Crack tips plane strain

Fracture toughness (plane strain) K c The crack extension resistance under conditions of crack tip plane strain. 

A second type of transition, termed an upper transition, corresponds to the size of the plastic zone reaching a maximum dimension characteristic of the test. An example here is the critical size of the plastic zone at the tip of a crack in plane strain, which gives... 

The strain-energy-release rate was expressed in terms of stresses around a crack tip by Inwin. He considered a crack under a plane stress loading of a , a symmetric stress relative to the crack, and x°° a skew-symmetric stress relative to the crack in Figure 6-12. The stresses have a superscript" because they are applied an infinite distance from the crack. The stress distribution very near the crack can be shown by use of classical elasticity theory to be, for example. 

This plastic deformation is localised around the crack tip and is present in all stressed engineering materials at normal temperatures. The shape and size of this plastic zone can be calculated using Westergaards analysis. The plastic zone has a characteristic butterfly shape (Fig. 8.83). There are two sizes of plastic zone. One is associated with plane stress conditions, e.g. thin sections of materials, and the other with plane strain conditions in thick sections-this zone is smaller than found under plane stress. 

We obtained full field solutions for an axial crack on the ID surface of a pipe under plane strain conditions. The pipe has an outside diameter of 40.64 cm and a wall thickness h = 9.52 mm. The depth of the axial crack is a = 1.9 mm (a/h = 0.2) (Fig. 2). In the absence of hydrogen, we assume a finite crack-tip opening displacement (CTOD), ha = 2 pm. 

The equations above are valid for plane-stress situations, i.e. for thin sheets. For thicker sheets plain-strain conditions occur at the crack tip. In that case for a thick plate of infinite width containing a crack of length 2a the fracture stress is... 

Fig. 8. Schematic view of the epsilon CTPZ region. The plane stress shear zones occur near the surface on either side of the crack opening (AA). The plane strain shear bands are internal, forming at the crack tip...

In PEI the DCG process, as in any polymer, is active. The epsilon CTPZ, however, was not observed. No plane strain shear bands have yet been observed. Some form of localized crack tip shear process can be activated, however, as evidenced by the inversion transition that occured at higher stresses (at the higher temperatures). The fracture surface did not show fracture to occur on a slanted 45 degree plane. The fracture plane was still normal to the leading direction. The fracture surface, however, was not smooth, as seen with craze fracture, but has a definite roughened texture which is associated with active localized shearing. This texture is often described as honeycomb or tufted. 

Equation (14.41) is obeyed under the condition of plane stresses at the tip of the crack ((T33 = 0), which occurs in laminae of narrow thickness (thin laminae). For thick laminae, the condition that occurs is that of plane strain ( 33 = 0), and this leads to (38) (see also Chapter 16)... 

Displacement in craze at crack tip 6 J Critical value of 6 at fracture K 3 — 4 V for plane strain... 

Since the stresses are singular at the crack tip, then clearly the yield oiterion is exceeded in some zone in the crack tip region. If this zone is assumed to be small, then it will not greatly disturb the elastic stress field so that the extent of the plastic zone will be defined by the elastic stresses. If it is assumed that the Von Mises yield criterion is applicable (a reasonable first approximation for polymers), then the shape and e of the plastic zone may be derived from the stresses given in Eq. (15). As-sumii a state of plane strain so that the transverse stress is given by 1/(0 + oee)> then for a yield stress of Oy, the dastic zone radius becomes ... 

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