Question

7.7 A thin plate is rigidly fixed at its edges (see Figure Ex7.7). The plate has a height L and thickness t (normal to the plane of the figure). A crack moves from left to right through the plate. Every time the crack moves a distance Ax, two things happen: 1. Two new surfaces (with specific surface energy) are created. 2. The stress falls to zero behind the advancing crack front in a certain volume of the material. Obtain an expression for the critical stress necessary for crack propagation in this case. Explain the physical significance of this expression. Assume the stress, σ, ahead of the crack is uniform.

Answer 1

solution:

Strain energy per unit volume = $\frac{\sigma ^{2}}{2E}$

Total strain energy released when the crack moves a distance Δ At the same time as the strain energy is released due to the crack propagation, two new surfaces are created which result in an increase in surface energy equal to γ.2Axt. 2E If U represents the change in energy, then for crack propagation to occur, we must have OAr 2E Therefore, oAx 2E Or, the critical stress for crack propagation is 4Ey

Significance

The thinner is the plate (i.e., the smaller is the L), the larger is thestress necessary for crack propagation. The situation is akin to that obtained in anadhesive joining to two parts. The thinnest possible adhesive layer will lead to thestrongest possible bond.