Expressions for predicting the elasticity modulus of materials reinforced by second-phase grains
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I.
INTRODUCTION
IT is of practical importance to study the effective elasticity modulus of the multiphase composite. This is not only a subject of the composite mechanics but one of the failure of the materials as well. The elastic percolation failure of materials is one of the basic types of failure. For a perfect material, the elasticity modulus has a well-defined value, much higher than zero. While the voids, which may be considered as another phase with zero modulus, are introduced into the material randomly, the effective elasticity modulus of the material containing the voids will decrease. When the volume fraction of the voids is increased up to a critical value, the so-called elastic percolation failure threshold, the modulus decreases down to zero. This phenomenon is called elastic percolation failure, which has been widely studied in recent years. There are three approaches to study the modulus of the granular composite so far. The first is the upper and lower limit approach, which is based on the symmetry of the structure of the system. Obviously, the accurate value of the modulus could not be determined by using the upper and lower limit approach, tl] The second is the digital simulation, which is based on the microelasticity theory. [21 Although the accurate value of the modulus could be obtained by the digital simulation, it is expensive and time consuming, and the value of the modulus will depend on the model used in the digital simulation.t2] The third is the effective medium model, which is derived from the solution of the typical problem in the mechanics of elasticity, f31 In principle, the effective medium model has some advantages: it is simple and convenient to use and can give the accurate value of the modulus. However, if the typical problem could not be selected properly, the advantages of the effective medium model would M. ZHENG is Doctoral Candidate with the Department of Materials Science and Engineering, Northwestern Polytechnical University, and Lecturer, with the Department of Physics, Xidian University, Xi'an, Shaanxi 710071, People's Republic of China. X. ZHENG, Professor, is with the Department of Materials Science and Engineering, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, People's Republic of China. Manuscript submitted January 30, 1990. METALLURGICALTRANSACTIONSA
disappear, t3j Recently, many effective medium models from the viewpoint of the static or the dynamic mechanics are reported in the literature, t3'41 but most of them could not be thought to be perfect. In the present study, a new effective medium model is developed based on the solution of the overmatching problem in two and three dimensions and substantiated by using the test results given in available literature.
II.
T W O - D I M E N S I O N A L PROBLEMS
A. Expression for Elasticity Modulus of Two-Dimensional Granular Composite Let a round plate of radius r~ embed into the hole of radius r2 in an infinite sheet in the two-dimensional (2-D) space. If r~ > r2, it may be called overmatching in
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