Arcady Dyskin

Mechanisms of brittle fracture of rock with pre-existing cracks in compression

Fracture of rocks containing a multitude of pre-existing cracks is considered from both theoretical and experimental points of view, paying attention mainly to the underlying mechanisms. The competition between a number of mechanisms in producing tear or shear type fractures is discussed in relation to the properties of the rock and the system of pre-existing cracks on the one hand and the type of loading on the other hand. First, 2-D theoretical models and experimental results aimed at the explanation and description of brittle fracture under compression are considered. Their insufficiency and the necessity to address the 3-D peculiarities of crack growth in rock are shown on the basis of new experimental results on 3-D crack propagation in transparent rock-like brittle materials under uniaxial compression. The results show that in contrast to the 2-D case, a single 3-D crack cannot propagate any appreciable distance and the loading results in dynamic, burst-like failure of the sample. Possible mechanisms of the routinely observed extensive fracture propagation in rock samples (splitting), as well as the possibility of shear (oblique) fracture in uniaxial compression, are discussed in connection with these experiments.

Fracture mechanisms and instability of openings in compression

When drilling circular openings (e.g., tunnels or boreholes) in brittle rock, the in-situ stress conditions are often such that failure is initiated at or near the wall of the opening. In this work, a mechanism of open hole instability is considered based on growth of pre-existing micro-fractures in the direction of greatest compression. The major factor enabling the pre-existing 3-D cracks to propagate extensively is the presence of the intermediate principal compression near the opening wall (in the direction of the opening axis). The unstable growth of wing cracks leads to separating thin rock plates (flakes) from the bulk of the rock mass followed by their buckling, separation, and exposure of the fresh surface. Then this process of rock surface spalling repeats itself eventually changing the shape of the hole. As the opening develops, its shape becomes elongated which, in turn, can affect this mechanism primarily through continuous changes in the stress concentrations around the opening. The sole cause of the unstable phase of crack propagation is the crack–boundary interaction. The opening develops if the unstable crack growth proceeds at least up to the buckling size. Otherwise, the opening shape gets stabilized. This approach also allows for determining the final stable cross-section, as well as its relationship to the applied boundary stresses. The extent of failure is primarily determined by the initial parameters of micro-crack distribution.

Asymptotic analysis of crack interaction with free boundary

Abstract This paper employs the beam and dipole asymptotic techniques for modelling interaction of a crack with parallel free boundaries. Two configurations are considered: (1) a crack in a half-plane and (2) a crack in the centre of an infinite strip. Both, the stress intensity factors and the areas of the crack opening are calculated. For the crack situated close to the boundary, the part of the material between the crack and the boundary is represented by a beam (plate in plane-strain). This allows calculating the area of the crack opening. The stress intensity factors are calculated by matching the beam approximation with Zlatin and Khrapkovs solution (Zlatin and Khrapkov, 1986) for a semi-infinite crack parallel to the boundary of a half-plane or with Entov and Salganiks solution (Entov and Salganik, 1965) for a central semi-infinite crack in a strip. It has been shown that this asymptotic method allows obtaining two leading terms for the SIFs and the crack opening area. When the distance between the crack and the free surface is large, the problem is treated in the far field approximation. This, dipole asymptotic method allows finding the leading asymptotic terms responsible for the crack–boundary interaction. For intermediate distances between the crack and the boundary, simple interpolating formulas are derived. Particular examples of cracks loaded by pair of concentrated forces and for uniform loading are considered. The obtained results are compared with available numerical solutions.

A study of the mechanism of flexural toppling failure of rock slopes

SummaryThe mechanism of flexural toppling failure of jointed rock slopes has been investigated through a series of centrifuge experiments conducted on models manufactured from two types of materials (brittle: a sand-gypsum mixture; and ductile: fibre-cement sheeting). The basal failure plane observed in the centrifuge models, has been found to emanate from the toe of the slope, and orient at an angle of 12 to 20° upward from the normal to the discontinuities. A theoretical model based on a limiting equilibrium approach (Aydan and Kawamoto, 1992) has been adopted to analyse the centrifuge test data. After calibration, the model was found to accurately predict the failure load for all the tests reported in this study. Using this model, a set of charts has been prepared to assist with the analysis of slopes susceptible to flexural toppling.

Influence of shape and locations of initial 3-D cracks on their growth in uniaxial compression

Abstract Mechanisms of 3-D crack development in uniaxial compression have been studied. Several tests undertaken on different transparent casting resin, cement and mortar samples with single internal cracks of different shapes and sizes, produced by different techniques, demonstrated that 3-D crack growth in compression was qualitatively different from the two-dimensional case. Unlike 2-D cracking, there were intrinsic limits on 3-D growth of wing cracks produced by a single pre-existing crack. This limitation was related to the wrapping (curving) of emerging wings around the initial crack. On the other hand, in samples with few initial cracks, a special crack arrangement existed which could produce a large tensile fracture parallel to the direction of loading. Particularly, for two coplanar initial cracks, the tensile fracture was produced only when the inter-crack spacing was below a critical value. In samples where the initial cracks were present in high concentrations, the influence of interaction always generated a number of tensile fractures parallel to the direction of loading.

Crack growth under biaxial compression

An experimental and analytical investigation into three-dimensional crack growth under biaxial compression is presented. Tests were carried out on different materials, including transparent resin samples, each with a single embedded disk-like crack. These cracks grew extensively parallel to the load directions causing splitting. This behaviour is markedly different from that observed under uniaxial compression where the crack growth is limited in size, and is not capable on its own to induce failure. The presence of the intermediate principal compressive stress radically changes the mechanism of crack growth. A model is proposed where the growing crack is represented as a disk-like crack oriented parallel to the loading direction and opened by a pair of concentrated forces at its centre. It is shown that the crack growth is stable until it reaches a size comparable to its distance from the free surface.

A Cosserat continuum model for layered materials

Abstract Modelling the behaviour of materials consisting of a large number of layers is often required in geomechanical applications. In this study, an equivalent continuum model suitable for describing the mechanical response of such layered materials is considered. The model is based on the Cosserat continuum theory and incorporates the moment (couple) stresses in its formulation. The layers are assumed to have equal thickness and equal mechanical properties with elastic behaviour. In contrast to the earlier Cosserat models, the possibility of layer interface (joint) plastic-slip as well as tensile-opening during loading is considered. The importance of moment stress in describing the behaviour of such materials is discussed and highlighted through an example. It is shown through numerous examples that when there is a possibility of inter-layer slip and subsequent layer bending, equivalent continuum models based on the conventional anisotropy theory may not represent a true response of the layered materials. The relationship between the large-scale (Cosserat) description of the layered material and the fine-scale (micro) description of the stress-strain state of an individual layer is determined. The model is incorporated into the finite element (FE) code AFENA and several examples of load-deflection problems in layered materials are analysed. The Cosserat model is verified against the explicit joint FE model. Comparison between the two models shows a remarkable agreement suggesting that the Cosserat model is capable of providing an accurate prediction of the load-deflection behaviour of layered materials.

Topological interlocking of platonic solids: A way to new materials and structures

The structural integrity of natural and engineered materials relies on chemical or mechanical bonding between the building blocks of which they consist. Materials whose building blocks are not joined, but rather interlocked topologically, possess remarkable mechanical and functional properties. We show that identical elements, in the shape of the five platonic solids, can be arranged into layer-like structures in which they are interlocked topologically. It is shown that truncated icosahedra (buckyballs) can also be arranged in a layer with topological interlocking. The geometrical possibility of such assemblies opens up interesting avenues in the design of structures and materials.

Toughening by fragmentation – How topology helps

In this brief overview, we present recent work on the novel concept of topological interlocking as a means of designing new materials and structures. Starting from a special self-supporting arrangement of tetrahedron-shaped elements that was discovered first, we proceed with the introduction of other shapes exhibiting topological interlocking. Unusual mechanical properties of assemblies of tetrahedrons that were studied both experimentally and theoretically are discussed. Possible applications can range from large scale mortar free construction in civil engineering to novel advanced materials based on microscale interlocking elements.

Crack growth criteria incorporating non-singular stresses: Size effect in apparent fracture toughness

Simple criteria accounting for the non-singular stresses at the crack tip are considered. They are based on the comparison of the local stress concentration with the material microstrength. The local stress concentration is estimated either as the magnitude of the conventional elastic stress ahead of the process zone, or by its averaging over the process zone length. When a criterion of this type is used to find the critical load and then the conventional fracture toughness, the latter will be dependent of the crack length. This size effect in fracture toughness manifests itself as an increase (if the non-singular part of the near-tip stress field is positive) or decrease (if the non-singular part is negative) in the apparent fracture toughness as the crack length increases. The obtained dependence is compared with available experimental data. It is also shown that when the load can be resolved into a superposition of elementary loads, the size effect can asymptotically (for long cracks) be presented as a weighted sum of the elementary size effects (i.e. the size effects associated with the elementary loads) with the weights equal to the relative contributions of the elementary loads into the total stress intensity factor.