In the last article, we talked about fracture mechanics, and the method of fracture toughness testing using fracture mechanics. In this article, we are going to discuss how fracture toughness can be used to solve fracture and fatigue problems.

In traditional fatigue analysis, stress concentration factors are used to calculate fatigue life. These stress concentration factors are holes or notches that locally increase the stress. In fracture mechanics, the stress intensity factor is calculated for a given crack and compared to the critical value, K_{Ic}, critical stress intensity factor. If the stress intensity factor is larger than the critical stress intensity factor, then the crack will propagate catastrophically. A comparison of traditional fatigue analysis and fracture mechanics is shown in Table 1.

From Paris [2], the propagation of a crack follows the basic form of:

where C and N are material constants, and

da/dN is the crack growth in mm/cycle. The values of C and N can vary due to heat treatment and condition of material, in general, so equations that are accepted [3]

For martensitic steels, this is:

For ferrite-pearlite steels:

And for aluminum [4]:

An example of a crack growth rate curve is shown in Figure 1 [5].

These basic equations can be used to calculate fatigue crack growth from some basic flaw size a to the critical flaw size ac, where catastrophic crack propagation can occur. The first thing that is needed is an understanding and relationship for the stress intensity factor, K.

In static loading, the stress intensity factor for a small crack can be expressed as a function of the stress, crack length a, and geometry, β:

In theory, if K_{I} is less than K_{Ic} the crack will not propagate. If K_{I} is equal to K_{Ic} the crack will propagate catastrophically. This can be used to calculate a critical flaw size (discussed later). The calculation of the stress intensity factor K is usually done currently using finite element analysis. Compilations of stress intensity factors are available in texts on fracture mechanics [6] or in numerous different handbooks [7].

The basic method used for calculating fatigue life of a component consists of the following:

- Find stress intensity factor for the geometry of interest.
- Determine critical flaw size where K
_{max}= K_{Ic }. - Verify that the linear elastic fracture mechanics (LEFM) requirements are fulfilled.
- Integrate the Paris law with the integration bounds from the assumed or detectible flaw size to the critical flaw size.
- Solve for the number of stress cycles until failure.

In this example, we have a pearlitic steel, with a yield strength of 165 ksi, with a fracture toughness, K_{Ic} of 65 ksi-√in. An infinite plate with uniform axial loading (Figure 2) is loaded to 80 ksi, then loading released back to zero. From non-destructive testing, the smallest flaw that the technique can detect is 0.040” in diameter, or a = 0.020”.

### Determination of the Critical Flaw Size

The first thing necessary to determine the Critical Flaw Size, a_{cr }, is to remember that [8]:

Rearranging:

At this loading, a flaw equal to the critical flaw size, a_{cr }, will result (in theory) of the material failing catastrophically.

Stress intensity factor will change as the geometry and type of loading changes. The specific condition should be evaluated using the proper geometry and load condition [9].

### Determination of fatigue life

From Paris [2], the crack life relationship or Paris Law, is:

Rearranging:

Since this is a pearlitic steel, for this example, c = 3.6 x 10^{-10} and n = 3.0. The exact coefficients should be verified for the alloy and loading conditions. Many of the materials are tabulated in [5]. The stress intensity factor for this loading is:

Substituting and integrating, with a_{cr} (0.21”) as the upper bound and the lower bound as the smallest detectible flaw size a (0.020”), the following integral is established:

There will be an incubation time that is dependent on the material, as well as some plasticity at the crack tip. However, this method is used extensively in aerospace to determine inspection cycles, and to establish life of a component.

### Conclusions

In this brief article, some basic principles of fracture mechanics were illustrated, as well as how fracture mechanics and fracture toughness measurements can be used directly in design. This is an enormously useful tool for determining the life of a component.

It is hoped that this series of articles on the application of mechanical testing has been informative and useful. In the next series of articles, we will be discussing the various non-destructive testing techniques to detect flaws. Should you have any comments or questions regarding this article, or have suggestions for further articles, please contact the writer or editor.

### References

- G. Jacoby, “Application of Microfratography to the Study of Crack Propagation Under Fatigue Stress,” 1966.
- P. C. Paris and F. Erogan, “A Critical Analysis of Crack Propagation Laws,” Journal of Basic Engineering, Transactions of the American Society of Mechanical Engineers, pp. 528-534, 1963.
- S. T. Rolfe and J. M. Barsom, Fatigue and Fracture Control in Structures, Englewood Cliffs, NJ: Prentice-Hall, 1977.
- N. E. Frost, K. J. Marsh and L. P. Pook, Metal Fatigue, London, England: Oxford University Press, 1974.
- Department of Defense, Metallic Materials and Elements for Aerospace Vehicle Structures, Department for Defense Handbook, 2003.
- T. L. Anderson, Fracture Mechanics Fundamentals and Applications, Boca Raton FL: CRC Press, 1995.
- H. Tada, P. C. Paris and G. R. Irwin, The Stress Analysis of Cracks Handbook, New York: ASME Press, 2000.
- G. R. Irwin, “Fracture,” in Encyclopedia of Physics, vol. VI, Heidelberg, Springer, 1958, p. 561.
- J. W. Faulpel and F. E. Fisher, Engineering Design, New York: Wiley, 1981.