PSI - Issue 18

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

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

2.3. Consideration of dilatancy factor

Alejano and Alonso (2005) reported a comprehensive review on considerations of dilatancy angle of di ff erent rocks. For practical applications it is argued that assumption of constant dilatancy angle is unrealistic and material couldn’t exhibit infinitesimal expansion in localization band. Thus, the dilatancy angle and factor must be the functions of accumulated inelastic strain and confinement stress. In this work we utilized a simple equation for dilatancy factor, we ignored the dependence of dilatancy factor on confinement stress and account only for dependence on accumulated inelastic strain: Λ = Λ r + ( Λ 0 − Λ r ) exp( − γ P γ ∗ ) (2) where Λ 0 = 0 . 0625 is a peak dilatancy factor, Λ r = 0 . 001 is a residual dilatancy factor, γ ∗ = 0 . 01 is critical value of intensity of accumulated inelastic strain when dilatancy factor decreases by e . Dilatancy factor turns to residual value when γ P ≈ 4 γ ∗ . The peak dilatancy factor Λ 0 was chosen according to recommendations by Hoek and Brown (1997) for weak rocks. Recommendations were developed initially for rock mass, but they are applicable for small scale rock specimens due to the aforementioned idea of finite dilatancy, which underlies the behavior of rocks. Pore size distribution shows that maximum valuable pore size is ≈ 240 µ m. In order to provide a volume repre sentativeness we took the width W of a computational domain 20 times bigger than the maximum pore size, height to width ratio is H / W = 2, where H is height of computational domain. We used a rectangular shaped grid with cell of 10 × 10 µ m 2 . We utilized the Finite-di ff erence method (FDM) based approach in order to simulate the deformation and failure of sandstone specimens. Numerical simulation is carried out in 2D plain strain formulation. Comprehensive formulation of applied method as well as basic equations could be found in Wilkins (1999). The model also includes equations for yield envelope, damage accumulation kinetics and fracture criterion. In this work we introduced several modifications to the kinetic equation of damage accumulation, e.g. Lode parameter was excluded from the equation. All other pa rameters are physical material constants, except the parameter t ∗ , which is chosen in order to fit the experimental data. Yield envelope is based on the modified Drucker and Prager (1952) model. Modification is related to the utilization of non-associated flow rule and dependence of materials strength constants on accumulated damage. Material is fractured if damage parameter D is equal to 1. A variant of applied model can be found elsewhere Makarov et al. (2014). The most common two types of boundary conditions in numerical simulation are uniaxial loading with restricted sliding and free sliding on the surface of load application. Experimental loading conditions lie between these two types of loading, thus, we performed simulation with two types of boundary conditions: • Free sliding. Nodal velocities are assigned to the mesh nodes belonging to the boundaries B1 and B3: v y = v , if x i ∈ B 1 and v y = − v , if x i ∈ B 3. • Confined sliding. The same as in free sliding, but further v x = 0, if x i ∈ B 1 , B 3. • Lateral boundaries. In both types of loading σ i j n j = 0, if x i ∈ B 2 , B 4. • Pores. σ i j n j = 0, if x i ∈ B 5. A technique of very slow load is applied in calculations in order to maintain the quasi-static conditions of loading. In the next section we provide the results of numerical simulation and discuss them. 3. Mathematical statement of the problem

4. Results of numerical simulation

Fig. 2 illustrates obtained fracture patterns as well as loading diagrams for two types of loading. Both fracture patterns are provided by two failure mechanisms - formation and propagation of tensile and shear cracks. In the case of confined sliding the upper part of specimen is poorly filled with tensile or shear cracks. It is known that near the loading boundary, in the case of restricted sliding, there is a zone of stagnated deformation. The distribution of pores