Verifying micromechanical predictions of the material properties of three-phase composites by means of finite element analysis

The prediction of mechanical properties of composite materials based on their constituent properties is an important step in using composites. In some metal matrix composites, calculation of these properties is complicated by the formation of a reaction zone between the fiber and the matrix. The problem of developing micromechanical models to predict this modulus based on constituent properties has been studied previously. Some models developed in these previous studies are presented briefly. Experimental verification of these models is costly. An alternative is to verify the validity of micromechanical models through the use of "numerical" experiments. This paper concentrates on the process of using finite element results a means of verifying micromechanical predictions. A plane stress analysis was conducted for the transverse properties. Average values of stress and strain are obtained, and an equivalent elastic modulus, Poisson's ratio, and shear modulus for the composite are calculated. Longitudinal properties require a threedimensional analysis. Differences in the results from the finite element analysis and the theoretical predictions are very small for the cases examined. This suggests that the finite element method offers a simple cost efficient way of verify micromechanical models. Associate Professor, Industrial and Engineering Technology ** Associate Professor, Civil Engineering and Mechanics, Member A IM , ASME, and ASC Co~vrieht @ 1993 American Institute of Aeronautics and Introduction Predicting the mechanical properties based on the known properties of its constituents is a necessary abilrty in modeling composite material behavior. This study concentrates on the properties of metal matrix composites where a third phase or reaction zone forms between the fiber and matrix. One example of this type of material is a tungsten fiber reinforced superalloy (TFRS). The properties of interest are the longitudinal and transverse elastic moduli, Ell and E, the out of plane shear modulus, G,,, and the Poisson's ratios, vl, and v,, A theoretical model, developed by Hopkins and Chamis' is used. Additional work on this model has been done by Craddock et. aL2, and Craddock and Koche? . Predictjons of material propehes based on the Hopkins-Chamis model are compared to values obtained by a finite element analysis. Results for the material properties are also obtained for several different situations. Reaction zone size and reaction zone modulus are all varied. The fiber volume ratio is also varied. Values of the properties are obtained from the theoretical model and a commercial finite element code named ANSYSQFor the finite element results two different method of computing the modulus are used. A comparison of the theoretical and numerical results is made. The difference in these results is small, showing that the theoretical models are valid. The methodology also suggests that a series of "numerical experiments" can be used in place of actual experimental measurements to verify micromechanical models for cornposite materials. This approach could result in a significant reduction in cost and effort in many instances. .. Astronautics, Inc. All rights reserved. 1638 Hopkins-Chamis Model A set of micromechanical models for predicting the properties of three-phase composites has been developed by Hopkins and Chamis'. The Hopkins-Chamis model is based on a square unit cell as shown in Figure 1. The reaction zone is assumed to a circular region growing inward from the matrix fiber interface toward the center of the fiber. In this diagram, Do is the original fiber diameter, D is the diameter of the unreacted portion of the fiber, and a, is the overall width of the cell. The size of the reaction zone is measured by D/D,. The volume fraction of the matrix is k,,,. The initial fiber volume fraction is k,. These quantities are given by: After the formation of the reaction zone, the volume fractions of the unreacted fiber and of the reaction zone are given by: where k, is the volume fraction of the reaction zone and k," is the volume fraction of the unreacted portion of the fiber. The expression for the longitudinal modulus of elasticity is given by: Figure 1. Unit cell Used in Micromechanical Model. where E,, is the longitudinal modulus of elasticrty of the composite, and E,, E m , and Em are the moduli of elasticity of the fiber, reaction, and matrix respectively. Similarly, the in-plane Poisson's ratio is given by: where v is the Poisson' ratio, and the subscripts are as above. For the transverse properties, the circular regions of the fiber and reaction zone are transformed into squares having areas equal to the respective circular regions. A simple analysis then yields the transverse modulus of elasticity, E, as: where the terms C, through C, are constants given by: In a like manner, the expression for G,, is given by: where G, is the out-of-plane shear modulus for the composite and G, , G,, and G, are the shear moduli for the matrix, fiber and reaction zone; and the terms B, through B, are: If transverse isotropy is then assumed, the out-ofplane Poisson's ratio is: where v,, is the out-of-plane Poisson's ratio.