effects for composites with highly conducting interface, J. mech. phys. solids, 45, 361-384, (1997) · Zbl 0969.74571
[25] Lipton, R.; Vernescu, B., Composites with imperfect interface, Proc. R. soc. London A, 452, 329-358, (1996) · Zbl 0872.73033
[26] Mal, B.; Bose, S.K., Dynamic elastic moduli of a suspension of imperfectly bonded spheres, Proc. Cambridge philos. soc., 76, 587-600, (1974) · Zbl 0309.73021
[27] Miller, R.E.; Shenoy, V.B., Size-dependent elastic properties of nano-sized structural elements, Nanotechnology, 11, 139-147, (2000)
[28] Miloh, T.; Benveniste, Y., On the effective conductivity of composites with ellipsoidal inhomogeneities and highly conducting interfaces, Proc. R. soc. London A, 455, 2687-2706, (1999) · Zbl 0937.74051
[29] Milton, G.W., The theory of composites, (2002), Cambridge University Press Cambridge · Zbl 0631.73011
[30] Movchan, A.B.; Bullough, R.; Willis, J.R., Two-dimensional lattice models of the Peierls type, Philos. mag., 83, 569-587, (2003)
[31] Murdoch, A.I., A thermodynamical theory of elastic material interfaces, Q. J. mech. appl. math., 29, 245-275, (1976) · Zbl 0398.73003
[32] Niklasson, A.J.; Datta, S.; Dunn, M.L., On ultrasonic guided waves in a thin anisotropic layer lying between two isotropic layers, J. acoust. soc. am., 91, 1875-1887, (2000)
[33] Niklasson, A.J.; Datta, S.; Dunn, M.L., On approximating guided waves in plates with thin anisotropic coatings by means of effective boundary conditions, J. acoust. soc. am., 108, 924-933, (2000)
[34] Olsson, P.; Datta, S.K.; Bostrom, A., Elastodynamic scattering from inclusions surrounded by thin interface layers, J. appl. mech., 112, 672-676, (1990)
[35] Pham Huy, H.; Sanchez-Palencia, E., Phénomènes de transmission a travers des couches minces de conductivité élevée, J. math. anal. appl., 47, 284-309, (1974) · Zbl 0286.35007
[36] Povstenko, Y.Z., Theoretical investigation of phenomena caused by heterogeneous surface tension in solids, J. mech. phys. solids, 41, 1499-1514, (1993) · Zbl 0784.73072
[37] Rokhlin, S.A.; Huang, W., Ultrasonic wave interaction with a thin anisotropic layer between two anisotropic solids: exact and asymptotic boundary-condition methods, J. acoust. soc. am., 92, 1729-1742, (1992)
[38] Rokhlin, S.A.; Huang, W., Ultrasonic wave interaction with a thin anisotropic layer between two anisotropic solids. II. second-order asymptotic boundary-condition methods, J. acoust. soc. am., 94, 3405-3420, (1993)
[39] Rokhlin, S.A.; Wang, Y.J., Equivalent boundary conditions for thin anisotropic layer between two solids: reflection, refraction, and interface waves, J. acoust. soc. am., 91, 1875-1887, (1992)
[40] Rubin, M.B.; Benveniste, Y., A Cosserat shell model for interphases in elastic media, J. mech. phys. solids, 52, 1023-1052, (2004) · Zbl 1112.74422
[41] Sanchez-Palencia, E., Comportement limite d’un problème de transmission à travers une plaque faiblement conductrice, C. R. acad. sci. Paris ser. A, 270, 1026-1028, (1970) · Zbl 0189.41503
[42] Sharma, P.; Ganti, S.; Bhate, N., Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities, Appl. phys. lett., 82, 535-537, (2003)
[43] Shuttleworth, R., The surface tension in solids, Proc. phys. soc. A, 63, 444-457, (1950)
[44] Steigmann, D.J.; Ogden, R.W., Plane deformations of elastic solids with intrinsic boundary elasticity, Proc. R. soc. London A, 453, 853-877, (1997) · Zbl 0938.74014
[45] Tai, C.-T., Generalized vector and dyadic analysis, (1997), IEEE Press New York
[46] Tiersten, H.F., Elastic surface waves guided by thin films, J. appl. phys., 40, 770-789, (1969)
[47] Torquato, S.; Rintoul, M.D., Effect of the interface on the properties of composite media, Phys. rev. lett., 75, 4067-4070, (1995)
[48] Van Bladel, J., Electromagnetic fields, (2006), IEEE Press New York, (First edition: Van Bladel, J., 1964. Electromagnetic Fields. McGraw Hill, NY)
[49] Wang, J.; Duan, H.L.; Zhang, Z.; Huang, Z.P., An anti-interpenetration model and connections between interphase and interface models in particle-reinforced composites, Int. J. mech. sci., 47, 701-718, (2005) · Zbl 1192.74084
[50] Weissmüller, J.; Cahn, J.W., Mean stresses in microstructures due to interface stress: a generalization of a capillary equation for solids, Acta mater., 45, 1899-1906, (1997)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.