= Bundle material area fraction
{title2=$\phi=A_m/A_e$}
The <bundle material area fraction> is the fraction of a stated bundle envelope occupied by strand material in a specified transverse section:
$$
\phi=\frac{A_m}{A_e},\qquad A_m=\sum_iA_i.
$$
Here $A_e$ includes spaces between <material fibres>, while $A_m$ excludes them. Count each strand area once and state how the envelope is drawn. A tighter envelope can increase $\phi$ without adding material. For parallel prismatic strands and a constant envelope over common length $L$, the corresponding <volumes> are $A_mL$ and $A_eL$, so the volume fraction equals $\phi$ under those assumptions.
An original two-strand example shows why an average diameter is insufficient. Take circular strand radii $r_1=1\,\mathrm{\mu m}$ and $r_2=3\,\mathrm{\mu m}$, inside a stipulated envelope of area $A_e=20\pi\,\mathrm{\mu m^2}$. Then
$$
A_m=\pi(r_1^2+r_2^2)=10\pi\,\mathrm{\mu m^2},\qquad\phi=\frac12.
$$
Replacing both radii by their <arithmetic mean> $2\,\mathrm{\mu m}$ would give $A_m=8\pi\,\mathrm{\mu m^2}$ and $\phi=0.4$, underestimating the material area by twenty percent. Area depends on the mean squared radius, not merely the square of the mean radius.
For straight, equally strained strands of a common <Young's modulus> $E$, the <axial stiffness> is $EA_m/L$. Dividing the measured rigidity by $A_e$ instead gives the envelope-based effective modulus $\phi E$, rather than the strand material's $E$. This uses the uniform-bar relation applied to parallel strands.<Structural Stiffness Concept by TU Delft OpenCourseWare>[Axial relation and distinction between structural geometry and material properties]{ref}
A physical estimate can image a transverse section, establish <measurement calibration> for image lengths, identify the strand boundaries and calculate both areas from the same section. State <section orientation>: an oblique image is not automatically a perpendicular-area measurement. Distinguish spaces between strands from internal voids within a strand if the intended quantity is solid material area. Changes of envelope convention or classification must not be mistaken for changes in material quantity.
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