Solution (Problem 4)
This question should take about 15 minutes to complete. It requires some thought about the specimen dimensions used in calculating bending stresses.
A rectangular perspex plate 600 mm by 300 mm by 6 mm thick is scribed into two equal squares by a knife, leaving a uniform cut of depth 0.3 mm.
What is the bending moment required to break the plate if the perspex has a work to fracture of 500 J/m2?
Note that E = 2.5 GPa for perspex.
Answer: 59.2 Nm
We have not been provided with a value for Poisson's ratio, so it is reasonable to assume plane stress conditions here, even though the plate is fairly thick and perspex is moderately brittle at ambient temperatures. This problem has two stages to the solution, firstly to calculate the Griffith fracture stress and, secondly, to find the bending moment that corresponds to this. Note that this
technique is often used in practice to fracture brittle and quasi-brittle materials, e.g. glass, tiles and polymers.
In calculating the answer to this question one has to make an assumption about the type of bend loading, i.e. whether it is three- or four-point bend. The load type impacts on calculation of nominal applied stress. In four-point bend, the stress is nominally uniform over the central portion of the plate between the two inner loading rollers. In this case, it seems reasonable to calculate the applied bending stress using the full depth of the plate before making the incision. The cut is then acting as a crack, and its effect should be taken into account in the Griffith's equation. However, in practice, if one was breaking this plate, it is likely that it would be placed with the cut at the edge of a solid surface.
This loads the plate in cantilever bend, and it seems more reasonable to calculate the stress using the reduced cross-section taking account of the depth of the cut. This is what has been done in the present case.
Recalling Griffith's equation as:
and noting that this is an edge crack, i.e. a = 0.3 mm, we can substitute in the values to get:
We can find the required bending moment from the simple bend equation:
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