Process-zone

The elastic stress cannot exceed the yield stress of the material, implying a region of local yielding at the crack tip. Nevertheless, to apply the simple framework of hnear elastic fracture mechanics, Irwin [J. Applied Mechanics, 24, 361 (1957)] proposed that this process zone size / be treated as an effective increase in crack length be. Fracture toughness is then given by... 

The process zone is a measure of the yield stress or plasticity of the material in comparison to its brittleness. Yielding within the process zone may take place either plastically or by dimise microcracking, depending on the brittleness of the material. For plastic yielding, / is also referred to as the plastic zone size. 

Figure 20-87 shows a typical production-size batch fluid-bed granulator. The air-handling unit dehumidifies and heats the inlet air. Heated fluidization air enters the processing zone through a distribu-... 

Rather than bearing an infinite stress at the crack tip, yielding occurs resulting in a volume of inelastically deformed material along the crack front called the process zone, as shown in Fig. 2. The size of the inelastic zone, r j , under a monotonic tensile stress, o , can be approximated by substituting o = Oj into eq. 2 for the horizontal plane, 0 = 0... 

The variability in fracture stress when small artificial flaws were controlling strength was particularly pronounced for H-451 graphite, as can be seen in Fig. 12. Here the crack dimensions and crack trip process zone dimensions are comparable to the microstructural dimensions. Consequently, local variations in microstmeture... 

The second physical quantity of interest is, r t = 90 pm, the critical crack tip stress field dimension. Irwin s analysis of the crack tip process zone dimension for an elastic-perfectly plastic material began with the perfectly elastic crack tip stress field solution of Eq. 1 and allowed for stress redistribution to account for the fact that the near crack tip field would be limited to Oj . The net result of this analysis is that the crack tip inelastic zone was nearly twice that predicted by Eq. 3, such that... 

If we take the nominal fracture toughness of IG-11 graphite to be 1 MPayin and the maximum stress in the process zone to be = 60.2 MPa according to the above analysis, we find that rj, = 88 pm. This value is virtually identical to r<, , = 90pm, the process zone dimension determined using Eq. 3. To summarize, the above analysis strongly supports a hypothesis that the maximum critical stress... 

The above results are derived from linear elastic fracture mechanics and are strictly valid for ideally brittle materials with the limit of the process zone size going to zero. In order to apply this simple framework of results, Irwin (1957) proposed that the process zone, r be treated as an effective increase in crack length, Sc. With this modification, the fracture toughness becomes... 

As discussed above, because materials often fail by the propagation of cracks, particularly brittle ones, it is difficult to measure the true yield stress, oy, of a material. Instead one measures a fracture stress, Op which is related to the size of the process zone and actual crack length by ... 

Hi) The process zone sizes are large and typical for polymeric materials... 

To summarize, we can say that granulated materials should behave in a brittle manner, similar to a ceramic, in the sense of having small yield strains and small critical displacements but their behavior should also be ductile, similarto polymers, in the sense of having large process zones. The consequence of having large process zones is important since it implies that... 

Several additional, non-microstructural, inputs are required for the fracture model (i) Particle critical stress intensity factor, KIc. Here, the value determined in a previous study (Klc = 0.285 MPa in )[3] was adopted for all four graphites studied. This value is significantly less than the bulk Klc of graphites (typically -0.8-1.2 MPa rn). However, as discussed in the previous section, when considering fracture occurring in volumes commensurate in size with the process zone a reduced value of Klc is appropriate (ii) the specimen volume, taken to be the stressed volume of the ASTM tensile test specimens specimen used to determine the tensile strength distributions and (iii) the specimen breadth, b, of a square section specimen. For cylindrical specimens, such as those used here, an equivalent breadth is calculated such that the specimen cross sectional area is identical, i.e.,... 

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