International Journal of Fracture 50: 199-219, 1991. © 1991 Kluwer Academic Publishers. Printed in the Netherlands. Modeling of fiber toughening in cementitious materials using an R-curve approach B. MOBASHER l, C. OUYANG 2 and S.P. SHAH 2 1USG Corporation, Research Center, Libertyville, Illinois 60048, USA; 2NSF Center for Science & Technology of Advanced Cement-Based Materials, Northwestern University, Evanston, Illinois 60208, USA Received 15 December 1989; accepted in revised form 8 July 1990 Abstract. This paper presents the theoretical formulation describing the role of fibers in enhancing the fracture toughness of quasi-brittle cement based materials. The formulation is based on the well known R-curve approach which correlates the increase of the apparent fracture toughness of a material with the existence of a pre-critical stable crack growth region. By assuming that the critical crack length in plain matrix is a function of an initial crack length ao, a formulation for the R-curves has recently been derived and applied to predict the response of positive and negative geometry specimens of various sizes and materials. This approach is further applied to uniaxial tensile specimens containing various fiber types. Fiber reinforcement is modeled by means of applying closing pressure on crack surfaces resulting in closure of the crack faces and a decrease in the stress intensity factor at the tip of the propagating crack. Incorporation of these two factors in the energy balance equations for crack growth results in increases in both the slope and the plateau value of the R-curve representing matrix response. Enhancement in material response is shown to occur only if precritical crack growth exists, causing fibers to convert the stable cracking process into an increase in load carrying capacity of the material. Fracture response of fiber reinforced composites can be predicted up to the bend-over-point. The theoretical predictions are compared with the experimental results of cement-based composites containing unidirectional, continuous glass, steel or polypropylene fibers. 1. Introduction Fiber reinforced cement-based materials are increasingly being used for building products. Based on the need to find a substitute for asbestos fibers, a significant amount of research and development has been conducted during the past two decades to incorporate polymeric, metallic and glass fibers. The current state of practice in using fiber reinforced concrete is limited to applications of low fiber contents, (less than 1 percent by volume) where the contribution of the fibers is apparent primarily in the post-peak region of the response of the composite. As the volume fraction of fibers increases and they become more uniformly distributed, the chance that they can hinder the growth of microcracks (prior to initiation of localization) through an arrest mechanism increases. Hence the matrix fracture toughness can increase significantly. The interaction between fibers and brittle matrices has been actively studied in various fields. Some aspects of the evolution and accumulation of damage and characterization of failure have been addressed by Bunsell et al. [1] using acoustic emission technique. Sato et al. [2] have used acoustic emission, scanning electron microscopy, and polarized transmission optical microscopy to study unidirectional carbon-fiber reinforced epoxy resin composites. Fracture mechanics based approaches in cementitious materials have been used by Foote [3], and a probabilistic analysis approach was used by Naaman et al. [4]. Mobasher, Stang, and Shah I-5,6], and Mobasher, Castro-Montero and Shah [7] used acoustic emission technique, quantitative fluorescent microscopy, and laser holographic interferometry to study damage accumulation in