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# Fracture Mechanics: Part Three

Abstract:

Linear elastic fracture mechanics (LEFM) principles are used to relate the stress magnitude and distribution near the crack tip to (1) remote stresses applied to the cracked component, (2) the crack size and shape, and (3) the material properties of the cracked component.

### LEFM Assumptions

Linear elastic fracture mechanics (LEFM) is based on the application of the theory of elasticity to bodies containing cracks or defects. The assumptions used in elasticity are also inherent in the theory of LEFM: small displacements and general linearity between stresses and strains.

In the general form of the LEFM equations a singularity exists such that as r, the distance from the crack tip, tends toward zero, the stresses go to infinity. Since materials plastically deform as the yield stress is exceeded, a plastic zone will form near the crack tip.

The basis of LEFM remains valid, though, if this region of plasticity remains small in relation to the overall dimensions of the crack and cracked body.

There are generally three modes of loading, which involve different crack surface displacements (see Fig. 3). The three modes are:
 Mode 1: opening or tensile mode (the crack faces are pulled apart) Mode 2: sliding or in-plane shear (the crack surfaces slide over each other) Mode 3: tearing or anti-plane shear (the crack surfaces move parallel to the leading edge of the crack and relative to each other)

The following discussion deals with Mode 1 since this is the predominant loading mode in most engineering applications. Similar treatments can readily be extended to Modes 2 and 3.

### Stress Intensity Factor

The stress intensity factor, K, which was introduced in Eq. 1, defines the magnitude of the local stresses around the crack tip. This factor depends on loading, crack size, crack shape, and geometric boundaries, with the general form given by

 (2)

where:

s = remote stress applied to component (not to be confused with the local stresses, sij, in Eq. 1)
a = crack length
f (a/w) = correction factor that depends on specimen and crack geometry

Figure 4 gives the stress intensity relationships for a few of the more common loading conditions. Stress intensity factors for a single loading mode can be added algebraically. Consequently, stress intensity factors for complex loading conditions of the same mode can be determined from the superposition of simpler results, such as those readily obtainable from handbooks.

 Figure 4. Stress intensity factor for (a) Center-cracked plate loaded in tension, (b) Edge-cracked plate loaded in tension, (c) Double-edge-cracked plate loaded in tension (d) Cracked beam in pure bending

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Total Materia Extended Range includes the largest database of fracture mechanics parameters for hundreds of metal alloys and heat treatments conditions. K1C, KC, crack growth and Paris law parameters are given, with the corresponding graph of crack growth.

Monotonic properties are added for the reference, as well as estimates of missing parameters based on monotonic properties where applicable.

Enter the material of interest into the quick search field. You can optionally narrow your search by specifying the country/standard of choice in the designated field and click Search.

After clicking the material from the resulting list, a list of subgroups that are standard specifications appears.

Because Total Materia Extended Range fracture mechanics parameters are neutral to standard specifications, you can review fracture mechanics data by clicking the appropriate link for any of the subgroups.

The data are given in a tabular format, with the Paris curve (Region II) where applicable. Explicit references to the data sources are given for each dataset.

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