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The Kirsch equations describe the stress tensor around a traction-free circular hole in an infinite linear elastic plate under uniform remote tension. For remote stress ${\sigma }_{\infty }$ along $x$, hole radius $a$ and polar coordinates $(r,\theta )$ measured from that direction, put $q=a^2/r^2$. In the classical isotropic plane stress model,The fields satisfy equilibrium and the prescribed boundary tractions. They can be obtained using an Airy stress function satisfying the biharmonic compatibility equation.^[ref]

At $r=a$, $q=1$, so ${\sigma }_{rr}={\sigma }_{r\theta }=0$ whileThe tangential stress is $3{\sigma }_{\infty }$ at the top and bottom of the hole and $-{\sigma }_{\infty }$ at its sides. A traction-free boundary does not require every stress component to vanish: the outward boundary normal is radial, so radial normal and shear tractions must vanish.

For an original distance comparison along $\theta =\pi /2$, the tangential stress is ${\sigma }_{\infty }(1+q/2+3q^2/2)$. At $r=2a$ it is $1.21875{\sigma }_{\infty }$, already far below the edge value three. At large distance the polar components approach those of the original uniform Cartesian tension.

A finite-width plate, internal hole pressure, plasticity or out-of-plane bending changes the boundary-value problem. The stress concentration factor three is therefore specific to the stated remote-tension model. The MIT laboratory uses local strain measurements and photoelasticity to examine how actual geometries compare with ideal concentration fields.^[ref]

A 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,At $\theta =\pi /2,3\pi /2$, the tangential normal stress reaches $3{\sigma }_{\infty }$, giving $K_t=3$ 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.^[ref]

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 conversionThis 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.^[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.^[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.

A stress concentration factor compares a specified local peak stress with a defined nominal reference:It is dimensionless. Both the stress component and the reference geometry belong in its definition; a number quoted without them can be ambiguous. In the infinite circular-hole Kirsch equations, the maximum tangential tension is three times the remote uniaxial stress, so that particular reference gives $K_t=3$.^[ref]

For an original finite-plate normalization example, let width $W=50\mathrm{mm}$, thickness $t=2\mathrm{mm}$, central-hole diameter $d=10\mathrm{mm}$ and tensile force $P=1000\mathrm{N}$. The gross and minimum-net reference stresses areIf the local peak is stipulated as $28\mathrm{MPa}$, the same specimen has $K_{tg}=2.8$ and $K_{tn}=2.24$. In general $K_{tn}=K_{tg}(1-d/W)$ for these reference definitions. This conversion does not solve the finite plate's stress field or predict its peak.

The reference can also be a nominal bending stress or another stress appropriate to the applied loading. A factor obtained for axial tension cannot automatically be reused for a bending problem with different boundary conditions.

An elastic concentration factor is distinct from the crack-tip stress intensity factor, which has units of stress times square root of length. Nor does $K_t$ by itself supply fatigue life or a failure load: yielding, defects, residual stress and the loading history require additional analysis. The factor describes amplification within its stated elastic model.