Ellipsoidal inhomogeneity with anisotropic incoherent interface. Multipole series solution and application to micromechanics

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Abstract

The ellipsoidal inhomogeneity with transversely isotropic incoherent interface is considered in the conductivity and elasticity context. The general imperfect interface is modeled by the first order approximation of thin transversely isotropic interphase layer. This model expands the Gurtin et al. (1998) theory of curved deformable interfaces in solids with a nanometer-scale microstructure to the incoherent interfaces between the dissimilar elastic materials and provides a certain insight into the interface moduli. The rigorous analytical solution to the conductivity and elasticity problems has been obtained by the multipole expansion method in terms of ellipsoidal solid harmonics. An accurate fulfillment of the imperfect interface conditions reduces the model boundary value problem to the linear algebraic system for multipole strengths. These results apply equally to the inhomogeneities with anisotropic interphases and nano level incoherent interfaces. The obtained solutions are valid for the non-uniform far loading and are readily incorporated in the many-particle (finite cluster or representative unit cell) model of heterogeneous solid with anisotropic incoherent interface. The tensors of effective conductivity and elastic stiffness of composite with ellipsoidal inhomogeneities are evaluated using Maxwell homogenization scheme. The obtained accurate numerical data indicate that the effective properties of composite may vary widely due to shape of inhomogeneities and the interface anisotropy ratio. Taking the incoherency of interface into account may increase reliability of predicting the behavior of nanostructured solids.

Introduction

The “ellipsoidal inhomogeneity in unbounded solid” problem is widely used in micromechanics as a model of the matrix type heterogeneous solid. For the comprehensive review on the problem, see e.g. Kachanov and Sevostianov (2018). This model is attractive for it provides considering a wide range of heterogeneous solids with round inclusions, short fibers, platelets, elliptical cracks, etc. within a unified formalism and, at the same time, allows the analytical solution of the corresponding boundary-value problem. The Fricke (1924) and Eshelby (1959) solutions for a single ellipsoidal inhomogeneity have gained much popularity and provided the theoretical background of several micromechanical theories. In these and numerous subsequent publications, the perfect thermal/elastic contact between the matrix and inhomogeneity was assumed. This questions their applicability to real-life heterogeneous solids whose interfaces are imperfect and play an important role in the transport processes. This is particularly true for the nanostructured materials that exhibit a very high interface area-to-volume ratio.

There is a limited number of publications where the ellipsoidal inhomogeneity with imperfect bonding to matrix is addressed. The first order approximations for the effective conductivity of composite with arbitrarily oriented ellipsoidal inclusions were obtained using the Mori–Tanaka (Dunn and Taya, 1993, Nan et al., 1997) and Maxwell (Duan & Karihaloo, 2007) averaging schemes. These and several analogous works rely on the “variable skin constant” hypothesis (Duschlbauer et al., 2003) taking an effect of the Kapitza type imperfect interface into account approximately, by means of fictitious equivalent inclusion perfectly bonded to matrix. The papers by Benveniste and Miloh (1986) and Miloh and Benveniste (1999) are the first works where the imperfect ellipsoidal interfaces (low conductive (LC) and highly conductive (HC), respectively) were properly taken into account. In (Kushch, 2017), a complete solution to the many-particle model of conductive composite with ellipsoidal inhomogeneities, anisotropic constituents and imperfect (both LC and HC) interfaces has been obtained by the multipole expansion method. In the elasticity context, the soft and hard imperfect interfaces are modeled usually by the spring (e.g., Hashin, 1991) and membrane (Gurtin & Murdoch, 1975) model, respectively. The approximate solutions are available in literature for the ellipsoidal inclusion (Qu, 1993) and inhomogeneity (Sharma & Wheeler, 2007) with imperfect interface. There, an effect of imperfect interface is taken into account in averaged sense. Recently, the complete solution has been obtained for a single (Kushch, 2019a) and multiple (Kushch, 2019b, Kushch, 2020a, Kushch, 2020b, Kushch, 2020c) ellipsoidal inhomogeneities with the spring-type imperfect interface.

Interface is crucial in modeling the nanoheterogeneous solids (e.g., Wang et al., 2011). Most publications on the subject use the coherent interface model by Gurtin and Murdoch (1975). In terms of interface conditions, coherency means the displacement vector continuity across the interface between the abutting solids. Applicability of the coherent interface model to the real-world nanostructured solids with predominantly incoherent interfaces/grain boundaries (e.g., Mittemeijer, 2011) is not obvious. A very few papers consider the material interface model with the displacement discontinuity. Gurtin et al. (1998) have developed a general theory of curved deformable interfaces in solids with a nanometer-scale microstructure. Dingreville et al. (2014) has derived a generalized continuum framework describing the elastic coherent and incoherent interfaces under general loading conditions. The common feature of these theories is an enlarged number (5 and 4, respectively) of the interface elastic constants. In Firooz et al. (2019), the self-consistent and Mori–Tanaka homogenization schemes are extended to the elastic particulate composite with general imperfect interface by taking three (two in-plane and one orthogonal) elastic moduli of interface into account.

In this work, the general incoherent interface between the matrix and ellipsoidal inhomogeneity is addressed. To the best author’s knowledge, the problem in this formulation has never been considered. The interface model we consider has been derived by Benveniste (2006) as the first order approximation of thin anisotropic interphase layer. In Section 3, this model is discussed in the conductivity and elasticity context. In the following two Sections, a rigorous analytical solution to the conductivity and elasticity boundary value problem in terms of ellipsoidal harmonics is derived. Their application to micromechanics in the framework of Maxwell homogenization scheme is discussed in Section 6. The background theory and auxiliary formulas are provided in Appendices to C.

Section snippets

Problem statement

We consider an unbounded isotropic solid, or matrix, containing an isotropic ellipsoidal inhomogeneity with the semiaxes a1>a2>a3 oriented along the corresponding axes of Cartesian coordinate system Ox1x2x3, Fig. 1.

For the ellipsoidal coordinates and other notations, see Appendix A. The ellipsoidal coordinates ρ,μ,ν, Eq. (A.2) are introduced in a way that the coordinate surface ρ=a1 coincides with the surface S of ellipsoid defined by the equation xiai21=0. The volume of ellipsoid is V=4π3a1a2

Model of incoherent interface

It was mentioned already that the general imperfect interface model we consider has been derived by Benveniste (2006) as the first order approximation of thin anisotropic interphase layer. Following Kushch, 2021a, Kushch, 2021b, we re-formulate the Benveniste model for the case of transverse isotropy in the compact form.

Multipole series expansion

Following Kushch (2017), we write the regular temperature field inside the inhomogeneity T(1) as a series expansion over a set of the interior solid harmonics Enm defined by Eq. (A.1) of Appendix A: T(1)x=n=0m=12n+1DnmEnmx.Here, Dnm are the inhomogeneity-related series expansion coefficients. The temperature field outside the inhomogeneity is written as a sum of the far field Tfar and perturbation field Tper (0 for x) caused by the inhomogeneity. In a general case, the series expansion

Solution to the elasticity problem

Following Kushch, 2019a, Kushch, 2019b, we use Papkovich–Neuber representation (e.g., Gurtin, 1973) of the displacement vector in terms of four scalar harmonic potentials Bi u=41νBgradxB+B0,B=Bjij.The corresponding stress tensor is expressed in terms of these potentials as σ=μgradu +graduT+λIdivu=12νgradB+gradBTgradgradBx+2νIdivBgradgradB0.

Application to micromechanics

Mechanics of materials is one of the application areas of the developed theory. Here, we incorporate it in the Maxwell homogenization scheme (e.g., Sevostianov et al., 2019) to evaluate the effective conductivity and stiffness of composite with general imperfect interface between the matrix and ellipsoidal inhomogeneities.

Conclusions

The findings of this work can be summarized as follows.

The general imperfect interface model is formulated using the first order approximation of thin transversely isotropic interphase layer. This model expands the Gurtin et al. (1998) theory of curved deformable interfaces in solids with a nanometer-scale microstructure on the incoherent interfaces between the dissimilar elastic materials and provides a certain insight into the interface elastic moduli. The additional material parameters

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      Citation Excerpt :

      To evaluate importance of interface effects, several continuum models have been studied. The theory proposed by Gurtin and Murdoch (1975b), and its generalization made by Steigmann and Ogden (1999) to incorporate flexural rigidity, are two of the most commonly-used theories accounting for interface-induced size effects (Argatov, 2022; Chhapadia, Mohammadi, & Sharma, 2011; Duan, Wang, Huang, & Karihaloo, 2005; Duan, Wang, Karihaloo, & Huang, 2006; Grekov & Sergeeva, 2020; Gurtin & Murdoch, 1978; Han, Mogilevskaya, & Schillinger, 2018; Jiang, Li, & Hu, 2022; Kushch, 2018, 2021; Li, Hu, & Ling, 2016; Li & Mi, 2019; Lu, He, Lee, & Lu, 2006; Mi, 2018; Mikhasev, Botogova, & Eremeyev, 2021; Wang, Yan, Dong, & Atluri, 2020; Zemlyanova & Mogilevskaya, 2018b; Zheng & Mi, 2021; Zheng, Zhang, & Mi, 2021; Zhu & Li, 2019). Both the Gurtin–Murdoch model and the Steigmann–Ogden model use a zero-thickness interface, and external forces per unit area come from stress discontinuity across the interface.

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