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September 8th, 2015
Bazant, Tabbara et al. (1990) used the random particle system described before (Figs. 14.4.lab and 14.4.2) to simulate tensile tests and bending tests on notched specimens. A similar model was used by Jirasek and Bazant (1995a, b) to relate the microscopic features of the model (such as the softening curve and the statistics of strength distribution) to the macroscopic properties, particularly size effect and fracture energy.
Fig. 14.4.6 shows direct tension specimens of various sizes studied computationally by Bazant, Tabbara et al. (1990), with the results displayed in Fig. 14.4.7 as the calculated curves ofload (axial force resultant)
Figure 14.4.6 Geometrically similar specimens of various sizes with randomly generated particles (adapted from Bazant, Tabbara et al. 1990). 
vs. relative displacement between the ends.
Fig. 14.4.7b gives the curves for several specimens of the smallest size from Fig. 14.4.6. Fig. 14.4.7c shows the curves for the medium size specimens and Fig. 14.4.7d the curve for one large size specimen. Fig. 14.4.7e shows, in relative coordinates, the average response curves calculated for the small, medium, and large specimens. Note that while the prepeak shape of the load displacement curve is size independent, the postpeak response curve is getting steeper with increasing size.
Fig. 14.4.8 shows the progressive spread ol’cracking in one of the smallest specimens from Fig. 14.4.6. The cracking patterns arc shown lor four different points on the load displacement diagram, as seen in Fig. 14.4.8a, the first point corresponding to the peak load. The dashed black lines are the normals to the links that undergo softening and correspond to partially formed cracks. The solid lines are normal to completely broken links and represent fully formed cracks. The gray dashed lines represent normals to the links that partly softened and then unloaded, and correspond to partially formed cracks that are closing. Note from Fig. 14.4.8 that the cracking is at first widely distributed, but then it progressively localizes.
Fig. 14.4.9 shows the calculated peak loads for the specimens from Fig. 14.4.6 in the usual size effect plot of the logarithm of nominal strength vs. logarithm of the size. Bazant, Tabbara et al. (1990) interpreted the results in terms of the classical siz. e effect law Eq. (1.4.10) with relatively good results. The recent results of Bazant explained in Section 9.1, particularly the size effect formula for failures at crack initiation from a smooth surface (Section 9.1.6) suggest that these results must be interpreted using Eq. 9.1.42. Thus the results of Bazant, Tabbara et al. (1990) have been fitted here by the simplest version of this curve (for which 7 — 0). Fig. 14.4.9 shows the resulting fit, which is excellent for the mean, values of the data. .
Threepointbend fracture specimens of three sizes in the ratio 1:2:4 wete simulated in the manner illustrated in Fig. 14.4.10a. Fig. 14.4.10b shows the size effect plot obtained from the three sizes of the specimens in Fig. 14.4.10a for three different materials. As can be seen, the calculated maximum loads can be well approximated by Bazant’s size effect law, Eq. (1.4.10).
By fitting the size effect law to the maximum load obtained by the lattice or particle model for similar specimens of different sizes, one can determine the macroscopic fracture energy C j of the particle system and the effective length Cf of the fracture process zone (see Chapter 6). It thus appears that fracture simulations with the lattice model or random particle system provide a further verification of the general applicability of the size effect law.







Figure 14.4.10 Simulation of three threepointbend tests by random particle model.(a) Thrccpointbend specimens with d = 36.72, and 144 mm and (b) corresponding size effect plot. (Adapted from BaiSant, Tabbara et al. 1990.)
At the same time, the size effect law is seen to be an effective approach for studying the relationship between the microscopic characteristics of the particle system, simulating the microscopic properties of the material, and the macroscopic fracture characteristics.
Such studies have been undertaken by Jirasek and Bazant (1995b). Fig. 14.4.11 show the results of a large number of such simulations, dealing with two dimensional threepointbend fracture specimens of different sizes. In these specimens, the microductility number, representing the ratio sc/ep, was varied (see Fig. 14.4.2b). The coefficient of variation of the microstrength of the particle length, used in random generation of the properties of the links, was also varied (the microstrength was assumed to have a normal distribution).
It was found that both the microductility and the coefficient of the microstrength of the links have a significant effect 011 the macroscopic fracture energy Gy and on the effective length (.7 of the process zone; see Fig. 14.4.1 la, c. Randomness of these plots is largely due to the fact that the number of simulations was not very large (the values in the plot are the averages of the values obtained in individual sets of simulations of specimens of different sizes). The plots in Fig. 14.4.1 la, c have been smoothed as shown in Fig. 14.4.1 lb, d by the following bilinear polynomials which provide optimum (its:
Figure 14.4.11 Normalized fracture energy and normalized effective process zone size as a function of two parameters: (a) and (c) computed, (b) and (d) fitted by bilinear functions (from Jirdsek and Baz. ant 1995b). 
Cf = 0.64 – I – 0.08wy + 0.097/ — 0.19u>/7/, (14.4.2)
in which the superimposed bars refer to average values, and cjj and 7/ are the coefficient of variation of microstrength and the microductility number. Obviously, the effect of various other microscopic characteristics of particle systems on their macroscopic fracture properties could also be studied in this manner, exploiting the size effect law.
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