Some rock fracture elastic nonlinear normal deformation constitutive models were reviewed and the significant deviation of these models from experimental results at intermediate stress was pointed out. A new concept named half-closure stress σ 1/2 was introduced to lay emphasis on analysis of the mathematic defects of both BB model and classical exponential model. The generalized exponential model proposed in Malama and Kulatilake (2003) was treated as an example for further statement of the necessary for model improving, meanwhile analyzing the deficiency of physical meaning of the initial normal stiffness of fracture Kni because of modification by accession of power function n on σn1/2. Based on the above research, the governing differential stress-displacement equation of fracture was established under monotonically increasing normal compressive loading and the concept of pseudo fracture maximum allowable closure Dma= ξ dma was defined. Then a new 3-parameter constitutive model which improved fracture elastic nonlinear normal deformation behavior was proposed. It was proved strictly that BB model and classical exponential model were two special cases of the new model. The new parameter was easy to determine, and the mathematic defects of BB model and classical exponential model were overcome. Counteracting the deviation of simulated results of these traditional nonlinear models with test results at intermediate stress, the physical meaning of Kni did not lose with the accession of the new parameter ξ. The predication results fitted the test results well in adopting the new model for the predication of others' test data, which verified the feasibility of the new model.


Fracture deformability under the action of normal compressive stress is of fundamental importance to the study of the hydraulic and mechanical behavior of fractured rock. Fracture deformation directly affects the aperture distribution, contact area distribution, and spatial connectivity of the apertures, factors that govern the hydraulic conductivity of single fractures. Since fracture networks govern the hydraulic behavior of fractured rock masses, it follows directly that deformability of single fractures due to the action of compressive stress would affect the hydraulic properties of a rock mass. It is also generally understood that the mechanical behavior of rock masses is dominated by the deformation of discontinuities.

Normal deformation behavior of rock fractures had been widely studied under quasistatic or cyclic loading conditions. Shehata (1971) used a semi-logarithmic empirical model to describe the relation. Goodman (1975) suggested a hyperbolic relation to describe fracture closure under normal compressive stress. And then Kulhaway (1975) proposed another simple hyperbolic model. Bandis et al. (1983) have, however, shown that plots of dn data against logarithmically scaled σ n do not show a linear fit across the entire range of normal stresses. Bandis et al. (1983) and Barton et al. (1985) presented a modification of Goodman's hyperbolic model, which was shown to provide a better fit to experimental data across the whole range of stress and closure values. Alternative models, based on Hertzian contact theory, have also been used to describe the non-linear stress deformation behavior.

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