= Inter-fibre sliding
<Inter-fibre sliding> is relative tangential motion between contacting <material fibres>. It is distinct from stretching both strands together: equal axial extension can occur with no relative motion at a contact, while bending can require strands to exchange longitudinal position. For local strand displacements $\mathbf u_1$ and $\mathbf u_2$ and a contact tangent $\mathbf t$, the <contact slip displacement> is
$$
\delta_s=(\mathbf u_1-\mathbf u_2)\cdot\mathbf t.
$$
The contact and the sign of $\mathbf t$ must be specified. Contact <friction> can oppose changing $\delta_s$ even in an <unbonded material-fibre bundle>.
An original bending geometry makes the required relative travel visible. Consider two nearby reference paths of radii $R$ and $R+h$, turning through the same angle $\theta$, measured in radians. Their arc lengths differ by
$$
\Delta s=(R+h)\theta-R\theta=h\theta.
$$
For chosen separation $h=0.50\,\mathrm{mm}$ and turn angle $30^\circ=\pi/6$, this mismatch is $0.262\,\mathrm{mm}$. If the strands remain inextensible while following those paths, material must feed through the ends to accommodate the difference. Fixing both ends of equal-length strands to remain aligned instead imposes additional strain or a change in the assumed geometry.
This explains the importance of a separate <neutral axis> for each strand in the freely sliding model for <bending rigidity of an unbonded fibre bundle>. Bonding imposes longitudinal compatibility and can require axial tension in some strands and compression in others. The usual beam model relates the resulting axial strain to position relative to a common <neutral axis>.<Stresses in Beams by David Roylance>[Normal Stresses, pp.2–4, equations 1–6]{ref}
A useful mechanical test marks neighbouring strands, bends the assembly while allowing end travel, and measures their relative mark positions. Repeating the bend with constrained ends tests a different loading condition. Record contact loading and bend history: an absence of observed slip at small applied force does not by itself establish a permanent bond.
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