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Micromechanics Modeling of Functionally Graded Materials Containing Multiple Heterogeneities

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Micromechanics Modeling of Functionally Graded Materials Containing Multiple Heterogeneities Vol. 26, No. 6, 392-397 (2013) DOI: http://dx.doi.org/10.7234/composres.2013.26.6.392 ISSN 2288-2103(Print), ISSN 2288-2111(Online) Paper Micromechanics Modeling of Functionally Graded Materials Containing Multiple Heterogeneities Jaesang Yu*†, Cheol-Min Yang**, Yong Chae Jung** ABSTRACT: Functionally graded materials graded continuously and discretely, and are modeled using modified Mori- Tanaka and self-consistent methods. The proposed micromechanics model accounts for multi-phase heterogeneity and arbitrary number of layers. The influence of geometries and distinct elastic material properties of each constituent and voids on the effective elastic properties of FGM is investigated. Numerical examples of different functionally graded materials are presented. The predicted elastic properties obtained from the current model agree well with experimental results from the literature. Key Words: functionally graded material (FGM), micromechanics, heterogeneity, composites 1. INTRODUCTION of three phase FGMs. Pindera et al. [10] used a computational micromechanical model, the generalized method of cells, to Due to their unique structure and multi-functionality, func- predict local stress in the fiber and matrix phases of FGMs. tionally graded materials (FGMs) are gaining attention and are Fang et al. [11], developed a micromechanics-based elasto- being proposed for many applications [1-3]. In FGMs, the dynamic model to predict the dynamic behavior of two-phase variation of material properties along a specific direction helps functionally graded materials. the material to have the desirable material properties of each Zuiker [12] reviewed the micromechanical modeling of constituent at a specific material point with structural conti- FGMs and concluded that the self-consistent method (SCM) nuity. For example, ceramic/metal FGMs have the important provided good estimates, with minimal effort, and with no material properties of metal, such as fracture toughness and need for empirical fitting of parameters for the silicon carbide high thermal conductivity, without losing the important prop- (SiC) - carbon (C) FGMs. Gasik [13] studied the efficiency of erties of a ceramic, such as hardness and thermal protection the simplest micromechanical models to provide the most [4-7]. A detailed review on the performance of FGM can be accurate estimates of FGM components with an arbitrary non- seen in the literature. linear three-dimensionally orientation of phases. Reiter and To accurately model FGM, knowing the effective or bulk Dvorak [14] used the transition function with Mori-Tanaka material properties as a function of individual material prop- method (MTM) and SCM to predict the thermo-mechanical erties and geometry especially at micromechanics level is properties in C/SiC FGMs. Yin et al., [15,16] used MTM, with essential. In the last few years, different models have been pro- the effect of particle distance interactions, to predict the elastic posed to estimate the effective properties of FGMs with and thermoelastic properties of TiC/Ni3Al and Mo/SiO2 respect to reinforcement volume fractions. Weng [8], inves- FGMs. Similar models are available for two phase functionally tigated the effective bulk moduli of two functionally graded graded materials that contain two different material systems. composites by means of change of the dependent variable. The actual functionally graded materials can have more than Rahman and Chakraborty [9] proposed a stochastic micro- two compositions or constituents, and a model that accounts mechanical model for predicting probabilistic characteristics for multiple heterogeneities is essential. Received 29 November 2013, accepted 26 December 2013 *†한국과학기술연구원 탄소융합소재연구센터, Corresponding author (E-mail: [email protected]) **한국과학기술연구원 탄소융합소재연구센터 Micromechanics Modeling of Functionally Graded Materials Containing Multiple Heterogeneities 393 In the present study a modified micromechanics model is zone. In this zone a linear phenomenological transition func- proposed and extended to estimate the effective elastic prop- tion introduced by Reiter and Dvorak [14] is adopted. erties of FGMs containing multiple heterogeneities with an arbitrary number of coating layers. Different numerical exam- 2.2 Micromechanics modeling ples are considered and compared with experimental results 2.2.1 Modified Mori-Tanaka method (MTM) for FGM with from the literature. multiple heterogeneities and multiple layers In MTM, the effective material properties for composites 2. PROBLEM FORMULATION containing arbitrary ellipsoidal heterogeneities can be obtained using the continuum-averaged stress and strain fields [17-20]. 2.1 Functionally graded materials (FGMs) For two phase composites, containing a matrix phase (0) and a Functionally graded materials can be idealized as a com- particle phase (1), the effective 4th order elastic stiffness ten- bination of a number of discreet layers of nano-composite, sor, L, can be expressed as where each of the layers has a defined microstructure that is a ()0 ()1 ()1 ()1 –1 function of the volume fraction of the constituents of the LL= :{}Ic+ 1()S –I :()A –S : material. From a micromechanics modeling point of view, ()1 ()1 ()1 –1 –1 these discrete graded layers can be categorized in three distinct {}Ic+ 1S :()A –S (1) zones. Let us consider a functionally graded material, where mate- where rial A, (A could be more than one material) with different ()1 ()0 ()1 –1 ()0 geometry (short fibers, platelets, spheres, etc) is reinforced in a A = ()L –L :L (2) base material B, and the volume fraction of A is increasing from one end to the other end as shown in Fig. 1. As shown in is the local strain concentration tensor for the particles, L(0) the figure, zone 1 is the region where the reinforcement, phase and L(1) are the 4th rank elastic stiffness tensors for the matrix A, volume fraction is lower than the base material, phase B, and particles, c1 is the particles volume fraction, S(1) is the 4th referred to as the phase B dominant zone. Zone 3 is the region rank Eshelby [19-21] tensor for the particles, and I is the 4th where the volume fraction of the reinforcement, phase A, is rank identity tensor. Here a colon “:” is used to denote the ten- higher than the base material, phase B, referred to as the phase sor double dot product. The Eshelby tensor (S(1)) accounts for A dominant zone. Zone 2 is the region where the volume frac- the influence of the aspect ratio/geometry of the particles on tion of phase A and phase B cannot be clearly defined, referred the local strain field. Eshelby tensors for specific reinforcement to as the transition zone. Each layer in Zone 1 and Zone 3 can, shapes (spheres, platelets, fibers, etc.) are readily available in in principle, be modeled similarly using a micromechanics the literature [19,20]. For FGMs, the effective elastic property approach as the volume fraction between the two phases are for each layer through the thickness direction can be estimated distinct enough to satisfy the main assumption. by employing Eq. (5) with either normalized thickness 0 ≤ x ≤ The main challenge is dealing with the transition zone, 1 or volume fractions of phase (0) and phase (1). where the matrix and the reinforcement are not sufficiently Consider a composite with m types of distinct ellipsoidal distinguishable. Recently, a different method has been pro- heterogeneities (k = 1, 2, ..., m) in the matrix (0), where each posed to overcome this problem and model the transition heterogeneities consists of n layers (αk = 1, 2, ..., nk) graded in the direction of thickness, having a normalized thickness 0 ≤ z ≤ 1. Each heterogeneity has a distinct elastic property, shape, and orientation distribution. The strain and stress fields at any location z with heterogeneities, with arbitrary number of layers, can be given by α ∞ * α 〈〉ε k = ε + :ε k k k S()k (3) α ()0 ∞ * α 〈〉σ k = :{}ε +()– :ε k k k L S()k I (4) * α where the homogenizing eigenstrain (ε k k ) is * α ()α –1 ∞ ε k k = ()k – :ε A()k S()k (5) Fig. 1. Schematic illustration of multi-phase functionally graded where is the local strain concentration tensor for the αk th materials (FGMs) layer of the kth heterogeneity (αk = 1, 2, ..., nk, k = 1, 2, ..., m) 394 Jaesang Yu, Cheol-Min Yang, Yong Chae Jung of each FGM layer through the thickness direction and can be malized thickness direction, 0 ≤ x ≤ 1, can be obtained from given as the MTM model by setting . Unlike MTM, however, in the SCM model the effective elastic tensor of the homogenized ()α ()0 ()α –1 ()0 k = ()– k : A()k L L()k L (6) matrix is unknown, and an iterative procedure is required. In this model, first an initial effective elastic tensor will be Here is the 4th rank elastic stiffness tensor for the αk th assumed, and with an iteration scheme the final tensor will be layer of the kth heterogeneity for FGM layers through the determined. Suppose that the effective homogeneous medium thickness direction. By substituting Eqs. (5-6) into Eqs. (3-4), with elastic stiffness tensor, L , contains m distinct types of the average strain and stress fields over the infinite domain can ellipsoidal heterogeneities (k = 1, 2, ..., m), each consisting of be obtained as nk layers (αk = 1, 2, ..., nk) in a matrix phase (0). Then the overall elasticity tensor, L , the complete formulation is given m nk ()α –1 ∞ k ∞ 〈〉ε = ε + ∑ ∑ c()αS():()A() –S() :ε (7) in the literature [22-24], can be expressed as D k k k k k = 1 α = 1 k k ()α k LfLS= (),,()A() ,c()α (10) ⎧⎫k k k k ⎪⎪m nk ()α –1 ()0 ∞ k ∞ 〈〉σ = L :⎨⎬ε + ∑ ∑ c()α()S()–I :()A() –S() :ε For the case of aligned, two-dimensional randomly oriented D k k k k k ⎪⎪= 1 α = 1 ⎩⎭k k and three-dimensional randomly oriented heterogeneities, the calculated elasticity tensor for each layer of FGMs will display (8) orthotropic, transversely isotropic, and isotropic material sym- The overall elasticity tensor, L , for a composite containing metries, respectively.
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