Abstract
Corroding reinforcement is thought to weaken by growing thinner. This paper shows that what breaks the concrete is not the thinning but the swelling—a loss of $\mathbf{0.001000}$ mm cracks the cover. No new theorem or law is claimed.
First, a loss of $0.1$ mm removes $\mathbf{4.995132}$ mm$^2$ of iron per unit length (Section 2).
Second, at a volume ratio of $4$ that becomes $\mathbf{19.980529}$ mm$^2$ of rust, a gain of $\mathbf{14.985397}$ (Section 2).
Third, this is the core of the paper. Cracking $30$ mm of cover takes a loss of only $\mathbf{0.001000}$ mm (Section 3).
Fourth, at a volume ratio of $1$ the swelling is exactly $\mathbf{0.000000000000}$ mm (Section 2).
Fifth, at a ratio of $2.0$ the required loss is $0.003001$ mm and at $6.4$ it is $\mathbf{0.000556}$ (Section 3).
Sixth, the separator is whether the section shrinks or the volume grows (Section 4).
Keywords: corrosion of reinforcing steel, expansion of rust, cracking of concrete, concrete cover, Pilling-Bedworth ratio.
Scope (what is not claimed)
No new theorem or law is claimed—the volume expansion of iron corrosion products and the cracking of concrete cover are all standard. No structure is assessed—capacity, remaining life and the need for repair are not the subject; real judgement rests on professional survey. Corrosion rate is excluded—how many years $0.001$ mm takes is not the subject, nor are carbonation, chlorides and electrochemistry. The mechanics are simplified—strain is taken as swelling divided by cover thickness; in reality the stress field is three-dimensional and the hoop tension governs. The numbers are representative—volume ratios of $2$ to $6.4$ and a tensile failure strain of $100$ microstrain are representative, not measured. Relation to earlier papers: Paper 475 showed that nine tenths of what is added never reaches the interface—there too, the same act was useful or wasteful depending on the aim. Paper 300 showed that common root and distinct root can be decided—“loss of section” and “growth of volume” are distinct roots by that criterion. Paper 454 showed that a cubic load still needs only a linear width—there too, geometry settled the answer. What is added is tabulating the volume ratio of four corrosion products, giving the iron lost and the rust formed, giving the radial swelling as $0.292768$ mm, confirming exactly zero swelling at a ratio of $1$, solving for the loss that cracks the cover as $0.001000$ mm and inverting, and placing the separator at section versus volume.
1. Notation and the Question
Setting: When a bar of radius $r$ loses a thickness $\delta$, the iron lost per unit length is
$$V_{\mathrm{lost}}=\pi\bigl(r^{2}-(r-\delta)^{2}\bigr)$$With a corrosion product of volume ratio $\kappa$, the rust occupies $\kappa V_{\mathrm{lost}}$. Taking the rust to accumulate outward, the outer radius $R$ follows from
$$\pi\bigl(R^{2}-(r-\delta)^{2}\bigr)=\kappa V_{\mathrm{lost}}$$and the radial swelling is $R-r$. Below, $r=8$ mm, the cover is $30$ mm and the tensile failure strain of concrete is $100$ microstrain.
Observation 1.1 Corroding reinforcement is thought to weaken by growing thinner. Less section, less capacity—a natural argument.
Observation 1.2 But the rust does not disappear. It stays in the same place, occupying more—and pushes what surrounds it.
Question: Which arrives first.
2. Rust Swells Fourfold
Computation 2.1: Tabulate the volume ratio of four corrosion products relative to iron.
| Product | Volume ratio |
|---|---|
| $\mathrm{Fe}_{3}\mathrm{O}_{4}$ | $2.0000$ |
| $\mathrm{Fe}_{2}\mathrm{O}_{3}$ | $2.2000$ |
| $\mathrm{Fe(OH)}_{3}$ | $\mathbf{4.0000}$ |
| Hydrated hydroxide | $\mathbf{6.4000}$ |
Observation 2.2 Every one exceeds $1$. From $2.0$ to $6.4$—iron always swells when it rusts.
Computation 2.3: Take the volumes for a bar of radius $8$ mm losing $0.1$ mm at a ratio of $4$.
| Quantity | Value [mm$^2$] |
|---|---|
| Iron lost | $\mathbf{4.995132}$ |
| Rust formed | $\mathbf{19.980529}$ |
| Increase | $\mathbf{14.985397}$ |
Observation 2.4 The increase is three times the iron lost. $\kappa-1=3$—matching the closed form exactly.
Computation 2.5: Convert that increase into a radial swelling.
$\mathbf{0.292768}$ mm.
Observation 2.6 Larger than the $0.1$ mm that was lost. By a factor of $2.93$—far from growing thinner, the outside is growing.
Observation 2.7 At a volume ratio of $1$ the swelling is exactly $\mathbf{0.000000000000}$ mm. Rust of the same volume as the iron leaves the radius unchanged—a boundary, and a check on the expression.
3. Cracking Takes a Loss of 0.001000 mm
Computation 3.1: Solve by bisection for the loss at which swelling over cover reaches $100$ microstrain.
$\mathbf{0.001000}$ mm (inverting returns a strain of $1.000000\times10^{-4}$).
Observation 3.2 One micrometre. Invisible—a loss of section of only $0.025$ per cent.
Observation 3.3 At that point the capacity has hardly fallen. $0.025$ per cent of section gone—and the concrete is already cracked.
Computation 3.4: Vary the volume ratio and take the loss that cracks the cover.
| Volume ratio | Loss to crack [mm] |
|---|---|
| $2.0$ | $\mathbf{0.003001}$ |
| $4.0$ | $0.001000$ |
| $6.4$ | $\mathbf{0.000556}$ |
Observation 3.5 The larger the ratio, the less loss is needed. A factor of $5.40$ between $2.0$ and $6.4$—which product forms decides the life.
Observation 3.6 And once cracked, water and oxygen enter more easily. Corrosion accelerates and swells further—a positive feedback.
Observation 3.7 So inspection should look at cracks in the concrete, not thinning of the bar. The thinning cannot be measured until the end—the cracks appear far sooner.
4. The Separator
The separator: does the section shrink, or does the volume grow.
Observation 4.1 If the section shrinks, weakening follows the amount lost. $0.025$ per cent gives $0.025$ per cent—it proceeds slowly.
Observation 4.2 If the volume grows, what it pushes against breaks. The cover thickness sets the ceiling—and it arrives far sooner.
Observation 4.3 These are distinct roots by the criterion of Paper 300. Both are called “deterioration by corrosion of the reinforcement”—one a quantity of iron, one of iron paired with the cover.
Observation 4.4 This stands where Paper 454 stands. There, no answer came without looking at both load and resistance—here both the lost side and the gained side must be seen. In both, one side alone misleads.
Observation 4.5 It overlaps Paper 475 as well. There, the same act was useful or wasteful depending on the aim—here the same corrosion is trivial for capacity and fatal for the cover.
Observation 4.6 And thickening the cover raises the loss required. Because strain is swelling over cover—so the cover is not only a barrier against corrosion but room to absorb the swelling.
Observation 4.7 To be honest—the mechanics were simplified. Strain was taken as swelling over cover, whereas the real field is three-dimensional and governed by hoop tension—$0.001000$ mm depends on that; that swelling occurs whenever the volume ratio exceeds $1$ follows from geometry alone.
5. Roll Call and Closing
- Volume ratios of $2.0$ to $6.4$ (Computation 2.1): all exceed $1$.
- $4.995132$ mm$^2$ of iron lost (Computation 2.3): for $0.1$ mm.
- $19.980529$ mm$^2$ of rust (Computation 2.3): a gain of $14.985397$.
- The gain is three times the loss (Observation 2.4): exactly $\kappa-1$.
- A radial swelling of $0.292768$ mm (Computation 2.5): $2.93$ times the loss.
- Exactly zero at a ratio of $1$ (Observation 2.7): a boundary confirmed.
- Cracking at $0.001000$ mm (Computation 3.1): the core.
- A section loss of $0.025$ % (Observation 3.2): capacity nearly intact.
- A factor of $5.40$ with the ratio (Observation 3.5): the product sets the life.
- The separator is section or volume (Observation 4.3): distinct roots by Paper 300.
Closing: Corroding reinforcement is thought to weaken by growing thinner—less section, less capacity, a natural argument. But the rust does not disappear: it stays in the same place occupying more, and pushes what surrounds it. Rust swells—corrosion products run from $2.0$ to $6.4$ times the volume of the iron, all exceeding $1$: iron always swells when it rusts. A bar of radius $8$ mm losing $0.1$ mm turns $\mathbf{4.995132}$ mm$^2$ of iron into $\mathbf{19.980529}$ mm$^2$ of rust at a ratio of $4$, a gain of $\mathbf{14.985397}$, which is three times the iron lost and matches $\kappa-1$ exactly. The radial swelling is $\mathbf{0.292768}$ mm, $2.93$ times the $0.1$ mm lost: far from growing thinner, the outside is growing. At a volume ratio of $1$ the swelling is exactly $\mathbf{0.000000000000}$ mm, since rust of the same volume leaves the radius unchanged—a boundary and a check on the expression. And the cracking arrives astonishingly early—cracking $30$ mm of cover takes a loss of only $\mathbf{0.001000}$ mm, one micrometre, with inversion returning a strain of $1.000000\times10^{-4}$. That is a loss of section of only $0.025$ per cent, at which point the capacity has hardly fallen—and the concrete is already cracked. Varying the volume ratio, the required loss is $\mathbf{0.003001}$ mm at $2.0$ and $\mathbf{0.000556}$ at $6.4$, a factor of $5.40$: which product forms decides the life. And once cracked, water and oxygen enter more easily, corrosion accelerates and it swells further—a positive feedback. So inspection should look at cracks in the concrete and not thinning of the bar: the thinning cannot be measured until the end, while the cracks appear far sooner. One thing separates them—does the section shrink, or does the volume grow. If the section shrinks, weakening follows the amount lost and proceeds slowly. If the volume grows, what it pushes against breaks, the cover thickness sets the ceiling, and it arrives far sooner. They are distinct roots by the criterion of Paper 300, one a quantity of iron and one of iron paired with the cover. This stands where Paper 454 stands, where no answer came without looking at both load and resistance: in both, one side alone misleads. It overlaps Paper 475 too, where the same act was useful or wasteful depending on the aim: here the same corrosion is trivial for capacity and fatal for the cover. And thickening the cover raises the loss required, since strain is swelling over cover—so the cover is not only a barrier against corrosion but room to absorb the swelling. To be honest—the mechanics were simplified, with strain taken as swelling over cover where the real field is three-dimensional and governed by hoop tension. $0.001000$ mm depends on that; that swelling occurs whenever the volume ratio exceeds $1$ follows from geometry alone.
References
- Pilling, N. B., & Bedworth, R. E. (1923). The Oxidation of Metals at High Temperatures. Journal of the Institute of Metals.
- Tuutti, K. (1982). Corrosion of Steel in Concrete. Swedish Cement and Concrete Research Institute.
- Broomfield, J. P. (2007). Corrosion of Steel in Concrete (2nd ed.). Taylor & Francis.
- 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 Water Load Grows as the Square, and the Dam Only as the First Power [Paper 454]. Zenodo. 10.5281/zenodo.22732989
- Yamagishi, Y. (2026). Nine Tenths of What You Add Never Reaches the Interface [Paper 475]. Zenodo. 10.5281/zenodo.22733613
Reference
Yamagishi, Y. (2026). What Breaks It Is Not the Iron Lost but the Rust That Takes Its Place [Paper 476]. YOI: https://www.yamagishiyuuki.com/yoi/476
DOI: 10.5281/zenodo.22733641
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 476. This paper is a preprint and has not been peer reviewed.