PSI - Issue 18

Mikhail Eremin / Procedia Structural Integrity 18 (2019) 135–141

137

Mikhail Eremin / Structural Integrity Procedia 00 (2019) 000–000

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Fig. 1. (a) log-normal law of pore size distribution; (b) simulated distribution of pores in computational domain.

2.2. Nonlinearity in the beginning of deformation

It is known that rocks usually exhibit non-linear behavior at initial stages of deformation. Di ff erent authors argue that nonlinearity is due to closure of preexisting microcracks in rocks and, sometimes, not proper parallelism of loading surfaces Hoek and Martin (2014); Farrokhrouz and Asef (2017); Stefanov (2018). When stresses are applied to the specimen, initial microcracks close, which causes an increase of bulk modulus. Short review of this problem could be found in Stefanov (2018). We will only give the equation which describes this phenomenon and provide the constants used in simulation. K = K 0 1 − a φ − ε φ P 0 − σ P 0 (1) where K 0 is a bulk modulus of material with closed preexisting microcracks, a < 1 is a constant, φ has a physical meaning of volume of closed cracks, ε = − ∆ V V is volumetric strain with reversal sign, P 0 is hydrostatic pressure causing closure of all preexisting cracks, σ > 0 is current hydrostatic pressure. The choice of parameters of Eq. 1, as well as other parameters in the model, is made on the basis of experimental data fitting. Uniaxial compression is quite simple type of loading and choice of model parameters might be made according to the following recommendations: • Hydrostatic pressure at uniaxial compression σ = − 1 3 σ 11 , where σ 11 is axial stress. If we take an experimental loading diagram and fix the point where initial nonlinearity is finished with axial stress σ 11 , then P 0 ≈ − 1 3 σ 11 . • The same logic we follow when we choose the value of φ . In the case of uniaxial compression volumetric strain θ = ε 11 (1 − 2 ν ), where ε 11 is axial strain, ν is the Poisson’s ratio. Then φ ≈ ε 11 (1 − 2 ν ) for the point where initial nonlinearity is finished. Obviously, such approach is suitable better for materials with very low porosity. Constants of Eq. 1 for materials with su ffi cient porosity need more adjustment to fit the experimental data. Table 1 summarizes all physical-mechanical properties of pore-free sandstone which are used in simulation.

Table 1. Physical-mechanical properties of pore-free sandstone. K 0 , GPa a φ P 0 , MPa ρ , g / cm 3

µ , GPa

T , MPa

σ C , MPa

Y , MPa

σ

α

15.276

0.95

0.0001

3.5

2.54

13.04

37.6

113.6

46

0.5