Cracks In Fractal Materials

The proposed analysis is based on modelling a material with self-similar structure by a continuum sequence of continua of increasing scales each determined by its own size of the averaging volume element. The scaling is represented by power laws with the exponents determined by the microstructure, but not necessarily by the material fractal dimension. The tensorial quantities must scale isotropically (the same exponent for all non-zero components) with only prefactors accounting for anisotropy. This is prescribed by the linear relationships between the tensor components in different coordinate sets and the fact that the power functions with different exponents are linearly independent. Stresses are defined in each continuum (and are measured in conventional units of stress) with the scaling law controlling the transition from one continuum to another, i.e. from one stress field to another. Within each continuum the cracks produce conventional stress singularities. However, as the point of singularity is approached, the transition to finer continua is necessary, resulting in apparent non-conventional stress singularity. A specific case considered is the collective growth of cracks which are already distributed self-similarly. In this regard we consider the crack growth under localized loading. Such a loading models two main situations: (1) tensile cracks growing under the action of heterogeneous stress field generated by material heterogeneities or residual strain and; (2) sliding zones (shear cracks) in a fault system produced by local losses of shear strength over time associated with either the increase in stress or the phenomenon of delayed fracture. If the cracks form self-similar sets then this mechanism of crack growth can either maintain or destroy the self-similarity. It was found that if the cracks are uniformly distributed and isotropically oriented, their self-similar size distribution is maintained. If, however, the cracks are all parallel to each other (transverse isotropic material) the crack growth destroys self-similarity. The situation drastically changes if the parallel cracks are localised in a narrow layer because then the crack growth maintains self-similarity. Thus in order to preserve self-similarity in parallel cracks a mechanism of localisation in multiple crack growth should be at work

Self-similar pattern formation and continuous mechanics of self-similar systems

International audienceIn many cases, the critical state of systems that reached the threshold is characterised by self-similar pattern formation. We produce an example of pattern formation of this kind ? formation of self-similar distribution of interacting fractures. The driving force of the fracture system formation is the crack growth due to the action of stress fluctuations. The importance of this mechanism is that even when the fluctuations have zero average the cracks generated by them could growth far beyond the scale of stress fluctuations. Further development of the fracture system is controlled by crack interaction, which in the case of isotropically oriented cracks leads to the emergence of self-similar distributions. The presence of self-similar distributions of fractures in a material poses a challenge in continuum modelling, since this material becomes discontinuous at any scale. We develop a continuum fractal mechanics to model mechanical behaviour of such materials. We introduce a continuous sequence of continua of increasing scales covering this range of scales. The continuum of each scale is specified by the representative volume elements of the corresponding size over which averaging is performed in the process of defining the field variables in the continuum. Subsequently, at each scale the material is modelled by a continuum that hides the cracks of smaller scales while explicitly introducing larger structural elements. The properties assigned to the continuum are effective characteristics accounting for the macroscopic effect of the hidden cracks. Using the developed formalism we investigate the stability of self-similar crack distributions with respect to crack growth and show that while the self-similar distribution of isotropically oriented cracks is stable, the distribution of parallel cracks is not. For the isotropically oriented cracks scaling of permeability is determined. For the crack distribution produced by the action of stress fluctuations permeability increases as cube of crack radius. This property could be used for detecting this specific mechanism of formation of self-similar crack distributions

Simulation Of Crack Trajectories In Materials With Weak Interfaces

Propagation of cracks in plane elastic bodies with two mutually orthogonal sets of pre-exiting weak interfaces is studied. It is assumed that crack follows the horizontal interfaces while the vertical ones arrest the crack and produce offsets in the crack path. The method of singular integral equations and the Gauss-Chebyshev quadrature for singular integrals is used with the modification that preserves positions of collocations on every crack segment when crack propagates. The stress intensity factors (SIFs) are calculated for different crack trajectories. Based on these calculations it is found that the preferable crack trajectories are the ones that inclined at an angle to the horizontal direction in which the crack were expected to propagate due to symmetry. One can call this phenomenon the symmetry breaking. The exponent of the power law describing the increase of the mode I SIF with the increasing crack length is close to 0.4 instead of 1/2 for the conventional straight crack

Multiscale Modelling Of The Stress Singularity At A Mode I Crack Tip

Direct modelling technique is used to study stresses at the tip of a mode I crack where domain of the material microstructure is explicitly discretised and represented by finite elements. Two cases of internal (at crack faces) and external uniform loading are considered. A multiscale asymptotic model accounting for both the external boundaries and the non-singular stresses at the crack tip are used to analyse the numerical results. It is shown that the parameters (coefficients) of the expansion of the stress concentration at the crack tip can be recovered for both cases of loading from the simulated stress distribution ahead of the crack tip. Applicability of this technique is validated by recovering the parameters for a wider range of heterogeneous material properties

A Theory Of Disclinations For Anisotropic Materials With Bending Stiffness

The paper considers a special type of failure in layered materials with sliding layers that develops as a progressive breakage of layers forming a narrow zone. This zone propagates as a "bending crack", i.e. a crack that can be represented as a distribution of disclinations. This situation is analysed using a 2D Cosserat continuum model. Edge dislocations (displacement discontinuities) and a disclination (the discontinuity in the derivative of layer deflection) are considered. The disclination does not create shear stresses along the axis perpendicular to the direction of layering, while the dislocation does not create a moment stress along the same axis. Semi-infinite and finite bending cracks normal to layering are considered. The moment stress concentration at the crack tip has a singularity of the power -1/4. The possibility to derive equilibrium conditions for cracks and disclinations from J-type path independent integrals is also pointed out

Mechanisms Of Fracturing In Structures Built From Topologically Interlocked Blocks

Failure of materials is in many cases associated with initiation and subsequent propagation of macroscopic fractures. Consequently, in order to increase the strength, one needs to inhibit either crack initiation or propagation. The principle of topological interlocking provides a unique opportunity to construct materials and structures in which both routes of the strength increase can be realised. Materials and structures built on the basis of this principle consist of many elements which are hold together by the special geometry of their shape, together with an external constrain. The absence of the binder phase between the elements allows the interfaces to arrest macroscopic crack propagation. In addition, with sufficiently small size of the elements an increase in local strength and, possibly, in the stress for crack initiation can be achieved by capitalising on the size effect. Furthermore, the ability of some interlocking structures to tolerate missing elements can serve to prevent the avalanche-type failure initiated by failure of one of the elements. In this paper, experimental results and a theoretical analysis with regard to this possibility are presented

Investigation of thin films MgAl2O4, deposited on the Si substrates by vacuum thermal evaporation

The article presents data on the study of X-ray structural and microstructural characteristics of thin films of aluminum-magnesium spinel MgAl2O4 deposited on Si substrates by vacuum thermal evaporation. MgAl2O4 films have a polycrystalline rhombic structure. The values of the unit cell parameters of MgAl2O4 are calculated. Scanning electron and atomic force microscopy showed that MgAl2O4 films have a densely packed structure without cracks. Physical characteristics and good adhesion of MgAl2O4 thin films to silicon substrates indicate their possibility of using in devices of opto- and microelectronics

A Criterion for Brittle Failure of Rocks Using the Theory of Critical Distances

This paper presents a new analytical criterion for brittle failure of rocks and heavily overconsolidated soils. Griffith’s model of a randomly oriented defect under a biaxial stress state is used to keep the criterion simple. The Griffith’s criterion is improved because the maximum tensile strength is not evaluated at the boundary of the defect but at a certain distance from the boundary, known as the critical distance. This fracture criterion is known as the Point Method, and is part of the Theory of Critical Distances, which is utilized in fracture mechanics. The proposed failure criterion has two parameters: the inherent tensile strength, ó0, and the ratio of the half-length of the initial crack/flaw to the critical distance, a/L. These parameters are difficult to measure but they may be correlated with the uniaxial compressive and tensile strengths, óc and ót. The proposed criterion is able to reproduce the common range of strength ratios for rocks and heavily overconsolidated soils (óc/ót=3-50) and the influence of several microstructural rock properties, such as texture and porosity. Good agreement with laboratory tests reported in the literature is found for tensile and low confining stresses.The work presented was initiated during a research project on “Structural integrity assessments of notch-type defects", for the Spanish Ministry of Science and Innovation (Ref.: MAT2010-15721)