Crack plane strain load

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. 

There can be no doubt as to the importance of plane strain conditions for the fracture of plastics especially where sharp notches and thick sections are concerned. Such conditions nearly always lead to brittle or semi-brittle fracture. Vincent has shown that the notch sensitivity in a braod range of amorphous and crystalline polymers is increased as the testing temperature is lowered and the loading rate is increased. Before fracture occurs, amorphous plastics often craze under these conditions. The complex questions of craze initiation, propagation and transformation into a crack have been treated extensively for amorphous polymers in the first three chapters of this book (see also The problem becomes more complicated when... 

Figures 7 and 8 respectively show the effect of a and 6 and a and kq on the normalized strain energy release rate in an orthotropic FGM with a crack of length 2a located along the x axis. The problem is one of mixed-mode, the external load is a uniform tension po perpendicular to the crack plane and Qo is the strain energy release rate for the corresponding homogeneous isotropic medium for which a = 0, kq = 1. = 1.= 0.3 [18].

The crack-driving force G may be estimated from energy considerations. Consider an arbitrarily shaped body containing a crack, with area A, loaded in tension by a force P applied in a direction perpendicular to the crack plane as illustrated in Fig. 2.6. For simplicity, the body is assumed to be pinned at the opposite end. Under load, the stresses in the body will be elastic, except in a small zone near the crack tip i.e., in the crack-tip plastic zone). If the zone of plastic deformation is small relative to the size of the crack and the dimensions of the body, a linear elastic analysis may be justihed as being a good approximation. The stressed body, then, may be characterized by an elastic strain energy function U that depends on the load P and the crack area A i.e., U = U(P, A)), and the elastic constants of the material. 

In this case, the crack size is much larger than the plane strain crack-tip plastic zone size. As such the effective crack length 2a = 2ao + 2rjy would be effectively equal to the initial (or physical) size of the crack 2ao. The load-displacement trace would be essentially hnear up to the point at which the specimen fractures abruptly (see Fig. 4.5a). The plane strain fracture toughness Kic can be computed directly from the maximum load Pmax or stress a ax i e., the load or stress at fracture) and the initial crack size ao using Eqn. (4.7). 

This stipulation on the allowable extent of plastic deformation at the crack tip cannot be used conveniently in fracture testing. The extent of deformation that would be allowed, however, is equivalent to a specification on the change in load-displacement curve at the maximum load point in relation to the initial slope ie., from elastic to elastic-plastic deformation). This change in slope can be readily measured and is used in plane strain fracture toughness testing. 

If the specimen is very thick i.e., with thickness B much greater than the plastic zone size, or Kic/oysY), the constraint condition along the crack front in the midthickness region is that of plane strain and is barely affected by plastic deformation near the surfaces. Abrupt fracture crack growth) will occur when the crack-tip stress intensity factor reaches the plane strain fracture toughness Kjc. The load-displacement record, similar to that of the penny-shaped crack, is depicted by Fig. 4.7a. 

For a very thin specimen i.e., with B (Kjc/ays) ), the influence of plastic deformation at the surfaces will relieve crack-tip constraint through the entire thickness of the specimen before Kj reaches Kjc. As such, the opening mode of fracture is suppressed in favor of local deformation and a tearing mode of fracture. The behavior is reflected in the load-displacement record by a gradual change in slope and final fracture, which could still be abrupt (see Fig. 4.7b), but the conditions of plane strain would not be achieved. 

For experimental purposes, the procedure must be cast in terms of measurable quantities from the load-displacement records (namely, the test data). As such, the physical quantities, such as crack increments, must be related to the changes in displacements, or in the slopes of the load-displacement records. To establish a permissible limit for AailUo, it is assumed that Aoi should not exceed the formally computed plane strain plastic zone correction term, namely... 

For the recommended range of values of Uo/W of 0.45 to 0.55, a value for H/50 of 0.05 has been adopted for the testing of single-edge-cracked specimens. This is embodied in the so-called hve percent slope offset method for establishing the pop-in load in ASTM Method E-399 for Plane Strain Fracture Toughness [2]. 

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