## Stress Field Around A ^HOT^ Crack Tip

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In fracture mechanics, the stress intensity factor (K) is used to predict the stress state ("stress intensity") near the tip of a crack or notch caused by a remote load or residual stresses.[1] It is a theoretical construct usually applied to a homogeneous, linear elastic material and is useful for providing a failure criterion for brittle materials, and is a critical technique in the discipline of damage tolerance. The concept can also be applied to materials that exhibit small-scale yielding at a crack tip.

Linear elastic theory predicts that the stress distribution ( Ïƒ i j \displaystyle \sigma _ij ) near the crack tip, in polar coordinates ( r , Î¸ \displaystyle r,\theta ) with origin at the crack tip, has the form [4]

In 1957, G. Irwin found that the stresses around a crack could be expressed in terms of a scaling factor called the stress intensity factor. He found that a crack subjected to any arbitrary loading could be resolved into three types of linearly independent cracking modes.[6] These load types are categorized as Mode I, II, or III as shown in the figure. Mode I is an opening (tensile) mode where the crack surfaces move directly apart. Mode II is a sliding (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 a tearing (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.

Different subscripts are used to designate the stress intensity factor for the three different modes. The stress intensity factor for mode I is designated K I \displaystyle K_\rm I and applied to the crack opening mode. The mode II stress intensity factor, K I I \displaystyle K_\rm II , applies to the crack sliding mode and the mode III stress intensity factor, K I I I \displaystyle K_\rm III , applies to the tearing mode. These factors are formally defined as:[7]

The stress intensity factor, K \displaystyle K , is a parameter that amplifies the magnitude of the applied stress that includes the geometrical parameter Y \displaystyle Y (load type). Stress intensity in any mode situation is directly proportional to the applied load on the material. If a very sharp crack, or a V-notch can be made in a material, the minimum value of K I \displaystyle K_\mathrm 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 ( K I c \displaystyle K_\mathrm Ic ) of the material. K I c \displaystyle K_\mathrm Ic has units of stress times the root of a distance (e.g. MN/m3/2). The units of K I c \displaystyle K_\mathrm Ic imply that the fracture stress of the material must be reached over some critical distance in order for K I c \displaystyle K_\mathrm Ic to be reached and crack propagation to occur. The Mode I critical stress intensity factor, K I c \displaystyle K_\mathrm Ic , 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.

The stress intensity factor for an assumed straight crack of length 2 a \displaystyle 2a perpendicular to the loading direction, in an infinite plane, having a uniform stress field Ïƒ \displaystyle \sigma is [5][7]

For a slanted crack of length 2 a \displaystyle 2a in a biaxial stress field with stress Ïƒ \displaystyle \sigma in the y \displaystyle y -direction and Î± Ïƒ \displaystyle \alpha \sigma in the x \displaystyle x -direction, the stress intensity factors are [7][11]

Abstract:We investigated the evolution of the strain fields around a fatigued crack tip between the steady- and overloaded-fatigue conditions using a nondestructive neutron diffraction technique. The two fat