Crack Influence Function in Three Dimensions

Подпись: 0 + Re о2
Подпись: О і — Re O2 Подпись: Ini <7.

The case of three dimensions (3D) is not difficult when the cracks are penny-shaped (i. e., circular) and the boundary is remote. The stresses around such cracks have traditionally been expressed as integrals of Bessel functions (Sneddon and Lowengrub 1969; Kassir and Sih 1975), which are however cumbersome for calculations. Recently, though, Fabrikant (1990) ingeniously derived the following closed-form expressions:

Подпись: (13.3.47)rxz – Re tz

in which

in which a = crack radius; r, 0, ф are the spherical coordinates (Fig. 13.3.8) attached to cartesian coor­dinates x, y, z at point £, with angle в measured from axis z which is normal to the crack at point £; r = distance between points x and £; р, ф,г are the cylindrical coordinates with origin at the crack center; and p, ф are polar coordinates in the crack plane, angle ф being measured from axis x.

Crack Influence Function in Three Dimensions Crack Influence Function in Three Dimensions Подпись: + (1 — 2u - 5 cos2 0) sin2 0 — (1 - 2u - 5 cos2 0) sin2 в Подпись: (13.3.49)

The long-range asymptotic form of the foregoing stress field has been derived (Bazant 1994b). The derivation is easy if one notes that, for large r, L иг – asin$, Ь2 к r | a sin# (see the mean­ing of L] and L2 in Fig. 13.3.8a), (i ~ asin#,/2 ~ r and, for r > a, arcsin(a/h) ~ [1 + {а2/ві)а/І2, /Ц — а2 и r[l – (a2/2r2)]. The result is the following long-range asymptotic field:

Подпись: а рф — a ф2 — 0apz — ak(r) sin 20 (4 — 5 sin2 0),

Crack Influence Function in Three Dimensions

Подпись: Figure 13.3.7 Crack influence function determined by Bazant and Jirasek (1994a)): (a) total crack influence function for the case of parallel source and target cracks, (b) analytical expression having the correct long-range asymptotic field, and (c) difference of the crack influence functions in (a) and (b). (From Bazant and Jirasek 1994a.)

Crack Influence Function in Three Dimensions

Crack Influence Function in Three Dimensions

in which, for three dimensions, k(r) — a3/(nr3). For the same reasons as those that led to Eq. (13.3.45), this expression may be replaced by

сз. ї.50)

(Fig. 13.3.8b) which is asymptotically correct for r —> oo and nonsingular at r — 0. The crack influence function based on (13.3.49) satisfies again the condition that its spatial average over every surface r = constant be zero.

For large distances r, the crack influence function in three dimensions asymptotically decays as r~3, whereas in two dimensions, it decays as r-2. Again, in contrast to the phenomenological models we expounded before, the weight function (crack influence function) is not axisymmctric (isotropic) but depends on the polar or spherical angles (i. e., is anisotropic).

Further note that one can again distinguish a shielding sector and an amplification sector. According to the change of sign of azz in Eq. (13.3.49), the boundary of these sectors is given by the angle

Bi, — arcsin 1/2/З = 54.736° (13.3.51)

or 90° — 0(, = 35.264°. Thus, the amplification sector в > Bb’is significantly narrower in three than in two dimensions.

In the case of a field that is translationally symmetric in г, one might wonder whether integration over 2 might yield the two-dimensional crack influence function. However, this is not so because the two-dimensional crack influence function represents in three dimensions the effect of an infinite strip (of thickness dx) at coordinate x of pressurized cracks aligned in the 2 direction on the stresses in a strip of glued cracks at coordinate £. This cannot yield the same properties as the field of one penny-shaped crack.