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Published equation contexts

σmax⁡=σ∞(1+2aρ)\sigma_{\max} = \sigma_\infty\left(1 + 2\sqrt{\frac{a}{\rho}}\right)

Why this formula appears here

The first half of the argument is purely elastic and predates Griffith. An elliptical hole in a stressed plate does not merely remove load-bearing material; it redistributes the stress that would have passed through that material into the region around the hole’s sharpest curvature. For an elliptical flaw of half-length a and tip radius of curvature ρ\rho in a plate under remote tension σ∞\sigma_\infty , the peak stress at the tip is approximately σmax⁡=σ∞(1+2aρ)\sigma_{\max} = \sigma_\infty\left(1 + 2\sqrt{\frac{a}{\rho}}\right). Three things follow. First, the concentration depends on the flaw’s aspect ratio, not on its absolute size — a long shallow scratch and a short sharp one can concentrate stress equally. Second, as the tip sharpens toward an…

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σmax⁡\sigma_{\max}

Symbol sigma_max

sigmama_max is part of the quantity the equation computes from the expression on the right.

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σ∞\sigma_\infty

Symbol sigma_infty

sigmaia_infty is one of the signed contributions combined to compute the quantity on the left.

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ρ\rho

Symbol ρ

ρ occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

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σmax⁡=σ∞(1+2aρ).\sigma_{\max} = \sigma_\infty\left(1 + 2\sqrt{\frac{a}{\rho}}\right).

Equation 4 · Materials Science

Why Materials Fail: Fracture, Flaws and the Inspection Interval

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

The first half of the argument is purely elastic and predates Griffith. An elliptical hole in a stressed plate does not merely remove load-bearing material; it redistributes the stress that would have passed through that material into the region around the hole’s sharpest curvature. For an elliptical flaw of half-length a and tip radius of curvature ρ\rho in a plate under remote tension σ∞\sigma_\infty , the peak stress at the tip is approximately σmax⁡=σ∞(1+2aρ)\sigma_{\max} = \sigma_\infty\left(1 + 2\sqrt{\frac{a}{\rho}}\right). Three things follow. First, the concentration depends on the flaw’s aspect ratio, not on its absolute size — a long shallow scratch and a short sharp one can concentrate stress equally. Second, as the tip sharpens toward an…

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