The mechanics of cracks and a statistical strength theory for rocks

doi:10.1016/S1365-1609(99)00074-X

International Journal of Rock Mechanics and Mining Sciences

Volume 37, Issue 3, 1 April 2000, Pages 477-488

https://doi.org/10.1016/S1365-1609(99)00074-X Get rights and content

Abstract

Features of the crack structure of intact rock are systematically investigated by using advanced video microscope, electronic microscope and graphic analysis technology. The fractal effects of the discontinuous structure on the macro-mechanical behavior are discussed. Damage evolution in rocks is described by means of statistical mathematics. Based on the fractal distribution of cracks and on the theory of the weakest link, a general statistical formula of rock strength under a complex stress state has been developed, of which the influence of orientation distribution of cracks and irregularity of crack growth on rock strength has been taken into account. The new expression for rock strength reflects the collective effects of the scale of crack distribution, crack orientation distribution and the irregularity of crack surfaces on rock strength. It relates the macro-strength of rocks to its intrinsic structural properties. It is shown that the Weibull strength formula is a special case of the statistical strength expression when the orientational distribution of cracks is neglected.

Introduction

Rock mechanics, as a branch of solid mechanics, mainly concentrates on the mechanical behavior and mechanical properties of rocks such as deformation, strength and failure under the effects of the surrounding physical conditions and engineering environment. At the present time, the authors consider that the physical and mechanical behavior of rocks cannot be effectively estimated because of the extreme complexity of rocks. The frequent catastrophes in rock engineering such as rockbursts, coal and gas bursts, roof caving, water bursts, slope failures, dam fracture, collapses of ground surface and earthquakes indicate that the understanding and knowledge of the physical and mechanical behavior of rocks still remains in the primary stage and cannot yet fully meet the needs of engineering design. As a result, the further understanding of the mechanisms of fracture and the laws of damage evolution in the micro- and meso-¹scale has theoretical and practical significance.

The complexity of rock mechanics arises from two aspects. One is the complexity of the rock structure; whilst the other is the complexity of the surrounding engineering environment. In terms of the former aspect, there exist a large number of defects in rocks such as dislocations, crystal boundaries, secondary phases, twin crystallites, fissures, pores, inclusions and precipitates. In fact, the rock is discontinuous, non-linear, inhomogeneous and anisotropic in its mechanical behavior — because of the extreme irregularity of the scale distribution and the spatial distribution of cracks. In terms of the latter aspect, the complexity of the geological environment, unfortunately, the geological stress cannot be estimated accurately because of the unknown factors of rocks. In addition, it is not easy for researchers to determine the spatial distribution of faults and cracks in rocks which obviously influence the mechanical behavior of rocks. In one sentence, the extreme complexity of rocks is a serious challenge for classical mechanical theories.

How does one describe quantitatively the discontinuous structure of rocks, how does one estimate the production, development and devolution of cracks and how does one establish the relation between the macro-behavior and the meso-structure of rocks? These are some of the current concerns of rock mechanics. The solutions of these difficulties will improve the state of solid mechanics applied to rocks in the 21st century.

The term ‘meso-mechanics’ is used here. It is encountered in many different subjects such as mechanics, physics, material science and chemistry, and is a new and active branch of mechanics. It is developed primarily on the basis of the theory of plasticity and creep fracture of metals. Rock meso-mechanics did not gain popularity until the 1960s. In the 1960s and 1970s, rock mechanics was limited in scope to deformation and failure of rocks in which the optical microscope was employed for measurements, especially in the structural geology area. For example, Ossibov [1] researched the influence of micro-structure on the strength of clay. Nicolis et al. [2] discussed the plastic characteristic and the creep behavior of metamorphic rock. Nolen-Hoeksema [3] researched the process of propagation of cracks in marble with increasing of applied loads.

In the 1970s, the wide use of the scanning electron microscope (SEM) and tunneling electron microscope (TEM) made it possible to measure the deformation and failure of materials on the micro-scale and improve the meso-mechanics understanding. Many achievements were made at home and abroad.

Perie [4] discussed the micro-mechanisms of crack propagation. Nagahama [5] proposed correlations for roughnesses of a fracture plane, particle size distribution and energy distribution. Sakellariou et al. [6] studied the self-affine property of fracture planes. Gentier et al. [7] studied the problem of fracture resulting from the evolution of pores. Dyskin and Germanovich [8] suggested a model of crack propagation for damaged rock. Yokoboyi et al. [9] proposed a concept of micro- and meso-coupled fracture. Wang and Scholz [10] discussed the scaling laws for the parameters of a fractal plane. Wu and Chudnovsky [11] discussed the influence of the distribution of micro-cracks on the stress factor of macro-cracks. Dahl and Dormagen [12] researched the micro-mechanisms of the initiation and propagation of cracks.

In China, many achievements have also been made. Zhang [13] proposed that the cracks in rocks appear to be extensive. Xu and Li [14] measured the whole process of crack propagation using an optical microscope. Xie employed fractal geometry to systematically investigate the micro-fracture of rocks, propagation and branching of cracks, rockbursts, roughness of rock joints, contact mechanics of joints and evolution of damage [15], [16], [17], [18], [24], [25], [26], [27], [28]. Hu and Qu [19] measured the micro-structure of clay and freezing soil [19]. Zhao and Wang [20] also measured the generation and propagation of micro-cracks and discussed some physical and mechanical phenomena associated with earthquakes.

However, the difficulty is to establish the bridge between the macro-mechanical behavior and the meso-structural property of rocks. Research work in this field is still at an early stage. A mature and universal theory for the meso-mechanics of rocks has not yet been proposed. Our research indicates that an effective approach is to deal with complicated rock meso-mechanics by using fractal geometry to characterize the irregularity of the rock meso-structure and use stochastic mathematics to describe the indeterminate rock mechanics behavior.

Section snippets

Dislocations in a rock crystal

Some rocks are a complex crystal material in which there exist a great number of defects with a complex spatial distribution. The TEM is employed to measure the micro-structure of thin pieces of different standard rock samples in the different stages of stress under uniaxial and triaxial compression. Our experimental results show that there are a great number of complex dislocations in rocks. The density of dislocations rapidly increases with the increase of the applied load. When a group of

Dislocation model for the micro-fracture of rock

The theory for crystal damage shows that if the stress exceeds the critical stress of the material, the dislocations are active and slide along the crystal interface. When the propagation of dislocations is encumbered by obstacles such as crystal boundaries, secondary phases and the other groups of dislocations, a dislocation pile-up will be produced and a high stress zone will be caused. With the increase of applied load, the high stress is released in two ways: the front dislocations produce

Stochastic model of crack propagation and its general solution

Due to the inhomogeneity of the meso-structure of rocks, the crack propagation in rocks is a stochastic process. It can be divided into two parts. One is the transition crack growth A(c, t), which is concerned with the average structure effect. The other is the fluctuation crack growth F(c, t), which is related to the irregularity of the rock meso-structure. Thus, the crack growth velocity can be written as: $c ̇ =A(c, t)+F(c, t)$ Because of the high degree of the irregularity of the rock

Classical statistical strength model

Suppose that the material is homogeneous and isotropic statistically, and unstable propagation at the most critical crack leads to total failure of the specimen, i.e. a weakest link model. Then the distribution of the fracture strengths of cracks can be approximated well by the three-parameter Weibull distribution. Its probability density function f(σ) is defined as $f(σ)= m σ_{0} σ−σ_{th} σ_{0}^{m−1} exp − σ−σ_{th} σ_{0}^{m} σ≥σ_{th} 0 σ<σ_{th}$ where σ_th is the threshold value of stress, usually taken to be zero for brittle

Conclusions

In this paper, the internal meso-structural features of rocks have been systematically investigated using the advanced video microscope, the electronic microscope and graphic analysis technology. Physical mechanisms for damage and fracture, and the stochastic damage evolution of rocks, have been analyzed. Based on the fractal distribution of meso-cracks in rocks, a general statistical formula of rock strength under a complex stress state has been deduced, which takes into account the influence

Acknowledgements

The present research is supported by the National Distinguished Youth’s Science Foundation of China, the Trans-Century Program for the Talents by the State Education Commission of China and the National Natural Science Foundation of China (19702020).

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The mechanics of cracks and a statistical strength theory for rocks

Abstract

Introduction

Section snippets

Dislocations in a rock crystal

Dislocation model for the micro-fracture of rock

Stochastic model of crack propagation and its general solution

Classical statistical strength model

Conclusions

Acknowledgements

Int. J. Rock Mech. Min. Sci. and Geomech. Abstr.

Int. J. Rock Mech. Min. Sci. and Geomech. Abstr.

Int. J. Rock Mech. Min. Sci. and Geomech. Abstr.

Int. J. Rock Mech. Min. Sci. and Geomech. Abstr.

Int. J. Rock Mech. Min. Sci. and Geomech. Abstr.

Int. J. Rock Mech. Min. Sci. and Geomech. Abstr.

Theor. and Appl. Fract. Mech.

Int. J. Rock Mech. Min. Sci.

Chaos, Solitons and Fractals

Essence of strength and deformation of clay and rock

Crystalline plastics and solid state flow in metamorphic rock

Determination of fracture mechanism by microscopic observation of crack

Int. J. Rock Mech. Min. Sci. and Geomech. Abstr.

Laboratory testing of the voids of fracture

Rock Mech. and Rock Engng

A concept of combined micro and macro fracture mechanics to brittle fracture

Int. J. Fract.