Int J Fract (2007) 146:295–299 DOI 10.1007/s10704-007-9170-6 © Springer Science+Business Media B.V. 2007 LETTERS IN FRACTURE AND MICROMECHANICS 123 ON THE EFFECTIVE ELASTIC PROPERTIES OF CRACKED SOLIDS – EDITOR’S COMMENTS Mark Kachanov Department of Mechanical Engineering Tufts University, Medford, MA 02155, USA Mark.Kachanov@tufts.edu The problem of effective elastic properties of cracked solids is a classical one. It has been extensively discussed since the pioneering works of Bristow (1960) (where the crack density parameter was introduced) and Walsh (1965a,b), motivated by microcracking in metals and in rocks, respectively. In these works, the effective constants were derived in the isotropic case of randomly oriented cracks of the circular (“penny”) shapes in the non-interaction approximation. These results, supposedly, lose accuracy at higher crack densities when interactions become significant. Actually, correctness of this conjecture – that may seem obvious – is not immediately clear. Indeed, crack interactions may either “stiffen” the elastic response (if they are of predominantly shielding nature) or “soften” it (if the amplifying interactions are dominant). This is controlled by the mutual positions of cracks, and examples of both situations are readily given by various periodic arrangements. For example, in a doubly periodic 2-D array of parallel cracks, increasing the ratio of the “vertical” spacing to the “horizontal” one changes the character of interactions from the predominantly shielding one to the predominantly amplifying one. It is entirely possible, therefore, that the competing interaction effects may largely balance one another, so that, whereas the interactions produce strong local effects (for example, on the stress intensity factors), their effect on the overall stiffness may be small. Several approximate schemes aimed at accounting for crack interactions have been proposed. The best known ones are the self-consistent scheme, SCS (Budiansky and O’Connell, 1976) where cracks are placed into the matrix with effective properties, the differential scheme, DS (Vavakin and Salganik, 1975, Hashin, 1988) where the same procedure is applied in increments and Mori- Tanaka’s (1973) scheme, MTS, formulated for the cracked materials by Benveniste (1986), where cracks are placed into the average, over the solid phase, stress. All these schemes place non- interacting cracks into some sort of “effective environment”. The predictions of these schemes diverge substantially as crack density increases. The SCS predicts strong softening effect of interactions, the DE – substantially milder softening effect and the MTS predicts no interaction effect at all (consistently with the fact that the presence of cracks does not change the average stress in the matrix). The Hashin-Shtrikman bounds provide no guidance, since they degenerate for cracks: arrays of very high crack density producing a negligible effect, as well as arrays of very small density reducing the effective stiffness to zero, can be constructed. Since the bounds are realizable, crack arrays exist that match predictions of any scheme, by appropriately selecting the mutual positions (spatial distribution) of cracks. Since the mentioned schemes use the non-interaction approximation (one crack problem) as the basic building block, they do not reflect mutual positions of cracks. This factor, however, may have a strong effect on the effective properties, as illustrated by the example of a doubly-periodic array