A New Rock Strength Criterion from Microcracking Mechanisms Which Provides Theoretical Evidence of Hybrid Failure

Brittle rocks, either in their natural state or subjected to external loads, contain a large number of randomly oriented and distributed microcracks. In tension, these cracks are mostly open and will grow in response to locally concentrated stresses particularly at the tips. In that event, crack growth usually experiences an unstable stage toward a brittle failure. However, under increasing compressive stresses, nonlinear mechanical behaviors of the material are essentially governed by local coupling between friction-induced inelastic deformation and cracking-related damage evolution. Proper account of the unilateral effects and damage-friction coupling allows to explain and model lots of nonlinear mechanical phenomena usually observed at laboratory.

Theoretical prediction of macroscopic fracture stresses (strength) has been of great interest in mechanics and engineering science. For quasi-brittle microcracked rocks, the final failure is mainly attributed to damage cumulation due to crack growth. In the literature, various efforts have been made to predict material failure. Firstly, empirical attempts were devoted to setting up failure criteria based on experimental results. We can mention among others the well-known Mohr–Coulomb criterion, the Drucker–Prager criterion (Drucker and Prager 1952) and the Hoek–Brown criterion (Hoek and Brown 1980). However, these criteria are nearly based on mathematically empirical approaches. Starting from experimental values of failure stresses obtained from various loading paths (uniaxial tension and compression, conventional triaxial compression, true triaxial compression, direct shear test, Brazilian indirect tension test, etc.), one tried to determine a reduced form of mathematical criteria which are generally functions of stress invariants and material symmetry properties. Therefore, strength prediction in such a way is completely separated from deformation analysis. Moreover, although capable of defining limiting conditions on failure stresses, these criteria concerned and told very little about physical mechanisms behind macroscopic phenomena as well as specific microstructure evolution closely related to potential failure modes for a given loading path. In particular, the transition between different failure modes, which has been experimentally evidenced by Ramsey and Chester (2004), has never been properly theoretically investigated. Because of the lack of consistent framework, the empirically obtained failure envelopes are often not smooth.

The Griffith failure theory proposed nearly a century ago remains up to now the cornerstone of studying initiation and propagation of a single crack. In his pioneering works, Griffith (1921, 1924) regarded initial cracks as the governing factor of brittle failure and then predicted the tensile strength by investigating the stress state at the tips of flat elliptical cracks. This theory succeeded to a large extent in incorporating material microstructure (cracks) into failure prediction. Later, the Griffith failure theory was extended to various aspects; for example, the extension to penny-shaped cracks (Keer 1966; Margolin 1984), the contributions for brittle fracture under compressive or shear stresses (McClintock and Walsh 1962; Hoek and Bieniawski 1965; Wang and Shrive 1993), the works for Griffith failure in three dimensions (Sack 1946; Kassir and Sih 1967; Hatzitrifon and Gdoutos 1988), the work by Brace (1960) to take into account pore pressure effect, just mention the representative ones. Comparisons of the Griffith theory with other well-known failure criteria have also been performed (Clausing 1959; Brace and Bombolakis 1963; Hoek and Martin 2014). It is worth noticing that in view of the importance of the Griffith theory, Patersson and Wong (2010) included in their book a whole chapter on this topic.

In the Mohr plane, the original Griffith failure function takes the formwith \(\tau\), \(\sigma _{n}\) and \(\sigma _{t}\) representing the norm of the shear stress vector, the normal stress and the tensile strength, respectively. As have been commented by McClintock and Walsh (1962), the criterion (1) presents some obvious deficiencies in predicting compressive strength, arising from the smaller increase in compressive strength as the confining pressure increases. Moreover, the predicted ratio of compressive strength to tensile strength is fixed at eight, but this ratio is found in experiment to vary for different rocks. In order to amend the discrepancy between theoretical predictions by the Griffith criterion and experimental observations, McClintock and Walsh (1962) made their own contributions by accounting for contact conditions and frictional sliding for closed cracks under local normal compression. To be definite, when \(\sigma _{n}<0\), the Griffith failure condition is modified aswhere \(\alpha\) is the coefficient of friction of crack surfaces. When \(\sigma _{n}>0\), the Griffith failure function was assumed to be still valid. It is worthy emphasizing that for the modified failure criterion, although the two parts always intersect at the \(\tau\)-axis, its smoothness takes place only in the particular case where \(\alpha =1\), implying a mathematical inconsistency in this modification.

A criterion describing when material fails is essential, but still far from a complete constitutive modeling of mechanical behaviors of the material. Given a single strength criterion, there is still a need to incorporate into it evolution laws of both crack growth and inelastic strain and to introduce into it a hardening/softening function for achieving a complete material response. Differing greatly from the above common process, Zhu (2016) recently realized strength prediction in combination with deformation analysis for brittle rocks where use has been made of an upscaling method but under isotropic assumptions upon both inelastic strain and damage. As a further extension along the line, the present paper is devoted to setting up a new strength criterion from microcracking mechanisms via a well-developed micromechanics-based anisotropic unilateral damage-friction coupling model (Pensée et al. 2002; Zhu et al. 2008a, b, 2016; Zhu and Shao 2015). The closed-form failure functions are the inherent consequence of a combined unilateral damage/friction process. Not only the purely tensile failure mode but also the tensile-shear failure mode and the compressive-shear failure mode which may present in microcracked rocks are taken into account in a unified and consistent framework. The new criterion offers an enrichment to the original Griffith failure function and also to its modified version at least on three critical aspects: (1) in compression regime, it arises from a local Coulomb-type friction criterion in combination with a strain energy release rate-based damage criterion, both formulated on micro scale. Friction effect is thus taken into account in a proper way; (2) theoretical demonstration for the hybrid failure mode as well as the transition from the tensile failure mode to the compressive-shear failure mode; (3) the smoothness at the transition from the tensile to compressive-shear failure mode is ensured theoretically.

This section briefly presents the micromechanics-based unilateral damage formulations, relying on which we will derive failure functions for microcracked brittle rocks.

As aforementioned, brittle rocks usually contain a large number of randomly oriented and distributed microcracks and experience a nonlinear dissipative process by crack growth, coalescence and localization into a thin band, finally forming a macroscopic shearing plane. Material failure takes place with the occurrence of one or several critical sliding planes inclined at a mathematically determined tip angle. Therefore, in order to derive the failure function, we have to follow such a procedure: find the critical plane and then formulate the failure function. To this end, we distinguish three loading paths usually envisaged in geomechanics.

This section is concerned with the evaluation on the derived failure criterion by comparison with experimental data from triaxial compression tests on typical brittle rocks. The calibration procedure is as follows: calibrate first the two parameters \(\alpha\) and \({\mathcal {R}}\left( d_{c}\right)\) of the compressive failure function because triaxial compression tests are more easily performed to obtain reliable data with respect to tensile tests; next, predict the tensile strength \(\sigma _t\) according to the ratio (58); finally, plot the whole strength envelope and compare it with experimental failure data. It is worthy emphasizing that complete and consistent testing data including the zone of tensile failure were rarely reported in literatures.

We proposed a new way to derive a mechanisms-based failure criterion for brittle rocks weakened by penny-shaped microcracks. Inherent combination of strength prediction with inelastic deformation analyses makes it thoroughly different from the existing ones. The closed-form failure functions corresponding to the different failure modes have been determined using an upscaling method. The strength envelope possesses the \({\mathcal {C}}_1\)-type continuity, which is independent on material properties and state variables. As one of the original contributions, the hybrid fracture and the smooth transition from tensile fracture to compressive-shear fracture have been demonstrated theoretically. More useful remarks are made as follows:

In the presented framework, theoretical derivation of failure criterion has been regarded as one of mandatory parts of developing a complete multiscale constitutive model for cracked rocks. In other words, the criterion we established is the natural result of a set of constitutive equations (e.g., damage and friction criteria, evolution laws of internal variables) but not the starting point to formulate such equations by incorporating evolutions laws as well as hardening/softening functions.

The failure mechanisms behind theoretical derivations are quite clear. The formulations involve two crack geometric states (open or closed), two nonlinear dissipation mechanisms (anisotropic unilateral damage by cracking and inelastic deformation by frictional sliding) and three brittle failure modes (purely tensile mode, tensile-shear model and compressive-shear mode). To the author’s knowledge, failure criteria with these salient features have not ever been reported in literatures.

The present contribution amends the original Griffith failure criterion by taking into account the effect of friction on closed cracks under normal compressive stress, and simultaneously improves the modified Griffith criterion in the sense that the new one ensures theoretically the smoothness at the transition of the different failure modes.

Up to now, the effect of the intermediate principal stress remains a theoretical issue largely open to the scientific community and is also beyond the scope of the present work. In both the damage driving force (10) for open cracks and the friction criterion (11) for closed cracks are involved two stress-fabric invariants: \({\varvec{\sigma }}: {\mathbb {T}}:{\varvec{ \sigma }}\) and \({\varvec{\sigma }}: {\mathbb {N}}:{\varvec{\sigma }}\). However, the fourth-order orientation-dependent tensors \({\mathbb {T}}\) and \({\mathbb {N}}\) do not provide a closed-form decomposition of the fourth-order identity tensor \({\mathbb {I}}\). Incorporation of the invariant \({\varvec{\sigma }}: \left( {\mathbb {I}}-{\mathbb {N}}-{\mathbb {T}} \right) :{\varvec{\sigma }}\) into the constitutive model may cast a new light on the investigation on the effect of the intermediate principal stress.

Future work along this line may include first the account of pore pressure effect on material strength. On this topic, some preliminary results have been achieved under isotropic simplification for both the damage and inelastic strain (Zhu 2016). Moreover, according to experimental results, the ratio of compressive strength to tensile strength may vary within a large range (Hoek and Martin 2014). For some rocks, prediction by Eq. (58) may underestimate this ratio. This could be partially caused by the neglect of initial crack opening in the disturbed samples. In the present model, according to the opening/closure transition condition of microcracks, tensile failure only occurs for open cracks and has to be triggered by applying a macroscopic tensile stress. Thus, further efforts can be made by taking into account an initial normal strain.