Development of a 3-D confinement-dependent dilation model for brittle rocks; Part 1, derivation of a Cartesian plastic strain increments ratios approach for non-potential flow rules

Abstract

The viability and success of a deep mining or tunnelling project is dependent on a safe, reliable support design. In highly-stressed brittle rock, this design must properly account for the excessive deformations and bulking that can occur due to stress-induced extensional fracturing (referred to as spalling or slabbing) near the excavation boundary. Important is the recognition that stress-induced brittle fracturing is dominated by an extensional fracture mode with a directional dilation component that is highly sensitive to 3-D confinement. This behaviour differs from that represented by popular elastoplastic yield and dilation models used in commercial numerical modelling software (e.g., Mohr-Coulomb) that have been developed based on experiences involving ductile behaviour and shear in soils and weak rocks. Recent experiences suggest that these are inappropriate for robust design in brittle rock and can be dangerously misleading. The importance of the dilation model and its implementation through a flow rule – which determines the relative incremental plastic straining under a specific stress state and at a specific plastic straining history – has particularly been overlooked. Numerical analyses in most cases adopt the oversimplifying assumption of a uniform dilation model (that is independent of confinement and plastic strain history), using as input a constant dilation angle ψ. This imposes significant limitations in modelling the extent of the yielding zone and the resulting displacements. More advanced dilation models exist. However, they suffer from two key deficiencies: i) most do not account for the influence of the intermediate principal stress, σ2, on the directionality and magnitude of dilation; and ii) many require numerous empirical parameters or variables that do not have physical meaning (neither phenomenologically, nor micro-mechanically), making it difficult for practitioners to intuitively understand the influence and sensitivity of each parameter on the modelled response. This paper, presented in two parts, reviews the mathematical expression of the flow rule and its physical meaning in relation to brittle fracturing, and addresses the deficiencies in current dilation models by developing a new non-potential flow rule that accounts for the directionality and 3-D confinement-dependency of dilation (i.e., both σ3- and σ2-dependency), and uses parameters/variables that have physical meaning. In Part 1, we present a new framework for understanding and modelling the 3-D directionality of plastic deformations in Cartesian coordinates using what we call Plastic Strain Increments Ratios (PSIRs). The 3-D PSIRs are phenomenologically meaningful as they describe relative plastic straining. More importantly, we show that they are micro-mechanically meaningful as they associate the 3-D directionality of dilation with the different 3-D fracturing modes observed in intact brittle rock under various polyaxial stress states. In the Part 2 companion paper, we use these PSIRs to develop and derive the formulation of a new dilation model involving a non-potential 3-D confinement-dependent flow rule.