Elastic Modulus

Symbol

Strain and stress

Strain $\varepsilon$, stress $\sigma$

Strain stress bulk modulus relationship

\[E = \frac{\sigma}{\varepsilon}\]

Average Elasticity and Composite

Composite

Assume $f_i = V_i / V$. We have $V \varepsilon = V_i \varepsilon_i$. Therefore,

\[\varepsilon = \frac{V_i}{V} \varepsilon_i = f_i \varepsilon_i\]

Uniform Strain and Voigt Average

Voigt (upper bound) average assuming uniform strain ( $\varepsilon_i = \varepsilon_0 = \varepsilon$ ):

\[\begin{aligned} &\hat{E}=\frac{\hat{\sigma}}{\hat{\varepsilon}}=\frac{\sum f_{i} \sigma_{i}}{\hat{\varepsilon}}=\frac{\sum f_{i}\left(\hat{\varepsilon} E_{i}\right)}{\hat{\varepsilon}}\\ &\hat{E}=\sum f_{i} E_{i} \end{aligned}\]

Therefore,

\[K_\text{V} = \sum_i f_i K_i\]

\[G_\text{V} = \sum_i f_i G_i\]

Uniform Stress and Reuss Average

Reuss (lower bound) average assuming uniform stress ( $\sigma_i = \sigma_0 = \sigma$ ):

\[\begin{array}{l} {\hat{E}=\frac{\hat{\sigma}}{\hat{\varepsilon}}=\frac{\hat{\sigma}}{\sum f_{i} \varepsilon_{i}}=\frac{\hat{\sigma}}{\sum f_{i}\left(\frac{\hat{\boldsymbol{\sigma}}}{E_{i}}\right)}} \\ {\frac{1}{\hat{E}}=\sum \frac{f_{i}}{E_{i}}} \end{array}\]

Therfore,

\[K^{-1}_\text{R} = \sum_i f_i K_i^{-1}\]

\[G^{-1}_\text{R} = \sum_i f_i G_i^{-1}\]

Body Wave Velocities

References