= Stress concentration
{wiki}
A \b[stress concentration] is a localized increase of <stress> caused by geometry or another disturbance of the load path. Holes, notches and abrupt section changes can make the largest local <normal stress> much greater than a stress calculated from the total <force> divided by a reference <cross-sectional area>. A <stress concentration factor> $K_t=\sigma_{\max}/\sigma_{\mathrm{ref}}$ is meaningful only after both the stress component and its reference have been specified.
The <Kirsch equations> give an instructive exact result for a circular hole of radius $a$ in an infinite, <isotropic> <linear elastic> <elastic plate> under remote tensile <stress> $\sigma_\infty$ in one direction, using a <plane stress> model. Measure the <polar angle> $\theta$ from the loading direction. At the hole boundary,
$$
\sigma_{rr}=\sigma_{r\theta}=0,\qquad
\sigma_{\theta\theta}(a,\theta)=\sigma_\infty(1-2\cos2\theta).
$$
At $\theta=\pi/2,3\pi/2$, the tangential <normal stress> reaches $3\sigma_\infty$, giving \b[$K_t=3$ relative to remote stress]. At $\theta=0,\pi$, it is $-\sigma_\infty$. A traction-free hole therefore does not require every component of its <stress tensor> to vanish: it requires zero <traction> on the radial boundary.<Isotropic Linear Elastic Stress Concentration by MIT Mechanical Engineering>[§4.1, equations 1–3]{ref}
\Image[/circular-hole-stress-concentration.png]
{title=Tangential <stress concentration> at a traction-free circular hole}
{description=Original rendering of the exact infinite-plate boundary formula, normalized by positive remote tensile stress. The loading is in the horizontal direction. The highest tangential tension occurs at the top and bottom of the hole; the side points have tangential compression. This is an analytical model, not experimental data.}
For a finite plate, distinguish gross and net reference areas. A plate of width $W$, thickness $t$ and central hole diameter $d$ transmits <force> $P$ through gross area $A_g=tW$ and minimum net area $A_n=t(W-d)$. Defining $\sigma_g=P/A_g$ and $\sigma_n=P/A_n$ gives the exact conversion
$$
K_{tn}=\frac{\sigma_{\max}}{\sigma_n}
=K_{tg}\frac{A_n}{A_g}
=K_{tg}\left(1-\frac dW\right).
$$
This is a change of normalization, not a solution for the finite plate's stress field. For an original numerical example, take $W=40\,\mathrm{mm}$, $t=2\,\mathrm{mm}$, $d=10\,\mathrm{mm}$ and $P=800\,\mathrm N$. The reference stresses are $10$ and $13.333\,\mathrm{MPa}$. If the local maximum is stipulated as $32\,\mathrm{MPa}$, the same physical maximum gives $K_{tg}=3.2$ and $K_{tn}=2.4$. Neither number changes the local stress by choosing a different denominator.
The distinction appears in real historical measurements. In 1939, <C. Dumont> loaded a perforated aluminum-alloy plate in out-of-plane <bending> and measured <strain> and deflection at two loads. The reported peak tangential stress was 1.59 times the computed net-section stress, but 1.85 times the computed stress for an unperforated plate under the same bending forces. These are bending results for that geometry, rather than the infinite-plate tensile factor three.<Stress Concentration Around an Open Circular Hole in a Plate Subjected to Bending Normal to the Plane of the Plate by C. Dumont>[NTRS abstract; NACA-TN-740]{ref}
Modern demonstrations can combine a metal plate instrumented with <strain gauges> and a transparent polymer viewed in a <polariscope>. The latter uses stress-induced <birefringence>; its fringes respond to differences of <principal stresses>. The MIT laboratory compares local and remote strain measurements with qualitative <photoelasticity> observations around holes and a slot.<Isotropic Linear Elastic Stress Concentration by MIT Mechanical Engineering>[§§6–7, Figures 6–9 and strain-gauge procedure]{ref}
The elastic <stress concentration factor> is dimensionless. It differs from the <stress intensity factor> used to describe a sharp crack's singular field. Local yielding or a crack changes the mechanical problem; multiplying a remote stress by the elastic $K_t$ does not, by itself, establish a failure load.
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