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Common Misconceptions on Rules of Mixtures

Wenbin Yu's picture

Please pardon me if I am preaching to the choir here. Rules of mixtures (ROM) are very simple mechanics models. Everybody on this site has a very good understanding of it. However, confusion and mistakes on ROM constantly appear in textbooks, journal articles, online learning materials, etc. See attached two wiki articles. The major confusing point is that vf*Ef+vm*Em is derived from the isostrain assumption and the upper bound. Both statements are incorrect. There might be two main reasons contributed to this mistake/confusion. First, vf*Ef+vm*Em was derived using isostrain assumption in the loading direction, and neglecting the Poisson's effects in other directions, which is equivalent to isostress assumption in other directions, thus destroying the bounding nature of this simple formula. Second, a simple, explicit formula for correct upper bound in terms of Young's moduli and Poisson's ratios in most literature.  

To clarify, I wrote a short note on this.  Yu, W. "Common Misconceptions on Rules of Mixtures for Predicting Elastic Properties of Composites" , AIAA Journal, 2024. A simple yet rigorous derivation is given for elastic properties based on both isostrain and isostress assumptions. Correct upper bound are given explicitly in terms of Young's moduli and Poisson's ratios. A hybrid rule of mixtures is used to derive f*Ef+vm*Em, emphasizing it is not a bound. 

Please let me know your comments, questions, criticisms. Also, if you are familiar with wiki, please help correct these two wiki articles.  Thanks a lot for your attention!
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