Mechanics of distributed cracking

As we know, failure of many materials involves propagation of a large system of densely distributed cracks rather than a single, precisely defined fracture. This is typical of concrete and rock, as well as stiff clays, ice, especially sea ice, filled elastomers, wood, particle board, paper, two-phase ceramic composites, fiber-reinforced polymers, fiber-reinforced concrete, asphalt and polymer concretes, and refractory concretes. The number of cracks or microcracks is extremely large, and their locations and orientations are random. Therefore, it is inevitable to treat the densely cracked material as a continuum. The constitutive relation must then exhibit strain-softening, a phenomenon where the matrix of iangential elastic moduli ceases to be positive definite. Unreasonable though such an approach might seem (84.40), it has nevertheless been proven useful provided that certain mathematical difficulties inherent to it are properly tackled. Some of these difficulties were already pointed out in 1903 by Hadamard (03.1) and others were further analyzed by Thomas, Maier, Mr6z, BaZant, Burt and Dougill, Sandler and Wright, Balant and Belytschko, Darvall, Read and Hegemier, Wu and Freund, and others (61.4, 76.2, 77.4, 84.40, 84.46, 84.58). Recently, due to large errors in finite element predictions of complete failures of concrete structures or rock masses subjected to blast, impact, earthquake, thermal loading, creep or shrinkage, the modeling of distributed cracking came to the center of attention. The literature has become so extensive that a complete coverage in one paper is impossible. Therefore, I will focus my review mainly on the work done at our Center at Northwestern University which was carried out jointly with T B Belytschko, L Cedolin, T P Chang, P Gambarova, J K Kim, F B Lin, B H Oh, P Pfeiffer, P Prat, and A Zubelewicz, under the sponsorship of the Air Force Office of Scientific Research (Grant No 85-00(9) and the National Science Foundation (Grants Nos CEE800-3148 and CEE821-1642). If I fail to mention some important contributions, their authors should realize that they will be in a good company. Let me begin by showing a micrograph of the fracture tip region in Portland cement mortar, obtained by means of scanning electron microscopy by Mindess and Diamond (80.22). Instead of a continuous crack we see many discontinuous microcracks which are not arranged along one line but are also

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