R - curve modeling of rate and size effects in quasibrittle fracture

The equivalent linear elastic fracture model based 0:;': an R-curve (a curve characterizing the variation of the critical energy release rate 'WIth the crack propagation length; IS generalized to describe both the rate effect and size effect observed in concrete. rock or other quasi brittle matenals. It is assumed that the crack propagation velocity depends on the ratio of the stress intensity factor to its criti~':!.l value based on the R-curve and that this dependence has the form of a power function with an exponent much largeT than l. The shape of the R-curve is determined as the envelope of the fracture equilibrium curves corresponding [.) the maximum load values for geometrically simIlar specimens of different sizes The creep in the bulk of a concrete specimen must be taken into account. which is done by replaCing the elastic constants in the iInear elastic :'~acture ~echanlcs ILEF\I) formulas with a linear viscoelastic operator In time Ifor rocks. \\ hlCh do not creep. thiS IS omitted L The experimental observation that the brittleness of concrete increases as the loading rate decreases (i.e. the response shifts in the size effect plot closer to LEFM) can be approximately described by assuming that stress relaxation causes the effective process zone length in the R-curve expression to decrease with a decreasing loading rate .. -\nother ?ower function is used to describe this. Good fits of test data for which the times to peak range from I sec to 2500C10 sec are demonstrated. Furthermore. the theory also describes the recently conducted relaxation tests. as well as the r~cently observed response to a sudden change of loading rate (both increase and decrease I. and particularly the fact thaI a sufficient rate increase in the post-peak range can produce a load-displacement response of positive slope leading: :0 a second peak. l. Introduction The rate of loading as well as the load duration is known to exert a strong influence on the fracture behavior of concrete. Much has been le::uned in the previous studies of Shah and Chandra [1]; Wittmann and Zaitsev [2]; Hughes and Watson [3J; Mindess [4]; Reinhardt [5J; Wittmann [6]: Darwin and Attiogbe [7]; Reinhard! [8]: Liu et al. [9J; Ross and Kuennen [10] and Harsh et al. [IIJ; in particular, it has been weLl! established that the strength as well as the fracture energy or fracture toughness increases wiTh increasing rate of loading, roughly as a power function of the loading rate. The previous ~[udies. however, focused mainly on the size effect under dynamic loading, at which the loading rates are very high. At such high rates. the rate effect is mainly due to the thermally activated process of bond ruptures. arising from the effect of stress on the Maxwell-Boltzmann distribution of thermal energies of atoms and molecules. In this study. we focus on the rate effect at static loading rates at which the creep properties of a material such as concrete begin to play also a significant role, aside from the thermal activation of bond ruptures. The rate effect at such Jaw rates, which is no doubt closely related to the effect of load duration, needs to be known for the design of civil engineering structures carrying high permanent loads or subjected to long rime thermal or shrinkage stresses. For such conditions (which are, for example, important for the fracture of dams), the rate effect in concrete '\\'alter P. \Iurphy Professor of Civil Engineaing. 356 Z.P. Ba:GIll and ,'vi. Jirilsek fracture has been essentially unexplored until the recent experimental studies of Baiant and Gettu [12-15]. The difficulty for materials such as concrete (which also includes rocks and tough ceramics) is that a nonlinear fracture model taking into account the existence of a large fracture process zone is required. Such materials. nowadays widely called quasibrittle. exhibit a transitional size effect in terms of their nominal strength: for small sizes. the behavior is close to plasticity, for which there is no size effect, while for very large sizes the behavior approaches linear elastic fracture mechanics (LEFM), for which the size effect is the strongest possible. As recently discovered (Bazant and Gettu [12-15J). the size effect plot. i.e. the plot of the nominal strength versus the characteristic structure size, is significantly influenced by the loading rate or loading duration. Generally, the loading rate or duration significantly influence the brittleness. Mathematical modeling of this phenomenon is the principal aim of this study. In previous work, the effect of loading rate on the size effect has been approximately described by quasielastic analysis, in which the behavior at each loading rate for all the specimen sizes is described according to LEFM with an elastic modulus that in effect represents the well-known effective modulus for creep. Such analysis brought to light the changes of brittleness: it. however, cannot be used as a general model if, e.g., the loading rate would vary with time. In this study, we will attempt a more general and fundamental model, which can be readily generalized to arbitrary loading histories. The model will represent an adaptation of quasilinear elastic fracture analysis by means of the so-called R-curves. The general principles of this approach, without any experimental verification, have already been suggested in Bazant [16,17]. In the present study we refine and extend this mathematical model and compare it to test data. The most general and fundamental approach for capturing both the size and rate effects in the fracture of concrete and other quasibrittle materials is of course a constitutive model for the evolution of damage in the fracture process zone, with an appropriate localization limiter. Such a model, which will be required for general finite element codes. should be the objective of future investigations. 2. Basic equations The R-curve (resistance curve) approach represents an attempt to describe the nonlinearity of the law of crack propagation in quasi brittle materials using an approximately equivalent linear model in which the fracture energy is considered to depend on the length of an equivalent linear elastic crack. This equi\ alent crack is defined as a crack in a linear elastic material having the same compliance as the actual specimen with a large nonlinear fracture process zone (Fig. I). Let us denote the initial crack length by Go and the current crack length by a. It is often more convenient to work with nondimensional quantities 10 = ao:d and 1 = G.d, where d is the total length of the ligament (Fig. I). According to LEFM, an applied load P causes a load-point displacement