= Unbonded material-fibre bundle
An <unbonded material-fibre bundle> collects separate <material fibres> without continuously bonding neighbouring strands together. Contact and <friction> may still transfer <force>, so unbonded does not automatically mean freely sliding. The response depends on which strands are gripped, their alignment, slack and whether <inter-fibre sliding> is permitted.
For straight strands of common length $L$, let both end fixtures impose the same small extension $\Delta L$ on every strand. Under <linear elasticity>, strand $i$ carries $F_i=E_iA_i\Delta L/L$, so
$$
F=\frac{\Delta L}{L}\sum_i E_iA_i,\qquad
K_{\rm ax}=\frac{1}{L}\sum_iE_iA_i.
$$
This follows by applying the uniform-bar <axial stiffness> relation to the strands in parallel.<Structural Stiffness Concept by TU Delft OpenCourseWare>[Axial member relation and material-versus-geometrical contributions]{ref} Equal extension here is an end-loading condition. It does not require continuous bonding, nor does it describe the separate freely sliding bending experiment in <bending rigidity of an unbonded fibre bundle>.
The material <cross-sectional area> $A_m=\sum_iA_i$ excludes gaps. An envelope enclosing the bundle can have larger area $A_e$. Define the <bundle material area fraction> $\phi=A_m/A_e$. For identical moduli $E$, inferring a modulus from the envelope and the measured <axial stiffness> gives
$$
E_{\rm envelope}=\frac{K_{\rm ax}L}{A_e}=\phi E.
$$
As an original example, four circular strands of radius $0.10\,\mathrm{mm}$ have $A_m=0.12566\,\mathrm{mm^2}$. Stipulate an envelope area $0.20\,\mathrm{mm^2}$, $E=100\,\mathrm{GPa}$ and $L=50\,\mathrm{mm}$. Then $\phi=0.6283$, $K_{\rm ax}=251.3\,\mathrm{N\,mm^{-1}}$ and $E_{\rm envelope}=62.83\,\mathrm{GPa}$. That lower effective value does not mean the strand material has changed.
A useful tensile comparison counts the engaged strands, measures their areas and checks that the fixtures actually impose the assumed extension. A loose or broken strand changes the load-bearing area even if the outer envelope barely changes. Individual strand properties can be characterized by a <single-fibre tensile test>; neither envelope dimensions nor a bundle slope alone identifies every strand's <Young's modulus>.
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