The Center Cracked Infinite Panel and the Near-Tip Fields

Consider a crack of length 2a in a two-dimensional infinite linear elastic isotropic solid, subjected to uniform normal stress ох at infinity in all directions (Fig. 2.2.1). The solution of this problem was obtained by Griffith (1921), as a particular solution of the panel with an elliptical hole obtained by Inglis (1913), and is derived in full in Chapter 4. Using the central axes shown in Fig. 2.2.1, the normal stresses

an along the uncracked part of the crack plane (xi = 0, x — a2 > 0) arc expressed as:

<%2Л)

The Center Cracked Infinite Panel and the Near-Tip Fields Подпись: 3r 5 r2 1 + 4^ ~ 32? + Подпись: (2.2.2)

This result shows that the stresses tend to infinity when the crack tips are approached from the solid. So the stress field has a singularity at the crack tip. In order to determine the asymptotic r! ear-tip stress field, we write the stresses as a function of the distance r to the right crack lip, that is, replacing x — a with r (and x with r + a, and x 4- a with r 4- 2a). Then, setting x] — a2 — (a-, 4- a)(xi – a), we get for <722 the following asymptotic approximation:

where the factor in square brackets shows the first three terms of the Taylor scries expansion of (1 4- г/а)/л/1 + г/2а. This factor obviously tends to 1 for r a. It is now customary to denote

Подпись: (2.2.3)КI — UoaSpita

Подпись: <7 22 Подпись: к, Подпись: (2.2.4)

and call it the stress intensity factor (Subscript 1 refers to the opening mode of fracture, or mode I, to be distinguished from the shear modes 11 and III whose discussion is deferred to Chapter 4). The near-tip (r/a -> 0) expression for <722 now becomes

which shows that the stress displays a singularity of order r 1/12 at the crack tip.

For the normal displacements щ along the crack faces (x — a2 < 0), the clastic solution delivers

ut – J“2 ~ xf ‘ P-2-5)

where uj and uf are, respectively, the vertical displacement of the upper and lower faces of the crack, and E’ is the effective elastic modulus defined as

E’ — E for plane stress

E’ = E/{ 1 — v2) for plane strain )

The crack opening w is the jump in displacement between the faces of the crack, w — — u2 ,-and

The Center Cracked Infinite Panel and the Near-Tip Fields Подпись: (2.2.7)

is obtained from Eq. (2.2.5) as

The Center Cracked Infinite Panel and the Near-Tip Fields
To see how the crack opening behaves in the neighborhood of the crack tip, we rewrite the last equation as a function of the distance r from the right crack tip (now r = a — X) and substitute ax = К;/у^іта to get

which shows that the profile of the deformed crack is parabolic (more precisely, a parabola of the second degree with its axis coincident with the crack line.)

Although the above near-tip results made use of a quite particular case, they remain valid for all the mode I loading cases. This will be further discussed next.