Mathematical Physics - Volume II - Numerical Methods

Chapter 6. Introduction to Computational Mechanics of Discontinua

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velocities of the grains; see the original reference [128] for details. The behavior of the parallel bond mimicking cement behavior is defined by five parameters: normal and shear stiffness coefficient ( ˆ k n , ˆ k t ), the tensile and shear strength ( ˆ σ m , ˆ τ m ), and multiplier ˆ λ which defines the diameter of the parallel bond depending on the diameter of the circular elements in contact. The interaction of the grains is represented by the total force and moment, ˆ f i j and ˆ M i j , transmitted by the parallel bond (Figure 6.20b). The force and moment can be projected in the directions of normal and tangent in the following way ˆ f i j = ˆ f n i j n i j + ˆ f t i j t i j , ˆ M i j = ˆ M n i j n i j + ˆ M t i j t i j (6.70) where ˆ f n i j , ˆ f t i j denote the normal and shear forces, and ˆ M n i j , ˆ M t i j the twisting and bending (rolling) moments (naturally, there are two of the later in 3D), respectively. (Note that ˆ M n i j ≡ 0 for 2D models and the bending moment acts in the out-of-plane direction.) At the initialization of the parallel bond, ˆ f i j and ˆ M i j are set to zero; each subsequent relative increment of translation and angle of rotation ( ∆ u n i j , ∆ u t i j , ∆ θ i j = ( ω j − ω i ) ∆ t ) lead to the corresponding increase in the components of force and moment (Figure 6.20) ∆ ˆ f n = ˆ k n A ∆ u n , ∆ ˆ f t = − ˆ k t A ∆ u t , ∆ ˆ M n = − ˆ k t J ∆ θ n , ∆ ˆ M t = − ˆ k n I ∆ θ t (6.71) which are added to the current values. The geometrical properties of the parallel-bond cross section in (6.71) — area ( A ), axial ( I ) and polar ( J ) moment of inertia — are defined by well-known expressions in terms of the parallel-bond radius, ˆ R , for the 2D ( PFC 2 D ) and 3D ( PFC 3 D ) models. The maximum normal and shear stresses acting on the circumferences of the parallel bond are calculated using the elementary beam theory ˆ σ max = − ˆ f n A + | ˆ M t | I ˆ R , ˆ τ max = − | ˆ f t | A + | ˆ M n | J ˆ R (6.72) If the value of the maximum normal stress (6.72) 1 exceeds the tensile strength ( ˆ σ max ≥ ˆ σ m ) or the maximum shear stress (6.72) 2 exceeds shear strength ( ˆ τ max ≥ ˆ τ m ), the parallel bond is broken and removed from the model, which corresponds to the nucleation of the tension/shear mesocracks. The cracking of the cement reduces the contact of the corresponding pair of grains (e.g., the glass beads in Figure 6.1a) to the usual non-cohesive interaction with friction. Potyondy and Cundall [128] demonstrated the ability of this model to reproduce a number of rock behavior characteristics such as fracture, damage-induced anisotropy, dilatation, softening, and confinement-driven strengthening. The evolution of damage is explicitly presented as a process of progressive accumulation of broken ties; “no empirical relations are needed to define damage or to quantify its effect on material behavior ” [128]. The obtained damage patterns agree well with the experimental observations and reveal some subtle details of the influence of lateral confinement. As for the model’s ability to reproduce the macroproperties of granite; the modulus of elasticity, Poisson’s ratio and uniaxial compressive strength were reproduced with satisfactory accuracy (especially in the case of 3D models). On the other hand, the tensile strength and friction angle obtained by the simulations show large discrepancies with the