Two-Parameter Fracture Model (TPFM)

Cement-based materials exhibit pre-peak crack growth, therefore linear elastic fracture mechanics (LEFMs) cannot be directly applied to these materials. Over the last decades, several experimental and theoretical approaches have been developed to determine reliable parameters that can represent fracture properties of cementitious composites which are able to account for the development of the fracture process zone (Hillerborg et al., 1976; Bazant Z.P., 1984; Tang T et al., 1996). One, probably the most cited, fracture model which has been developed to account for the pre-critical crack growth for cement-based materials is the two- parameter fracture model (TPFM), proposed by Jenq and Shah (1985), which is based on the simple premise that a change in specimen compliance can be correlated to the length of the effective crack at the point when the critical (i.e. peak) load is reached. To introduce this model first the load CMOD (crack mouth opening displacement) of a notched specimen is considered.

A Linear elastic response up to a load corresponding approximately to Pmax / 2; that is, the

induced LEFM (linear elastic fracture model) KI (Stress intensity factor) is less than KIC/2.

During this stage the CTOD (crack tip opening displacement) is zero as predicted by LEFM. During the second stage, significant inelastic deformation takes place. This is caused by the formation of the process zone ahead of the crack tip (the existing crack being pre-notched or precast, and not the result of some prior crack nucleation/extension, for which a process zone first has to be developed). This process zone formation has also been referred as slow crack growth (Savastano H et al., 2009). As a result of this micro-cracking, the crack tip starts to open in a fashion similar to the blunting of sharp cracks in metals due to yielding. At the peak load, there are two conditions which are simultaneously satisfied:

At the peak load and for unstable geometries (i.e. increasing KI with increasing crack length

a), there are two conditions which are simultaneously satisfied:

KI = and CTOD = (2.1)

Where the parameters on the right hand side are considered to be material properties and CTOD refers to the notch opening and not the crack opening.

The results of the fracture test using this model are specimen size-independent, hence the critical values, and are size independent which is one of the major advantages

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of using two parameter fracture model; another requirement of this mode is the analytical expression of the COD and KI along the crack for the geometry considered.

2.8.2.1 Difficulties Using TFPM Based on Compliance Method

According to Tang and colleagues (1996), there is a concern of overestimating KIC and

CTODC whilst carrying out experiment procedure because the crack can extend between the

time of peak load and the time of unloading which will eventually result in recording larger measured change in compliance other than those associated with change exactly at the peak load.

Due to the large/rapid decrease in strength of material immediately after the peak load especially in highly brittle materials such as paste and high strength concrete, it has become a challenge to unload within 95% or exactly 95% of peak load. (Even though RELIM specifies that unloading should be done when peak load is decreased within 95% of the peak load). Some of these difficulties are summarised hence:

The response rate of dependency of the testing equipment results in variations in measurements taken between different laboratories. Close-loop method may not be available for most users.

The initial and unloading compliances (Ci ; Cu) are very sensitive to minor variations of

slopes; therefore extra care should be taken when measuring these parameters.

The use of 3-point bend beam is not always convenient especially for field application. The Modulus of elasticity is calculated from:

(2.2) (2.3) (2.3.1) (2.3.2) (2.3.3)

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Stress intensity factor can be calculated with the equation below:

(2.4)

Critical crack tip opening displacement can be calculated with the equation below:

(2.5)

= Initial compliance = unloading compliance = Initial crack length

= Critical effective-elastic crack length = Geometrical function

= thickness of specimen (beam)

= Critical stress intensity factor

= Critical crack tip opening displacement HO= Height of knife edge

In this research, the calculation of Cu, unloading flexibility was not achieved using the Instron

machine in the labs, and in order to get the unloading compliance two approaches were examined. In light of this, Krason and Jirsa (1969) found a band of points on the stress-strain plane which controls the degradation of the concrete under continued load cycles. The band is reduced to a single curve which is then called the locus of common points. The common point limit shows the maximum stress at which a reloading curve may intersect the original unloading curve, this means that the assumption made about the unloading flexibility Cu is

valid, the initial flexibility of the second cycle is taken as the unloading flexibility of the first cycle, the flexibilities of the concrete is the gradient of the curve produced by the Inston machine. This approach was assumed by the author, and the reason behind it is that the unloading was assumed to be nothing but the removal of the load cell from the beam; hence it will unload at the same rate to the loading of the second cycle. Because this assumption was

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not strong enough, the second approach was also used, which was to calculate the unloading compliance (flexibility) using the following formula in Eq 2.6 (Zhao et al, 2005; Zhao et al, 2007).

Cu = CMODc / Pmax. (2.6)

Where Pmax is the maximum load recorded during the test by Instron machine, and CMODc is

its corresponding crack opening displacement.

In the linear elastic fracture model this formula was also used, the unloading flexibility was assumed to be a straight line and hence is calculated by dividing the maximum load be the corresponding CMOD, to get the gradient which is the value of Cu (See Figure 2.18).

As aforementioned, in order to obtain the E, a linear regression function was fit between 5% and 40% of the peak load from the initial slope of the Load vs. CMOD curve, as shown in Figure 2.18; this is also known as the initial loading compliance (Ci).

Figure 2.18—Typical curves for fracture toughness of concrete specimen using TPFM: Load versus CMOD curve