Cracks In Solids Research Articles

The contact problem for an open penny-shaped crack under normally incident tension-compression wave

This paper concerns fracture dynamic problems for elastic cracked solids with allowance for crack faces contact interaction. The contact problem for a penny-shaped crack with an initial opening under normally incident tension-compression wave is solved by the method of boundary integral equations. The contact forces and the displacement discontinuity of the crack faces are studied. The solution is compared with those obtained without allowance for crack faces contact interaction for various shapes of the initial opening.

Effect of Contact Interaction on the Stress Intensity Factors for a Crack under Harmonic Loading

This paper concerns fracture dynamic problems for elastic cracked solids with allowance for crack faces contact interaction. The contact problem for a penny-shaped crack with an initial opening under normally incident tension-compression wave is solved by the method of boundary integral equations. The solution is compared with those obtained without allowance for crack faces contact interaction for various values of the initial opening.

The influence of crack shape on the effective elasticity of fractured rocks

A circle is the basic fracture shape adopted by conventional effective media theories to describe the overall elasticity of cracked solids. Fractures in rocks do not resemble circles, so it is important to find out to what extent the available theoretical results apply to realistic fracture shapes. To address this issue, we conduct 3D numerical experiments on the so-called digital rocks containing irregular cracks that might be partially closed and might intersect each other. Despite profound deviations of our fracture geometries from circles, we find that basic theoretical results originally developed for penny-shaped cracks remain valid for arbitrary fracture shapes. Based on a series of finite-element computations, we show that as far as the effective elasticity is concerned, flat fractures with random in-plane irregularities are represented accurately by circular ones. We also show that approximate effective elliptical orthotropy established for multiple sets of dry, penny-shaped cracks embedded in isotropic host rock holds with the same accuracy for irregular, possibly intersecting fractures.

Two-dimensional time-harmonic BEM for cracked anisotropic solids

A mixed time-harmonic boundary element procedure for the analysis of two-dimensional dynamic problems in cracked solids of general anisotropy is presented. To the author's knowledge, no previous BE approach for time-harmonic two-dimensional crack problems in anisotropic solids exists. In the present work, the fundamental solution is split into the static singular part plus dynamic regular terms. Hypersingular integrals associated to the singular part in the traction boundary integral equation are transformed, by means of a simple change of variable, into regular ones plus very simple singular integrals with known analytical solution. Subsequently, only regular (frequency dependent) terms have to be added to the regularized static fundamental solution in order to solve the dynamic problem. The generality of this procedure permits the use of general straight or curved quadratic boundary elements. In particular, discontinuous quarter-point elements are used to represent the crack-tip behavior. Stress intensity factors are accurately computed from the nodal crack opening displacements at discontinuous quarter-point elements. The efficiency and robustness of the present time-harmonic BEM are verified numerically by several test examples. Results are also obtained for more complex configurations, not previously studied in the literature. They include curved crack geometry.

Mechanical behaviour of a solid with many stress corrosion growing cracks

The theory of effective elastic properties of cracked solids is used to predict the degradation of mechanical properties of a stainless steel induced by stress corrosion growing cracks. The material is considered as an evolutionary composite with a cracked external volume (damaged elastic behaviour) and a safe internal volume (elastoplastic behaviour). From continuous surface observation of cracks by in situ videomicroscopy, the influence of interactions between multiple cracks is particularly studied. We show that coalescence of cracks can be neglected and that amplification and shielding effects counterbalance each other. This experimental study allows to choose the most suitable method to determine the effective elastic properties. Then, we propose a modelling to represent the overall mechanical behaviour of a solid with many surface cracks. Simulated and experimental slow strain rate curves are compared.

Modification of the fracture criterion for V-shaped notches (plane problem). Relationship between toughness and strength and structural parameters

A fracture criterion of the type of the Neuber-Novozhilov criterion is proposed to describe the fracture in the vicinity of the tip of a V-shaped notch under tensile and shear loading. In the proposed criterion, the limits of averaging of the stresses along the notch axis depend on the presence, location, and size of the initial defects in the material. The crystal lattice parameter of the initial material is chosen for the characteristic linear size. For a V-shaped notch subjected to tension and shear, simple equations are obtained that relate the stress intensity factors for the modified singularity coefficients, the singularity coefficients themselves, and the theoretical tensile and shear strengths of a single crystal of the material taking into account the damage to the material in the vicinity of the notch tip. The equations obtained allow a passage to the limit from a notch to a crack. It is shown that the classical critical stress intensity factor used in the strength analysis of cracked solids is not a material constant.

Surface waves in a general anisotropic poroelastic solid half-space

SUMMARY A method is introduced for studying surface waves in a general anisotropic poroelastic medium. The method is analogous to the one used for isotropic media and derives a complex secular equation to represent the propagation of surface waves at the stress-free plane surface of a nondissipative porous medium. The point of importance is that the derived equation is, analytically, separable into real and imaginary parts and hence can be solved by iterative numerical methods. A root of this secular equation represents the existence of surface waves and calculates the apparent phase velocity along a given direction on the surface. Numerical work is carried out for the model of a crustal rock. The propagation of surface waves is studied numerically for the top three anisotropies (i.e. triclinic, monoclinic, orthorhombic). 1I N T R O DUCTION The dynamic behaviour of fluid-saturated porous media is attracting wide attention due to its importance in oil exploration, earthquake engineering, structural engineering, soil dynamics and hydrology. The dynamic equations formulated by Biot (1956) have long been regarded as standard and have formed the basis for the study of wave propagation problems in poroelastic media. Biot (1955) discussed the stress‐strain relations for anisotropic poroelasticity. Anisotropy in porous solids may be the result of propagation through a distribution of aligned cracks, microcracks and preferentially oriented pore space. Occasionally, anisotropy may be due some other phenomenon such as rock foliation or crystal alignment. Crampin (1994) reviewed the various studies related to the observation of shear wave splitting and confirmed the presence of anisotropy in almost all the rocks in the uppermost half of the crust. Thomsen (1995) related the anisotropy to pore properties and crack parameters in a porous rock. This work was supported by an experimental study of the anisotropy of sandstone with controlled crack geometry by Rathore et al. (1995). Sharma (1996) discussed the coexistence of cracks and pores and its effect on surface wave propagation. Hudson et al. (1996) studied the effects of connection between cracks and of small-scale porosity within the solid material on the overall elastic properties of cracked solids. The presence of mineral orientations, microfracturing, thin layering or combinations of these in a material results in a general anisotropy of arbitrary symmetry. The absence of symmetry in aligned microcracks or pore space also results in general anisotropy. The propagation of energy in such a general anisotropic medium is, in fact, a 3-D phenomenon. Wave motion in such a material is not confined to a particular plane. This explains the out-of-sagittal-plane motion of particles in Rayleigh waves. Apart from its well-known seismic importance, surface wave propagation in anisotropic media has widespread applications in the non-destructive evaluation of materials (Buden & Datta 1990; W uW Ting 2002) to study the anisotropic propagation of surface waves considers motion in an arbitrary but fixed plane. The Christoffel equation, in a polynomial form, is solved for vertical slowness values representing different quasiwaves. In the case of a general anisotropic porous solid, an eighth-order polynomial equation is required to be solved for complex values of vertical slowness. This gives a complex secular equation to represent surface waves in the medium. It is not surprising that there is no analytical method available to solve this system. Even the numerical solution may be very difficult. Moreover, researchers in this field may expect a method that is analogous to that used to study Rayleigh waves in an isotropic medium. This means that the secular equation should involve the elastic constants of the medium and the velocities (instead of the vertical slownesses) of the various body waves propagating in the medium. In this paper such a secular equation is derived for the surface waves in an anisotropic porous solid with arbitrary symmetry.

Interrelation between toughness and both strength and structural parameters of material*

The Neuber-Novozhilov fracture criterion is proposed for a description of tensile and shear fracture in the vicinity of the tip of a V-shaped cut. In the criterion proposed, the averaged limits of stresses on the cut axis depend on the presence, size and location of defects in the initial material. The parameter of the crystal lattice of material is chosen for the specific linear size. For V-shaped cuts, simple relations for tension and shear have been obtained that relate stress intensity factors at modified singularity coefficients; also obtained are these singularity coefficients and theoretical tensile and shear strengths of single crystal material with consideration of damage of the material in the vicinity of the cut tip. In the relations obtained, the passage to the limit by the angle from the V-shaped cut to a crack is possible. It has been shown that the critical stress intensity factor that is used when evaluating the strength of cracked solids is not a material constant.

Effects of Parallel Crack Distributions on Effective Elastic Properties - a Numerical Study

This paper is concerned with numerical studies of effective elastic properties of cracked solids. We concentrate on two dimensional media containing different patterns of parallel crack distributions. We use the Rotated Staggered Grid (RSG) which allows one to simulate elastic wave propagation very accurately in fractured media. Our aim is to compare the predictions given by several effective medium theories to the effective properties we derive from our numerical experiments. Namely, these are the ``Non-interaction approximation (NIA)'', the ``Differential scheme (DS)'' and an extension of the DS (EDS). According to our results, the DS theory and its extension perform well. Simulations of media containing very few cracks prove that for our setup the effective properties stabilize at low numbers of cracks. Finally, we studied parallel cracks clustered in stacked columns. We found that, as expected, the shielding effects dominate in such patterns.

A Boundary Element Analysis of Quasi-Brittle Solids Containing Cracks

Quasi-brittle solids usually contain many cracks. When numerical methods are used, it is difficult to simulate all the cracks’ behaviors, especially when the crack propagation is included. To surmount the difficulty, a new procedure combining the subdomain boundary element method and the generalized self-consistent scheme is presented. The behavior of a single crack is analyzed first. The obtained information is then used to predict the overall behavior of the cracked solids. The relationship between the overall response of the cracked solids and the growth of wing cracks is established numerically. A cohesive-crack model is used to simulate the fracture process zone of the quasi-brittle solids. Besides, the advancement of the wing crack tip is used as a control variable. An incremental iteration algorithm is developed to trace the growth of the wing (secondary) cracks, which is not known a priori. The influence of the crack angle and the friction coefficient on the effective moduli is also investigated. Finally, a gypsum square plate containing orderly distributed cracks is analyzed to illustrate the adequacy and efficiency of the present procedure.

A fundamental solutions for transversely isotropic, piezoelectric solids under electrically irrotational approximation

A time-domain boundary element method (TDBEM) for transient dynamic analysis of two-dimensional (2D), homogeneous, anisotropic and linear piezoelectric cracked solids is presented in this paper. The present analysis uses a combination of the strongly singular displacement boundary integral equations (BIEs) and the hypersingular traction boundary integral equations. The spatial discretization is performed by a Galerkin-method, while a collocation method is implemented for the temporal discretization. Both temporal and spatial integrations are carried out analytically. In this way, only the line integrals over a unit circle arising in the time-domain fundamental solutions are computed numerically by standard Gaussian quadrature. An explicit time-stepping scheme is developed to compute the unknown boundary data including the generalized crack-opening-displacements (CODs) numerically. Special crack-tip elements are adopted to ensure a direct and an accurate computation of the dynamic field intensity factors (IFs) from the CODs. Several numerical examples involving stationary cracks in both infinite and finite solids under impact loading are presented to show the accuracy and the efficiency of the developed hypersingular time-domain BEM.

Reconstrucción de las curvas de esfuerzo-deformación a partir de datos de conductividad eléctrica en experimentos de deformación triaxial en rocas

Measurements of complex electrical conductivity as a function of frequency is an extremely sensitive probe for changes in pore and crack volume, crack connectivity, and crack surface topography. Such measurements have been made as a function of pore fluid chemistry, hydrostatic codning pressure, as well as uniaxial and tnaxial deformation. This paper develops a method for using such experimental conductivity data to model the mechanical properties of saturated rocks undergoing triaxial deformation. It does so by defming a new direction sensitive crack damage parameter denved from the electrical data on four samples of Darley Dale sandstone. More specifically this paper will: 1) review the theoretical concepts necessary to understand the modelling procedure (crack density parameter and effective elastic properties of cracked solids), 2) derive a new crack damage parameter from electrical conductivity data, 3) use the electrical data to model the mechanical stress-strain behavior of porous rocks, and 4) compare the modelled results with the stress-strain behavior measured during the deformation. The resultant synthetic stress-strain curves show good agreement with the experimentally derived stress-strain curves, and the estimated values of model parameters also compare well with their experimental counterparts. This modelling is an improvement on similar curves produced using isotropic crack damage parameters, such as those denved from acoustic emission data. The improvement is likely to be due to the directional sensitivity of the electrical conductivity measurement, and its ability to discriminate between the formation of isolated cracks, and those cracks that contribute to the interconnected crack space, i.e., those cracks upon which both the transport properties of the rock such as electrical conductivity, and the mechanical properties depend most critically during triaxial deformation.

A Dislocation Model for the Directional Anisotropy of Grain-Boundary Fracture

That fracture is governed by processes occurring over a wide range of length scales has been recognized since the earliest developments of modern fracture mechanics. Griffitha's study of the strength of cracked solids 1,2 is perhaps the earliest example of such multiscale thinking, predating by several decades the first attempts to apply atomistically grounded traction-separation laws to fracture (e.g., the Orowan-Gilman model3,4). Griffith recognized the critical condition for crack extension to be a statement of thermodynamic equilibrium of a cracked solid, representing a balance between the mechanical energy decrease upon crack extension and the corresponding increase in energy due to the newly created crack surface. Griffith determined the elastic strain energy of the cracked body using the continuum solution of the stress field about an ellipse5 and recognized that the potential energy associated with the cleavage surfaces of the crack was directly proportional to the surface energy, the latter deriving from the cohesive molecular forces of the solid. The Irwin-Orowan extension of Griffith mechanics to include plastic dissipation,6-9 which is known to occur on the mesoscopic length scale (~1–100 μm), provides yet a further example of multiscale thinking in the early community of fracture researchers. In fact, the interaction of length scales is of central importance in most problems of fracture.

Approximate Evaluation for Effective Elastic Moduli of Cracked Solids

A system of equations for effective elastic moduli of 2-D cracked solids is presented by combining the energy balance equation proposed by Shen and Yi (2000) with the integral equations which control the problem of an infinite solid with a finite number of cracks in a sub-region. Then, using Kachanov's method (Kachanov, 1987) for the solutions of the integral equations, 2-D effective bulk and shear moduli for solids with randomly distributed cracks are evaluated.

Determination and representation of the stress coefficient terms by path-independent integrals in anisotropic cracked solids

Because the elastic T-stress and other coefficients of the higher-order terms play an important role in fracture mechanics such as the stability of crack kinking, crack path, and two-parameter characterization of elastic-plastic crack tip fields, determination of all the coefficients in the crack tip field expansion in an anisotropic linear elastic solid is presented in this paper. Utilizing conservation laws of elasticity and Betti's reciprocal theorem, together with selected auxiliary fields, T-stress and third-order stress coefficients near the crack tip are evaluated first from path-independent line integrals. To determine the T-stress terms using the J-integral and Betti's reciprocal work theorem, auxiliary fields under a concentrated force and moment acting at the crack tip are used respectively. Through the use of Stroh formalism in anisotropic elasticity, analytical expressions for all the coefficients including the stress intensity factors are derived in a compact form that has surprisingly simple structure in terms of one of the Barnett-Lothe tensors, L. The solution forms for degenerated materials, monoclinic, orthotropic, and isotropic materials are also presented.

Influence of stress-induced and thermal cracking on physical properties and microstructure of La Peyratte granite

Abstract New data are presented on the influence of thermal cracking and stress-induced cracking on several physical properties (porosity, acoustic velocity, attenuation, electrical conductivity, permeability) of a fine-grained granite from La Peyratte, France. Two sets of samples were studied: the `TC' set was made of samples in which thermal cracking was induced, the `MC' set included samples in which cracks were generated at increasing stress levels in mechanical tests. Good correlations were found between the evolution of the physical properties and the amount of damage induced. For the mechanical cracking process, we show that acoustic emissions recording was the key point in order to control in real time the evolution of the cracking processes. The strong anisotropy in the acoustic velocity data found only in the MC samples is clearly due to the anisotropy of the crack distribution, as shown by microstructure analyses and 3D reconstructions of crack networks from confocal scanning laser microscopy images. A model for the propagation of elastic waves in cracked solids was used in order to infer the crack density and the mean crack aspect ratio for the granite samples. Our results show that the aspect ratio of artificially induced cracks is significantly smaller than that of cracks in `natural' rocks. Network modelling for fluid flow in porous media was done using data from mercury injection tests and available information on the crack network geometry as input data. Although the numerical model gives the correct order of magnitude for the permeability, it could not account for relative permeability evolution with induced damage, presumably because of oversimplified assumptions in the model. All in all, our study confirms the strong control of the rock microstructure on physical properties. The data set presented in this paper might be very useful for studies on safety assessment for potential nuclear waste storage in geological environment.

Inelastic Effects of Microcracking: A New Continuum Damage Model

Microcracking is an important mechanism of damage accumulation in stressed solids. Two contributions to deformation can be distinguished: modification of the bulk effective elastic behaviour as cracks are introduced; and accumulation of an inelastic offset for each crack on formation. Equivalent medium approximations consider the former. In this study we have developed a new formulation to model the latter. The change in strain energy at the instant of formation of each individual microcrack is used to calculate the incremental inelastic strain increase during monotonic loading. This leads to integral equations describing the damage state, in which stress and strain are functions of damage. The macroscopic behaviour is path dependent. We obtain an explicit form of the stress-strain relation as a function of both intrinsic and extrinsic variables (damage and loading history) in order to test the model using experimental data for uniaxial tests on rock samples. Quantitative damage estimation is made through monitoring acoustic emissions emitted as microcracks form, which are related to crack dimensions through an independently determined scaling relation. The new model successfully reproduces the experimental stress-strain behaviour through failure, and also allows secondary effects, such as residual strain or hysteresis to be modelled, which is not possible with the equivalent elastic medium approach. The results suggest that (a) the incremental damage approach is needed to fully model behaviour in compression; (b) inelastic effects arising from cracking must be considered as well as modifications to the sample stiffness; and (c) acoustic emission parameters can be used in quantitative modelling of inelastic behaviour of cracked solids.

Brittle Fracture Of Cracked Solids As Affected By Surfactants

The comparative study of breaking cracked solids, with or without a surfactant in a crack, is in progress. Chemically intensified fracture as well as the fracture retardation due to formation of new force bonds at the crack tip are considered. Experiments with breaking chromite (FeCVzĈ ) have been performed in a planetary mill. Solid-phase additives measuring about 4% of chromite by weight are used for surfactants. Estimates for the critical stress intensity factor (SIF) and the critical length for cracks of opening mode are obtained. For considered cases the critical crack lengths, as a rule, differ by one order. The critical values of loading for fixed crack lengths for these cases differ markedly.

A continuum damage constitutive law for brittle rocks

The non-linear mechanical behaviour of two different granites was investigated. Short term triaxial tests have been performed in order to study the short term behaviour (instantaneous growth of cracks) of the rock. A continuous damage model is then proposed. The internal damage variable is directly related to the second order crack density tensor used by Kachanov in the study of effective elastic properties of cracked solids. An instantaneous crack growth criterion has been used to describe the time-independent evolution of the damage tensor. The verification of the model has been performed for various loading paths. The results of a numerical modelling show that it is possible to reproduce most of the observable characteristics of damaged zone around an underground opening excavated into brittle rocks.

Acoustoelasticity in cracked solids

SUMMARY A new model that accounts for the stress dependence of the phase velocity of elastodynamic waves propagating in a cracked solid under compression is presented. The phase velocities of longitudinal and shear waves are derived from the effective elastic properties of a cracked solid, which are evaluated within the framework of Kachanov’s approach. Following Kachanov, the extra-compliance tensor of the cracked solid is related to the crack compliances, which display a marked non-linear behaviour when subjected to a compressive load. Such non-linear behaviour is shown to be derived from the elastic interaction between the contacting crack faces under compression. This work does not address the effect of mutual interaction among cracks and the generation of higher harmonics due to the medium non-linearity. Numerical examples are presented that illustrate the phase velocity changes occurring in a solid with a random distribution of parallel cracks as a function of an external compressive load. A distinctive feature of the acoustoelastic effect in solids with large parallel fractures and in solids with distributions of aligned microcracks is also illustrated.