Solution
Stress corrosion cracking (SCC) is another sub-critical crack growth mechanism which is very suitable for LEFM treatment, because applied stress intensity values are usually low in this cracking regime. One of the problems with SCC is that crack growth velocities increase very sharply with increase in K, and the lifetime may be correspondingly short. Hence, if the combination of likely defect size and applied, or residual, tensile stress cause the threshold for stress corrosion cracking (K1SCC) to be exceeded (see the theory box), it is usually necessary to avoid the possibility of SCC. This is likely to require either a change in alloy, or surface protection. Nonetheless, it is useful to illustrate the application of fracture mechanics to SCC lifetime determination.
This first problem is straightforward and shows a typical application that might arise in, for example, failure analysis, although the question is phrased as a manufacturer's assessment of life. It should take perhaps 20 minutes to complete.
A glass shelf in a bookcase can be considered as simply supported beam subject to a uniformly distributed load. A particular manufacturer produces bookcases with shelves that are 1.5 m in length (L), have a width (b) of 200 mm and a thickness (t) of 10 mm. The design load on the shelves is estimated as w = 100 N/m2.
The manufacturer knows that handling of the glass shelves during fabrication may cause surface flaws to be present in them, which can be considered as semi-elliptic in shape with a maximum depth of 0.1 mm. He is also aware of the fact that a moist air environment may cause stress corrosion cracking to occur in stressed glass. He has therefore approached you, as a fracture mechanics/failure consultant to advise him whether he should offer a one year guarantee on these bookcases.
In your calculation of the lifetime of a typical shelf under these conditions, you may assume no incubation period is required before the crack extends. You have access to the following information:
1. The maximum bending stress in a simply supported beam subject to a distributed load is given by:
2. The shelves are to be made from soda glass with a modulus of elasticity E = 70 GPa and a work to fracture R = 0.01 kJ/m2.& A crack velocity (v-K) curve for soda glass in a moist air environment is given below.
3. The stress intensity factor for a semi-elliptic surface defect can be calculated from:
where a is maximum depth of defect.
Answer: Nf The crack growth lifetime of a shelf is about 167 days.
Solution of this problem requires substitution of the value of the fracture stress (which is equal to the applied stress) into the stress intensity equation. This will enable us to find the critical defect size and hence we can calculate the initial and final values of K which provide limits on the integration of the v-K equation, i.e.:
We can then express crack velocity as da/dt = (da/dK).(dK/dt) separate variables and hence obtain the time to failure from integrating with respect to time t.
The given information includes work to fracture R and the elastic modulus E, which implies that the Griffith equation should be used to calculate the critical crack size causing fracture. The applied stress is found by rearranging the bending equation and substituting in for M and I:
Rearranging the Griffith equation gives the critical crack size:
Hence we can now calculate values for Ki and Kf.
To find the constants in the v-K curve is quite simple, as the plot is linear on a log-log scale, i.e.:
Taking two points on the curve, e.g. 10 -2 m/s where K = 0.68 MPa m1/2, and 10-8 m/s where K = 0.29 MPa m1/2, gives n = 16.21. The constant D can be obtained by substituting into the v-K equation.
Checking this value with the other chosen point confirms that it is correct, and the v-K equation is therefore:
It is useful to perform the integration symbolically, as this facilitates finding errors, and substituting actual values into the final form.
Integrating this with respect to t gives:
As this is less than 6 months, the one year guarantee does not seem like a good option. In reality, there would be an incubation period before crack growth started and the shelves may well last the required period of time. Nonetheless, a prudent manufacturer might look to reducing stresses through, for example, shorter shelves or reducing the initial defect size by polishing the shelves! It should also be noted that that typical scratches, although around 0.1 mm in depth would not act like sharp cracks, which is why glass shelves can be routinely found in bookcases.
