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A crack is an internal separation surface, or a curve in a two-dimensional reduction, across which a displacement can have distinct traces. In the ideal traction-free model its two faces carry zero imposed traction. A crack tip or front terminates this separation and can produce a singular stress field. The stress intensity factor measures the leading singular amplitude, while the energy release rate measures the reduction in equilibrium potential energy per increment of crack extent.

David Roylance developed an engineering treatment of fracture mechanics connecting cracks, loading and a material's resistance to failure. His notes derive an energy-balance description and introduce the stress intensity factor as a measure of crack-tip loading.^[ref]

In a simple linear-elastic mode-I geometry, $K_I=Y\sigma \sqrt{\pi a}$, with crack scale $a$ and dimensionless geometry coefficient $Y$. Comparing $K_I$ with the relevant fracture toughness shows why a nominally moderate bulk stress can be critical when a sufficiently large flaw is present. The geometry and elastic regime are essential assumptions.

Use antiplane energy $\frac{\mu }{2}\int |\nabla u{|}^2$ and normalize the tip singularity as $u=k\sqrt{r}\sin (\theta /2)$ plus a less singular term. The energy release rate is the small-circle flux $\int [\mathcal{W}{n}_x-\mu u_r{u}_x]rd\theta$. Here $\mathcal{W}=\mu k^2/(8r)$, $u_r=k\sin (\theta /2)/(2\sqrt{r})$ and $u_x=-k\sin (\theta /2)/(2\sqrt{r})$. Integrating over $(-\pi ,\pi )$ gives the displayed coefficient. The full three-dimensional antiplane stress has squared Frobenius norm $2{\mu }^2|\nabla u{|}^2$, so its leading norm ahead of the tip is $\mu |k|/\sqrt{2r}$. These constants depend on the normalization of the stress intensity factor.

Fracture mechanics studies the initiation and growth of cracks under loading. It connects a defect's geometry with the local driving force for crack extension and the material's resistance. A component's average stress alone cannot describe the concentration of deformation near a crack tip.^[ref]

In linear elastic fracture mechanics, a mode-I stress intensity factor often has the form $K_I=Y\sigma \sqrt{\pi a}$, where $a$ is a specified crack dimension and $Y$ accounts for geometry. Doubling $a$ increases $K_I$ by $\sqrt{2}$ at fixed nominal stress. Comparing the appropriate intensity with measured fracture toughness is meaningful only when the loading, geometry and small-scale yielding assumptions of the model apply.

Fracture toughness measures resistance to crack advance under specified loading and environmental conditions. In linear elasticity, $K_c$ is the critical stress intensity factor. For opening-mode loading, a valid plane strain measurement is denoted $K_{Ic}$, with units $\mathrm{Pa}\sqrt{\mathrm{m}}$, commonly $\mathrm{MPa}\sqrt{\mathrm{m}}$. It differs from a tensile yield strength: a crack changes the near-tip stress concentration.

For a fixed geometry with $K_I=Y\sigma \sqrt{\pi a}$, the threshold $K_I=K_{Ic}$ givesFor example, $K_{Ic}=10\mathrm{MPa}\sqrt{\mathrm{m}}$, $\sigma =100\mathrm{MPa}$ and $Y=1$ give $a_c\simeq 3.18\mathrm{mm}$ in this idealized geometry. The crack-tip plastic zone must remain small for the elastic intensity description to apply; extensive plasticity in continuum mechanics requires a different characterization.^[ref]

A thin elastic plate model may instead prescribe a critical front curvature ${\kappa }_f$. Such a model parameter cannot be substituted numerically for $K_{Ic}$ without a relation specific to that fracture model.

Linear elastic fracture mechanics predicts the loading of a crack using linear elasticity outside a small region near its tip. The near-tip stress field has a characteristic $r^{-1/2}$ dependence; its amplitude is the stress intensity factor. For an opening crack under nominal tensile stress $\sigma$,where $a$ is a specified crack-length parameter and $Y$ a dimensionless geometry factor. A central crack in an effectively infinite plate has total length $2a$ and $Y=1$. Crack advance is assessed by comparing $K_I$ with the appropriate measured fracture toughness.^[ref]

For $a=1\mathrm{mm}$ and $\sigma =100\mathrm{MPa}$ in that ideal geometry, $K_I\simeq 5.60\mathrm{MPa}\sqrt{\mathrm{m}}$. Increasing the crack's half-length fourfold doubles $K_I$ without changing the applied stress. This explains why a structure can fail at a stress below the strength of an uncracked specimen.

A toughness measurement loads a precracked specimen of known geometry and records the force associated with crack extension. Thickness, remaining ligament and crack length must be large relative to the crack-tip plastic region before the plane-strain value $K_{Ic}$ is transferable in this model. Large-scale plastic deformation requires a different description; the elastic singularity is a useful outer approximation, not an assertion of physically infinite stress inside the material.^[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.