Vertex Effects

There is another important property that is exhibited by the microplane model, and not, for example, by macroscopic plasticity models. For a nonproportional path with an abrupt change of direction such that the load increment in the ay space is directed parallel to the yield surface, the response of a plasticity model is perfectly elastic, unless this change of direction happens at a corner of the yield surface. But

Vertex Effects

> uniaxial iensioVQ4,o.. (Petersson 1981)

axial strain є, (rnrn/m)

 

Vertex Effects

Figure 14.2.2 Experimental results from various sources and best fits with the microphtne model (after Bazant, Xiang et al. 1996): (a) uniaxial compression tests by van Mier (1984); (b) uniaxial compression tests by Hognestad, Hanson and McHenry (1955; (c) uniaxial tension tests by Petersson (1981); (d) confined compression test (uniaxial strain) of Bazant, Bishop and Chang (1986); (e) hydrostatic compression test by Green and Swanson (1973); (f) triaxial test data (increasing axial compression at constant lateral confining pressure) by Balmer (1949); (g) uniaxial cyclic compression tests of Sinha, Gerstle and Tulin (1964). (Adapted from Bazant, Xiang et al. 1996.)

 

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Vertex Effects

Figure 14.2.3 Fitting of (a) shear compression failure envelope (in torsion) measured by Bresler and Pisler (1958), (b) biaxial failure envelope measured by Kupfer, Hilsdorf and Riisch (1969), and (c) failure envelopes in hydrostatic planes at various pressures measured by Launay and Gachon (1971). (Adapted from Bazant, Xiang el al. 1996.)

Vertex Effects

Figure 14.2.4 Vertex effect: (a) preloading in the <тц-£ц space at increasing єц and zero shear strain; (b) in the Є11-Є12 space preloading corresponds to segment 01 and further tangent loading to segment 12; (c) the further tangent loading in the СГ12-С12 diagram corresponds to segments 03 in classical plasticity models (fully clastic loading) and to segment 04 when vertex effect is present (after Bazant, Xiang et al. 1996).

in reality, for all materials, this response is softer, in fact much softer, than elastic. It is as if a corner or vertex of the yield surface traveled with the state point along the path.

This effect, called the vertex effect (see Sec. 10.7 in Bazant and Cedolin 1991), is automatically described by the microplane model, but is very hard to model with the usual plastic or plastic-fracturing models. It can be described only by models with many simultaneous yield surfaces, which are prohibitively difficult in the (Jij space. The microplane model is, in effect, equivalent to a set of many simultaneous yield surfaces, one for each microplane component (although these surfaces are described in the space of microplane stress components rather than in the а;у space).

This is one important advantage of the microplane approach, [t is, for example, important for obtaining the correct incremental stiffness for the case when a dc^-increment (segment 12 in Fig. 14.2.4b) is superimposed on a large strain (segment 01) in the inelastic range. Segment 03 in Fig. 14.2.4c is the predicted response according to all classical macroscopic models with yield surfaces, which is elastic, and segment 04 is the prediction of microplane model, which is much softer than elastic (i. e., dcr^/dt’is < 2G where G = elastic shear modulus). Fig. 14.2.4c shows the incremental stiffness 04 calculated for the case of the present reference parameters and Єц 0.005. Indeed, the slope 04 is almost 1/5 of the slope 0.3 which would be predicted by plasticity with a simple yield surface.