In Cases of Two Material States

The two simplest models, characterizing the effective properties of a statistically homogeneous mixture composed of two materials in different states, are the rule 

These two simple models have been used to predict different thermal and mechanical properties of a two-state material such as thermal conductivity and elastic modulus, where they are also called parallel and series models respectively. These two models are further able to define the upper and lower bounds for the effective properties of the mixture [1], as follows  

In case three or more than three states are involved in the mixture, the rule and inverse rule of mixture can be easily extended for multiple states, as shown in Eq. 3.11 and Eq. 3.12  

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