Effects of particle size on residual stresses of metal matrix composites

YAN Yi-wu(晏义伍), GENG Lin(耿 林), LI Ai-bin(李爱滨)

School of Materials Science and Engineering, Harbin Institute of Technology, Harbin 150001, China

Received 28 July 2006; accepted 15 September 2006

Abstract: A finite element analysis was carried out on the development of residual stresses during the cooling process from the fabrication temperature in the SiC_p reinforced Al matrix composites. In the simulation, the two-dimensional and random distribution multi-particle unit cell model and plane strain conditions were used. By incorporating the Taylor-based nonlocal plasticity theory, the effect of particle size on the nature, magnitude and distribution of residual stresses of the composites was studied. The magnitude thermal-stress-induced plastic deformation during cooling was also calculated. The results show similarities in the patterns of thermal residual stress and strain distributions for all ranges of particle size. However, they show differences in magnitude of thermal residual stress as a result of strain gradient effect. The average thermal residual stress increases with decreasing particle size, and the residual plastic strain decreases with decreasing particle size.

Key words: metal matrix composite; thermal residual stress; particle size effects; non-local theory; finite element analysis

1 Introduction

The residual stresses are inherent in SiC_p/Al composites owing to the mismatch in thermal expansion between aluminum and silicon carbide. The stresses develop during the cooling process from the fabrication temperature. The subject of magnitudes of the residual stresses has been studied analytically by utilizing analytical method[1-2] as well as by numerical method[3-4]. However, the research on the residual stress of SiC_p/Al composites predicted by these continuum-based models exhibits dependence on particle volume fraction, geometry and distribution, but not on particle size.

To predict the particle size dependence, dislocation models[5-6] and the strain gradient plasticity theory [7-9] have been developed. However, the application of the dislocation models becomes cumbersome once the particle spacing becomes much greater than the average dislocation spacing. Strain gradient plasticity theory postulates that the stress depends not only upon strain but also upon the strain gradient in regions exhibiting plasticity in homogeneity. Some phenomenological strain gradient plasticity theories have been developed by FLECK et al[7-8]. Quite dissimilar to the phenomeno- logical strain gradient plasticity theories, GAO et al[10] proposed a mechanism-based theory of strain gradient plasticity (MSG) based on a multiscale framework linking the microscale notion of statistically stored and geometrically necessary dislocations to the mesoscale notion of plastic strain and strain gradient.

Recently, GAO and HUANG [11] developed a Taylor based nonlocal theory (TNT) of plasticity to account for the size dependence of plastic deformation of SiC_p/Al composites. The theory does not require additional boundary conditions and the length scale is introduced into the constitutive equations via nonlocal variables which are expressed as an integral of local variables over all the material points in the body. Compared to the MSG theory, the advantage of TNT is the simplicity of its governing equations and no higher-order stresses and strains are involved.

In this study, the finite element method is used to investigate the effects of particle size on the nature, magnitude and distribution of residual stresses of the SiC_p/Al composites resulting form the cooling process. For modeling the matrix behavior of the composites, the Taylor-based nonlocal theory (TNT) of plasticity is used. The SiC particle is characterized as an isotropic elastic material. Perfect bonding between the matrix and the particle is assumed.

2 Finite element model

In order to investigate the effects of the particle size on residual stresses of the SiC_p/Al composites, the two-dimensional plane strain and random distribution multi-particle unit cell model is designed, as shown in Fig.1. A program is developed to automatically generate the position of particles. At first, the center coordinates (x₀, y₀) of a particle are assigned. Let L denote the width and height of the unit cell (Fig.1), a random number r₀  (0＜r₀＜1) is generated by the computer, and then the product of r₀ and L is used as x₀. In the same way, y₀ is obtained. The program also judges if a particle overlaps with others. The algorithm is to calculate the nearest distance between the two particles. If the distance is smaller than the sum of their radii, the particles intersect; otherwise, they do not. When a particle is found to overlap with any other one, a new position will be assigned to it. In the unit cell, the particle volume fraction is 20%.

Fig.1 Unit cell model(a) and finite element mesh(b) of model

The boundary conditions are defined as: u_x=0 at the left surface, u_y=0 at the bottom surface; all nodes at the upper surface have a common unknown u_y; all nodes at the right surface have a common unknown u_x. A 400 ℃ temperature drop is applied to stimulate a decrease from 420 ℃ processing temperature to the 20 ℃ working temperature. This temperature drop is applied as a uniform body load through the analysis. The average residual stresses in the y-direction are obtained by averaging the stresses in the unit cell.

The SiC particle is assumed to be linearly elastic and isotropic, with elastic modulus E_SiC=450 GPa, and Poisson ratio v_SiC=0.17. The uniaxial stress-strain behavior for unreinforced pure Al is well represented by the Ramburg-Osgood relation[12]:

                   (1)

where  the Ramburg-Osgood factor β is taken as 3/7; E_m and σ_m0 are the elastic modulus and yield strength of the matrix; n is the strain hardening exponent. The parameters of σ_m0 and n can be readily determined from the experimental stress—strain curve for the unrein- forced pure Al. They are 108 MPa and 0.06, respectively. The elastic modulus and Poisson ratio of pure Al are E_Al=70 GPa and v_Al=0.33, respectively. These properties are assumed to be constant over the entire temperature range.

It is well known that the microstructure and properties of the composite matrix may be significantly different from those of the unreinforced matrix because cooling from elevated temperature can result in local yielding and work hardening[13]. In order to model the matrix behavior of the composites, the Taylor-based nonlocal theory of plasticity is used. In the TNT plasticity, the density of geometrically necessary dislocations is treated as a nonlocal variable expressed in terms of an integral average of plastic strain, in which no higher-order strains need to be introduced. The flow stress of the matrix is given by[11]

            (2)

where  a is the radius of spherical particle,  f is the particle volume fraction, μ is the shear modulus, α is an empirical material constant, b is the Burgers vector. By inserting Eqn.(1) into Eqn.(2), the flow behavior of the matrix can be determined.

The numerical model is constructed using two-dimensional 6-nodel triangular elements. The adequacy of the finite element discretization is checked by repeating a calculation using refined meshes until a satisfactory computational accuracy is obtained.

3 Results and discussion

The contour plots of von Mises effective stress in aluminum matrix for the 20%SiC_p/Al composites are shown in Fig.2. It is apparent that the distribution of von Mises effective stress in matrix is very inhomogeneous due to the mismatch of thermal expansion between particle and matrix. The results show that the particle size has no influence on the distribution of von Mises effective stress. The largest stress of the matrix is located at the region between neighboring particles. This is attributed to the fact that the interaction of neighboring particles makes the stress at the matrix region between these particles much higher. It is evident from Fig.2 that the maximum von Mises effective stress of the matrix decreases with increasing of the particle size, suggesting that particle size influences the effective stress of the matrix. As particle size increases from 1 μm to 56 μm, the maximum von Mises effective stress of the matrix decreases from 204 MPa to 143 MPa.

Fig.3 shows the y-direction average thermal residual stresses in matrix and particles of composites. It can be seen from Fig.3 that the particles stress is in compression and matrix stress is in tension. The average thermal residual stresses in matrix and particles both increase with decreasing particle size.

The total amount of equivalent plastic deformation predicted to occur during the cooling process from 420 ℃ to 20 ℃ is shown in Fig.4. It can be seen from Fig.4 that there is a great plastic strain gradient in the matrix. The equivalent plastic strain of the matrix increases with increasing of the particle size. As particle size increases from 1 μm to 56 μm, the maximum equivalent plastic strain of the matrix increases from 0.03 to 0.032.

For SiC_p/Al composites, the presence of particles induces an inhomogeneous deformation and high geometrically necessary dislocation density in the composite matrix due to the elastic modulus mismatch and thermal expansion mismatch. The gradient of deformation (shown in Fig.4) requires that geometrically necessary dislocations be stored and also that glide dislocations supplying the bulk strain must cut through these secondary dislocations, and overcome the remaining long-range back stress from the particles in order to move. Thus, for a given plastic flow, the smaller particle size, which induces smaller interparticle distance, should lead to a greater strain gradient in the composites matrix, which, in turn, results in a higher geometrically necessary dislocation density. For small particle, the increased geometrically necessary dislocation density leads to a higher work hardening in the matrix. In TNT theory, the density of geometrically necessary dislocations is treated as a nonlocal variable expressed in terms of an integral average of plastic strain. The flow stress of the matrix is given by Eqn.(2). From this equation, it is shown that the flow stress of the matrix increases with decreasing the particle size.

Fig.2 Contours of von Mises equivalent stress in matrix for 20%SiC_p/Al composites with various particle sizes: (a) 1 μm; (b) 5 μm; (c) 20 μm; (d) 56 μm

Fig.3 y-direction average residual stresses of matrix(a) and particles(b)

Fig.4 Contours of von Mises equivalent plastic strain in matrix for 20%SiC_p/Al composites with various particle sizes: (a) 1 μm;   (b) 5 μm; (c) 20 μm; (d) 56 μm

It is generally known that considerable amount of thermal mismatch stresses would generate upon cooling from fabrication temperature to ambient temperature due to mismatch in the coefficient of thermal expansion (CTE) between the metal matrix and ceramic reinforcement. Once the stress is larger than the yield stress of the matrix, the stress can be released by creating a plastic deformation zone[14]. For small particle size, the relaxation of thermal stress would be much harder due to its higher flow stress and higher work hardening. Therefore, the average thermal residual stress of matrix and particle both increase with decreasing particle size, and the thermal residual plastic strain in the matrix increases with increasing particle size.

4 Conclusions

1) The magnitude of residual stresses developed during cooling of composites is strongly influenced by particle size. The average residual stresses of matrix and particle both increase with decreasing particle size.

2) The distribution of thermal residual stress and strain is not influenced by particle size.

3) The residual plastic deformation increases with increasing particle size.

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(Edited by YUAN Sai-qian)

Foundation item: Project(NCET-04-0324) supported by the Program for New Century Excellent Talents in University

Corresponding author: YAN Yi-wu; Tel: +86-451-86419625; E-mail: yanyiwuhit@yahoo.com.cn