Stress fields of a spheroidal inhomogeneity with an interphase in an infinite medium under remote loadings

This paper presents the elastostatic solution of the problem of an arbitrarily oriented spheroidal inhomogeneity with an interphase embedded in an infinite medium. The latter is under a remote axisymmetric loading. The complete solution of this problem requires three fundamental solutions, which are obtained by the Papkovich–Neuber displacement potentials and the expansion formulae for spheroidal harmonics. New displacement potentials are given when the remote loading is a longitudinal shear. The influence of the orientation and aspect ratio of the inhomogeneity, and of the remote stress ratio on the stress concentrations at the interfaces and the von Mises equivalent stress in the inhomogeneity, are studied. It is found that the interphase between the inhomogeneity and the surrounding medium significantly alters the stress distribution in, and around, the inhomogeneity. In addition to the general solution for an inhomogeneity with an interphase, the stress field exterior to a spheroidal inhomogeneity without an interphase (the Eshelby problem) is presented in a simple form. It is pointed out that the solution of a spheroidal inhomogeneity with an interphase in an infinite medium subjected to an arbitrary uniform eigenstrain, or a combination of a uniform eigenstrain and an arbitrary remote mechanical loading, can be obtained using the procedure developed in this paper.

[1]  J. D. Eshelby.  The determination of the elastic field of an ellipsoidal inclusion, and related problems , 1957, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[2]  Jianxiang Wang,et al.  Prediction of the overall moduli of layered silicate-reinforced nanocomposites—part I: basic theory and formulas , 2004 .

[3]  M. Taya,et al.  Thermal Stress in a Coated Short Fibre Composite , 1986 .

[4]  Minoru Taya,et al.  Stress Field in and Around a Coated Short Fiber in an Infinite Matrix Subjected to Uniaxial and Biaxial Loadings , 1985 .

[5]  Quanshui Zheng,et al.  An explicit and universally applicable estimate for the effective properties of multiphase composites which accounts for inclusion distribution , 2001 .

[6]  Toshio Mura,et al.  The Elastic Field Outside an Ellipsoidal Inclusion , 1977 .

[7]  J. D. Eshelby.  Elastic inclusions and inhomogeneities , 1961 .

[8]  F. Montheillet,et al.  A generalized self-consistent method for solids containing randomly oriented spheroidal inclusions , 1999 .

[9]  I. Jasiuk,et al.  Composites with functionally graded interphases: Mesocontinuum concept and effective transverse conductivity , 1996 .

[10]  M. Cherkaoui,et al.  Elastic composites with coated reinforcements: A micromechanical approach for nonhomothetic topology , 1995 .

[11]  Jianxiang Wang,et al.  Prediction of the overall moduli of layered silicate-reinforced nanocomposites—part II: analyses , 2004 .

[12]  K. Reifsnider,et al.  Residual Stresses in a Composite with Continuously Varying Young's Modulus in the Fiber/Matrix Interphase , 1992 .

[13]  Jiann-Wen Ju,et al.  A Novel Formulation for the Exterior-Point Eshelby's Tensor of an Ellipsoidal Inclusion , 1999 .

[14]  Toshio Mura,et al.  The sliding inclusion under shear , 1987 .

[15]  Z. Zhong,et al.  On the Imperfectly Bonded Spherical Inclusion Problem , 1999 .

[16]  Javier Segurado,et al.  A numerical approximation to the elastic properties of sphere-reinforced composites , 2002 .

[17]  Toshio Kurauchi,et al.  Mechanical properties of nylon 6-clay hybrid , 1993 .

[18]  T. Mura,et al.  The Stress Field in an Elastic Half Space Having a Spheroidal Inhomogeneity Under All-Around Tension Parallel to the Plane Boundary , 1983 .

[19]  Toshio Mura,et al.  The stress field of a sliding inclusion , 1985 .

[20]  C. Q. Ru,et al.  Column buckling of multiwalled carbon nanotubes with interlayer radial displacements , 2000 .

[21]  R. Christensen,et al.  Solutions for effective shear properties in three phase sphere and cylinder models , 1979 .

[22]  W. T. Chen.  Stress Concentration Around Spheroidal Inclusions and Cavities in a Transversely Isotropic Material Under Pure Shear , 1970 .

[23]  Toshio Mura,et al.  Micromechanics of defects in solids , 1982 .

[24]  N. C. Huang,et al.  Interfacial debonding of a spherical inclusion embedded in an infinite medium under remote stress , 2001 .

[25]  Yijun Liu,et al.  Evaluations of the effective material properties of carbon nanotube-based composites using a nanoscale representative volume element , 2003 .

[26]  E. Hobson.  The Theory of Spherical and Ellipsoidal Harmonics , 1955 .

[27]  Tsu-Wei Chou,et al.  Multiscale modeling of carbon nanotube reinforced polymer composites. , 2003, Journal of nanoscience and nanotechnology.

[28]  I. Jasiuk,et al.  The elastic stress field in a half-space containing a prolate spheroidal inhomogeneity subject to pure shear eigenstrain , 2000 .

[29]  Philippe H. Geubelle,et al.  The elastic modulus of single-wall carbon nanotubes: a continuum analysis incorporating interatomic potentials , 2002 .

[30]  Chunyu Li,et al.  Single-walled carbon nanotubes as ultrahigh frequency nanomechanical resonators , 2003 .

[31]  Maurizio Prato,et al.  Molecular design of strong single-wall carbon nanotube/polyelectrolyte multilayer composites , 2002, Nature materials.

[32]  A. Love.  A treatise on the mathematical theory of elasticity , 1892 .

[33]  Hossein M. Shodja,et al.  Elastic Fields in Double Inhomogeneity by the Equivalent Inclusion Method , 2001 .