These modes are merely a man-made device to help organize unwieldy phenomena. In real life, cracks rarely appear in the three simple modes. Actual cracks are a complex combination of all three. |

**Yung S. Oh**

Consultant

O.M. Engineering Materials

Grayslake, Ill.

Parts break due to bad designs, material flaws, unexpected loads, manufacturing errors, and a host of other reasons that are often complicated and not fully understood. But getting to the root cause of a failure is critical if you want to avoid repeating the same mistake.

There are several guiding principles in failure analysis. Assuming the material is sound, parts often fail prematurely because applied loads exceed those expected for normal operation. Or normal operating loads are within spec, but impact loads induce forces large enough to break the part.

Shock and impulse loads are often overlooked, perhaps because they are not easy to calculate. Estimating applied kinetic energy is relatively straightforward but does not automatically provide the crucial impact force, which depends on impact duration. Impact duration is difficult to estimate, so there is often no choice but to experimentally measure impact forces.

After determining all the forces acting on a part, stress analysis can begin. Linear-static FEA is usually the first step. Many engineers routinely perform this fairly simple and painless analysis, and many vendors bundle CAD with FEA as a complete engineering tool. But FEA is by no means a panacea.

Linear-static FEA produces numbers and only numbers. They do not indicate whether or not a part will fail and are only good for the initial elastic range of the part. Therefore, it is often prudent to let analysts interpret data and draw meaningful conclusions. In fact, while vendors like to tout the features and user friendliness of their FEA programs, the most critical part of the analysis usually lies in the technical competence of the analyst.

As force increases material behavior changes from elastic to plastic, usually described by a material flow curve. Flow stress depends on strain, strain rate, temperature, and other factors. There are published curves for common engineering materials, although data on materials sensitive to high strain rate and temperature may not be readily available. Again, this information may have to be determined experimentally.

The J-integral characterizes fractures in ductile materials. |

Analyzing parts in the plastic deformation range requires nonlinear-material FEA. Nonlinear analysis involves careful consideration of mesh density, loading conditions, convergence errors, and so on. Sloppy model building may mean the FEA does not converge (no solution) or give bad results. In other words, nonlinear FEA is less forgiving than its linear counterpart. But nonlinear FEA is a must when designs involve part deformation under large loads. Once again, analysts should interpret results and draw conclusions.

FE analysis provides enough data to apply various fracture criteria. For example, designers often use Cockcroft-Latham criterion for chevron fracture in ductile materials. Other options include McClintock, Oh-Chen-Kobayashi, and Hofmanner fracture criteria.

However, physical flaws, such as a hairline crack in the part, require a completely different analysis method and make FEA irrelevant. This is because stress at the tip of the crack becomes infinite in linear analysis and extremely large in nonlinear FEA. Thus, stress becomes meaningless as a failure parameter. A different method -- fracture mechanics -- is needed.

Unfortunately, fracture mechanics has a reputation of being highly mathematical and too difficult for novices. But it is too important a tool to dismiss and should be part of every mechanical engineer's arsenal.

Fractured surfaces are complicated, but cracks can be simplified into three basic modes (See "Basic crack modes") that describe fracture and shear conditions. From this, one calculates stresses near the crack-tip region. For example, for a linear-elastic material, stress along the *X* axis in the tip region of Mode 1 is:

where = stress at a distance *r* and angle from the crack tip.

While the equation is complex, all the stresses in all three modes share this same form. They all have a constant (*K*_{I}, *K*_{II}, or *K*_{III}) and are a function of *r* and . Therefore, knowing the constant *K*, one can determine stresses in the crack region.

Remember *K* does not represent an obvious physical quantity like stress, but characterizes the general "state" of the stresses in the region. This is how fracture mechanics differs from conventional stress analysis. Values for *K*, termed the stress intensity factor, are available in fracture-mechanics reference literature for various geometries and load conditions.

This method assumes linear-elastic material behavior, but stresses at the tip of the crack exceed the elastic limit. So in actual applications the stress intensity factor is modified by a small plastic-correction factor added to the crack length. Various theories calculate the factors, such as Irwin, Dugdale, Barenblatt, and McClintock.

The *K* theory works well for brittle fracture -- mostly elastic behavior with little plastic deformation. Ductile materials, on the other hand, experience non-negligible plastic deformation before fracture. Thus, an analysis of ductile material uses the so-called *J*-integral defined as:

Here, *W* = strain-energy density where

and *T* = a traction vector, *u* = a displacement vector, and = a contour from the lower to the upper crack surface, surrounding the crack tip.

Like the stress intensity factor, *J* is defined by another complex equation. Without worrying too much about the technical details, two components are critical. First, the strain-energy density *W* contains data on a material's plastic behavior. (For linear-elastic material, *W* is simply the elastic potential.)

Second, the *J*-integral does not depend on any particular contour as long as it remains within the part domain. These two components make the *J*-integral a versatile tool, although the method does not work well for cyclic loading. Like *K*, *J* does not measure any obvious quantity, although some interpret it in terms of crack energy per unit length.

In practice, *K* and *J* are meaningless unless they relate to actual materials. Fortunately values have been experimentally established and are available in the literature. Note that experimental values at fracture are called critical values, denoted by *K _{c}* and

*J*.

_{c}

To use the theory in failure analysis, first calculate *K* or *J*. Normally, it is advisable to look up solutions in references because they are by no means trivial equations. Next, compare *K* or *J* with the critical values.

If *K*K_{c} or *J*J_{c}, there is likely little danger of fracture. But if *K* or *J* exceed the critical values, corrective measures may be necessary to prevent fracture.

Finally, don't forget to examine fractured surfaces under a microscope, especially for beach markings, ratchet lines, and nucleation sites. This additional information can be valuable for a better understanding of the fracture and perhaps a better choice of material.