Simplified prediction model for elastic modulus of particulate reinforced metal matrix composites

WANG Wen-ming(王文明)^{1, 2}, PAN Fu-sheng(潘复生)^{1, 2}, LU Yun(鲁 云)³, ZENG Su-min(曾苏民)⁴

1. Chongqing Engineering Research Center for Magnesium Alloys, Chongqing University,

Chongqing 400044, China;

2. College of Materials Science and Engineering, Chongqing University, Chongqing 400044, China;

3. Faculty of Engineering, Chiba University, Chiba, Japan;

4. Southwest Aluminum Industry (Group) Co. Ltd, Chongqing 401326, China

Received 28 July 2006; accepted 15 September 2006

Abstract: Some structural parameters of the metal matrix composite, including particulate shape and distribution do not influence the elastic modulus. A prediction model for the elastic modulus of particulate reinforced metal matrix Al composite was developed and improved. Expressions of rigidity and flexibility of the rule of mixing were proposed. A five-zone model for elasticity performance calculation of the composite was proposed. The five-zone model is thought to be able to reflect the effects of the MMC interface on elastic modulus of the composite. The model overcomes limitations of the currently-understood rigidity and flexibility of the rule of mixing. The original idea of a five-zone model is to propose particulate/interface interactive zone and matrix/interface interactive zone. By integrating organically with the law of mixing, the new model is found to be capable of predicting the engineering elastic constants of the MMC composite.

Key words: particulate reinforced metal matrix composite; elastic modulus; prediction model

1 Introduction

The elastic modulus is an important index to reflect the ability of particulate reinforced metal matrix composites to resist deformation under an applied force. However, the elastic modulus(E) measurement of a composite is restricted by many factors including the required dimensional specification of the test specimen and fabrication cost. The rule of mixing does not completely define the factors that influence the E of a composite or that can be measured wholly by actual specimen preparation and by mechanical testing. Several predictive analytical and numerical models of E for a composite have been developed[1-3].

Macro-micro structure unified constitutive model of the composite and integrated analytical procedure is a kind of comparatively new approach which couples microscopic and macroscopic fields. It is not only able to fully consider the effects of components, volume fraction, and geometric microstructure on macroscopic behavior, but it also is able to obtain macroscopic as well as microscopic stress-strain field during structural analysis. No previous data have been presented previously either within China or abroad regarding the adoption of a macro-micro structure unified constitutive model to predict the elastic modulus of particulate reinforced metal matrix composite. The current macro-micro structure unified constitutive model is comparatively ideal to predict the elastic modulus of particulate reinforced aluminum matrix composite when the interfacial bonding is good[4-5]. However, the predicted value is higher than the experimental when the interfacial bonding is weak. It cannot explain why the elastic modulus of a composite is sometimes lower than that of unreinforced matrix. Therefore the model should be modified.

Comparatively detailed derivation and establish- ment of macro-micro structure unified constitutive model has previously been published[4-5]. In this paper we have simplified and improved the model reasonably based on experimental generalization of many previous predictive models.

  2 Simplified and improved micromechanics model of composite

Because some structural parameters (such as particulate shape, arrangement pattern and dimensional variance mode) of the composite have no obvious influences on its elastic modulus, it is possible to ignore their influences during a simplified calculation[6]. According to the macro-micro structure unified constitutive model, the formula that defines the rigidity matrix of the composite is

                 (1)

Definitions of symbols (h, l, h_β, l_γ, β, γ) in Eqn.(1) can be referred to Fig.1. C^{(β, γ)} is the rigidity matrix of the subcell (β, γ).  is the shape matrix of the subcell (β, γ). Ignoring the effect of some structural parameters, the following simplified mixed rigidity matrix of the composite can be attained.

                              (2)

where  V_i and C_i are volume fraction and rigidity matrix of the ith phase respectively, and N is the total number of phases composed of the composite. Inversing the rigidity matrix of the composite gives the flexibility matrix of the composite. The elastic modulus of the composite can be attained according to the relationship between the elastic modulus and the flexibility matrix. The elastic modulus derived from Eqn.(2) is the maximum. In this paper Eqn.(2) is considered ‘the rigidity mixed law expression of the composite’. Transforming Eqn.(2), the simplified mixed flexibility matrix of the composite can be attained:

                              (3)

where  V_i and S_i are the volume fraction and flexibility matrix of the ith phase respectively, and N is the total number of phases that make up the composite. The elastic modulus derived from Eqn.(3) is a minimum. In this paper Eqn.(3) is termed ‘the flexibility mixed law expression of the composite’.

Fig.1 N_β×N_γ subcell

Because the predictive results of Eqn.(2) and Eqn.(3) are the maximum and minimum value of the elastic modulus respectively, the value between them cannot be predicted. Therefore some modifications have been imposed on Eqns.(2) and (3). Eqns.(2) and (3) have separated matrix phases, reinforced phases and interfacial phases of the composite. In this paper the model ignoring the interaction of components is called a three-zone model, as shown in Fig.2 (a). We propose a five-zone model as shown in Fig.2(b). It is found by modifying Eqns.(2) and (3) that a weak interface will influence the stress-strain distribution of the matrix and reinforcement near the interface. It is hypothesized that the interface and the particulate interact via a partial disturbance to the particulate adjacent to the interface. The volume fraction of the section of particulate disturbance that occupies V_ip, rigidity matrix and flexibility matrix respectively, grow into C_ip and S_ip. The interface and matrix interact thereby bring partial disturbance to the particulate adjacent to the interface. The volume fraction of the portion of the matrix disturbance that occupies V_im, the rigidity matrix and the flexibility matrix respectively, grow into C_im and S_im. V_ip, C_ip, S_ip as well as V_im, C_im, S_im is determined below.

Fig.2 Sketches of three-zone model (a) and five-zone model (b)

Given that rigidity matrix, flexibility matrix and volume fraction of matrix, reinforced phase and interface phase are C_m, S_m, V_m; C_p, S_p, V_p and C_i, S_i, V_i, respectively.

First, consider the interaction between interface and particulate.

                  (4)

Inversing C_ip1 gives the corresponding flexibility matrix, and allows us to calculate the elastic modulus E_max.

                  (5)

The value of elastic modulus E_min can be attained from S_ip. V_ip and S_ip are determined by the following expressions:

                  (6)

Inversing S_ip gives the corresponding rigidity matrix C_ip.

Next, consider the interaction between the interface and matrix.

                 (7)

Inversing C_im1 gives the corresponding flexibility matrix, and the elastic modulus E_max.

                 (8)

The value of elastic modulus E_min can be attained from S_im1; V_im and S_im are determined by the following expressions:

                 (9)

Inversing S_im can get the corresponding rigidity matrix C_im.

V_ip, C_ip, S_ip as well as V_im, C_im, S_im are defined above. The final rigidity matrix of the composite is calculated according to the following expression:

                  (10)                    

The curves of elastic modulus of the composite in response to the interface modulus calculated by Eqns.(2), (3) and (10) are shown in Fig.3. The five-zone model is able to reflect the effects of the interface modulus on the elastic modulus of the composite. The original idea is to propose a particulate/interface interactive zone and a matrix/interface interactive zone. This connects organically with a rigidity mixed law and a flexibility mixed law. The model is able to predict the engineering elastic constant of the composite quite exactly.

Fig.3 Curves of elastic modulus vs interface modulus

3 Demonstration of five-zone model to predict elastic modulus of composite

The calculated and predictive values are shown in Fig.4, comparing with the previous models. After comparing different kinds of predicted values and experimental values[7-8], it can be seen that the upper and lower limits have a wider range than predicted from Voigt-Reuss upper and lower limits model or the Hashin-Shrikman upper and lower limits model. By contrast, the predicted precision is more exact from the five-zone model when the appropriate interface/matrix modulus ratio is selected. The selection of an appropriate interface/matrix modulus ratio can be calculated according to the given elastic modulus of the composite.

Thereafter the elastic modulus values of the other composites prepared from the identical process flow can be calculated. Voigt-Reuss upper and lower limits model and Hashin-Shrikman upper and lower limits model are based on the hypothesis of equivalent strain and stress. If the actual situation is not equivalent strain or stress, the predicted value will offset the real value. The five-zone model is able to predict the elastic modulus closer to the real value when selecting the proper interface/matrix modulus ratio. If the interface/matrix modulus ratio is not properly selected, the predicted value will also offset the real value.

Fig.4 Demonstration of five-zone model to predict elastic modulus of composite: (a) Al₂O₃/6061Al; (b) SiC/6061Al

4 Conclusions

1) The five-zone model is able to reflect the effects of the interface modulus on the elastic modulus of the composite.

2) The original basis of the model proposes a particulate/interface interactive zone and a matrix/ interface interactive zone.

3) The new model is an improvement over previous theoretical models.

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(Edited by HE Xue-feng)

Foundation item: Project(7884, CSTC2004DE4002) supported by the Chongqing Science and Technology Commission

Corresponding author: WANG Wen-ming; Tel: +86-23-65111520; E-mail: welcome163email@163.com