Crack Influence Function in Two Dimensions
Consider now a crack in an infinite solid, subjected to uniform pressure a (Fig. 13.3.3b). According to Westergaard’s solution (Chapter 4) the stress distribution can be written as
xx — Re 5/ — у Im Z‘ — a, Oyy — ReZ + у I in Z’— a, rxy — ~yRcZ’ (13.3.37)
Here 2a = crack length, і2 — –1, Z’ — dZ/dz, and г, ф = polar coordinates with origin at the crack center and angle ф measured from the crack direction x. For r » a we have the approximation:
ZJ — o(—a2z~2 + …) у Im Z’ — aa2r sin ф Im — – aa2r~2 sin 0 (— sin 30)
Substituting this into (13.3.37) and using the formulas for products of trigonometric functions, we get the following simple result for the long-range (r » a) asymptotic field (Bazant 1992b, 1994b):
. , . cos4$ , . . / „ cos4</>
<Jxx = ск(г) —~2~’ °уу = ak^T’ Vcos2<*>——————- 2—
, sin 4<p – sin 2ф тху – crk(r) –
where k(r) — a2/r2. Subscripts x, у refer to cartesian coordinates with origin at point £ coinciding with the crack center and axis у normal to the crack; oxx and ayy are the normal stresses, тху is the shear stress; and ф are polar coordinates with origin at the crack center, with the polar angle ф measured from axis X. The principal stresses cr^!’ and a^ and the first principal stress direction ф^~> are given by:
tan 2ф^ — — cot Ъф
The foregoing expressions describe the long-range form of function Л(х, £). It docs not matter that they have a r"2 singularity at the crack center, because they are invalid for not too large r. Note that the average of each expression over the circle r = constant is zero, which is, in fact, a necessary properly.
By virtue of considering only principal stress directions, Л(х, £) is a scalar. All the information on the relative crack orientations is embedded in the values of this function. The principal stress dir ection at point
which can be regarded as the domi nant crack direction at that location (Fig. 13.3.4a), is all the directional information needed to calculate the stress components at point x; sec (40), in which r = || x—£ |j = distance between points x and The value of Л(х,£), needed for (13.3.31) or (9), may be determined as the projection of the stress tensor at point x onto the principal inelastic stress direction at that point. According toMohrcircle: 2Л(х,£) — (crxx +oyy) – f (axx ~оуу)соъ2(ф~в)—2тхунп2(ф– 0) in which 0, ф – angles of the principal inelastic stress directions at points £, x, respectively, with the line connecting these two points (i. e., with the vector x – £). Substituting here for axx, etc., the expressions from (13.3.41), one obtains a trigonometric expression which can be brought by trigonometric transformations (Planas 1992) to tile form:
Л(х,£) = —[ cos26 + cos2ф + cos2(0 + ф)] (13.3.43)
where 0 — 90° — ф. Note that the function Л(х, £) is symmetric. This is, of course, a necessary consequence of the fact that the body is elastic.
Two properties contrasting with the classical nonlocal formulations explained before should be noted:
(1) the crack influence function is not isotropic but depends on the polar angle (i. e., is anisotropic), and
(2) it exhibits a shielding sector and an amplification sector. We may define the amplification sector as the sector in which ayy (the stress component normal to the crack plane) is positive, and the shielding sector as the sector in which ayy is negative. The amplification sector ayy > 0 , according to (13.3.41), is the sector ф < фь where
фь = 55.740° (13.3.44)
The sector in which the volumetric stress axx + ayy (first stress invariant) is positive is ф < 45°. The sector in which axx > 0 is ф < 22.5° and ф > 67.5°. The sector in which 2 rmax — crxx — ayy > 0 is ф < 45°. The maximum principal stress cr^1 -* is positive for all angles ip, and the minimum principal stress cri2i is positive for ф < 21.471°.
The consequence of the anisotropic nature of the crack influence function is that interactions between adjacent cracks depend on the direction of damage propagation with respect to the orientation of the maximum principal inelastic stress. In a cracking band that is macroscopically of mode I (Fig. 13.3.5a), propagating in the dominant direction of the microcracks, the microcracks assist each other in growing because they lie in each other’s amplification sectors. In a cracking band that is macroscopically of mode II (Fig. 13.3.5b), the microcracks are mutually in the transition between their amplification and shielding sectors, and thus interact little. Under compression, a band of axial splitting cracks may propagate
(a) (b) (C) (d) (Є) (f) (g)
Figure 13.3.5 Crack bands and cracks near boundary (from Bazant 1994b).
sideways (Fig. 13.3.5c), and in that case, the microcracks inhibit each other’s growth because they lie in each other’s shielding sectors. Differences in the kind of interaction may explain why good fitting of test data with the previous nonlocal microplane model required using a different material characteristic length for different types of problems (e. g., mode 1 fracture specimens vs. diagonal shear failure of reinforced beam).
For small r, function Л(х, £) is a result of interactions in all directions. As the first approximation, these interactions may be assumed to cancel each other. Accordingly, we replace function k(r) — a2/r2 by a simple function of the same asymptotic properties for r —* 00 which does not have a singularity at r — 0 and for r 0 approaches 0 with a horizontal tangent:
Here к is an empirical constant such that k£ roughly represents the average or effective crack size a for the macro-continuum; і is a certain constant representing what may be called the characteristic distance of crack interactions (it represents the radial distance to the peak in Fig. 13.3.6). This length may be identified with what has been called the characteristic length of the nonlocal continuum. It reflects the dominant spacing of the microcracks, which in turn is determined by the size and spacing of the dominant inhomogeneities such as aggregates in concrete.
The foregoing expressions give the crack influence function which is exact asymptotically for r —> oo but is only a crude approximation for small r. It is now convenient to represent the complete crack influence function Л in the form:
Л(0,£) – Ax(Z, v) +-Л,(С?7) (13.3.46)
where Лі represents a difference that is decaying to infinity faster (i. e., as a higher power of r) than and can, therefore, be neglected for sufficient distances r from the center of the source crack.
The complete function Л was determined by numerical integration of Fq. (13.3.35) using a dense square mesh; see 13.3.7a (Bazant and Jirasek 1994a). The target crack was considered parallel to the source crack, and a/s — 0.25. The asymptotically correct analytical expression for the crack influence
function from Eq. (13.3.45) is plotted in Fig. 13.3.7b. After its subtraction from the values, one obtains the plot of the difference Лі shown in Fig. 13.3.7c. A table of numerical values of Л| was reported in Bazant and Jirasek (1994a)).
Function Лі (x, у) obviously depends on the relative crack size a/s. However, it lias been found that it depends on a/s only very little when a/s > 0.25. For smaller a/s, the crack interactions are probably unimportant. So perhaps a single crack influence function expression could be used for all the cases.
A statistical definition of Л in three dimensions that is analogous to Eq. (13.3.35) can obviously be written, too.