Calibration of Microplane Model and Comparison with Test Data
The microplane model we described has been calibrated and compared to the typical test data available in the literature (Bazant, Xiang et al. 1996). They included: (1) uniaxial compression tests by van Mier (1984, 1986; Fig. 14.2.2a), for different specimen lengths and with lateral strains and volume changes measured, and by Hognestad, Hanson and McHenry (1955; Fig. 14.2.2b); (2) uniaxial direct tension tests by Petersson (1981; Fig. 14.2.2c); (3) uniaxial strain compression tests of Bazant, Bishop and Chang (1986; Fig. 14.2.2d); (4) hydrostatic compression tests by Green and Swanson (1973; Fig. 14.2.2e); (5) standard triaxial compression tests (hydrostatic loading followed by increase of one principal stress) by Balmer (1949; Fig. 14.2.2Q; (6) uniaxial cyclic compression tests of Sinha, Gerstle and Tulin (1964; Fig. 14.2.2g). (7) tests of shear-compression failure envelopes under torsion by Bresler and Pister (1958) and Goode and Helmy (1967; Fig. 14.2.3a); (8) tests of biaxial failure envelope by Kupfer, Hilsdorf and Rilsch (1969; Fig. 14.2.3b); and (9) failure envelopes from triaxial tests in octahedral plane (7r-projection) by Launay and Gachon (1971; Fig. 14.2.3c).
As seen from the figures, good fits of test data can be achieved with the microplane model. In Fig. 14.2.2a it should be noted that the uniaxial compression stress-strain diagrams are well represented for three specimens lengths, l — 5, 10, and 20 cm (it was already shown that the series coupling describes well the length effect in these tests; sec Bazant and Cedoiin 1991, Sec. 13.2). Fig. 14.2.2d serves as the basis for calibrating the volumetric stress-strain boundary, and a good fit is seen to be achieved for these enormous compressive stresses (up to 300 ksi or 2 GPa). Fig. 14.2.2f shows that the large effect of the confining pressure in standard triaxial tests can also be captured.
In Fig. 14.2.2g, note that the subsequent stress peaks in cycles reaching into the softening range are modeled quite correctly, and so are the initial unloading slopes. Significant differences, however, appear at the bottom of the cyclic loops, which is due to the fact that the unloading modulus is, in the present model, kept constant (a refinement would be possible by changing the constant unloading slope on the microplane level to a gradually decreasing slope, of course, with soine loss of simplicity). It should also be noted that the loading in these tests was quite slow and much of the curvature may have been due to relaxation caused by creep.
In Fig. 14.2.3c note that the model predicts well the shape of the failure envelopes, which is noncircular and nonhexagonal, corresponding to rounded irregular hexagons squashed from three sides. Fig. 14.2.3b shows that the ratio of uniaxial and biaxial compression strengths found in these tests can be modeled.
It must be emphasized that all the solid curves plotted in the figures are the curves that are predicted by the nticroplane model. The dotted curves in Fig. 14.2.2 arc those after correction according to the series coupling model. The dashed curves in Fig. 14.2.2c are those after correction according to the size effect law, and the dotted curves are those after a further correction according to the series coupling model.
Note that only six parameters need to be adjusted if a complete set of uniaxial, biaxial, and triaxial test data is available, and two of them can be determined separately in advance from the volumetric compression curve. If the data are limited, fewer parameters need to be adjusted. The parameters are formulated in such a manner that two of them represent scaling by affinity transformation. Normally only these two parameters need to be adjusted, which can be done by simple closed-form formulas. Thus, we can conclude that the model may be efficiently used to describe concrete behavior in uniaxial, biaxial, and triaxial situations.