MAT-44

Rule of mixtures

Isostrain composite modulus.

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GoverningRule of mixtures

Governing equation

E=E_f V_f+E_m(1-V_f)

$Vf \in \left[0.05, 0.8\right]$

where

Ef: Ef (GPa)
Em: Em (GPa)
Vf: Vf (—)
E: Rule of mixtures (GPa)

The figure, part by part

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①

Governing law — red

E=E_f V_f+E_m(1-V_f)

$Vf \in \left[0.05, 0.8\right]$

$Vf = 0.05$ $E = 14.82$

Along the curve

②

Domain of the variable — blue

Vf_{\min} = 0.05,\quad Vf_{\max} = 0.8

$Vf \in \left[0.05, 0.8\right]$

$Vf = 0.05$ $Vf = 0.05$

Along the curve

③

Response curve — green

E=E_f V_f+E_m(1-V_f)

$Vf \in \left[0.05, 0.8\right]$

$Vf = 0.05$ $E = 14.82$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}E}{\mathrm{d}Vf}

$Vf \in \left[0.05, 0.8\right]$

$Vf = 0.055$ $E^{\prime} = 226.5$

Along the curve

Lecture brief

Historical brief

Hooke (1678), Hall–Petch grain size, Arrhenius activation and Paris fatigue-crack growth are the spine of materials selection. The sheets relate stress, microstructure and life. This sheet (MAT-44 — Rule of mixtures) is the form associated with Rule of mixtures. Working symbols: $Ef$ , $Em$ , $Vf$ $\rightarrow$ $E$ . Isostrain composite modulus. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

E

from

Ef

Em

Vf

in Materials via

E=E_f V_f+E_m(1-V_f)

Isostrain composite modulus. Use it when a real materials question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

Ef = 230.000\,\mathrm{GPa}

Em = 3.500\,\mathrm{GPa}

Vf = 0.500\,\mathrm{—}

, the governing relation

E=E_f V_f+E_m(1-V_f)

yields

E = 116.750\,\mathrm{GPa}

. One governing identity, SI units, a single sweep on the sheet. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Ef Ef230 GPaEm Em3.5 GPaVf Vf0.5 —

Outputs

Rule of mixtures E116.750 GPa

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Narration of this film

One governing identity, SI units, a single sweep on the sheet.

Isostrain composite modulus. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube