Effect of crack orientation statistics on effective stiffness of mircocracked solid

Abstract

a b s t r a c tThis paper addresses the problem of calculating effective elastic properties of a solid containing multiple cracks with prescribed orientation statistics. To do so, the representative unit cell approach has been used. The microgeometry of a cracked solid is modeled by a periodic structure with a unit cell containing multiple cracks; a sufficient number is taken to account for the microstructure statistics. The developed method combines the superposition principle, the technique of complex potentials and certain new results in the theory of special functions. A proper choice of potentials provides reducing the boundary-value problem to an ordinary, well-posed set of linear algebraic equations. The exact finite form expression of the effective stiffness tensor has been obtained by analytical averaging the strain and stress fields. The convergence study has been performed; the statistically meaningful results obtained show dependence of the effective elastic stiffness on angular scattering of cracks. Comparison has been made with the selected simple micromechanical models, namely, non-interaction approximation, differential scheme and modified differential scheme. It is found that, among these models, the differential scheme provides the best fit of the numerical data.

Universal and Nonuniversal Aperture‐to‐Length Scaling of Opening Mode Fractures Developing in a Particle‐Based Lattice Solid Model

“…For a solid containing a set of randomly distributed parallel cracks aligned with the x direction, all three theories predict that E x = E, whereas E y = E∕(1 + 2 ) (noninteraction approximation), E = E exp(−2π ) (differential scheme), and E = E∕[1+2π exp(π )] (extension of the differential scheme), with being the fracture density as defined by equation (8). The normalized effective moduli E y ∕E obtained from the lattice models illustrate that, on average, the differential schemes provide the best fit, a result consistent with existing studies (Kushch et al, 2009;Orlowsky et al, 2003). The scatter is however significant, given the relatively small number of fractures per model and the large range of fracture lengths (see Figure 4).…”

“…where E x and E y are the effective Young's moduli in the x and y directions (which are planes of symmetry in the orthotropic solid) and E and are the elastic constants of the unfractured material (matrix; e.g., Kushch et al, 2009;Orlowsky et al, 2003). The effective moduli are determined from the pre-fractured models in the same fashion as for the intact lattice (as described in section A1), and the properties of the intact solid are computed using the plane strain analytical solutions (equation (A2)).…”

Effect of stress interactions on anisotropic P‐SV‐wave dispersion and attenuation for closely spaced cracks in saturated porous media

“…It is interesting to compare the effect of stress interactions in elastic and poroelastic media, respectively. In elastic media, stress shielding stiffens media, while amplification reduces the elasticity (Kushch et al, 2009). In poroelastic media, the stress shielding not only stiffens the media but also leads to smaller attenuation, while the amplification corresponds to the softer media and thus greater attenuation.…”

A Fast and Stable Solver for Singular Integral Equations on Piecewise Smooth Curves

“…The literature on integral equation methods for crack problems is rich; see the references in [3]. Further examples include [8,18,22,30] on internal cracks, [17,26] on interface cracks, [10] on edge cracks, and [5,21] on contacting cracks.…”