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Saturday, October 24, 2009

Creep Failure

As a follow-up to the dispersive failure analysis I found a potential behavior that illustrates the early failure profile in a less abstract and perhaps more realistic manner.

Due to dispersion, a slow linear rate of growth in the breakdown process will lead to an early spike in the failure (or hazard) rate. This becomes the characteristic leading downward slope in the bathtub curve.

Figure 1: Linear into accelerating growth function
The enhanced early probability of failure arises purely through the spread in the failures through disorder mechanisms just as in the popcorn popping experiment. This in general leads to a bathtub shape with an analogy to the life-span of the human body (infant mortality, maturity, old age). For mechanical equipment, the shape in (b) provides a more realistic portrayal according to some.

Figure 1: Empirically observed bathtub curves
(a) electronics components (b) mechanical components

I earlier had abstracted away the velocity of the failure stimulus and integrity of the component to generate a parametric expression for the failure rate.
r(t) = dg(t)/dt / (tau + g(t))
In a real situation the velocity/integrity abstraction might occur as a stress/strain pairing particularly in a mechanical component.
We scale the integrity of the component as a physical dimension; it could be a formally defined measure such as strain, but we leave it as an abstract length for the sake of argument. The process acting on this abstraction becomes a velocity; again this could be a real force, such as the real measure of stress. [link]
In this case, a mechanism called creep can play a big role in determining the failure dynamics. Creep happens to a load under a constant stress condition over a period of time. This leads to a curve as shown below, which demonstrates a relatively quick rise in strain (i.e. deformation) before entering a linear regime and then an exponential as the final wear-out mechanism becomes too great.

Figure 3: Creep curve which physically realizes the growth function of Figure 1.

Although the stress/strain relationship can get quite complex, to first-order, we can compare the curve in Figure 3 to the general curve in Figure 1. The monotonic increase in the abstract growth term, g(t), remains the same in both cases, with both a linear and exponential regime noticeable in the middle (secondary) and late (tertiary) periods. The big difference lies in the early (primary) part of the curve, where due to elastic and plastic deformation (particularly in a viscoelastic material) the growth increases rapidly before settling into the linear regime. This fast "settling-in" regime intuitively provides a pathway to an earlier failure potential than a purely accumulating process would.

One can approximate the general trend in the primary part of the growth either by a rising damped exponential or by a parabolic/diffusive growth that rises with the n'th root of time. The following figure uses a combination of a square root and the exponential growth to model the creep growth:
g(t)= A*sqrt(kt) + B*(e-ct -1)

Figure 4: Modeled creep growth rate

Applying this growth rate to the dispersive failure rate, g(t), we get the following bathtub curve.

Figure 5: Bathtub curve for growth rate exhibiting physical creep has a sharper initial failure rate fall-off due to earlier deformation.

Clearly, the early part of the curve becomes accentuated relative to the linear growth mechanism, and a more asymmetric v-shape results, characteristic of the mechanical failure mode of Figure 1(b).

References:
1. The physics of creep: creep and creep-resistant alloys, F. Nabarro, H. De Villiers, CRC Press,1995.