A micropolar model for elastic properties in functionally graded materials

Abstract

By considering the description of phase volume fractions, a micromechanics model is presented for predicting the elastic mechanical properties of isotropic two-phase functionally graded materials. The particle size dependence of the overall elasticity of functionally graded materials is not generally considered by classical continuum micromechanics; however, being based on micropolar theory, the presented micromechanics model is able to study such size effects. As the effective material properties vary gradually along the gradation direction, a functionally graded material can be divided into two distinct zones: the particle–matrix zone and the transition zone. In the particle–matrix zone, the matrix material is idealized as a micropolar continuum and Mori–Tanaka’s method is extended to the micropolar medium to evaluate the effective elastic properties. The effective properties across the gradation forms are further derived and the effects of particle size on the effective properties of a functionally graded materials are also studied. The validity and effectiveness of the present model is demonstrated by comparing the model results with other model outputs and experimental data.

Introduction

Functionally graded materials (FGMs) are composite materials that are microscopically inhomogeneous, and their overall properties vary continuously in one (or more) direction(s) to eliminate singular stresses, relax residual stresses, and enhance bonding strength. Therefore, aerospace, automotive, marine, and biological industries are taking advantage of the special characteristics of FGMs. ¹ However, the extent to which FGMs can be tailored to produce a target performance, that is, the design of FGMs, strongly depends on the resultant effective properties and, more importantly, on how these properties relate to microstructure. Therefore, an important aspect of the design of FGMs is prediction of the mechanical, thermal, and other properties based on the given microstructure and its spatial distribution.

Many models and approaches have been proposed in the literature for the evaluation of the various effective properties. Classical micromechanic approaches such as the Mori–Tanaka method, ² the self-consistent method, ³ and a variety of other models have been directly applied to estimate the effective elastic properties of FGMs.^4–8 Aboudi et al. ⁹ developed a higher order theory for FGM application; this theory can explicitly couple the local microstructure and global response. Yin et al.^10,11 employed an elastic model that included the pairwise particle interaction and gradient effects of phase volume fraction to investigate the mechanical and thermal-elastic behavior of FGMs. The effective properties of FGMs have also been determined by the micromechanical finite element method.^12,13 In order to determine the effects of the stochastic behavior of FGMs, Rahman and Chakraborty^14,15 presented a stochastic micromechanical model for predicting the probabilistic characteristics of the elastic mechanical properties of isotropic FGMs; they also developed a concurrent multiscale model for the fracture behavior of FGMs. Guo et al. ¹⁶ studied the fracture problem of FGMs by combining the stochastic micromechanics model with the extended piecewise-exponential method.

FGMs are essentially heterogeneous in nature; homogenization of the material depends on length scales. The above-mentioned FGM model based on classical elasticity did not directly include size effects owing to the lack of a material length scale parameter. This motivated the development of microstructure-dependent models using higher order continuum theories that are able to describe such size effects. ¹⁷ In recent years, the micropolar elasticity theory has gained great importance due to the large-scale development and utilization of composite, reinforced, and coarse-grained materials. Forrest and Sab ¹⁸ proposed a micro–macro transition method by examining periodic boundary conditions on a representative volume element (RVE). Cheng and He^19,20 derived the analytical solutions for a homogeneous micropolar material in which a cylindrical or a spherical region is subjected to both a uniform eigenstrain and uniform eigentorsion. Ma et al. ²¹ proposed a microstructure-dependent Timoshenko beam model based on the modified couple stress theory. Reddy ²² used the modified couple stress theory to develop a model for the FGM beam that considered the size effect. Ostoja-Starzewski et al. ²³ studied the effective properties and size effect of a two-phase composite material based on couple stress theory. Hu and Colleagues^24–26 presented an analytical homogenization method to derive the higher order material parameters of micropolar theory and explain the observed particle size dependence in composite materials.

Although there is much work devoted to micromechanics models for FGMs, there is at present no analytical continuum micromechanical method in the framework of micropolar theory that enables one to determine the microstructure parameters for FGMs.

This article presents a simple analytical micromechanical method based on micropolar theory. In this method, an average equivalent inclusion method is advanced and the Mori–Tanaka method is extended to a micropolar medium to evaluate the effective elastic properties. The studied FGM consists of two constituents and its microstructure is assumed to be solid spherical particles that are spatially distributed in a homogeneous matrix. Numerical results are presented to show the effects of both gradient form and particles size upon the effective properties of the FGM. The predictive capability of the proposed model is illustrated by comparison with experimental results and other theoretical results.

Micromechanics for the micropolar medium

Basic relations of micropolar theory

In this section, we present the basic equations of the micropolar elasticity theory. The classical Cauchy continuum mechanics is based on the assumption that the interaction between any two continuum particles across an elementary area lying within the body occurs solely through the force vector. ²⁷ While in micropolar theory, it allows the points in the continuum to rotate as well as translate, and the continuum supports couple per unit area as well as force per unit area. In micropolar theory, it is assumed that not only forces but also moments can be transmitted across the surface element of a body. In addition to the displacements, three extra degrees of freedom are introduced, characterizing the microrotation of each material point. In the absence of body force and body moment, the governing equations of in-plane micropolar theory are as follows (with proper boundary conditions) ²⁶

$${\epsilon }_{ij}=u_{j,i}-e_{kij}{\phi }_k,k_{ij}={\phi }_{j,i}$$

(1)

$${\sigma }_{ij,i}=0,m_{ij,i}+e_{jik}{\sigma }_{ik}=0$$

(2)

$${\sigma }_{\mathrm{ij}}={\delta }_{\mathrm{ij}}\lambda {\epsilon }_{\mathrm{kk}}+(\mu +\kappa ){\epsilon }_{\mathrm{ij}}+(\mu -\kappa ){\epsilon }_{\mathrm{ji}}$$

(3)

$$m_{\mathrm{ij}}={\delta }_{\mathrm{ij}}\alpha {\phi }_{k,k}+\beta {\phi }_{j,i}+\gamma {\phi }_{i,j}$$

(4)

where ${\epsilon }_{\mathrm{ij}}$ , ${\sigma }_{\mathrm{ij}}$ , and $m_{\mathrm{ij}}$ are strain tensors, stress tensors, and couple stress tensors, respectively; $u_i$ and ${\phi }_i$ are displacement and microrotation vectors, respectively; e_ijk is the permutation tensor; ${\delta }_{\mathrm{ij}}$ is the Kronecker delta; $\mu$ and $\lambda$ are classical Lame’s constants; and $\kappa ,\gamma ,\beta ,\alpha$ are the new elastic constants introduced in the micropolar theory. The constants $\mu ,\lambda ,\kappa$ have the dimension of force per unit area and $\gamma ,\beta ,\alpha$ have the dimension of force. The following relations for the Young’s modulus, Poisson’s ratio, and bulk modulus $E,\nu ,\kappa$ , respectively, still hold

$$E=\frac{\mu (3\lambda +2\mu )}{\lambda +\mu },v=\frac{\lambda }{2(\lambda +\mu )},\kappa =\lambda +\frac{2\mu }{3}$$

(5)

These constants relate the symmetric components of stress and strain, as in a Cauchy medium. ²⁶

In two-dimensional Cartesian coordinates, the above equations for the plane strain condition considered in this article become^28–30

$$\left(\begin{array}{c}{\epsilon }_{\mathrm{xx}}\hfill \\ \multicolumn{1}{c}{{\epsilon }_{\mathrm{yy}}}\\ \multicolumn{1}{c}{{\epsilon }_{\mathrm{yx}}}\\ \multicolumn{1}{c}{\begin{array}{c}{\epsilon }_{\mathrm{xy}}\hfill \\ \multicolumn{1}{c}{k_{\mathrm{xz}}}\\ \multicolumn{1}{c}{k_{\mathrm{yz}}}\end{array}}\end{array}\right)=\left(\begin{array}{c}{u}_{x,x}\hfill \\ \multicolumn{1}{c}{u_{y,y}}\\ \multicolumn{1}{c}{u_{x,y}-{\phi }_z}\\ \multicolumn{1}{c}{\begin{array}{c}{u}_{y,x}+{\phi }_z\hfill \\ \multicolumn{1}{c}{{\phi }_{z,x}}\\ \multicolumn{1}{c}{{\phi }_{z,y}}\end{array}}\end{array}\right)$$

(6)

$${\sigma }_{\mathrm{xx},x}+{\sigma }_{\mathrm{yx},y}=0,{\sigma }_{\mathrm{xy},x}+{\sigma }_{\mathrm{yy},y}=0$$

(7)

$$m_{\mathrm{xz},x}+m_{\mathrm{yz},y}+{\sigma }_{\mathrm{xy}}-{\sigma }_{\mathrm{yx}}=0$$

(8)

$${\sigma }_{\mathrm{ij}}=\lambda {\epsilon }_{\mathrm{nn}}{\delta }_{\mathrm{ij}}+(\mu +\kappa ){\epsilon }_{\mathrm{ij}}+(\mu -\kappa ){\epsilon }_{\mathrm{ji}},m_{\mathrm{iz}}=\beta {\phi }_{z,i}$$

(9)

where the subscript i, j varies from 1 to 2 and the repeated indices denote summation over the range (k = 1,2). In this case, the plane bulk modulus is defined as

$$\kappa =\lambda +\mu$$

(10)

The following conditions must be defined on the boundary in order to establish a well-posed problem

$${\sigma }_{\mathrm{ji}}{n}_i=p_j...m_{\mathrm{ji}}{n}_i=z_j...\mathrm{on}{\Gamma }^{\sigma }$$

(11)

$$u_i=u_i^b...{\phi }_i={\phi }_i^b...\mathrm{on}{\Gamma }^u$$

(12)

where p_j and z_j are surface force and moment vectors, respectively, and n_i is the exterior unit normal.

It should be noted that $\mu ,\lambda ,\kappa$ are the classical Lame’s constants and the bulk modulus, respectively, and they relate traditional stresses and strains. $\gamma ,\beta ,\alpha$ are the extra independent constants, which relate higher order couple stresses and torsions. The difference in the dimension between $\mu ,\lambda ,\kappa$ and $\gamma ,\beta ,\alpha$ do exist by length squared, and these define some intrinsic characteristic lengths in a micropolar material. Following the approach used in Liu and Hu, ²⁵ the characteristic length can be defined as

$$l_1^2=\frac{\gamma }{\mu },...l_2^2=\frac{\beta }{\mu },...l_3^2=\frac{\alpha }{\mu }$$

(13)

Micro–macro transition principle for a micropolar composite

In a classical Cauchy medium, the local elastic field for a single ellipsoidal inclusion in an infinite medium under a far-field strain is obtained by the Eshelby inclusion through the so-called equivalent inclusion method. ²⁷ The essence of this model is that the particle–matrix heterogeneous domain is replaced by a homogeneous domain that has the same matrix material but with an eigenstrain acting on the particle phase to represent heterogeneity. For a micropolar medium, Cheng and He^19,20 obtained the exact closed-form solutions for the cases of spherical and long cylindrical inclusions, respectively. Based on their method, there is one particle with radius a embedded in the infinite matrix at a certain x under the following traditional boundary condition on the boundary of an RVE: ²⁶

$$u_j=E_{(\mathrm{ij})}{x}_i,{\phi }_j=0$$

(14)

where $E_{(\mathrm{ij})}$ is symmetric and constant over the RVE. The volume average of the internal energy over the RVE can be expressed in terms of the statically balanced local stresses ( ${\sigma }_{\mathrm{ij}},m_{\mathrm{ij}}$ ) and geometrically compatible local strain fields ( ${\epsilon }_{\mathrm{ij}},k_{\mathrm{ij}}$ )

$$\begin{array}{c}\multicolumn{1}{c}{〈{\sigma }_{\mathrm{ij}}{\epsilon }_{\mathrm{ij}}+m_{\mathrm{ij}}{k}_{\mathrm{ij}}〉=〈{\sigma }_{\mathrm{ij}}(u_{j,i}-e_{\mathrm{kij}}{\varphi }_k)〉+m_{\mathrm{ij}}{\varphi }_{j,i}}\\ \multicolumn{1}{c}{=\frac{1}{V}{\int }_{\partial \mathrm{RVE}}{\sigma }_{\mathrm{ij}}{u}_j{n}_i\mathrm{dS}+\frac{1}{V}{\int }_{\partial \mathrm{RVE}}{m}_{\mathrm{ij}}{\varphi }_j{n}_i\mathrm{dS}}\\ \multicolumn{1}{c}{=E_{(\mathrm{mj})}\frac{1}{V}{\int }_{\partial \mathrm{RVE}}{\sigma }_{\mathrm{ij}}{x}_m{n}_i\mathrm{dS}=E_{(\mathrm{mj})}〈{\sigma }_{\mathrm{mj}}〉=E_{(\mathrm{mj})}〈{\sigma }_{(\mathrm{mj})}〉}\end{array}$$

(15)

where 〈·〉 and $n_i$ denote the volume average of quantity over the RVE and the outer normal of the boundary of the RVE, respectively.

If a stress boundary condition is applied on the boundary of the RVE, then

$${\sigma }_{\mathrm{ij}}{n}_i=\sum _{(\mathrm{ij})}{n}_i,m_{\mathrm{ij}}{n}_i=0$$

(16)

Through a similar process as presented in Equation (15), the energy equivalence can be expressed as

$$〈{\sigma }_{\mathrm{ij}}{\epsilon }_{\mathrm{ij}}+m_{\mathrm{ij}}{k}_{\mathrm{ij}}〉=\sum _{(\mathrm{ij})}〈{\epsilon }_{\mathrm{ij}}〉$$

(17)

with $〈{\sigma }_{(\mathrm{ij})}〉=\sum _{(\mathrm{ij})}$

Thus, the classical effective modulus tensor $\mathbf{C}$ and effective compliance tensor $\mathbf{S}$ can be expressed as

$$〈\mathbf{\sigma }\mathbf{;}\mathbf{:}\mathbf{\epsilon }+\mathbf{m}\mathbf{:}\mathbf{k}〉={\mathbf{E}}^{\mathbf{s}}\mathbf{:}\mathbf{C}\mathbf{:}{\mathbf{E}}^{\mathbf{s}}={\Sigma }^{\mathbf{s}}\mathbf{:}\mathbf{S}\mathbf{:}{\Sigma }^{\mathbf{s}}$$

(18)

where the superscript “s” denotes a symmetric tensor.

It is assumed that the eigenstrain ${\mathbf{\epsilon }}^{*}$ and eigentorsion ${\mathbf{k}}^{*}$ are uniform in the circular region; thus, the solution for the resulting strain and torsion in the whole medium can be expressed based on the work of Cheng and He^19,20

$$\mathbf{\epsilon }=\mathbf{K}(\mathbf{x})\mathbf{:}{\mathbf{\epsilon }}^{*}+\mathbf{L}(\mathbf{x}){\mathbf{:}\mathbf{k}}^{*}$$

(19)

$$\mathbf{k}=\hat{\mathbf{K}}(\mathbf{x})\mathbf{:}{\mathbf{\epsilon }}^{*}+\hat{\mathbf{L}}(\mathbf{x}){\mathbf{:}\mathbf{k}}^{*}$$

(20)

where $\mathbf{K},\hat{\mathbf{K}},\mathbf{L},\hat{\mathbf{L}}$ are Eshelby-like tensors; these are given in Cheng and He.^19,20x is a position vector.

It should be noted that for a micropolar medium, the resulting strain and torsion in the inclusion domain are not uniform, even for a circular inclusion. Similar to the work of Hu and Colleagues,^24–26 only for the average stress and couple stress in a heterogeneity will the considered to determine the effective property in an elastic case, the equivalent inclusion method could be used in an average sense for a micropolar composite. Based on the average equivalent inclusion method, the following equations of a spherical particle embedded into a micropolar matrix under a remote loading ${\mathbf{E}}_0$ , ${\mathbf{K}}_0$ can be obtained as

$${\mathbf{C}}_1:({\mathbf{E}}_0+〈\mathbf{\epsilon }{〉}_I)={\mathbf{C}}_0:({\mathbf{E}}_0+〈\mathbf{\epsilon }{〉}_I-〈{\mathbf{\epsilon }}^{+}{〉}_I)$$

(21)

$${\mathbf{D}}_1:({\mathbf{K}}_0+〈\mathbf{k}{〉}_I)={\mathbf{D}}_0:({\mathbf{K}}_0+〈\mathbf{k}{〉}_I-〈{\mathbf{k}}^{+}{〉}_I)$$

(22)

where ${\mathbf{C}}_0$ , ${\mathbf{D}}_0$ and ${\mathbf{C}}_1$ , ${\mathbf{D}}_1$ denote the modular of the micropolar matrix and spherical particle, respectively. ${\epsilon }^{+}$ and ${\mathrm{k}}^{+}$ are the prescribed uniform eigenstrain and eigentorsion for an isolated spherical inclusion in an infinite uniform micropolar medium, respectively. $〈·{〉}_P(P=A\mathrm{or}B)$ is the volume average over the inclusion domain. Using Equations (20) and (21), the average of the particle can be obtained as

$${〈\mathbf{\epsilon }〉}_I={〈\mathbf{K}〉}_I:{〈{\mathbf{\epsilon }}^{+}〉}_I+{〈\mathbf{L}〉}_I〈{\mathbf{k}}^{+}〉I$$

(23)

$${〈\mathbf{k}〉}_I={〈\hat{\mathbf{K}}〉}_I:{〈{\mathbf{\epsilon }}^{+}〉}_I+{〈\hat{\mathbf{L}}〉}_I{〈{\mathbf{k}}^{+}〉}_I$$

(24)

It should be noted that the volume averages of the Eshelby-like tensors $\mathbf{K},\hat{\mathbf{K}},\mathbf{L},\hat{\mathbf{L}}$ over the particle must be obtained to solve the problem. The details of this method are not presented here but can be found in the papers by Hu and Colleagues.^24–26 It is then found that

$${〈\mathbf{L}〉}_B=0\mathrm{and}{〈\hat{\mathbf{K}}〉}_B=0$$

(25)

This implies that on average, the eigenstrain and the eigentorsion are uncoupled, that is, on average, an eigenstrain produces only strain and an eigentorsion leads only to an average torsion deformation.

Micromechanics analysis of the FGM

Consider a particulate isotropic FGM (Figure 1) containing two phases A and B with elastic stiffnesses ${\mathbf{C}}_A$ and ${\mathbf{C}}_B$ , respectively. Either of them represents an isotropic or linear-elastic material. We use ${\phi }_A(y)$ and ${\phi }_B(y)$ to denote the volume fraction of Phases A and B, which both gradually change in the gradation direction y. Each of them is bounded between 0 and 1. The constraint condition is

$${\phi }_A(y)+{\phi }_B(y)=1$$

(26)

where the subscripts “A” and “B” refer to the phases A and B, respectively.

Figure 1.

Schematic of the microstructure of a two-phase FGM.

According to Rahman and Chakraborty ¹⁴ and Guo et al., ¹⁶ an RVE at one point of an FGM characterizes the material heterogeneity in the microscopic length scale. As shown in Figure 1, the particle and the matrix zones could be well defined when ${\phi }_A(y)$ and ${\phi }_B(y)$ were close to 0 or 1. Region I comprises Phase B as particles embedded into Phase A. The particle–matrix roles are reversed in Region III where the particles and matrix are Phase A and Phase B, respectively. A skeletal transition zone (Region II), in which it is difficult to differentiate the particle and matrix phases, normally exists in the intermediate region. The transition zone is determined by the fabrication process of the FGM, which is directly related to ${\phi }_A(y)$ and ${\phi }_B(y)$ . For convenience, we use $\phi (y)$ to denote the volume fraction of Phase B.

As mentioned above, when only a remote uniform symmetric stress (without couple stress) is applied, the effect of the eigenstrain and eigentorsion is uncoupled. Then, to calculate the effective FGM elastic stiffnesses $\overline{\mathbf{C}}(y)$ and $\overline{\mathbf{D}}(y)$ , a far-field loading ${\sigma }^0$ is first applied on the FGM y boundary. In this article, we consider the plane strain problem. Accordingly, based on the equilibrium condition, the constitutive equation for the FGM can be expressed as

$${\mathbf{\sigma }}_{\mathrm{ij}}^0=\overline{\mathbf{C}}(y)\mathbf{:}〈{\mathbf{\epsilon }}_{\mathrm{ij}}〉(y)$$

(27)

where the subscripts $i,j$ range from 1 to 2. $〈{\mathbf{\epsilon }}_{\mathrm{ij}}〉(y)$ is the average strain. The homogenization relations between the average stress $〈{\mathbf{\sigma }}_{\mathrm{ij}}〉(y)$ and average strain $〈{\mathbf{\epsilon }}_{\mathrm{ij}}〉(y)$ can be further written as

$${\sigma }_{\mathrm{ij}}^0=\phi (y){\mathbf{C}}_B\mathbf{:}{〈\mathbf{\epsilon }〉}_B(y)+[1-\phi (y)]{\mathbf{C}}_A\mathbf{:}{〈\mathbf{\epsilon }〉}_A(y)$$

(28)

$$〈{\mathbf{\epsilon }}_{\mathrm{ij}}〉(y)=\phi (y){〈\mathbf{\epsilon }〉}_B(y)+[1-\phi (y)]{〈\mathbf{\epsilon }〉}_A(y)$$

(29)

Equations (21) and (22) can be written as

$${\mathbf{C}}_B:({\mathbf{E}}_0+〈\mathbf{\epsilon }{〉}_B(y))={\mathbf{C}}_A:({\mathbf{E}}_0+〈\mathbf{\epsilon }{〉}_B(y)-〈{\mathbf{\epsilon }}^{+}{〉}_B(y))$$

(30)

$${\mathbf{D}}_B:({\mathbf{K}}_0+〈\mathbf{k}{〉}_B(y))={\mathbf{D}}_A:({\mathbf{K}}_0+〈\mathbf{k}{〉}_B(y)-〈{\mathbf{k}}^{+}{〉}_B(y))$$

(31)

With the solution of a single inclusion in a micropolar matrix, Equation (18) can be used to determine the effective modulus tensor. For the plane problem, Equation (30) can be written separately for the symmetric and anti-symmetric components. For a spherical inclusion in an isotropic micropolar matrix, the Eshelby-like tensor is an isotropic fourth-order tensor with the following form

$$K_{\mathrm{ijkl}}=T_1{\delta }_{\mathrm{ij}}{\delta }_{\mathrm{kl}}+(T_2+T_3){\delta }_{\mathrm{ik}}{\delta }_{\mathrm{jl}}+(T_2-T_3){\delta }_{\mathrm{il}}{\delta }_{\mathrm{jk}}$$

(32)

where

$$\begin{array}{c}{T}_1=\frac{{\gamma }_A-{\mu }_A}{4({\lambda }_A+{\mu }_A)}+\frac{{\kappa }_A}{2({\kappa }_A+{\mu }_A)}{I}_1(r_B/g){\varphi }_1(r_B/g)\hfill \\ T_2=\frac{{\gamma }_A+3{\mu }_A}{4({\lambda }_A+2{\mu }_A)}-\frac{{\kappa }_A}{2(k_A+{\mu }_A)}{I}_1(r_B/g){\varphi }_1(r_B/g)\hfill \\ T_3=\frac{4{\kappa }_A+{\mu }_A}{2{\mu }_A}-\frac{2{(2{\kappa }_A+{\mu }_A)}^2}{{\mu }_A({\kappa }_A+{\mu }_A)}{I}_1(r_B/g){\varphi }_1(r_B/g)\hfill \\ g^2=\frac{({\mu }_A+{\kappa }_A){\beta }_A}{4{\mu }_A{\kappa }_A}\hfill \end{array}$$

(32)

and the subscripts $i,j,k,l$ range from 1 to 2; $I_1(r_B/g)$ and ${\varphi }_1(r_B/g)$ are the modified Bessel function of the first and second kind, respectively; and $r_B$ denotes the radius of particle B. Thus, the volume average of the micropolar Eshelby-like relations can be decomposed into their symmetric and anti-symmetric components, respectively, as

$${〈{\epsilon }_{(\mathrm{ij})}〉}_B={〈K_{\mathrm{ijkl}}^S〉}_B:{〈{\epsilon }_{\mathrm{kl}}^{+}〉}_B,{〈{\epsilon }_{(\mathrm{ij})}〉}_B={〈K_{\mathrm{ijkl}}^A〉}_B:{〈{\epsilon }_{\mathrm{kl}}^{+}〉}_B$$

(33)

where $K_{\mathrm{ijkl}}^S=(K_{\mathrm{ijkl}}+K_{\mathrm{ijlk}})/2,K_{\mathrm{ijkl}}^A=(K_{\mathrm{ijkl}}-K_{\mathrm{ijlk}})/2$

This approach uses Equations (29)–(31) and (33), and at the same time decomposes Equation (31) into the symmetric and an anti-symmetric components. Thus, the effective modular $\overline{\mathbf{C}}$ for a micropolar FGM can be obtained by following the procedure presented by Hu et al.:^24–26

$$\overline{\mu }(y)={\mu }_A(1+\frac{\phi (y)}{2(1-\phi (y)){〈K_{1212}^S〉}_B+[{\mu }_A/({\mu }_B-{\mu }_A)]})$$

(34)

$$\overline{K}(y)=K_A(1+\frac{\phi (y)}{[(1-\phi (y)){〈K_{\mathrm{iijj}}^S〉}_B/2]+[K_A/(K_B-K_A)]})$$

(35)

$$\begin{array}{c}\multicolumn{1}{c}{{〈K_{1212}^S〉}_B=\frac{3(K_A+2{\mu }_A)}{5(3K_A+4{\mu }_A)}-\frac{3g(r_B+g){\kappa }_A}{5r_B^3({\kappa }_A+2{\mu }_A)}}\\ e^{-r_B/g}[r_B\cosh \frac{r_B}{g}-g\sinh \frac{r_B}{g}]\hfill \end{array}$$

(36)

$${〈K_{\mathrm{iijj}}^S〉}_B=\frac{9K_A}{3K_A+4{\mu }_A}$$

(37)

where $g=\sqrt{\frac{{\gamma }_A(2{\mu }_A+{\kappa }_A)}{4{\mu }_A{\kappa }_A}}$ ; $K_A,K_B,\overline{K}$ are the bulk modular terms of Phase A, Phase B, and FGM, respectively; and ${\mu }_A,{\mu }_B,\overline{\mu }$ are the corresponding shear modular terms of Phase A, Phase B, and FGM, respectively. From Equations (34)–(37), it can be observed that scale information is implicitly included in the Eshelby-like tensor through ${〈K_{1212}^S〉}_B$ and ${〈K_{\mathrm{iijj}}^S〉}_B$ . When particle size is increased, or the micropolar effect is neglected, ${〈K_{1212}^S〉}_B$ is reduced to the classical Eshelby tensor and the estimates of the effective properties $\overline{K}$ and $\overline{\mu }$ of the FGM are also reduced to those of a classical Cauchy medium.

The averaged strain field in Region III can also be calculated through a similar process by switching the matrix and particle phases because the particle and matrix zones in this region can be well defined.

In the transition zone (Region II), it is not easy to identify the particle and matrix phases when (1) both the volume fractions of the two phases are in the vicinity of 0.5 and (2) the two phases have interpenetrated to form a connected network. In this case, because the transition zone (Region II) for a practical FGM lies between Regions I and III and is also much smaller than these regions, the averaged elastic fields cannot explicitly be determined through the micromechanics framework. Following Reitor and Dvorak ⁸ and Yin et al., ¹⁰ a phenomenological transition function is introduced as

$$f(y)=[1-2\frac{\phi (y)-\phi (d_1)}{\phi (d_1)-\phi (d_2)}]{\left[\frac{\phi (y)-\phi (d_2)}{\phi (d_1)-\phi (d_2)}\right]}^2$$

(38)

Then, the averaged strain of each phase (A or B) in transition Region II can be estimated as a cubic Hermite function appropriately contributed by the averaged strain of the same phase (A or B) from two particle–matrix zones (Regions I and III) as

$${〈\mathbf{\epsilon }〉}_{\mathrm{zone}\mathrm{II}}^{\mathrm{A}\mathrm{or}\mathrm{B}}(y)=f(y){〈\mathbf{\epsilon }〉}_{\mathrm{zone}\mathrm{I}}^{\mathrm{A}\mathrm{or}\mathrm{B}}(y)+[1-f(y)]{〈\mathbf{\epsilon }〉}_{\mathrm{zone}\mathrm{III}}^{\mathrm{A}\mathrm{or}\mathrm{B}}(y)$$

(39)

Using Equations (29), (30), (38), and (39), the overall averaged strain tensor at each point in the transition zone can be calculated.

If a uniform far-field loading ${\sigma }_{\mathrm{yy}}^0$ acts on the lower and upper FGM boundaries, the effective bulk modulus $\overline{K}$ and shear $\overline{\mu }$ of the FGM as a micropolar composite can be derived from Equations (34)–(37). Accordingly, the effective Young’s modulus $\overline{E}$ and Poisson ratio $\overline{\nu }$ of the FGM can also be determined.

Numerical applications and discussion

In this section, the effectiveness of the proposed micromechanical FGM model will be illustrated using numerical examples. The geometry of the problem being examined is shown in Figure 1. The microstructure of the FGM includes two distinct material phases. The macro properties vary along the y direction. As in Yin et al., ¹⁰ the lower and upper bounds d₁ and d₂ are conveniently selected where the corresponding volume fractions are 40% and 60%.

Before performing the parametric study, the above solution was verified with an existing solution similar to the proposed problem. As mentioned above, when the micropolar effect is neglected, the estimates of the effective properties $\overline{K}$ and $\overline{\mu }$ of the FGM are reduced to those of a classical Cauchy medium. Thus, simulations of the effective properties of the FGM using the proposed model without the micropolar effect were compared with those from available experimental data and other simulation models. Initially, the FGM comprised TiC as Phase B and Ni₃Al as Phase A. Their elastic parameters were obtained from Zhai et al. ³¹ as $E_{\mathrm{TiC}}=460\mathrm{GPa}$ , ${\nu }_{\mathrm{TiC}}=0.19$ , $E_{\mathrm{N}{\mathrm{i}}_3\mathrm{Al}}=199\mathrm{GPa}$ , and ${\nu }_{\mathrm{N}{\mathrm{i}}_3\mathrm{Al}}=0.295$ . The transition zone was bounded by $\phi (d_1)=40\mathrm{\%}$ and $\phi (d_2)=60\mathrm{\%}$ . Three volume fraction distribution cases were considered and the related variations in the effective Young’s modulus and Poisson’s ratio with volume fraction distribution of Phase B (TiC; as shown in Equation (40)) from the present model are shown in Figure 2(a) and (b), respectively. It can be observed that the effective Young’s modulus ranges from $E_{\mathrm{N}{\mathrm{i}}_3\mathrm{Al}}$ to $E_{\mathrm{TiC}}$ , and the effective Poisson’s ratio ranges from ${\nu }_{\mathrm{N}{\mathrm{i}}_3\mathrm{Al}}$ to ${\nu }_{\mathrm{TiC}}$ . It can also be seen that both the phase volume fraction and its distribution form have a strong influence on the effective properties of the FGM. This indicates that the overall mechanical behavior of the FGM can be optimized by selecting the phase distribution pattern to obtain the desired FGM material and structure

$$\phi (y)={\left(\frac{y}{t}\right)}^2,\phi (y)=\frac{y}{t},\phi (y)=\sin \left(\frac{\pi y}{2t}\right)$$

(40)

Figure 2.

Effect of phase volume fraction distribution on elastic properties of the FGM: (a) Young’s modulus and (b) Poisson’s ratio.

To demonstrate the validity of the present model, a further example case was conducted by comparing the present model with experimental results and the Mori–Tanaka method. As in Yin et al., ¹⁰ we compared the results from the present model with the experimental data in Eshelby. ³² Both the effective Young’s modulus and Poisson’s ratio of the TiC/Ni₃Al FGM with phase material properties of $E_{\mathrm{TiC}}=460\mathrm{GPa}$ , ${\nu }_{\mathrm{TiC}}=0.19$ , $E_{\mathrm{N}{\mathrm{i}}_3\mathrm{Al}}=199\mathrm{GPa}$ , and ${\nu }_{\mathrm{N}{\mathrm{i}}_3\mathrm{Al}}=0.295$ were estimated and these are shown in Figure 3(a) and (b), respectively. It is assumed that the volume fraction distribution form is a linear function and the transition zone is included with the lower and upper bounds d₁ and d₂ selected to correspond with the volume fractions of 40% and 60%, respectively. It can be seen that the results from the present model are very close to those of the experiment; in particular, the obtained results of the present model and those of Mori–Tanaka are almost identical. It is worthy of note that if the micropolar effect is neglected, the present model becomes the standard Mori–Tanaka method.

Figure 3.

Comparison of the effective properties of the FGM with a linear volume fraction between simulation models and experimental data: (a) Young’s modulus and (b) Poisson’s ratio.

The next example demonstrates the particular usefulness of the method in analyzing the size effect in relation to the effective properties of the FGM. The conventional micromechanical model for FGMs based on a Cauchy medium do not take into account the effect of particle size in the FGM; however, the model presented in this article is based on micropolar theory and the size effect is directly considered. We consider a SiC particle reinforced metal matrix FGM (SiC/Al); the material constants of the two phases are listed in Table 1. The SiC particle reinforced metal matrix composite (not the FGM) was analyzed by Liu and Hu ²⁵ and Lloyd, ³³ respectively. For simplicity, we assumed the elastic characteristic lengths of Phase A to be $l_1=l_2=l_3$ . Next, we discuss the size effect of the effective properties of the FGM comprising the two constituents. In this article, it is assumed that the matrix (Phase A) is a micropolar material, and the particles (Phase B) are considered as a Cauchy material; that is, the corresponding micropolar modular of the particles is set to zero. The estimated effective Young’s modulus curves of the FGM with linear volume fraction distributions of the particles for different particles sizes are shown in Figure 4 (a) $r_B/l_A>1$ and (b) $r_B/l_A<1$ . For comparison, the results from the standard Mori–Tanaka method are also plotted in the figure. It can be observed that both the particle volume fraction and the particle size have significant effects on the effective Young’s modulus of the FGM. For hard particles ( $E_B>E_A$ ), the Young’s modulus of the FGM increases with an increase in the volume fraction of the particle and its effects increase with a decrease in its radius $r_B$ . It should be noted that when $r_B=0.05l_A$ or less, the effects of the particle become stable. It can also be observed that when the radius of the particle is large, especially $r_B=50l_A$ or greater, the effective Young’s modulus of the FGM predicted by micropolar theory coincides with that predicted by Cauchy theory and the size effects can be neglected. This reveals that the effective properties of the FGM can be optimized by choosing an appropriate size for the reinforcement phase.

Table 1.

Material constants of each constituent used in the FGM.

	Al356(T4)	SiC
E (GPa)	68.3	490
$\nu$	0.33	0.17
$\mu$	26	209
$\lambda$	50	108
$l_A$ ( $\mathrm{\mu }\mathrm{m}$ )	10
G	173

Figure 4.

Effects of particle size on the effective Young’s modulus of the FGM linear volume fraction distribution: (a) $r_B/l_A>1$ and (b) $r_B/l_A<1$ .

The influence of the volume fraction distribution form of Phase B is also examined. Figure 5 displays the variation in the effective Young’s modulus of the FGM with particle radius for the case $\phi =(y/t{)}^2$ . As in the previous case, the Young’s modulus of the FGM increases with an increase in the volume fraction of the particle, and its effect increases with a decrease in its radius $r_B$ .

Figure 5.

Effects of particle size on the effective Young’s modulus of the FGM with volume fraction distribution $\phi =(y/t{)}^2$ .

Conclusion

In this article, a micromechanics-based elastic model set in a framework of micropolar theory is presented for two-phase FGMs to investigate size effects on the effective properties of the FGM. The method is based on the average equivalent inclusion method for a micropolar material, and Mori–Tanaka’s method is extended to a micropolar FGM to evaluate the effective elastic stiffness tensor. The effects of volume fraction distribution forms, together with the size and properties of the particle, on effective properties of the FGM are examined. The size dependence is more pronounced for the case where the particle size approaches the matrix characteristic length and where the stiffness of the particle is large. When the particle size is much larger than the matrix characteristic length, the prediction based on micropolar theory is similar to the classical results and the size effects can be neglected. Comparisons are conducted between the present model and the available simulation and experimental data to validate the proposed micromechanics model.