The present paper responds to the poor treatment given to dilatancy in classic rock mechanics post-failure problems such as tunnel or mine pillar design. An empirical based formulation to estimate the dilatancy angle for rock and rock masses is proposed. It includes marked dependencies on confining stress and plastic parameter - plasticity already suffered by the rock - following different trends coinciding with those already observed in compressive tests and rock masses by different authors. It also includes an indirect scale effect. The model presents the advantage that dilatancy is estimated by only one parameter, taking however into account plasticity and confinement stress dependency. In this way the modeler must include among their inputs a reasonable approach of this parameter instead of a constant, and then more or less unrealistic, value of the dilatancy angle.

1.1 Initial considerations

The main aim of this study is to provide a consistent approach to estimate the dilatancy of a rock mass, which could be reliably applied by rock engineers. Our purpose is neither to obtain highly accurate values or to give a mirror image of reality, nor to contemplate all its elements in their exact proportions, but rather to focus the significant elements of dilatancy in its engineering use (Muir Wood, 1990).

In the last decades with the increase of the use of numerical modeling in rock engineering, many excavation designs rely, at least partially, on numerical studies. A literature review on this topic indicates that the dilatancy angle (ψ) is a parameter seldom considered, and when it is, a poor and simplistic approach is commonly used, consisting on considering either an associated flow-rule (Ф = ψ) or a non-associated flow-rule with ψ = 0°. Recently, Hock & Brown (1997) proposal of Ф = ψ/4, Ф = ψ/8 and ψ = 0 for good, average and bad quality rock masses is starting to be widely used.

Nevertheless, Vermeer & de Borst (1984) have remarked that an associated now rule does not represent post failure rock behavior and Detournay (1986) warned the rock engineers on the possible error observed in many studies when assuming constant dilatancy and proposed a shear plastic strain dependent formulation. It is also a fact the confining stress dependent nature of dilatancy (Shih-Che Yuan & Harrison, 2004).

Starting from a literature review, it is clear that dilatancy is highly dependant on the plasticity suffered by the material, on the confinement, and it seems that scale may play a non-negligible role on the topic (Elliot & Brown, 1986; Medhurst & Brown, 1998; Farmer, 1993).

1.2 Constitutive models

A constitutive model of a rock or a rock mass incorporates a series of stress-strain relationships that mark their stress-strain behavior. Irrespective of the simple elastic part and based on the incremental theory of plasticity, a material is characterized by a failure criterion f and a plastic potential g:

(Equation in full paper)

In the broadest sense - to include hardening or softening - the failure criterion.

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