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The bending rigidity of an unbonded fibre bundle relates total bending moment to curvature in a stated bending model. If slender material fibres slide longitudinally without appreciable friction, each bends about its own neutral axis. With the same imposed small curvature $\kappa$ and no additional axial tension, strand $i$ contributes $M_i=E_i{I}_i\kappa$, givingHere $I_i$ is that strand's second moment of area about its own bending axis. The underlying linear elasticity relation is the beam moment-curvature law.^[ref] Adding offset-area terms as though all strands shared one neutral axis would impose a different mechanical constraint.

An original two-strand comparison makes the difference quantitative. Take two identical circular strands of radius $r$, with centres vertically separated by $2r$, bending in the plane containing their centres. Each has $A=\pi r^2$ and $I=\pi r^4/4$. In the freely sliding model,For a separate idealization, bond the strands perfectly along their contact so they share a common neutral axis halfway between the centres. Neglect the bond's own area and compliance. Integrating squared distance from that common axis givesFor stipulated $E=2\mathrm{GPa}$ and $r=1\mathrm{mm}$, these are approximately $3142$ and $15708\mathrm{N}\mathrm{m}{\mathrm{m}}^2$. At $\kappa =0.001\mathrm{m}{\mathrm{m}}^{-1}$, the corresponding moments are $3.142$ and $15.708\mathrm{N}\mathrm{mm}$. Both specimens contain the same strand material; their ability to exchange longitudinal force differs.

A physical comparison would impose end rotations, permit the relative end travel required for inter-fibre sliding, and measure bending moment and curvature. Clamping every strand to identical end displacements, adding axial tension or allowing friction can change the response. The factor five above belongs to this two-strand geometry and ideal bond, not to every bundle. The equal-material subdivision model in material fibre makes a different comparison with a solid circular rod.

The contact slip displacement is signed relative tangential displacement at a specified contact. For small motions in a fixed local frame, let $\mathbf{t}$ be a tangential unit vector and let ${\mathbf{u}}_1,{\mathbf{u}}_2$ describe the two contacting surfaces. ThenThis scalar product extracts relative motion along the contact. The corresponding relative normal displacement is a separate quantity controlling opening or closure. This local distinction matters for inter-fibre sliding: an entire bundle can move without adjacent material fibres slipping against one another.

As an original local example, take $\mathbf{t}=(1,0)$, ${\mathbf{u}}_1=(0.30,0.02)\mathrm{mm}$ and ${\mathbf{u}}_2=(0.10,0.02)\mathrm{mm}$. The contact slip displacement is $0.20\mathrm{mm}$ and the relative normal displacement is 0. Adding the same translation to both surfaces changes neither result. Reversing the orientation of $\mathbf{t}$ reverses the sign of ${\delta }_s$ while preserving its magnitude. If the contact frame rotates substantially or different material points enter contact, relative tangential velocity must be evaluated in the current contact frame and integrated consistently; projecting one final global displacement difference onto a final tangent vector is not a general finite-motion formula.

Under the ideal dry-friction model, static friction can transmit tangential force while the local slip rate is 0. Sliding introduces kinetic friction; with constant normal force $N$, its magnitude is ${\mu }_{k}N$, where ${\mu }_k$ is the coefficient of friction.^[ref] The accumulated contact slip and the frictional dissipation from contact slip are therefore different from net signed displacement:The second expression follows from integrating the opposing frictional force over each increment of relative sliding; it assumes constant $N$ and ${\mu }_k$. For a prescribed out-and-back slip from 0 to $1\mathrm{mm}$ and back to 0, the final ${\delta }_s$ is 0 but ${\ell }_s=2\mathrm{mm}$. With stipulated $N=0.20\mathrm{N}$ and ${\mu }_k=0.25$, the dissipated energy is $100\mu \mathrm{J}$. Zero net slip does not imply zero frictional dissipation.

A simple measurement marks both surfaces close to a maintained contact, images their relative positions while tangential loading is varied, and resolves motion along the local contact direction. Measurement calibration converts image distances to lengths. Marks far from the contact can also move because the strands stretch or the fixtures deform; a grip-travel readout alone does not isolate contact slip displacement. Record both the signed motion and accumulated contact slip when comparing repeated bending cycles of an unbonded material-fibre bundle.

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 isThe 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 byFor 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.^[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.

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 $F_i=E_i{A}_i\Delta L/L$, soThis follows by applying the uniform-bar axial stiffness relation to the strands in parallel.^[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 $A_m={\sum }_i{A}_i$ excludes gaps. An envelope enclosing the bundle can have larger area $A_e$. Define the bundle material area fraction $\varphi =A_m/A_e$. For identical moduli $E$, inferring a modulus from the envelope and the measured axial stiffness givesAs an original example, four circular strands of radius $0.10\mathrm{mm}$ have $A_m=0.12566\mathrm{m}{\mathrm{m}}^2$. Stipulate an envelope area $0.20\mathrm{m}{\mathrm{m}}^2$, $E=100\mathrm{GPa}$ and $L=50\mathrm{mm}$. Then $\varphi =0.6283$, $K_{\mathrm{ax}}=251.3\mathrm{N}\mathrm{m}{\mathrm{m}}^{-1}$ and $E_{\mathrm{envelope}}=62.83\mathrm{GPa}$. 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.