Longitudinal Strength⚠ unverified
Mechanical / Composites · Compute the longitudinal tensile strength of a unidirectional composite
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| Xf | Xf | Pa | 1.0 | Fibre tensile strength |
| Xm | Xm | Pa | 1.0 | Matrix tensile strength (stress carried by matrix at fibre failure) |
| Vf | Vf | — | 1.0 | Fibre volume fraction (dimensionless), between 0 and 1 |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | X | Pa | Longitudinal strength, in pascals (Pa) |
The science & history
Understanding the Parameters
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Fiber strength $X_f$ — dominates the composite strength; because $X_f V_f$ is the large term, strength scales nearly linearly with fiber content and fiber quality.
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Matrix contribution $X_m$ — critically, this is the matrix stress at the strain where the fibers break ($X_m = E_m\varepsilon_{f,\text{ult}}$), not the matrix ultimate strength. Since the matrix is far more compliant, this is a small contribution — the matrix mainly transfers load into fibers rather than carrying it.
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Failure sequence — in a well-designed composite the stiff fibers reach their failure strain first; at that instant the composite fails (the matrix alone cannot carry the released load above a minimum $V_f$). This is why $X_f$ is evaluated at fiber fracture.
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Idealisation — the rule of mixtures assumes all fibers fail simultaneously at a single strength. Real fibers have strength scatter (Weibull statistics), so failure is progressive and the actual strength is somewhat lower; the ROM is an upper estimate.
Derivation (Approaching a Proof)
Under longitudinal tension, fiber and matrix share the same strain $\varepsilon$ (iso-strain, as for the modulus). The composite stress is the volume-weighted sum of the phase stresses:
$$\sigma_1(\varepsilon) = \sigma_f(\varepsilon) V_f + \sigma_m(\varepsilon)(1-V_f).$$
For fiber-dominated composites the fibers fail first, at strain $\varepsilon_{f,\text{ult}}$ where $\sigma_f = X_f$. Evaluating the composite stress at that instant, with the matrix carrying $X_m = \sigma_m (\varepsilon_{f,\text{ult}})$:
$$X = X_f V_f + X_m(1-V_f).$$
(Below a minimum fiber fraction the matrix could momentarily carry the load after fiber failure, giving a different, matrix-controlled branch; above it — the practical regime — fiber failure is catastrophic and this formula governs.)
Dimensional check. $X = X_f V_f + X_m(1-V_f) = \text{Pa}\cdot(\text{–}) = \text{Pa}$ — a strength, as required.
History and Development
Rule-of-mixtures strength is the strength companion to the stiffness rule of mixtures (Rule of Mixtures longitudinal), central to composites design since the 1960s. Its refinement — using the matrix stress at fiber failure, and correcting for Weibull fiber-strength scatter and the critical fiber volume fraction — is the content of composite strength theory (Kelly, Rosen). It sets the fiber-direction allowable that, with transverse and shear strengths, feeds failure criteria like Tsai Hill Criterion.
Related Concepts: Transverse Strength, Tsai Hill Criterion, Rule of Mixtures longitudinal, Fatigue Life Composite, Specific Strength, Composite Laminate Theory
Notes: $X_m$ is the matrix stress at the fiber failure strain ($E_m\varepsilon_{f,\text{ult}}$), not matrix ultimate — fibers fail first. Fiber-dominated ($X_f V_f$ term). Idealised (all fibers fail at once) — real strength lower due to Weibull scatter → upper estimate. Provides $X$ for Tsai Hill Criterion.