58 H. T. Nguyen et al. Fig. 8.2 (a) Effect of parallel stresses at different magnitudes and (b)Gf as a function of σyy →∞, the final asymptotic slope of LEFM, −1/2, is approached more quickly. The mechanisms that cause these phenomena were suggested to increase the volumetric stress on microcracks and expanding slip, see Fig. 8.2a. 8.4 Conclusion Moderate crack-parallel compressive stress drastically increases, even doubles, the Mode I fracture energy Gf (or fracture toughness KIc) of concrete, and probably also rock and other similar quasibrittle materials. Intuitive explanation: mainly the effect of increased hydrostatic pressure, simply described by the Drucker-Prager strength criterion. However, the drastic decrease of Gf at high crack-parallel compression requires a different damage constitutive model—the microplane model. High crack-parallel compressive stress drastically decreases Gf and eventually reduces it to zero. Intuitive explanation: mainly the effect of the compression cap on the Drucker-Pager criterion. The cohesive crack model, as a line crack model with a scalar relation between crack-bridging stress and relative displacement of crack faces, cannot capture these phenomena, since the crack-parallel stress σyy and strain are not the basic variables in this model. Soσyy must be used as parameters, which is only an approximation since it cannot capture the effect of the history of triaxial stress and strain. Linear elastic fracture mechanics, as well as fracture models with scalar damage at the front, which include the scalar phase-field model, cannot capture the effect of crack-parallel compression. To capture the experimentally evidenced effect of crack-parallel compression in general, a tensorial constitutive model for softening damage (which is based on vectorial description at mesoscale cracks) must be used. Most convenient is to use the crack band model coupled with the microplane constitutive law, which can reproduce the inelastic frictional slips separately on planes of different orientations. An effective method for fracture tests with crack-parallel compression is to generate the compression by plastic pads capable of perfectly plastic yielding and to install the supports with a gap such that they engage in contact only after the pads yield. The results are of particular interest for the shear failure of reinforced concrete beams and punching of slabs, as well as for the fracture of prestressed concrete, and to hydraulic fracturing of shale, at which the overburden and tectonic stress introduce significant crack-parallel compression. Acknowledgment Partial preliminary funding under NSF Grant No. CMMI-2029641 to Northwestern University is gratefully acknowledged.
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