Numerical simulations were executed using COMSOL Multiphysics. The fundamental local force balance (equilibrium) equation was used based on the continuum mechanics concepts as follows: ∇σ+F=0 (1 where σ is the Cauchy stress tensor and F represents the body/external force per unit volume. For an isotropic solid exhibiting linear elasticity, the governing stress–strain interaction is formally described by Hooke’s law, which provides the mathematical foundation for relating applied stresses to the resulting deformations. σ=C:ε (2 where C corresponds to the elasticity tensor of fourth order, which encapsulates the stiffness characteristics of the material, while denotes the infinitesimal strain tensor. The strain tensor is formulated directly from the displacement field , serving as the mathematical expression of how small variations in displacement translate into local deformations of the medium. It is formally defined as: ε= 1 2[(∇u)+(∇u)T] (3 For a linear, isotropic, elastic medium, the stress–strain relation is governed by Hooke’s law: σ= E(1+ν ) + Eν (1+ν )(1-2ν ) tr ( ε ) I (4 where E is the Young’s modulus, ν is the Poisson’s ratio, tr(ε) represents the volumetric strain, and I is the identity tensor. For the purposes of numerical analysis, the kimberlite deposit was designed as a cylindrical body with a diameter of 137.5 m and a vertical extent of 400 m. This ore body was positioned at the center of a larger granite block measuring 400 m by 400 m in plan and 450 m in depth, thereby creating a symmetric host rock environment around the modeled deposit, as illustrated in Figure 1.Within this configuration, two vertical shafts were introduced. The first shaft was excavated through the central axis of the kimberlite body, while the second was located adjacent to it, enabling the study of excavation interactions and the evaluation of stability when applying the VCM.
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