Mark Kachanov

Elastic Solids with Many Cracks and Related Problems

Publisher Summary This chapter discusses some basic problems in mechanics of elastic solids containing multiple cracks. A number of mathematical aspects that frequently constitute fields of their own (like various numerical techniques) are discussed very briefly in the chapter. The focus is on physically important effects produced by crack interactions and to present results in the simplest form possible. The problems considered in this chapter can be divided into two groups: (1) The impact of interactions on individual cracks, particularly on the stress intensity factors (SIFs), and (2) the effective elastic properties of solids with many cracks. Problems of the first group are, generally, relevant for the fracture-related considerations; solutions are sensitive to the positions of individual cracks. Problems of the second group deal with the volume average quantities; they are relatively insensitive to the information on individual cracks. The chapter discusses, in this connection, whether correlations exist between these two groups of quantities; in particular, whether microcracking can be reliably monitored by measuring changes in the effective elastic moduli.

Effective elastic properties of cracked solids: Critical review of some basic concepts

The problem of effective moduli of cracked solids is critically reviewed. Various approaches to the problem are discussed; they are further assessed by comparing their predictions to results for sample deterministic arrays. These computer experiments indicate that the approximation of noninteracting cracks has a wider than expected range of applicability. Some of the deficiencies of various approximate schemes seem to be related to inadequacy of the conventionally used crack density parameter (insensitive to mutual positions of cracks). An alternative parameter that has this sensitivity, is suggested. Finally, the problem of effective moduli is discussed in the context of {open_quotes}damage mechanics{close_quotes}. It is argued that, contrary to the spirit of many damage models, there is no direct quantitative correlation between progression of a microcracking solid towards fracture and deterioration of its stiffness; thus, the effective moduli may not always serve as a reliable indicator of damage. 84 refs., 14 figs.

Elastic solids with many cracks: A simple method of analysis

Abstract A simple method of stress analysis in elastic solids with many cracks is proposed. It is based on the superposition technique and the ideas of self-consistency applied to the average tractions on individual cracks. The method is applicable to both two- and three-dimensional crack arrays of arbitrary geometry. It yields approximate analytical solutions for the stress intensity factors (SIFs) accurate up to quite close distances between cracks. It is also suggested how a full stress field can be approximately constructed. Applications to a configuration “crack-microcrack array” and to a problem of effective elastic properties of a solid with cracks are considered.

Microcrack-induced elastic wave anisotropy of brittle rocks

The failure of brittle rocks during compression is preceded by the formation, growth, and coalescence of microcracks. Elastic wave velocities are reduced in the presence of open microcracks and fractures and may therefore be used to monitor the progressive damage of the rock. In general, these microcracks are not randomly oriented, and the rock displays an elastic anisotropy. The elastic anisotropy due to cracks can be expressed in terms of a second-rank and fourth-rank crack density tensor. For open cracks the contribution of the fourth-rank crack density tensor to the elastic wave velocities is small. These results are compared with recent measurements of the ultrasonic compressional and shear wave velocities for propagation parallel and perpendicular to an increasing axial stress applied at constant confining stress to Berea sandstone. Inversion of the velocity measurements indicates that the microcracks propagate parallel to the maximum compressive stress, in agreement with current rock mechanics theory. A reasonable fit to the data is obtained using only the second-rank crack density tensor even though, at high confining stress, the cracks are expected to be in partial contact along their length. This is consistent with the model of elastic wave propagation in a medium containing partially contacting fractures published by White. However, measurements of off-axis wave velocities are required to fully quantify the contribution of the fourth-rank crack density tensor.

A simple technique for finding effective elastic constants of cracked solids for arbitrary crack orientation statistics

Abstract The effective elastic properties of cracked solids are anisotropic if the cracks have preferred orientations. In this paper a simple scheme for evaluating the elastic stiffness tensor for an arbitrary orientation distribution of cracks at finite crack densities is presented. The approach is based on a tensorial transformation of the effective elastic constants for isotropic orientation statistics through the use of a second-order crack density tensor.

Nanoelectromechanics of Piezoresponse Force Microscopy

To achieve quantitative interpretation of piezoresponse force microscopy (PFM), including resolution limits, tip bias- and strain-induced phenomena and spectroscopy, analytical representations for tip-induced electroelastic fields inside the material are derived for the cases of weak and strong indentation. In the weak indentation case, electrostatic field distribution is calculated using an image charge model. In the strong indentation case, the solution of the coupled electroelastic problem for piezoelectric indentation is used to obtain the electric field and strain distribution in the ferroelectric material. This establishes a complete continuum mechanics description of the PFM contact mechanics and imaging mechanism. The electroelastic field distribution allows signal generation volume in PFM to be determined. These rigorous solutions are compared with the electrostatic point-charge and sphere-plane models, and the applicability limits for asymptotic point-charge and point-force models are established. The implications of these results for ferroelectric polarization switching processes are analyzed.

Explicit cross-property correlations for anisotropic two-phase composite materials

Abstract Explicit correlations between two groups of anisotropic effective properties—conductivity and elasticity—are established for two-phase composite materials with anisotropic microstructures (non-randomly oriented inclusions of non-spherical shapes). The correlations are derived in the framework of the non-interaction approximation. The elasticity tensor is expressed in terms of the conductivity tensor in closed form. Applications to realistic microstructures, containing mixtures of diverse inclusion shapes are given. Compliance/stiffness contribution tensors of an inclusion, that characterize the inclusions contribution to the overall elastic response, are derived in the course of analysis; these results are of interest on their own.

Effective elasticity of rocks with closely spaced and intersecting cracks

The noninteraction approximation (NIA) is the simplest effective media theory that describes the overall elasticity of fractured rocks. If the NIA is used for fracture characterization, its accuracy and the range of applicability must be estimated. We do it by performing a series of 3D finite element simulations of effective elasticity for models that contain several sets of fractures embedded in otherwise isotropic host rock. We intentionally place the cracks close to each other to create strong interactions in their local stress fields. In addition, we allow the cracks to intersect in such a way that they do not break a rock specimen apart. Perhaps surprisingly, we find that fracture interactions and intersections have little influence on the effective elasticity, and the NIA performs well in all cases. While it has a tendency to slightly underestimate the effective stiffnesses, the incurred errors are small; their typical magnitudes are just a few percent in the entire range of the crack densities expe...

Interaction of a crack with certain microcrack arrays

Abstract Interaction of a crack with microcracks (modelling “damage”) can significantly alter the stress concentration at the crack tip. Certain important effects of interaction—shielding effect (“toughening by microcracking”), amplification effect, influence of the orientations of microcracking and irregularities in its patterns, change of the character of interaction (shielding to amplification and vice versa) with change of the mode of loading, etc.—are demonstrated on several microcrack systems. It appears that these relatively simple systems exhibit the essential features of the crack-damage interactions. Consideration is based on the method of analysis of elastic solids with many cracks proposed recently [Kachanov, Int. J. Fracture 28 , R11–R19 (1985); Kachanov, Int. J. Solids Structures (in press)] and briefly presented here.

Elastic interaction of a crack with a microcrack array—I. Formulation of the problem and general form of the solution

Abstract Elastic interactions of a crack with an array of microcracks located near the tip is considered. The analysis is based on the potential representations (known also as representation of cracks by dislocations) and approximation of tractions on the microcracks by polynomials.