Stress intensity factor

Stress Intensity Factor, K, is used in fracture mechanics to more accurately predict the stress state ("stress intensity") near the tip of a crack caused by a remote load or residual stresses. It is a theoretical construct applicable to a homogeneous elastic material, It is useful for providing a failure criterion for brittle materials.

The magnitude of K depends on sample geometry, the size and location of the crack, and the magnitude and the modal distribution of loads on the material.

Linear elastic theory predicts a variation, for mode I, near the crack tip of the form

$\sigma _{ij}={\frac {K_{I}f_{ij}(\theta )}{\sqrt {(2\pi r)}}}$

This breaks down very close to the tip (small r), where plastic distortion typically occurs

Critical Stress Intensity

Mode-I, Mode-II, and Mode-III cracks.

Stress Intensity, K, is a parameter that amplifies the magnitude of the applied stress that includes the geometrical parameter Y (load type). These load types are categorized as Mode-I, -II, or -III. Stress intensity (K) in any mode situation is directly proportional to the applied load on the material. If a very sharp crack can be made in a material, the minimum value of K_I can be empirically determined, which is the critical value of stress intensity required to propagate the crack. This critical value determined for mode I loading in plane-strain is referred to as the critical fracture toughness (KIc) of the material. KIc has units of stress times the root of a distance. The units of KIc infer that the fracture stress of the material must be reached over some critical distance in order for KIc to be reached and crack propagation to occur. The Mode-I critical stress intensity factor, KIc is the most often used engineering design parameter in fracture mechanics and hence must be understood if we are to design fracture tolerant materials used in bridges, buildings, aircraft, or even bells. Polishing just won't do if we detect a crack. Typically for most materials if a crack can be seen it is very close to the critical stress state predicted by the "Stress Intensity Factor".

Mode I is opening or tensile mode where the crack surfaces move directly apart. Mode II is sliding or in-plane shear mode where the crack surfaces slide over one another in a direction perpendicular to the leading edge of the crack. Mode III is tearing and antiplane shear mode where the crack surfaces move relative to one another and parallel to the leading edge of the crack. Mode I is the most common load type encountered in engineering design.

Three different basic modes are shown in the figure.

Simple case

The stress intensity factor for a through crack of length 2a, at right angles, in an infinite plane, to a uniform stress field σ is

$K_{I}=\sigma {\sqrt {\pi a}}$

References

• A compendium of stress intensity factors for various load and specimen geometries is found in The Stress Analysis of Cracks Handbook by Hiroshi Tada, P. C. Paris, and George Rankine Irwin; American Society of Mechanical Engineers; 3rd edition (February, 2000).
• Software to calculate stress intensity is available in a program called pc-CRACK from Structural Integrity Associates