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Axial stiffness measures resistance to extension or shortening along a specified loading axis. For a proportional response, $F=K_{\mathrm{ax}}\Delta L$; for a nonlinear response, specify whether the reported value is a secant stiffness or tangent stiffness. A straight uniform bar in small-strain linear elasticity haswhere $E$ is Young's modulus, $A$ is the load-bearing cross-sectional area and $L$ the deforming length.^[ref] Axial rigidity $EA$ has units of force; axial stiffness $EA/L$ has units of force per length.

An original stepped-bar example shows why a single area is sometimes insufficient. Join two ideal straight segments end to end, with no distributed axial loading. Both carry the same force $F$, so their extensions add:Choose $E_1=E_2=70\mathrm{GPa}$, $L_1=20\mathrm{mm}$, $L_2=30\mathrm{mm}$, $A_1=1\mathrm{m}{\mathrm{m}}^2$ and $A_2=2\mathrm{m}{\mathrm{m}}^2$. The total mechanical compliance is $0.0005\mathrm{mm}{\mathrm{N}}^{-1}$, giving $K_{\mathrm{ax}}=2000\mathrm{N}\mathrm{m}{\mathrm{m}}^{-1}$. A $100\mathrm{N}$ force extends the bar by $0.050\mathrm{mm}$, divided into $0.0286$ and $0.0214\mathrm{mm}$ in the two segments. The thinner segment extends more despite being shorter.

In parallel, components sharing the same extension instead add their axial stiffnesses, as in a straight unbonded material-fibre bundle with every strand engaged. These are different constraints, described by series and parallel spring stiffness.

A physical test should measure extension between defined specimen points. Crosshead displacement can include machine compliance and grip motion, so the slope of the complete apparatus is not automatically the specimen's axial stiffness. Structural geometry and loading constraints must be separated from the material's Young's modulus.

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.

A material fibre is a long, slender physical strand: its length is large compared with its transverse dimensions. The geometry is shared by materials with different roles. Carbon fibres and glass fibres can reinforce a composite material, while an optical fibre guides light. The material's properties, the strand dimensions and the way strands are assembled all affect its response.

A single-fibre tensile test grips one strand, measures its axial force and extension over a specified gauge length, and uses its cross-sectional area to obtain stress and strain. A 2020 study used a semi-automated instrument to characterize glass fibres and carbon fibres through their stress-strain curves and inferred moduli.^[ref] Its public dataset supplies individual tensile curves at several gauge lengths, summary sheets and calculations of test-setup machine compliance.^[ref] This matters because crosshead displacement can include deformation of the apparatus as well as extension of the material fibre; its slope alone need not give the specimen's Young's modulus.

An original ideal model shows why making thinner strands changes some responses more strongly than others. Keep total material cross-sectional area $A$, length $L$ and Young's modulus $E$ fixed. Compare one solid circular rod with $N$ identical circular fibres, each of radiusFor axial loading, grip every strand so that all undergo the same extension $\Delta L$. Each supplies force $E(A/N)\Delta L/L$, and adding the forces gives the axial stiffnessindependent of $N$ under these assumptions. This is the uniform-bar linear elasticity relation applied to parallel strands.

Now make a separate pure-bending comparison of an unbonded material-fibre bundle. Allow longitudinal sliding, neglect friction, and let each strand bend about its own neutral axis with the same curvature $\kappa$. A circular strand has second moment of area $I_N=\pi r_N^4/4$ and filament bending modulus $E{I}_N$. Its bending moment is $E{I}_N\kappa$ in the small-deformation model.^[ref] Adding those moments gives the bending rigidity of an unbonded fibre bundle:Each material fibre has its own neutral axis, which is crucial to this calculation. Bonding strands together transfers shear stress and couples their axial deformation; the independent-strand sum then ceases to describe the assembly. In a reinforced composite material, that coupling is part of fibre-matrix load transfer.

For chosen $A=1{\mathrm{mm}}^2$, $L=100\mathrm{mm}$ and $E=70\mathrm{GPa}$, both axial specimens have $K_{\mathrm{ax}}=700\mathrm{N}\mathrm{m}{\mathrm{m}}^{-1}$. At $N=100$, each radius is approximately $0.0564\mathrm{mm}$, and the independent-bending value is $55.7\mathrm{N}\mathrm{m}{\mathrm{m}}^2$ compared with $5570.4\mathrm{N}\mathrm{m}{\mathrm{m}}^2$ for the solid rod. The parameters are stipulated for this model; $A$ counts material and excludes spaces between strands.

A useful mechanical comparison therefore records which material fibres are gripped, whether they can slide, their arrangement and the loading direction. Equal material quantity alone does not determine an assembly's stiffness, and small filament bending modulus does not imply small axial Young's modulus.

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.