Rule of mixtures

Rule of mixtures / Voigt model / equal strain Consider a composite material under uniaxial tension \sigma_\infty. Under the Voigt assumption, we model the strain in the two constituents as equal. In the context of fiber-reinforced composites, one might interpret this as an applied strain along the fiber direction. Hooke's law for uniaxial tension gives {{NumBlk|:|\frac{\sigma_1}{E_1} = \epsilon_1 = \overline{\epsilon} = \epsilon_2 = \frac{\sigma_2}{E_2}|}} where \sigma_1 and \sigma_2 are the stresses of constituents 1 and 2 respectively, and we define the homogenized strain as \overline{\epsilon}. Noting stress to be a force per unit area, a force balance gives that Equations and can be combined to give :\overline{E}_V \overline{\epsilon} = fE_1\epsilon_1 + \left(1-f\right)E_2\epsilon_2. Finally, since \overline{\epsilon} = \epsilon_1 = \epsilon_2, the overall elastic modulus of the composite can be expressed as : \overline{E}_V = fE_1 + \left(1-f\right)E_2. Assuming the Poisson's ratio of the two materials is the same, this represents the upper bound of the composite's elastic modulus. Inverse rule of mixtures / Reuss model / equal stress Alternatively, we can assume that the stress in the two constituents is equal, i.e. \sigma_\infty = \sigma_1 = \sigma_2. This corresponds to the two constituents being loaded in series, which in the context of fiber-reinforced composites roughly corresponds to transverse loading. In this case, the overall strain is distributed according to :\overline{\epsilon} = f\epsilon_1 + \left(1-f\right)\epsilon_2. The overall modulus in the material is then given by :\overline{E}_R = \frac{\sigma_\infty}{\overline{\epsilon}} = \frac{\sigma_1}{f\epsilon_1 + \left(1-f\right)\epsilon_2} = \left(\frac{f}{E_1} + \frac{1-f}{E_2}\right)^{-1} since \sigma_1=E_1\epsilon_1, \sigma_2=E_2\epsilon_2. == Other properties ==

Similar derivations give the rules of mixtures for • mass density:\left(\frac{f}{\rho_1}+\frac{1-f}{\rho_2}\right)^{-1} \leq\ \overline{\rho} \leq f \rho_1 + (1-f) \rho_2 • thermal conductivity:\left(\frac{1}{k_1} + \frac{1-f}{k_2}\right)^{-1} \leq \overline{k} \leq f k_1 + \left(1-f\right)k_2 • electrical conductivity:\left(\frac{f}{\sigma_1} + \frac{1-f}{\sigma_2}\right)^{-1} \leq \overline{\sigma} \leq f\sigma_1 + \left(1-f\right)\sigma_2 Notably, these forms are typically not applicable to strength-related properties. Consider the case where the strain in both phases is equal. Then, the rule of mixtures would only be applicable when both materials fail at the same strain. If this is not the case, the least ductile material will fail first. Thus, the strength of the composite would be determined not as a mix of constituent strengths, but rather as a function of the strength of a single constituent. In turn, this supposes that another failure mode, such as debonding, does not occur before either phase fails. == Generalizations ==

Some proportion of rule of mixtures and inverse rule of mixtures A generalized equation for any loading condition between isostrain and isostress can be written as: : (\overline{E})^k = f(E_1)^k + (1-f)(E_2)^k where k is a value between 1 and −1. More than 2 materials For a composite containing a mixture of n different materials, each with a material property E_i and volume fraction V_i , where : \sum_{i = 1}^{n}{V_i} = 1 , then the rule of mixtures can be shown to give: : \overline{E} = \sum_{i = 1}^{n}{V_i E_i} and the inverse rule of mixtures can be shown to give: : \frac{1}{\overline{E}} = \sum_{i = 1}^{n}{\frac{V_i}{E_i}} . Finally, generalizing to some combination of the rule of mixtures and inverse rule of mixtures for an n-component system gives: :(\overline{E})^k = \sum_{i=1}^{n}V_i(E_i)^k == See also ==