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# Stress Field Around A Crack Tip REPACK

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. 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.

## Stress Field Around A Crack Tip

Linear elastic theory predicts that the stress distribution ( σ i j \displaystyle \sigma _ij ) near the crack tip, in polar coordinates ( r , θ \displaystyle r,\theta ) with origin at the crack tip, has the form 

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. 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:

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 σ \displaystyle \sigma is 

For a slanted crack of length 2 a \displaystyle 2a in a biaxial stress field with stress σ \displaystyle \sigma in the y \displaystyle y -direction and α σ \displaystyle \alpha \sigma in the x \displaystyle x -direction, the stress intensity factors are 

Fracture mechanics is a methodology that is used to predict and diagnose failure of a part with an existing crack or flaw. The presence of a crack in a part magnifies the stress in the vicinity of the crack and may result in failure prior to that predicted using traditional strength-of-materials methods.

In fracture mechanics, a stress intensity factor is calculated as a function of applied stress, crack size, and part geometry. Failure occurs once the stress intensity factor exceeds the material's fracture toughness. At this point the crack will grow in a rapid and unstable manner until fracture.

The image below shows the SS Schenectady tanker, one of the World War II Liberty Ships and one of the most iconic fracture failures. The Liberty ships all had a tendency to crack during cold weather and rough seas, and multiple ships were lost. Approximately half of the cracks initiated at the corners of the square hatch covers which acted as stress risers. The SS Schenectady split in two while sitting at dock. An understanding of fracture mechanics would have prevented these losses.

As the radius of the crack tip approaches zero, the theoretical stress approaches infinity. This infinite stress is known as a stress singularity and is not physically possible. Instead, the stress distributes over the surrounding material, resulting in plastic deformation in the material at some distance from the crack tip. This region of plastic deformation is called the plastic zone and is discussed in a later section. The plastic deformation causes blunting of the crack tip which increases the radius of curvature and brings the stresses back to finite levels.

Because of the stress singularity issues that arise when using the stress concentration approach, and because of the plastic zone that develops around the crack tip which renders the stress concentration approach invalid, other methods have been developed for characterizing the stresses near the tip of the crack. The most prevalent method in use today is to calculate a stress intensity factor, as discussed in a later section.

The figure above shows the three primary modes of crack loading. Mode I is called the opening mode and involves a tensile stress pulling the crack faces apart. Mode II is the sliding mode and involves a shear stress sliding the crack faces in the direction parallel to the primary crack dimension. Mode III is the tearing mode and involves a shear stress sliding the crack faces in the direction perpendicular to the primary crack dimension.

For Mode I loading, the linear-elastic stresses in the direction of applied loading near an ideally sharp crack tip can be calculated as a function of the location with respect to the crack tip expressed in polar coordinates:

The stress intensity factor for a Mode I crack is written as K I. (From this point forward, it is assumed that all stress intensity factors are Mode I for reasons discussed previously, so the stress intensity will be denoted simply as K. Using the equation for the stress intensity factor, the original equation for stress near the ideally sharp crack tip can be re-written as:

It is important to note that because equations describing the linear-elastic stress field were used to develop the stress intensity factor relationship above, the concept of the stress intensity factor is only valid if the region of plastic deformation near the crack tip is small. This will be discussed in more detail in a later section.

The difficult part of calculating the stress intensity factor for a specific situation is finding the appropriate value of the dimensionless geometry factor, Y. This geometry factor is dependent on the geometry of the crack, the geometry of the part, and the loading configuration. A classic case is plate with a crack through the center, as shown below:

Because the concept of the stress intensity factor assumes linear elastic material behavior, the stress intensity factor solutions can be combined by superposition to find solutions to more complex problems. For example, the stress intensity factor solution for a single edge cracked plate in tension can be combined with the solution for a single edge cracked plate in bending, as shown in the figure below.

where σt is the applied tensile stress, σb is the applied bending stress, Yt is the geometry factor for the plate in tension, Yb is the geometry factor for the plate in bending, and a is the crack length.

A material can resist applied stress intensity up to a certain critical value above which the crack will grow in an unstable manner and failure will occur. This critical stress intensity is the fracture toughness of the material. The fracture toughness of a material is dependent on many factors including environmental temperature, environmental composition (e.g., air, fresh water, salt water, etc.), loading rate, material thickness, material processing, and crack orientation to grain direction. It is important to keep these factors in mind when selecting a fracture toughness value to assume during design and analysis.

Cracks and crack-like flaws are common in engineering materials. Cracks will typically form around pre-existing flaws which act as stress concentrations and which, upon high stress or fatigue, develop into full-fledged cracks. Many flaws are serious enough that they should be treated as cracks, and these include deep scratches, inclusions of foreign particles, and grain boundaries. In addition to material flaws, geometric features in a part which act as stress concentrations can lead to crack initiation, including notches, holes, grooves, and threads. Cracks can also initiate from flaws introduced through other failure mechanisms, such as from pitting due to corrosion or from abrasion due to galling. 350c69d7ab