A heuristic approach to microcracking and fracture for ceramics with statistical consideration A. Brencich, A. Carpinteri * Department of Structural Engineering, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italy Abstract Microcracking damage and toughening are examined for ceramics. These eects have been found to depend on the material microstructure and macrocrack growth. Isotropic damage, attributed to random distribution of microcrack location, length and orientation can be associated with a disordered microstructure and a non-uniform residual stress ®eld. When the applied stress is the main cause of cracking, the microcrack distribution is no longer random such as a system of quasi-parallel cracks. To highlight the eect of crack interaction, discrete models are advanced where damage is simulated by a distribution of microcracks. The dilute concentration assumption is invoked to simplify the analysis. The two-dimensional discrete model is based on a phenomenological approach that is statistical in character. In- teractions of microcracks and with a macrocrack are considered by means of a boundary element technique (A. Brencich, A. Carpinteri, Int. J. Fracture 76 (1996) 373±389; A. Brencich, A. Carpinteri, Eng. Fract. Mech. 59 (1998) 797±814) where both isotropic and anisotropic damage could be treated. Comparisons with other results are made to show that the model can be applied to analyse the fracture behaviour of dierent materials. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Microcrack damage; Crack interaction; Statistical model; Microcrack toughening 1. Introduction Brittle materials are known to contain extensive microcracks. Such a region is known as process zone. It is developed in front of a macrocrack [3± 7]. This occurs in ceramics, rocks and concrete-like materials. Microcracking damage tends to toughen the material at the macroscopic scale level for stationary and steadily growing cracks [5,8±10]. That is the load level at which a crack propagates is increased when compared with the estimated limit load for the undamaged material. With reference to the material microstructure, two dierent distributions could be identi®ed in- side the process zone. For a two-phase ceramic system, such as zirconia toughened alumina, mi- crocracks are nucleated at grain boundaries in the form of intergranular [8] and radial cracks [9]. Due to the random distribution of the second phase particles and grain facets, the microcracks are randomly distributed and the damaged zone ex- hibits an isotropic behaviour. Other ceramics, such as lithium±alumino-silicate glass ceramics [5] or alumina±silicon carbide composites [10] and con- crete-like materials [6,7], may be regarded as a homogeneous matrix containing dispersed second phase particles. For these materials, the micro- crack pattern resembles the principal stress www.elsevier.com/locate/tafmec Theoretical and Applied Fracture Mechanics 33 (2000) 135±143 * Corresponding author. Tel.: +39-11-564-4850; fax: +39-11- 564-4899. E-mail address: carpinteri@polito.it (A. Carpinteri). 0167-8442/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 8 4 4 2 ( 0 0 ) 0 0 0 0 8 - 2