6.3 Discussion of the Elastostatic State Near the Crack Front
6.3.1 Field Quantities Through the Plate Thickness
Figures6.11, 6.12, and 6.13 show the variations of the stresses and strains through the plate thickness along the plane θ = 45° for different values ofr/t. The nine values ofr/t are: 0.022, 0.066, 0.110, 0.155, 0.199, 0.243, 0.287, 0.331, and 0.535.
Note that the through thickness shear strain distribution is exactly the same as the shear stress distribution shown in figure 6.12. The plots of the stresses show that the in-plane stresses σn and σ22 are almost constant through the plate thickness except for a sharp decrease nearthe plate surface. These stresses appear to increase without bound as the crack front is approached (see also sections 6.3.2 and 6.3.3). At the free surface the extrapolated values are unclear. The reduced amplitudes of the in-plane tensile stressesnear the free surface are consistentwith the experimentally observed tendency of cracks in plates to grow initially at the center, and are also in agreement with the decayin the energy release rate as the free surface is approached observed in [14]. The value of σ33 at the free surface (excluding the crack-surface intersection point) is seen to be its required value of zero. However, σ33 at other points through the plate thickness increases toward
the crack front, apparently without limit. These results seem to agree well with the finite element calculations (using singular elements) reported in [39]. Note that for rft ~ 0.5, <Tn and σ22 are constant and <733 is zero through the plate thickness, which indicates that generalized plane stress conditions exist at this distance from the crack front.
The plots ofthe shear stresses in figure 6.12 show that through the plate thickness, σ12 is almost constant apart from a variation near the free surface. It, too, appears toincrease without bound towards the crack front. The out-of-plane shear stresses, σ23 and σ3ι, have a value of zero at the center plane of the plate.
This is due to symmetry of the problem. These stresses also have a value of zero at the plate’s free surface (x3ft = 0.5), since this surface is traction free.
Between these two planes the behavior is somewhat complicated. The σ23 stress component may have an infinite spike at the crack-surface intersection. This is discussedfurther in section 6.3.3. The behavior of σ31 is even more complicated.
A mesh having a finer discretization near the crack-surface intersection would provide a more detailed description of the nature of the σ23 and U31 distribution in thatregion. It should be noted that both oftheseout-of-planeshear stresses are smallcompared to the normalstresses, and so are sometimesconsidered negligible through the plate’s thickness, as is stated in [39].
The plots of the normal strains shown in figure 6.13 show that eil and e22 behave similarly to σ11 and σ22 except for the curious increase in e11 before it decreasesnear the free surface. Both of these quantities are constant through the thickness at r/t ~ 0.5. The magnitude of e33 reaches a maximum value and then decreases slightly as the crack front is approached in the interior of the
plate. These values have not yet converged to a constant value. Closer tothe free surface, e33 increases as the crack front is approached, apparently without limit.
The observation that, at the crack front, e33 may have finite, but nonzero, values through the plate thickness means that a degree of plane strain (see ap pendix A) of one at the crack tip in the interior of the plate does not imply that plane strain conditions exist there. This will be important when the behavior of the degree of plane strain is examined near the crack front in section 6.3.4.
Figure 6.14 shows the through thickness variations of the normal stresses presentin the solution of the residual problemofplane strain (see appendix A) for the same values of r∣t used in figures 6.11, 6.12, and 6.13. The residual problem can be thought of as containing all the three dimensional effects present in the original cracked plate problem. The in-plane normal stresses, σ"1 and σ22, appear to increase without limit toward the crack front (see also section 6.3.2), and, interestingly, the through thickness distribution changes sign as the free surface is approached. In the interior of the plate, the normal in-plane stresses of the residual problem are tensile, whereas near the surface, they becomecompressive.
This is again consistent with the experimentally observed tendency of cracks to grow in the center (the tunneling effect). As is evident from figure 6.14, the changes in sign of σ"1 and σ22 occur at different values of r3∕f that depend on r∕t. Possibly, the crossover point approaches the free surface as the crack front is neared, producinga complicated singularity.
The other normal stress, σ33, should become unbounded in tension in the plate’s interior at the crack front since σ"1 and σ22 appear to be unbounded in tension in this region and e"3 (which is the same as e33) appears to be finite,
although this behavior of σ"3 is not yet obvious in figure 6.14. At the free surface on the crack front, σ33 is unbounded in compression, as is required bythetraction boundary condition of the residual problem.
An important point that should be stressed when examining the solution to the residual problem of plane strain is that the results shown in figure 6.14 indicate that three dimensionaleffects are present at the crack front in the original problem, even near the center plane of the plate.