FRC Team 224

We know that a material which contains a crack may experience a very high stress concentration factor at the crack tip, allowing the crack to propagate extremely rapidly under the right circumstances. We’d like to have some way of understanding the kinds of stresses at the crack tip and predicting when this will happen.

Let’s examine the geometry of a crack in a little more detail. We know that there can be very large stresses generated at a crack. In a ductile material, this means that there will be a zone around the crack where the stresses are high enough that the material behaves plastically, deforming in response to the stress. We’re going to make the assumption that this zone of deformation is small in comparison to the overall part. This allows us to analyze the situation using the methods we’ve developed for looking at stress and strain in elastic materials, since that’s how the bulk of the part will behave. If the zone of local yielding is not small, you need a more complicated approach that can take into account the nonlinear behavior of plastic materials.

(Image from Machine Design: An Integrated Approach, 4th Ed., by Robert L. Norton.)

We’re also going to make the assumption that we’re dealing with a crack in tension. Mode I in the image below is what we’re looking at.

(Image from Machine Design: An Integrated Approach, 4th Ed., by Robert L. Norton.)

Anyways, the stresses in the zone around the crack tip vary with distance, r, and angle, θ. If the material in the immediate vicinity of the crack were elastic, we could analyze it with the regular equations for stress and strain and get an expression in cylindrical coordinates to characterize the stress field.

However, since we’re dealing with plastic deformation, we need some way to account for this. We’ll do this by including a stress intensity factor, K, whose value depends on the geometry of the part and the crack and the nominal stress the part would be subjected to if the crack weren’t there. This is basically another kind of stress concentration factor. For example, for a plate in tension with a crack in the center, where the crack width (2a) is small compared to the width of the plate (2b), the stress intensity factor is:

If the crack width is pretty large compared to the width of the plate, or the part is more complicated than just a plate, there’s an additional factor, β, to correct for that. This isn’t something you’d have to calculate - like stress concentration factors, you’d get it from tabulated values in a handbook.

For every material, there is a critical value of K, K_c, the fracture toughness. If a crack produces a K greater than the fracture toughness, the crack will experience sudden, rapid propagation of the sort that generally results in catastrophic failure. Generally speaking, more ductile materials (that is, materials with a lower modulus of elasticity) will tend to have a higher fracture toughness (i.e. if cracked, it will take more stress for them to fail than it would for a brittle material with the same crack). This fracture toughness is what you use to calculate safety factor for cracks.