Using the Pietruszczak-Mroz anisotropic failure criterion to model the strength of stratified rocks

doi:10.1016/j.ijrmms.2020.104312

International Journal of Rock Mechanics and Mining Sciences

Volume 130, June 2020, 104312

https://doi.org/10.1016/j.ijrmms.2020.104312 Get rights and content

Abstract

The strength of stratified rocks is closely related to their inherent structural anisotropy, which cannot be accurately represented by the isotropic failure criteria. In this study, the Pietruszczak-Mroz (PM) anisotropic failure criterion was used to describe the directional dependency of strength of stratified rocks. In the meridian plane, the frictional coefficient was established by incorporating an anisotropic variable defined as a joint invariant of fabric and stress tensors. In the deviatoric plane, the anisotropic failure surface was characterized by a smooth shape function of Lode's angle. The material parameters for the failure criterion can be determined using the triaxial test. The PM anisotropic criterion was utilized to predict the anisotropic strength of specific stratified rocks. Analysis of the experimental data validated the assertion that PM anisotropic criterion has excellent ability to model the anisotropic strength features of stratified rocks.

Introduction

Naturally, sedimentation of flaky and lathy mineral grains has produced rocks exhibiting inherent stratified structure, which is the main cause of rock anisotropy.¹^,² Typically, stratified rocks are considered to be transversely isotropic (cross-anisotropic).³ Unlike the isotropic materials, stratified rocks display different strength behaviors related to the loading direction, with respect to the bedding plane.⁴ Thus, the classical isotropic failure criteria, such as Mohr–Coulomb, Drucker-Prager, and Hoek–Brown, are not suitable for describing the anisotropic strength of stratified rocks. In predicting the direction-dependent strength of stratified rocks in the practice of geotechnical engineering, it is crucial to develop anisotropic failure criterion.

The conventional approach was to modify an existing isotropic criterion by adopting different cohesion and friction angles to describe the orientation-dependent strength behaviors of anisotropic geomaterials. With this approach, the cohesion and friction angle of the anisotropic criteria were frequently regarded as function of orientation of bedding planes.5, 6, 7, 8, 9, 10, 11, 12 These functions, however, were largely empirical and lacked the essential support of physical implications.

As revealed by a significant number of physical experiments, it is the existence of a bedding structure that results in the strength discrepancy associated with loading direction.⁹^,¹⁰^,13, 14, 15, 16 Consequently, researchers⁴^,17, 18, 19, 20, 21, 22 have attempted to construct their own anisotropic failure criteria with serious mathematic consideration on the mechanical effects of bedding planes. For instance, Pietruszczak and Mroz¹⁷^,¹⁸ introduced a fabric (microstructure) tensor to describe the bedding structure of geomaterials and defined an anisotropic variable as a joint invariant of the fabric and stress tensors. The anisotropic variable could comprehensively reflect the structural and mechanical effects of the bedding structure. An anisotropic strength theory based on the anisotropic variable was then developed by Pietruszczak and Mroz.¹⁷^,¹⁸ Within this theoretical framework, Pietruszczak and his co-authors17, 18, 19, 20 proposed a series of Pietruszczak-Mroz (PM) anisotropic failure criteria.

Inspired by the concept of fabric tensor, Lade²³ proposed a general 3D anisotropic failure criterion for cross-anisotropic frictional materials; Xiao et al.²⁴ and Kong et al.²⁵ developed the SMP (Spatially Mobilized Plane) cross-anisotropic failure criteria; Gao et al.²⁶ proposed a generalized anisotropic failure criterion for geomaterials; Parisio and Laloui²⁷ combined the Van Eekelen model with PM anisotropic variable to formulate a VEPM anisotropic failure criterion. These criteria have been successfully used for predicting the strengths of a variety of anisotropic geomaterials.

PM anisotropic strength theory has now become a foundation for building the anisotropic constitutive models. Pietruszczak, Lydzzba and Shao²⁸^,²⁹ proposed anisotropic elastoplastic and creep models for inherently anisotropic rocks and can be traced back to related researches. Based on the concept of fabric tensor, Chen et al.³⁰ proposed anisotropic constitutive models to explore the coupled elastoplastic damage behaviors of anisotropic rocks. Moreover, on the micromechanical level, Shen and Shao³¹ presented a micromechanical model for modelling the elastoplastic behavior of anisotropic sedimentary rocks. Pietruszczak and Oboudi³² and Chen et al.³³ had studied the induced anisotropy and damage behaviors of soils and rocks, respectively.

The aforementioned researchers28, 29, 30, 31^,³³ mainly conducted their constitutive modelling on a type of Tournemire shale in France. In terms of practical utilization, Parisio and Laloui²⁷ adopted PM anisotropic failure criterion to simulate the conventional triaxial compressive strengths of a variety of stratified rocks. Although they extended applications of the PM anisotropic failure criterion, the prediction of true triaxial compression strength under complex stress-loading states remains to be developed.

This is exactly the aspect of applied research that this paper is focused on. This study reviewed the PM anisotropic failure criterion to present the combined effects of stress and bedding structure on the anisotropic strength. Then, applicability of the PM anisotropic failure criterion in predicting the anisotropic strengths was verified by comparing with the conventional and true triaxial tests on several stratified rocks. More practically, this study can provide an example for the engineering application of the PM anisotropic failure criterion.

Section snippets

Fabric tensor and loading direction

Pietruszczak and Mroz¹⁷^,¹⁸ suggested a fabric tensor $a_{i j}$ to quantitatively describe the anisotropic structure in geomaterials. As shown in Fig. 1a, the unit vectors $v_{i}$ , $s_{i}$ and $t_{i}$ ( $i = 1, 2, 3$ ) denote the three principal axes of a stratified rock material. Then, using a spectral decomposition, $a_{i j}$ can be expressed as follows¹⁷^,¹⁸: $a_{i j} = a_{v} v_{i} v_{j} + a_{s} s_{i} s_{j} + a_{t} t_{i} t_{j}$ where, $a_{v}$ , $a_{s}$ and $a_{t}$ are the three principal values of $a_{i j}$ .

Stress space and loading direction

As shown in Fig. 1a, $σ_{i j}$ represents the stress tensor at a point. Then, the magnitudes of

Formulations of the PM anisotropic failure criterion

In this study, the mathematical expression of the PM anisotropic failure criterion is presented in Eq. (18), which is formulated by Pietruszczak and Haghighat.¹⁹ $F = q - k_{η} g (θ) (p + C) = 0$ where $p$ , $q$ and $θ$ are the mean stress, deviatoric stress and Lode's angle, respectively. They are defined as follows: $p = \frac{1}{3} σ_{k k}, q = \sqrt{3 S_{i j} S_{i j} / 2} and θ = \frac{1}{3} \sin^{- 1} (\frac{27 J_{3}}{2 q^{3}})$ where, $S_{i j} = σ_{i j} - p δ_{i j}$ denotes the deviatoric stress tensor. $J_{3} = \det S_{i j}$ is the third invariant of the deviatoric stress tensor.

Additionally, in Eq. (18), $g (θ)$ describes

Evaluations of the PM anisotropic failure criterion

A series of previously published test data for different stratified rocks were used to evaluate the PM anisotropic failure criterion. Before evaluations, the material parameters required in the PM anisotropic failure criterion had been firstly determined. With the material parameters, this criterion was then used to predict the anisotropic strengths of the stratified rocks. Modelling results were plotted using solid or dashed curves and compared against the experimental data, which were shown

Discussion and conclusion

Under the conventional triaxial condition ( $σ_{2} = σ_{3}$ ), the PM anisotropic failure criterion can appropriately predict the anisotropic strength of stratified rocks influenced by confining pressure $σ_{3}$ and bedding plane dip angle $β$ . The modelling results agree well with the conventional triaxial data.

As for the true triaxial tests with constant $σ_{3}$ , the PM anisotropic failure criterion adequately captures the effects of bedding plane dip direction $α$ , dip angle $β$ and intermediate principal stress $σ_{2}$ on

Declaration of competing interest

We solemnly declare: No conflict of interest exits in the submission of this manuscript.

Acknowledgements

The work presented in this paper was financially supported by the National Natural Science Foundation of China (Grant No. 51609070, 51708199) and by Fundamental Research Funds for the Central Universities (Grant No. 531118010069). We would like to express gratitude to the anonymous reviewers for their valuable comments, which have greatly improved this research.

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Layered rocks (LR) are composed of intact rock matrix (IRM) and various oriented rough weakness planes (e.g., fractures, bedding planes, and foliations). New microcracks can also be created and propagate in IRM during subsequent loading history. Effects of induced microcracks and pre-existing weakness planes on macroscopic mechanical behavior of LR are usually investigated separately. The novelty of the present work is to develop an original multi-scale damage approach to estimate the macroscopic anisotropic elastic properties and failure strength of LR. Two material scales are taken into consideration. At the macroscopic laboratory scale, a set of parallel rough weakness planes are embedded in the IRM, while at the microscopic scale, the IRM contains an elastic solid phase and randomly distributed microcracks. By means of a two-step homogenization technique, the effective elastic properties of LR is first determined. A multi-scale friction-damage model (MFDM) is further developed, by accounting for isotropic growth of closed microcracks and anisotropic evolution of rough weakness planes at two different scales. Subsequently, with the concept of critical damage, an analytical macroscopic strength criterion incorporating failure mechanisms of IRM and weakness planes is derived as an inherent result of the MFDM. Compared with experimental data, it shows that the macroscopic anisotropic elastic properties of LR are well estimated. However, the derived macroscopic strength criterion only provides a qualitative prediction of the anisotropic failure strength of LR for different orientations of weakness planes in conventional triaxial compression tests. By taking advantage of the fabric tensor approach, the strength criterion is further improved to take into account interaction between weakness planes and microcracks. The improved strength criterion is able to quantitatively capture the failure property of LR.
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Citation Excerpt :
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Using the Pietruszczak-Mroz anisotropic failure criterion to model the strength of stratified rocks

Abstract

Introduction

Section snippets

Fabric tensor and loading direction

Stress space and loading direction

Formulations of the PM anisotropic failure criterion

Evaluations of the PM anisotropic failure criterion

Discussion and conclusion

Declaration of competing interest

Acknowledgements

Anisotropy is everywhere, to see, to measure, and to model

Rock Mech Rock Eng

Experimental investigation of the strength and failure behavior of layered sandstone under uniaxial compression and Brazilian testing

Acta Geophys

Assessment of inherent anisotropy and confining pressure influences on mechanical behavior of anisotropic foliated rocks under triaxial compression

Rock Mech Rock Eng

An anisotropic strength criterion for jointed rock masses and its application in wellbore stability analyses

Int J Numer Anal Methods GeoMech

Assessment of some failure criteria for strongly anisotropic geomaterials

Mech Cohesive-Frict Mater

Shear failure of anistropic rocks

Geol Mag

A modified single plane of weakness theory for the failure of highly stratified rocks

Int J Rock Mech Min Sci

A failure criterion for transversely isotropic rocks

Int J Rock Mech Min Sci

The mechanical behavior of anisotropic sedimentary rocks

Journal of Engineering for Industry

A nonlinear criterion for triaxial strength of inherently anisotropic rocks

Rock Mech Rock Eng

New empirical polyaxial criterion for rock strength

Int J Rock Mech Min Sci

A modified empirical criterion for strength of transversely anisotropic rocks with metamorphic origin

Bull Eng Geol Environ

Mechanical and elastic properties of transversely isotropic slate

Rock Mech Rock Eng

Anisotropic strength and deformational behavior of Himalayan schists

Int J Rock Mech Min Sci

Laboratory investigation of the mechanical behaviour of Tournemire shale

Int J Rock Mech Min Sci

Experimental Rock Mechanics

Formulation of anisotropic failure criteria incorporating a microstructure tensor

Comput Geotech

On failure criteria for anisotropic cohesive-frictional materials

Int J Numer Anal Methods GeoMech