 Bundle material area fraction

The bundle material area fraction is the fraction of a stated bundle envelope occupied by strand material in a specified transverse section: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 $\varphi$ without adding material. For parallel prismatic strands and a constant envelope over common length $L$, the corresponding volumes are $A_{m}L$ and $A_{e}L$, so the volume fraction equals $\varphi$ under those assumptions.

An original two-strand example shows why an average diameter is insufficient. Take circular strand radii $r_1=1\mu \mathrm{m}$ and $r_2=3\mu \mathrm{m}$, inside a stipulated envelope of area $A_e=20\pi \mu {\mathrm{m}}^2$. ThenReplacing both radii by their arithmetic mean $2\mu \mathrm{m}$ would give $A_m=8\pi \mu {\mathrm{m}}^2$ and $\varphi =0.4$, 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 $E{A}_m/L$. Dividing the measured rigidity by $A_e$ instead gives the envelope-based effective modulus $\varphi E$, rather than the strand material's $E$. This uses the uniform-bar relation applied to parallel strands.^[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.