Abstract
The first strain gradient linear elasticity theory (FSGLET) is invoked to study an inhomogeneity problem. In the setting of this centrosymmetric problem, a set of concentric hollow spheres made of different materials is considered to serve as a model for a multi-coated nanometer-sized particle which in turn can be introduced as a representative volume element (RVE) of a nanocomposite. Perfect bonding is assumed between each layer of the RVE, and this system is subjected to uniform internal and external pressures. In the context of the FSGLET, the boundary value problem (BVP) for this model is formulated rigorously in spherical coordinates and its solution is obtained by means of analytical and numerical techniques. Employing the solutions of this BVP, we examine the important cases of a two-phase RVE consisting only of filler and matrix phases and a three-phase RVE where a homogeneous or a functionally graded interphase is also taken into account. We determine the associated elastic fields inside the RVE for each of these cases to study the impact of size effects, presence or absence of an interphase and its material properties distribution on the elastic fields variations. Furthermore, based on a simple and natural definition for the effective bulk modulus of the RVE, this effective modulus is calculated in each of the aforementioned cases at different filler volume fractions and characteristic lengths, and relevant comparisons are made.
Similar content being viewed by others
References
Zhou, K., Hoh, H.J., Wang, X., Keer, L.M., Pang, J.H.L., Song, B., Wang, Q.J.: A review of recent works on inclusions. Mech. Mater. 60, 144–158 (2013)
Mura, T.: Micromechanics of Defects in Solids, 2nd edn. Martinus Nijhoff, Leiden (1987)
Eshelby, J.D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. A Math. Phys. Eng. Sci. 241(1226), 376–396 (1957)
Eshelby, J.D.: The elastic field outside an ellipsoidal inclusion. Proc. R. Soc. A Math. Phys. Eng. Sci. 252(1271), 561–569 (1959)
Shodja, H.M., Sarvestani, A.S.: Elastic fields in double inhomogeneity by the equivalent inclusion method. J. Appl. Mech. 68(1), 3 (2001)
Moschovidis, Z.A., Mura, T.: Two-ellipsoidal inhomogeneities by the equivalent inclusion method. J. Appl. Mech. 42(4), 847–852 (1975)
Hori, M., Nemat-Nasser, S.: Double-inclusion model and overall moduli of multi-phase composites. Mech. Mater. 14(3), 189–206 (1993)
Fond, C., Riccardi, A., Schirrer, R., Montheillet, F.: Mechanical interaction between spherical inhomogeneities: an assessment of a method based on the equivalent inclusion. Eur. J. Mech. A Solids 20(1), 59–75 (2001)
Cheng, Z.Q., He, L.H.: Micropolar elastic fields due to a spherical inclusion. Int. J. Eng. Sci. 33(3), 389–397 (1995)
Li, S., Sauer, R.A., Wang, G.: A circular inclusion in a finite domain I. The Dirichlet–Eshelby problem. Acta Mech. 179(1–2), 67–90 (2005)
Li, S., Sauer, R., Wang, G.: A circular inclusion in a finite domain II. The Neumann–Eshelby problem. Acta Mech. 179(1–2), 91–110 (2005)
Li, S., Sauer, R.A., Wang, G.: The Eshelby tensors in a finite spherical domain—part I: theoretical formulations. J. Appl. Mech. 74(4), 770 (2007)
Christensen, R.M., Lo, K.H.: Solutions for effective shear properties in three phase sphere and cylinder models. J. Mech. Phys. Solids 27(4), 315–330 (1979)
Luo, H.A., Weng, G.J.: On Eshelby’s inclusion problem in a three-phase spherically concentric solid, and a modification of Mori–Tanaka’s method. Mech. Mater. 6(4), 347–361 (1987)
Mikata, Y., Taya, M.: Stress field in and around a coated short fiber in an infinite matrix subjected to uniaxial and biaxial loadings. J. Appl. Mech. 52(1), 19 (1985)
Khorshidi, A., Shodja, H.M.: A spectral theory formulation for elastostatics by means of tensor spherical harmonics. J. Elast. 111(1), 67–89 (2013)
Pan, C., Yu, Q.: Investigation of an arbitrarily shaped inclusion embedded in a two-dimensional finite domain. Int. J. Mech. Sci. 126, 142–150 (2017)
Zou, W.N., He, Q.C., Zheng, Q.S.: Inclusions in a finite elastic body. Int. J. Solids Struct. 49(13), 1627–1636 (2012)
Dai, M., Sun, H.: Thermo-elastic analysis of a finite plate containing multiple elliptical inclusions. Int. J. Mech. Sci. 75, 337–344 (2013)
Lubarda, V.A.: Circular inclusions in anti-plane strain couple stress elasticity. Int. J. Solids Struct. 40(15), 3827–3851 (2003)
Xun, F., Hu, G., Huang, Z.: Effective in plane moduli of composites with a micropolar matrix and coated fibers. Int. J. Solids Struct. 41(1), 247–265 (2004)
Haftbaradaran, H., Shodja, H.M.: Elliptic inhomogeneities and inclusions in anti-plane couple stress elasticity with application to nano-composites. Int. J. Solids Struct. 46(16), 2978–2987 (2009)
Alemi, B., Shodja, H.M.: Effective shear modulus of solids reinforced by randomly oriented-aligned-elliptic multi-coated nanofibers in micropolar elasticity. Compos. Part B Eng. 143, 197–206 (2018)
Shodja, H.M., Alemi, B.: Effective shear modulus of solids reinforced by randomly oriented-/aligned-elliptic nanofibers in couple stress elasticity. Compos. Part B Eng. 117, 150–164 (2017)
Zibaei, I., Rahnama, H., Taheri-Behrooz, F., Shokrieh, M.M.: First strain gradient elasticity solution for nanotube-reinforced matrix problem. Compos. Struct. 112, 273–282 (2014)
Sidhardh, S., Ray, M.C.: Exact solutions for elastic response in micro- and nano-beams considering strain gradient elasticity. Math. Mech. Solids 24, 895–918 (2018)
Mindlin, R.D., Eshel, N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4(1), 109–124 (1968)
Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51–78 (1964)
Mindlin, R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1(4), 417–438 (1965)
Altan, S.B., Aifantis, E.C.: On the structure of the mode III crack-tip in gradient elasticity. Scr. Metall. Mater. 26(2), 319–324 (1992)
Lam, D.C.C., Yang, F., Chong, A.C., Wang, J., Tong, P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51(8), 1477–1508 (2003)
Dell’Isola, F., Sciarra, G., Vidoli, S.: Generalized Hooke’s law for isotropic second gradient materials. Proc. R. Soc. A Math. Phys. Eng. Sci. 465(2107), 2177–2196 (2009)
Auffray, N., Le Quang, H., He, Q.C.: Matrix representations for 3D strain-gradient elasticity. J. Mech. Phys. Solids 61(5), 1202–1223 (2013)
Rahnama, H.: Micromechanical stress analysis of long fiber composites. M.Sc. Thesis, Iran University of Science and Technology (2015)
Rahnama, H., Shokrieh, M.M.: Axisymmetric equilibrium of an isotropic elastic solid circular finite cylinder. Math. Mech. Solids 24, 996–1029 (2018)
Gao, X.L., Park, S.K.: Analytical solution for a pressurized thick-walled spherical shell based on a simplified strain gradient elasticity theory. Math. Mech. Solids 14(9), 747–758 (2009)
Maple 15. Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario
Zare, Y.: Assumption of interphase properties in classical Christensen–Lo model for Young’s modulus of polymer nanocomposites reinforced with spherical nanoparticles. RSC Adv. 5(116), 95532–95538 (2015)
Ramezani, S.: Nonlinear vibration analysis of micro-plates based on strain gradient elasticity theory. Nonlinear Dyn. 73(3), 1399–1421 (2013)
Sadeghi, H., Baghani, M., Naghdabadi, R.: Strain gradient elasticity solution for functionally graded micro-cylinders. Int. J. Eng. Sci. 50(1), 22–30 (2012)
Chu, L., Dui, G.: Exact solutions for functionally graded micro-cylinders in first gradient elasticity. Int. J. Mech. Sci. 148(April), 366–373 (2018)
Acknowledgements
The first author is indebted to his colleague and dear friend Iman Zibaei, who introduced him to the strange world of generalized continuum theories. Moreover, his unfailing kindness and support during preparing this work, useful comments and fruitful discussions on first versions of the manuscript are greatly acknowledged.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Rahnama, H., Salehi, S.D. & Taheri-Behrooz, F. Centrosymmetric equilibrium of nested spherical inhomogeneities in first strain gradient elasticity. Acta Mech 231, 1377–1402 (2020). https://doi.org/10.1007/s00707-019-02570-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-019-02570-0