#### Installation — business terrible - 1 part

September 8th, 2015

Consider the center cracked panel of Fig. 3.2.2a subjected to remote stress r;,x – perpendicular to the crack plane, and assume that the crack length is much less than the remaining dimensions of the plate. The principal stress trajectories in Fig. 2.1.1a reveal that the formation of a crack causes stress relief in the shaded, approximately triangular regions next to the crack. As an approximation, one may suppose the stress relief region to be limited by lines of some constant slope к (Fig. 3.2.2a), called the “stress diffusion” lines, and further assume that under constant bt^tndary displacements the stresses inside the stress relief region drop to zero while remaining unchanged outside. Eased on this assumption, the total loss of strain energy due to the formation of a crack of length 2a at controlled (fixed) boundary displacements is AU = —2кагЬ(аІя/2Е) where o1azj2E is the initial strain energy density. Writing that CTqo = Ей/L where и and L are the panel elongation and length, we cart rewrite the loss of strain energy as AU = — 2ка2ЬЕ(гі2/21?) and, therefore, the energy release rate per crack tip is

where in reaching the last equality we assume that after crack formation the relationship <jx = EujL is still approximately valid, an assumption that we will show to hold later.

Figure 3.2.2 Approximate zones of stress relief: (a) for a center cracked panel, (b) for a penny-shaped crack.

The foregoing approximate result is in exact agreement with Griffith solution (1924) if one assumes that к — тг/2 = 1.571. Even if к is unknown, the form of this equation obtained by the stress relief argument is correct. If the stress relief zone is assumed to be a circle of radius a passing through the crack tips, the result also happens to be exact. The same is true when the zone is taken as a rectangle of width 7a and height rra/2, or any geometrical figure whose area is tra2.

For the penny-shaped crack in an infinite elastic space subject to remote tension a0Q (Fig. 3.2.2b), the stress relief region may be taken to consist of two cones of base 7ra2 and height ka. Therefore, AW = ~2^-кагка(а10/2Е) = — утска2 Ей2/I2). Also, Q = ~[дІШ ld(ita2)u = — [dAU/dau/2ita, i. e.,

Again, this equation is of the correct,.form and is in exact agreement with the analytical result (Sneddon 1946) if one assumes that к — 8/тг. The exact value also results if the stress relief zone is assumed to be a rotational ellipsoid of minor semiaxes a, a and %а/тт or any geometrical figure whose volume is 16a3/3.

The approximate method of stress relief zones can be applied in diverse situations for a quick estimate of Q. The value of к depends on geometry and its order of magnitude is 1 (except in the case of high orthotropy). The error in intuitive estimations of к can be substantial; however, the form of the equation obtained for Q is correct.

There is a dichotomy in the method of stress relief zone which one must be aware of. Since the stress relief zone in Fig. 3.2.2 does not reach the top and bottom boundaries, the stress a at top and bottom can remain constant and equal to (Joo – Since there is a continuous zone of constant stress a = aoo connecting the top and bottom boundaries, the displacements at top and bottom also remain constant during the crack extension at constant oz*,.

However, from Eq. (2.1.32), it follows that a non-zero Q implies an increase of the compliance due to the presence of the crack. This, in turn, implies that the stress cannot remain constant while the crack extends at constant displacement. The variation of compliance due to crack extension may be obtained from Eq. (3.2.1) and from this the stress variations at constant displacement may be inferred. Let D be the width of the panel, L its length, и the relative displacement between the top and bottom boundaries. The resultant load and initial (uncracked) compliance then are:

P — OcabD

Inserting the foregoing expression for P into Eq. (2.1.32) (taking care to change the total crack length

a to 2a) and equating the result to F. q. (3.2.1), one easily finds

dC{a) 8 ka

da = bD2E

which integrates to

C{a) = Co + i. piy (3.2.6)

From this, one sees that the variation of compliance contains the factor (a/D)2 whose absolute value tends to vanish when the size of the pane] is much larger than the crack length. The remote stress drop due to crack extension at constant displacement is also shown to vanish as (a/D)2. Indeed, by differentiating the relation u — C(a)P one finds the first-order approximation for the remote stress drop as

where the higher order terms in Aka2/LD have been neglected.

Henceforth, the initial contradiction between the hypotheses of both constant remote stress and displacement exists at the theoretical level, but is resolved at the approximation level hecause it has been proved a posteriori that, in this case, the stress drop is vanishingly small when the crack is small relative to the size of the panel.