Abstract
Mix carbon fibre and epoxy half and half. This paper asks what the stiffness of that composite is——the answer is that it is not one number but a range of $\mathbf{16.932376}$. No new mathematical theorem and no new law is claimed.
First, at $v_f=0.5$ the parallel bound is $116.7500$ GPa and the series bound $6.8951$ GPa (Section 2).
Second, this is the core of the paper. The ratio of the bounds has a closed form and is exactly symmetric in $v_f$ and $1-v_f$ (Section 3).
Third, the ratio is $v_f^2+(1-v_f)^2+v_f(1-v_f)(E_f/E_m+E_m/E_f)$ (Section 3).
Fourth, it is maximal at $v_f=0.5$ with the value $\frac{1}{2}+\frac{1}{4}(E_f/E_m+E_m/E_f)=\mathbf{16.932376}$ (Section 3).
Fifth, the two materials enter only through the single symmetric combination $E_f/E_m+E_m/E_f$. Which one is called the fibre does not move the ratio (Section 4).
Sixth, the separator is whether strain is shared or stress is (Section 5).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed——the Voigt (parallel, iso-strain) and Reuss (series, iso-stress) averages, the rule of mixtures, and the Hashin-Shtrikman bounds are all standard. We do not build mechanics of materials——the fibre-matrix interface, debonding, and residual stress are not entered. They are mentioned and no more. We do not treat strength——only stiffness (Young’s modulus) is treated; fracture strength and fatigue are not the subject. Strength belongs to Paper 344. We claim no optimality for the bounds——Voigt and Reuss are loose bounds; for an isotropic two-phase material the Hashin-Shtrikman bounds are narrower. The two loose ones are what is treated here. We design no laminate——classical laminate theory and weave patterns are not entered. We do not treat anisotropy in general——only the longitudinal and transverse directions of a unidirectional material. Relation to earlier papers: Paper 326 showed the square-cube “law” to be an identity and not a law——the ratio here is likewise not something measured but something that follows from algebra. The same move, performed on materials. Paper 330 showed the squaring of the condition number to be an identity——there too what was taken for “it usually comes out this way” followed from a definition. Paper 344 shows strength to be a function of volume and not a constant of the material——this paper says stiffness is not one number, and 344 says the same of strength. They make a pair. Paper 300 showed whether two things share a root is decidable——parallel and series have distinct roots by that test. What is added is deriving the closed form and showing the symmetry and the maximum algebraically, noting that the two materials enter only through one symmetric combination, confirming as a counter-case that $E_f=E_m$ gives a ratio of $1$ for every $v_f$, and putting the separator on what is shared.
1. Notation and the Question
Setting: Mix fibre of modulus $E_f$ with matrix of modulus $E_m$ at volume fraction $v_f$. There are two extremes of mixing.
| What is shared | Modulus of the composite | |
|---|---|---|
| Voigt (parallel) | strain | $E=v_fE_f+(1-v_f)E_m$ |
| Reuss (series) | stress | $1/E=v_f/E_f+(1-v_f)/E_m$ |
Observation 1.1 Parallel means pulling along the fibres. Fibre and matrix stretch by the same amount——series means pulling across them, with one stress passing through both.
Observation 1.2 What was mixed is the same. The same two materials at the same volume fraction——only the direction of the load changed.
Observation 1.3 The way of measuring is fixed first. Carbon fibre $E_f=230$ GPa and epoxy $E_m=3.5$ GPa——a ratio $E_f/E_m=65.714286$.
Question: How far does direction alone open it?
2. At $v_f=0.5$, $116.7500$ Against $6.8951$
Calculation 2.1: Compute both against volume fraction (computed in this paper).
| $v_f$ | Voigt (parallel) [GPa] | Reuss (series) [GPa] | Upper/lower |
|---|---|---|---|
| $0.00$ | $3.5000$ | $3.5000$ | $\mathbf{1.0000}$ |
| $0.20$ | $48.8000$ | $4.3584$ | $11.1967$ |
| $0.40$ | $94.1000$ | $5.7747$ | $\mathbf{16.2951}$ |
| $\mathbf{0.50}$ | $\mathbf{116.7500}$ | $\mathbf{6.8951}$ | $\mathbf{16.9324}$ |
| $0.60$ | $139.4000$ | $8.5547$ | $\mathbf{16.2951}$ |
| $0.70$ | $162.0500$ | $11.2666$ | $14.3832$ |
| $1.00$ | $230.0000$ | $230.0000$ | $\mathbf{1.0000}$ |
Observation 2.2 A check——at $v_f=0$ both give $3.5000$ and at $v_f=1$ both give $230.0000$. At the unmixed extremes the manner of mixing means nothing, so they must agree. They do.
Observation 2.3 At $v_f=0.5$, $116.7500$ against $6.8951$. The same materials in the same proportion, apart by $16.9324$——“what is the Young’s modulus of this composite” has no single answer.
Observation 2.4 The ratios at $v_f=0.4$ and $v_f=0.6$ agree to four decimals. Both are $16.2951$——it does not look accidental.
Observation 2.5 And at both ends the ratio returns to $1$. $1.0000$ at $v_f=0$ and at $v_f=1$——the ratio is a hill with its peak in the middle.
3. The Core: The Ratio Has a Closed Form, and It Is Symmetric
Setting: Write the ratio in symbols.
Observation 3.1 Put $a=E_f$, $b=E_m$, $u=v_f$; then
$$R(u)=\bigl(ua+(1-u)b\bigr)\Bigl(\frac{u}{a}+\frac{1-u}{b}\Bigr) =u^2+(1-u)^2+u(1-u)\Bigl(\frac{a}{b}+\frac{b}{a}\Bigr)$$——expanded and collected, with no approximation.
Observation 3.2 This is the core of the paper. All three terms are invariant under swapping $u$ for $1-u$——both $u^2+(1-u)^2$ and $u(1-u)$ are symmetric. So $R(u)=R(1-u)$ holds exactly.
Observation 3.3 So the agreement at $v_f=0.4$ and $0.6$ was no accident. What looked equal to four decimals in Section 2 is equal to eight——$16.29508075$ against $16.29508075$, because it is an identity.
Calculation 3.4: Check the closed form against direct computation (computed in this paper).
| $u$ | Direct | Closed form |
|---|---|---|
| $0.10$ | $6.73565528$ | $6.73565528$ |
| $0.30$ | $14.38319565$ | $14.38319565$ |
| $0.40$ | $16.29508075$ | $16.29508075$ |
| $\mathbf{0.50}$ | $\mathbf{16.93237578}$ | $\mathbf{16.93237578}$ |
| $0.60$ | $16.29508075$ | $16.29508075$ |
| $0.90$ | $6.73565528$ | $6.73565528$ |
Observation 3.5 Sweeping $u$ from $0$ to $1$ in steps of $0.01$, the largest discrepancy is $7.11\times10^{-15}$. Within double-precision rounding——the identity holds.
Observation 3.6 A symmetric quadratic has its maximum in the middle. Setting $u=1/2$,
$$R_{\max}=\frac{1}{2}+\frac{1}{4}\Bigl(\frac{a}{b}+\frac{b}{a}\Bigr)=\mathbf{16.932376}$$——the $16.9324$ read off the table in Section 2 has come out of the algebra.
Observation 3.7 This is the move of Paper 326. Paper 326 wrote that the square-cube “law” is a geometric identity——$16.932376$ here is likewise not a number obtained by measuring materials but one that follows from the definitions of two averages. It is not something to confirm by experiment.
4. The Two Materials Enter Through One Symmetric Combination
Observation 4.1 In the closed form the two materials appear only as $a/b+b/a$. Swap $a$ and $b$ and that combination is unchanged——which one is called “fibre” and which “matrix” does not move the ratio at all.
Calculation 4.2: Does the closed form hold for other pairs (computed in this paper)?
| Pair | $a/b$ | $R_{\max}$ (closed form) | Direct at $u=0.5$ |
|---|---|---|---|
| Carbon fibre/epoxy | $65.714286$ | $16.932376$ | $16.932376$ |
| $200/70$ GPa | $2.857143$ | $1.301786$ | $1.301786$ |
| $1000/1$ GPa | $1000.000000$ | $250.500250$ | $250.500250$ |
Observation 4.3 For materials of similar stiffness there is almost no range. $200$ against $70$ GPa gives $1.301786$, only thirty per cent——direction matters only when the two stiffnesses differ by decades.
Observation 4.4 If $E_f=E_m$ the ratio is $1$ for every $v_f$. With $a/b+b/a=2$, $R(u)=u^2+(1-u)^2+2u(1-u)=(u+(1-u))^2=1$——mix a material with itself and direction cannot matter. Obvious, and the formula says it.
Observation 4.5 When $a/b$ is large, $R_{\max}\approx a/(4b)$. Because $b/a$ drops out: $1000/1$ gives $250.500250$, nearly $1000/4=250$——the range is set by a quarter of the stiffness ratio.
Observation 4.6 To be stated honestly——Voigt and Reuss are loose bounds. For an isotropic two-phase material the Hashin-Shtrikman bounds are narrower——the $16.932376$ here is not “the range achievable by direction” but “the range spanned by two simple averages”. Measured longitudinal and transverse values fall inside it without reaching either end.
5. The Separator: Is Strain Shared, or Stress?
Observation 5.1 The separator——whether what the two phases share is strain or stress.
| Shared | Direction of load | $E$ at $v_f=0.5$ [GPa] | |
|---|---|---|---|
| Voigt | strain | along the fibres | $116.7500$ |
| Reuss | stress | across the fibres | $6.8951$ |
Observation 5.2 The separator is not on the side of the material. The same panel, the same composition, the same volume fraction——what makes the difference is only which way one pulls.
Observation 5.3 So “the Young’s modulus of this material” is an incomplete question. Without a direction the answer swings by $16.932376$——a word that could be omitted for isotropic materials cannot be omitted here.
Observation 5.4 The separator can be moved. Weave the fibres, or stack plies at angles, and any point between the extremes can be chosen——design is the act of deciding where in this $16.932376$ range to sit.
Observation 5.5 It pairs with Paper 344. Paper 344 says strength is a function of volume and not a constant of the material——this paper says stiffness is a function of direction and not a constant of the material. What is called “a constant of the material” turned out twice not to be one.
Observation 5.6 One last thing——the upper bound $116.7500$ is about half the fibre’s own $230$. Half of it is fibre, so that looks natural; but the lower bound $6.8951$ is not even twice the matrix’s own $3.5$——placed in series the fibre barely works. The same quantity of carbon fibre is very nearly wasted, according to direction.
6. The Census and the Close
- $116.7500$ against $6.8951$ at $v_f=0.5$ (Calculation 2.1): a factor of $16.9324$.
- Both agree at $v_f=0$ and $1$ (Observation 2.2): unmixed, mixing means nothing.
- $v_f=0.4$ and $0.6$ agree (Observation 2.4): not an accident.
- $R=u^2+(1-u)^2+u(1-u)(a/b+b/a)$ (Observation 3.1): no approximation.
- $R(u)=R(1-u)$ exactly (Observation 3.2): all three terms symmetric.
- $16.29508075$ against $16.29508075$ (Observation 3.3): equal to eight decimals.
- $R_{\max}=\frac{1}{2}+\frac{1}{4}(a/b+b/a)=16.932376$ (Observation 3.6): out of the algebra.
- Materials enter only as $a/b+b/a$ (Observation 4.1): which is the fibre does not matter.
- $E_f=E_m$ gives $1$ for every $v_f$ (Observation 4.4): the formula says it.
- The lower bound is not twice the matrix (Observation 5.6): the fibre barely works.
Closing: Mix carbon fibre and epoxy half and half, ask what the Young’s modulus is, and there is no single answer——along the fibres $\mathbf{116.7500}$ GPa, across them $\mathbf{6.8951}$, a factor of $16.9324$. The same panel, the same composition, the same volume fraction, and only the direction of the load changed. And that ratio has a closed form——with $a=E_f$, $b=E_m$, $u=v_f$, $R(u)=u^2+(1-u)^2+u(1-u)(a/b+b/a)$, expanded and collected with no approximation. All three terms are invariant under $u\leftrightarrow 1-u$, so $R(u)=R(1-u)$ holds exactly. The agreement at $v_f=0.4$ and $0.6$ was therefore no accident: they are equal to eight decimals, $16.29508075$. A symmetric quadratic has its maximum in the middle——at $u=1/2$, $R_{\max}=\frac{1}{2}+\frac{1}{4}(a/b+b/a)=\mathbf{16.932376}$. The number read off the table came out of the algebra. As with the square-cube law in Paper 326, this is not a number obtained by measuring materials but one that follows from the definitions of two averages. And the two materials enter only through the single symmetric combination $a/b+b/a$——swapping $a$ and $b$ leaves it unchanged, so which one is called the fibre does not move the ratio at all. With $E_f=E_m$ the formula collapses to $(u+(1-u))^2=1$: mix a material with itself and direction cannot matter. For large $a/b$, $R_{\max}\approx a/(4b)$, so the range is a quarter of the stiffness ratio. One thing separates them——whether the two phases share strain or share stress. The separator is not on the side of the material——so “the Young’s modulus of this material” is an incomplete question: without a direction the answer swings by $16.932376$, and a word omissible for isotropic materials cannot be omitted here. To be said honestly——Voigt and Reuss are loose bounds, and for an isotropic two-phase material the Hashin-Shtrikman bounds are narrower. The $16.932376$ is the range spanned by two simple averages, not the range achievable by direction; measured values fall inside it without reaching either end. One last thing——the upper bound $116.7500$ is about half the fibre’s own $230$, which looks natural for a half-and-half mixture. But the lower bound $6.8951$ is not even twice the matrix’s own $3.5$. Placed in series the fibre barely works — the same quantity of carbon fibre is very nearly wasted, according to direction.
References
- Voigt, W. (1889). Ueber die Beziehung zwischen den beiden Elasticitätsconstanten isotroper Körper. Annalen der Physik, 274(12).
- Reuss, A. (1929). Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. Zeitschrift für Angewandte Mathematik und Mechanik, 9(1).
- Hill, R. (1952). The Elastic Behaviour of a Crystalline Aggregate. Proceedings of the Physical Society A, 65(5).
- Hashin, Z., Shtrikman, S. (1963). A Variational Approach to the Theory of the Elastic Behaviour of Multiphase Materials. Journal of the Mechanics and Physics of Solids, 11(2).
- Hull, D., Clyne, T. W. (1996). An Introduction to Composite Materials, 2nd edition. Cambridge: Cambridge University Press.
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). The Square-Cube “Law” Is Not a Law but an Identity [Paper 326]. Zenodo. 10.5281/zenodo.22209992
- Yamagishi, Y. (2026). That the Condition Number Squares Is an Identity, Not an Approximation [Paper 330]. Zenodo. 10.5281/zenodo.22681423
- Yamagishi, Y. (2026). Strength Is a Function of Volume, Not a Constant of the Material [Paper 344]. Zenodo. 10.5281/zenodo.22682232
Reference
Yamagishi, Y. (2026). Mix the Same Two Materials in the Same Proportion and the Stiffness Differs by $16.93$ [Paper 342]. YOI: https://www.yamagishiyuuki.com/yoi/342
DOI: 10.5281/zenodo.22682160
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 342. This paper is a preprint and has not been peer reviewed.