The construction of a

The evaluation of the fracture toughness of a material, as a measure of its resistance to crack extension, has been a relevant issue for the past five decades. Both in linear elastic as well as in elastic-plastic fracture, the interest in producing a value for a fracture-related property has prompted an enormous body of research. Recently, Zhu and Joyce (_{Ic}

The construction of _{i} – a_{o}_{i}_{o}_{Q}_{y}(_{y}_{Q}_{Ic}

Many of the classical methods for developing the _{Ic}_{Q}_{Q}

In this context, Donoso and Landes (_{o}_{f}

As an addition to the methodologies discussed above, this work will show how to construct a “key curve” based on the Common and Concise (C&C) formats to help in the process of generating crack extension data. The method rests upon the construction of the _{o}

The key curve concept has been used, among others, by Joyce (

_{o} = W – a_{o}_{o}_{o}

Experimental curve and key curve for A508.

The key curve is constructed by adding the elastic and the plastic displacement at any given value of force, both displacements being dependent on crack (ligament) size. Since both a blunt-notch specimen as well as one undergoing stable crack extension usually display a plastic displacement larger than the elastic component, the generation of the key curve is carried out with the Common Format Equation, CFE (Donoso and Landes, _{pl}

In Eq. (_{pl}_{o}_{o}

When there is stable crack extension, however, the ligament size _{pl}_{pl}

In Eq. (_{o}_{1}_{o}_{1}_{o} – b

Substitution of Eq. (

Thus, Eq. (_{o}_{o}/W

Let us assume now that in Eq. (_{o}_{1}_{o}

In _{o}_{f}

In Eq. (_{o}_{f}

The fact that the final _{i}, P_{i}, a_{i}

Thus, the task at hand is to show that if at the same fixed total displacement, the elastic displacements for the experimental curve and for the blunt notch curve with _{o}_{i}

at any point with total load line displacement _{i}

The proof proceeds as follows: the elastic components may be calculated from the compliance values corresponding to the crack (ligament) sizes at points

And at the corrected curve point

In equations (9) and (10), _{i}_{o}_{i}_{o}_{j}

In Eq. (_{e}_{e}

for the test curve, and

for the key curve.

However,

Multiplying numerator and denominator of the right hand side of Eq. (_{i}/W)^{μ} and rearranging, leads to:

By substituting Eq. (

Where

From the Concise Format (Donoso and Landes, _{f}_{f}

Thus, at any point _{i}

The significance of this finding is far from trivial. Let us assume now that an experiment has been performed in which there is crack extension, and a large number of points along the curve are known as

In order to produce crack (ligament) sizes at any _{Ic}

The procedure was used on the A508 specimen, and the results are shown in Figs. 2 and 3. Computed crack sizes

The resulting crack sizes from application of Eq. (

The result is quite good, considering that the key curve is a continuous, “perfect” curve, and the experimental curve may show some ripples, this being the main cause for the difference in crack sizes at any point on the target curve, as may be inferred from _{o}_{1}

Experimental, key and C&C curves for A508.

The next step is the construction of the _{Y}_{limit}_{limit}^{−2} and 1.8 mm respectively. The number of experimental _{1}(^{C2} defined in E1820. For the crack extension limits given in the standard, such curve is characterized by the parameters: _{1}_{2}^{2} = 0.9998.

The J-R curves for the experimental and key-curve derived A508 data.

The value of _{Q}^{-2}, and the quantity defined in E1820 as _{Q}/_{Y}_{o}._{Q} = J_{Ic}_{JIc}

This result is quite encouraging, and the key curve procedure to generate crack sizes may be summarized as follows: run fracture tests on two identical C(T) specimens, one with a sharp crack of a given size, and another with a blunt notch of the same size. Application of Eq. (

One interesting case of a P-v curve with limited number of crack sizes available corresponds to a TWIP (Twinning Induced Plasticity) steel C(T) specimen tested at room temperature (De Barbieri,

The TWIP C(T) specimen has dimensions _{o}_{f}

Experimental curve, key curve and final point for the TWIP specimen.

The Key Curve for the TWIP specimen was generated in such a way that the experimental points with measured crack sizes could have a matching point on the Key Curve at the same total displacement. Thus, the Key Curve generated crack sizes were compared to the experimentally measured crack sizes for those same points. Initial and final crack sizes are also included; the result is shown in _{o}_{1}^{2} = 0.998.

Key curve generated crack sizes vs experimental values for the TWIP specimen.

The

C&C and experimental curves for the TWIP specimen.

Regression analysis for the C&C and the experimental J-R curves.

The next step is the _{Y}_{min}_{limit}_{limit}^{−2}]. For clarity purposes, the maximum value of ^{-2}].

It should be noted that the number of the experimental points within the _{Q}

The regression analysis for the experimental and the C&C _{1}(^{C2} obtained with the experimental points has _{1}_{2}^{2} = 0.9227. The line for the C&C points has _{1}_{2}^{2} = 0.9998. With the latter, a _{Q}^{-2}] is obtained at Δ_{Q}_{Q}/_{Y}_{o}_{Q}_{Y}

Having fulfilled the E1820 validity criteria that deal with number and spacing of data points; the quality of the correlation concerning _{2}^{2}; the size validity concerning _{o}_{Q}_{Ic}^{-2}] should lead to a value of _{JIc}

The construction and use of a Key Curve in the evaluation of crack size data has been studied by Joyce (

_{i}/W)^{m}, which changes as the ligament _{i}_{o} = CBW(b_{o}/W)^{m}, and are shown by the open circles. From the results shown in

Normalized curves (force axis) for the A508 and TWIP specimens.

The equality of Eq. (_{o}

After removing all common terms, Eq. (

And

Summarizing, the steps required to determine crack sizes with the aid of the C&C derived Key Curve, are:

Conduct a fracture test on a pre-cracked 1T-C(T) specimen with the usual dimensions suggested by E1820-15 (2015), and initial crack size _{o}/W_{o}_{f}

Run a similar test on a blunt-notch 1T-C(T) specimen with the same dimensions and initial crack size equal to the average _{o}

From the force value of the final point

If it is not possible to run a blunt-notch test, construct the Key Curve with the aid of Eq. (_{o}_{el}_{pl}

At any given displacement larger than that for which the experimental and the key curve diverge (like point

The calculations of step e above should yield a sizable number of data points for the experimental curve of the type {

Use the elastic compliance to obtain the elastic displacement for the experimental curve at any _{o}_{1}

Use Eq. (

Now a _{1}_{2}_{Q}

A Common and Concise Formats approach to a Key Curve construction for generating crack sizes in C(T) specimens, when there are not enough data to validate a

The construction of the Key Curve is based on the Common and Concise (C&C) Formats, and therefore, it has analytical support. On this basis, the elastic compliances of both the Key Curve and the experimental curve may be matched not only at the end of the actual test (final points

It has also been shown that at any given total displacement, the experimental force normalized by the variable quantity _{o}_{o}

The Key Curve method presented here has some advantages over the previous methodologies developed by Donoso and coworkers, i.e., the Crack Growth Law (Donoso et al.,

As explained earlier, these methodologies — the Crack Growth Law, the Intercept Method and the present C&C based Key Curve method — have been used up to now with data solely from C(T) steel specimens, with W = 50 mm. Work on other geometries, materials and sizes should be encouraged in order to improve one or more of these methodologies in order to generate crack sizes where they are missing, by an adequate use of blunt-notch tests, or by using the C&C Formats to construct Key Curves.

The support of the Metallurgical and Materials Engineering Department at Universidad Santa Maria for JRD is gratefully acknowledged. FDB acknowledges FONDECYT Project 1140241. FDB also acknowledges the use of facilities by Professor Juan Perez Ipiña at Universidad Nacional del Comahue, Argentina.

_{Ic}from the load versus load-line displacement record