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 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.

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.

Frictional dissipation from contact slip is the irreversible loss of mechanical energy associated with relative sliding at a contact. In the ideal kinetic friction model, friction opposes the local slip velocity and has magnitude ${\mu }_{k}N$, with nonnegative coefficient ${\mu }_k$ and normal force $N$. Its dissipated energy isThe work calculation and the distinction between signed and accumulated motion are developed in contact slip displacement. The kinetic-friction force law is an ideal physical model, rather than a universal description of every interface.^[ref]

An original two-stage example keeps the loading history explicit. Let ${\mu }_k=0.20$. Slide forward $0.40\mathrm{mm}$ while $N=0.50\mathrm{N}$, then backward $0.60\mathrm{mm}$ while $N=0.10\mathrm{N}$. The final signed slip is $-0.20\mathrm{mm}$, the accumulated contact slip is $1.00\mathrm{mm}$, andThe two stages contribute $40$ and $12\mu \mathrm{J}$. Using the unweighted mean of the two normal loads would instead give $60\mu \mathrm{J}$; the correct weighting follows sliding distance.

A physical measurement records relative motion close to the contact together with tangential force and normal loading. Integrating the opposing tangential force over the resolved sliding increments estimates the lost mechanical energy under the stated model. Force transmitted during ideal sticking, where local slip rate is zero, contributes no slip dissipation in that model, even if other parts of the apparatus deform. Grip travel and fixture mechanical compliance therefore must be distinguished from local sliding.

Accumulated contact slip measures the total distance slid at a specified contact, counting motion in either direction positively. For an absolutely continuous signed contact slip displacement in a consistently tracked local frame,It is the total variation of a function applied to the slip history. It differs from net slip ${\delta }_s(t)-{\delta }_s(0)$, which can lose information through cancellation. Contact slip displacement explains the contact-frame convention and the connection to frictional dissipation from contact slip.

For an original piecewise-linear trajectory, suppose successive slip values are $0,0.30,-0.10,0.20\mathrm{mm}$. Its net slip is $0.20\mathrm{mm}$, whileThe middle reversal contributes $0.40\mathrm{mm}$ even though it moves toward smaller signed coordinates. Recording only the first and last values would report $0.20\mathrm{mm}$ and miss eighty percent of the travelled distance in this stipulated history.

For exact samples at ordered times, the sum of absolute increments is a lower bound on the true accumulated motion; unobserved reversals can add distance between samples. The triangle inequality explains why inserting additional exact observations cannot decrease this sum. It equals the integral when the observed intervals contain no unresolved reversals and the path model is appropriate.

Measurement noise changes that conclusion about observed data. A truly stationary contact read as $+\eta ,-\eta ,+\eta$ would acquire an apparent distance $4\eta$ for positive $\eta$. More samples can reveal real reversals while also accumulating readout noise. A useful imaging study therefore reports time resolution, length measurement calibration and the treatment of noise, rather than interpreting every absolute increment as physical sliding.