Asymptotic analysis of boundary effects on concrete fracture

The recently-developed boundary effect concept and associated asymptotic model are used to analyse the various size-dependent fracture behaviours of concrete specimens. It is shown that the dependence of concrete fracture on specimen size, crack and/or ligament length is due to the same mechanism, i.e. the interactions between the crack tip fracture process zone and specimen boundaries. By introducing an equivalent crack ae, all these dimension dependent fracture behaviours become equivalent to a large plate with a small edge crack, and therefore, follow the same relationship of the “net” nominal strength σn versus ae. As a result, the size dependent fracture can be predicted using the data measured on different types of specimens. To demonstrate the flexibility and effectiveness of the asymptotic boundary effect model, three previous experimental results available in the literature are analysed, and excellent agreements are achieved. Furthermore, the implications behind the successful predictions are discussed. the FPZ is far away from specimen boundaries, the specimen strength is determined by the linear elastic fracture mechanics (LEFM) criterion. This paper aims to explain the asymptotic boundary effect model. For this purpose, the influence of the specimen boundaries on concrete fracture is firstly analysed by considering the role of the boundaries in dominating the failure mode. Then, three sets of fracture testing data available in the literature are analysed using the asymptotic boundary effect model. It will be shown that the predictions from the asymptotic model agree very well with those measured data, and the implications behind this agreement are discussed. 2 BOUNDARY EFFECT CONCEPT AND ASYMPTOTIC MODEL FOR FRACTURE STRENGTH OF CONCRETE The boundary effect concept is explained in Figure 1 where a finite-sized plate with a single edge crack is subjected to a remote applied load. Depending on the closeness of the crack tip FPZ to either boundary, the strength of the plate can be determined by either the material tensile strength ft or fracture toughness KIC. When the crack tip is close to the front boundary (short crack) as shown in Figure 1a, the crack will be shielded before specimen failure, and therefore, the plate strength is determined by the maximum tensile stress condition (ft). On the other hand, when the FPZ is close to the back boundary (short ligament, Fig. 1c), the ligament will “yield” prior to the final fracture, the failure of the plate is again controlled by ft. Only when the crack tip locates in the middle of the plate where it is far away from both boundaries, is the plate strength decided by the LEFM criterion (Fig. 1b). Figure 1. The dependence of failure mode on the crack-tip-toboundary distance: (a) a short crack is likely shielded before specimen failure; (b) crack tip is far away from both boundaries, and the LEFM dominates failure; and (c) a short ligament “yields” before final failure Recently, an asymptotic boundary effect model has been developed to describe the afore-discussed relationship between the distance of the crack tip FPZ to specimen boundaries and the strength behaviours of fracture mechanics specimens (Duan & Hu 2002, 2004, Duan et al. 2006). This asymptotic solution for finite-sized specimens is based on the equivalent relationship between finite-sized specimens and a large plate with a small edge crack, and includes the σf-condition as asymptote for short crack or ligament and KIC-criterion for a specimen with the crack tip away from boundaries. The large plate case can be obtained when the specimen width W >> crack size a in Figure 1, i.e. the ratio of the crack length a over the plate width W is equal to zero (α = a/W = 0), and as a result, only specimen front boundary will influence specimen strength. The asymptotic solution for the simple large plate case was proposed by Hu and Wittmann (Hu & Wittmann 2000), and is given by: * 1 ∞ + = a a ft n σ (1) in which, the reference crack a∞ represents the crack length where the two asymptotic limits intersects, and is given by: