Applied Mathematics and Mechanics >
Two interacting harmonic non-elliptical compressible liquid inclusions
Received date: 2025-06-04
Revised date: 2025-09-05
Online published: 2025-09-30
Supported by
Project supported by the Natural Sciences and Engineering Research Council of Canada (No. RGPIN-2023-03227 Schiavo)
Copyright
We present the design of two interacting harmonic non-elliptical compressible liquid inclusions embedded in an infinite isotropic elastic matrix subjected to uniform remote in-plane stresses. The original constant mean stress (or the first invariant of the stress tensor) in the matrix remains undisturbed in the presence of the two harmonic liquid inclusions. The two non-elliptical liquid-solid interfaces are described by a four-parameter conformal mapping function that maps the doubly connected domain occupied by the matrix onto an annulus in the image plane. The closed-form expressions for the internal uniform hydrostatic stress fields within the two liquid inclusions are obtained. The hoop stresses are uniformly distributed along the two liquid-solid interfaces on the matrix side.
Xu WANG , P. SCHIAVONE . Two interacting harmonic non-elliptical compressible liquid inclusions[J]. Applied Mathematics and Mechanics, 2025 , 46(10) : 1955 -1966 . DOI: 10.1007/s10483-025-3309-8
| [1] | BJORKMAN, G. S. and RICHARDS, R. Harmonic holes?—?an inverse problem in elasticity. ASME Journal of Applied Mechanics, 43, 414–418 (1976) |
| [2] | BJORKMAN, G. S. and RICHARDS, R. Harmonic holes for nonconstant fields. ASME Journal of Applied Mechanics, 46, 573–576 (1979) |
| [3] | RICHARDS, R. and BJORKMAN, G. S. Harmonic shapes and optimum design. Journal of the Engineering Mechanics Division, 106, 1125–1134 (1980) |
| [4] | WHEELER, L. T. and KUNIN, I. A. On voids of minimum stress concentration. International Journal of Solids and Structures, 18, 85–89 (1982) |
| [5] | ELDIWANY, B. H. and WHEELER, L. T. A three-dimensional inverse problem for inhomogeneities in elastic solids. Journal of Elasticity, 16, 201–211 (1986) |
| [6] | BJORKMAN, G. S. Design of pressure vessel pads and attachments to minimize global stress concentrations. Transactions, SMiRT 19, Toronto (2007) |
| [7] | RU, C. Q. Three-phase elliptical inclusions with internal uniform hydrostatic stresses. Journal of the Mechanics and Physics of Solids, 47, 259–273 (1999) |
| [8] | RU, C. Q. A new method for an inhomogeneity with stepwise graded interphase under thermomechanical loadings. Journal of Elasticity, 56, 107–127 (1999) |
| [9] | WANG, G. F., SCHIAVONE, P., and RU, C. Q. Harmonic shapes in finite elasticity under nonuniform loading. ASME Journal of Appliled Mechanics, 72, 691–694 (2005) |
| [10] | WANG, X. Uniform fields inside two non-elliptical inclusions. Mathematics and Mechanics of Solids, 17, 736–761 (2012) |
| [11] | WANG, X. and SCHIAVONE, P. Harmonic ellipsoidal elastic solid or liquid inclusions. Journal of Elasticity, 157, 61 (2025) |
| [12] | STYLE, R. W., WETTLAUFER, J. S., and DUFRESNE, E. R. Surface tension and the mechanics of liquid inclusions in compliant solids. Soft Matter, 11, 672–679 (2015) |
| [13] | STYLE, R. W., BOLTYANSKIY, R., ALLEN, B., JENSEN, K. E., FOOTE, H. P., WETTLAUFER, J. S., and DUFRESNE, E. R. Stiffening solids with liquid inclusions. Nature Physics, 11, 82–87 (2015) |
| [14] | MANCARELLA, F., STYLE, R. W., and WETTLAUFER, J. S. Interfacial tension and a three-phase generalized self-consistent theory of non-dilute soft composite solids. Soft Matter, 12, 2744–2750 (2016) |
| [15] | WU, J., RU, C. Q., and ZHANG, L. An elliptical liquid inclusion in an infinite elastic plane. Proceedings of the Royal Society A, 474(2215), 20170813 (2018) |
| [16] | CHEN, X., LI, M. X., YANG, M., LIU, S. B., GENIN, G. M., XU, F., and LU, T. J. The elastic fields of a compressible liquid inclusion. Extreme Mechanics Letters, 22, 122–130 (2018) |
| [17] | CHEN, X., LI, M., LIU, S., HE, W., and LU, T. J. Mechanics tuning of liquid inclusions via bio-coating. Extreme Mechanics Letters, 41, 101049 (2020) |
| [18] | KRICHEN, S., LIU, L. P., and SHARMA, P. Liquid inclusions in soft materials: capillary effect, mechanical stiffening and enhanced electromechanical response. Journal of the Mechanics and Physics of Solids, 127, 332–357 (2019) |
| [19] | DAI, M., HUA, J., and SCHIAVONE, P. Compressible liquid/gas inclusion with high initial pressure in plane deformation: modified boundary conditions and related analytical solutions. European Journal of Mechanics A-Solids, 82, 104000 (2020) |
| [20] | DAI, M., HUANG, C., and SCHIAVONE, P. Modified closed-form solutions for three-dimensional elastic deformations of a composite structure containing macro-scale spherical gas/liquid inclusions. Applied Mathematical Modelling, 97, 57–68 (2021) |
| [21] | TI, F., CHEN, X., YANG, H., LIU, S., and LU, T. J. A theory of mechanobiological sensation: strain amplification/attenuation of coated liquid inclusion with surface tension. Acta Mechanica Sinica, 37(1), 145–155 (2021) |
| [22] | TI, F., CHEN, X., LI, M. X., SUN, X. C., LIU, S. B., and LU, T. J. Cylindrical compressible liquid inclusion with surface effects. Journal of the Mechanics and Physics of Solids, 161, 104813 (2022) |
| [23] | GHOSH, K. and LOPEZ-PAMIES, O. Elastomers filled with liquid inclusions: theory, numerical implementation, and some basic results. Journal of the Mechanics and Physics of Solids, 166, 104930 (2022) |
| [24] | GHOSH, K., LEFèVRE, V., and LOPEZ-PAMIES, O. The effective shear modulus of a random isotropic suspension of monodisperse liquid n-spheres: from the dilute limit to the percolation threshold. Soft Matter, 19, 208–224 (2023) |
| [25] | GHOSH, K., LEFEVRE, V., and LOPEZ-PAMIES, O. Homogenization of elastomers filled with liquid inclusions: the small-deformation limit. Journal of Elasticity, 154, 235–253 (2023) |
| [26] | WANG, X. and SCHIAVONE, P. An edge dislocation interacting with an elliptical incompressible liquid inclusion. Journal of Mechanics of Materials and Structures, 19(1), 131–140 (2024) |
| [27] | WANG, X. and SCHIAVONE, P. An edge dislocation interacting with a hypotrochoidal compressible liquid inclusion. Acta Mechanica, 235, 3211–3218 (2024) |
| [28] | CHEREPANOV, G. P. Inverse problem of the plane theory of elasticity. Journal of Applied Mathematics and Mechanics, 38, 963–979 (1974) |
| [29] | MUSKHELISHVILI, N. I. Some Basic Problems of the Mathematical Theory of Elasticity, P. Noordhoff Ltd., Groningen (1953) |
| [30] | TING, T. C. T. Anisotropic Elasticity: Theory and Applications, Oxford University Press, New York (1996) |
| [31] | CHERAGHI, D. Geometric Complex Analysis, Lecture Notes, Imperial College London (2017) |
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