. The General Near-Tip Fields and Stress Intensity Factors

The results of the previous section may be generalized to any mode I loaded crack using various math­ematical formulations described in Chapter 4 to come. The analysis shows that the stresses <T;y(r, в) at any distance r of the crack tip may be written as are given by

0) = ~= %(r, 0) (2.2.10)

у/гттг

where К і is the mode l stress intensity factor proportional to the load, and functions Sij(r. 0) of polar coordinates (r, 0) —Fig. 2.2.2— are regular everywhere, except at load points, other crack tips and reentrant corners. These functions are dimensionless, and thus independent of structure size and load, but they depend on the geometry of the structure and of the loading. When the crack tip is approached (r 0), the general near-tip expression may be written as

Of/M) = -7==- (2.2.1 1 )

V 2717′

where К і is proportional lothe load and the dimensionless functions S(j(0) = SXj (0,в) are independent of geometry and the same for all mode I situations. They are given in Section 4.3.2, Eqs. (4.3.18)-(4.3.19).

This result means that two different linear elastic cracked bodies (different sizes, shapes, and material) subjected to mode l loading have identical stress distribution close and around the crack tip if the values of the stress intensity factors K[ are the same for both of them.

Подпись: <722 Подпись: Н-A^+ft^ + ... + A^ Подпись: +... Подпись: (2.2.12)
. The General Near-Tip Fields and Stress Intensity Factors

When r is not very small, the expression (2.2.11) represents the first term (the dominant one) of the series expansion in powers of r of the general expression (2.2.10). To illustrate this, the general power series expansion for the <722 stress component along the crack plane, which is analogous to that for the center cracked panel, Eq. (2.2.2), is

where P is a characteristic dimension of the structure (which may be, but in general need not be, chosen as the crack length a, as it was for the center-crackcd panel). The dimensionless coefficients pm, depend on the details of geometry and loading.

In the case of the center-cracked panel subjected to equiaxial remote stress, the series expansion is given by Eq. (2.2.2). For this geometry the first term is dominant, with error under within 3%, at distances r < 0.04a. For other geometries, the first term of the above series is identical, but the subsequent terms may differ appreciably (Wilson 1966; Knott 1973). However, if the size of the fracture process zone is much less than the /O-dominated zone (a few percent of the crack size, in general) the remaining terms can be neglected and LKFM holds. If the fracture zone is too large, some inelastic fracture theory must be introduced. This is the case for concrete in most practical situations, and the main concern of this book.

Similar conclusions are reached if the displacement field around the crack tip is analyzed. The general

Подпись: Figure 2.2.3 Near-tip stresses and crack openings in crack propagation: (a) initial situation, (b) after crack advanced an amount Sa.

solution using polar coordinates at the crack tip is of the form

‘Mr, в) = (2-2.13)

where the dimensionless functions Dj(r, в) are regular and depend on the geometry and loading. The near-tip distribution for r —» 0, however, is geometry independent:

D{{6) (2.2.14)

The dimensionless functions D[ (0) = D,(0, в) are given in Section 4.3.2, Eq. (4.3.20).

. The General Near-Tip Fields and Stress Intensity Factors Подпись: l+7,J+7^ + ... + 7^n Подпись: (2.2.15)

An important consequence of Eq. (2.2.13) is the expression of the crack opening profile. The upper crack face corresponds to 0 = it and the lower crack face to в = —тг, so we can writer/) — гіг(л it)—U2{v, — тг). Thus, using (2.2.13) and expanding the resulting expression in power series of r we get an expression similar to Eq. (2.2.8) for the center cracked panel:

where D is the characteristic dimension of the structure previously introduced in Eq. (2.2.12), and the dimensionless coefficients •ym depend again on the geometry of structure and loading. It can be proved (see Chapter 4) that they are related to coefficients /3m of the stress expansion as follows:

Подпись: (2.2.16)(~l)m 0

7m 0 . , Mm

2m + 1