P. Pallav, PhD

Standardized Specimen Design

Materials testing is generally carried out in a tensilometer. K_I or G_I is inferred from the load versus extension plot. This requires that the mathematical relationship between the stress intensity (K_I), crack length (a), and the applied stress (s) or force (P) is established for the shape of the sample.

When the relationship with the extension (D) and the compliance (C=D/P) is known too, it is possible to infer K_I versus crack size (a), provided the load-extension plot is not a straight line.

The ASTM proposes several specimen shapes and has published the mathematical solutions for most of these. Much paper work can be saved by using one of the ASTM specimen shapes. 
The sketch to the right is a disc-shaped compact specimen, left above is called a Double Edge Notched Tension (DENT) panel.

Pre-Cracking

The initial crack is generally produced in two steps. The first and largest part is often made by sawing. However, in order to obtain a valid measurement of K_I, the initial crack tip should be sharp, preferably sharper than during  crack propagation. Therefore a second step is required, with the purpose of creating a very small tip radius (r).

With relatively soft materials such as plastics and perhaps composites a razor blade may be pushed into the crack to create a tip radius, similar to the cutting edge of the blade.
With stronger materials this is not possible and much more effort is needed. Often the specimen is exposed to cyclic (fatigue) loading, so that in this way the sawed crack is extended with a fatigue crack. In other cases a Vickers Hardness indentation may be used to create a sharp pre-crack, which arises at the tip of the diamond indenter.

R-curve

This is a plot of either

• The resistance (R) of the material to fracture (either K_Ic or G_Ic) versus crack length (a), or
• The driving force for fracture (either K_I or G_I) versus crack length (a) at constant applied stress (s).

K_Ic versus Crack Length

Theoretically with absolutely brittle materials, a plot of the fracture toughness versus crack length is a horizontal line, because fracture toughness (K_Ic or G_Ic) is a material property.

With plastic deformation however the first part of crack growth will round off the crack tip (increase r), increasing the toughness of the material. This follows from in equation 1a in Fracture Mechanics.

As long as the tip radius keeps increasing with crack extension, the R-curve will be rising. At the same time the size of the plastic zone ahead and around the crack tip increases. The more plastic deformation, the longer (greater crack length) the R-curve will be rising.

Any assessment of K_Ic is valid only when the R-curve is horizontal enough because this more or less proves that

• Any plastic deformation has fully developed, and
• is limited to a small area around the crack tip, i.e. small compared to the width (W) and the crack length (a).

With plastically deformable materials ultimately the entire specimen may show plastic deformation and the R-curve will keep rising until complete failure. In such a case the specimens should be much larger. High toughness dental gold alloys may require huge specimen sizes in the order of a meter! Of course this only repeats that fracture mechanics is not relevant to the fracture of gold alloys in dentistry.

K_I versus Crack Length

As explained in Fracture Mechanics, the size of the crack (a) is a measure of the amount of stress, which gets concentrated at the crack tip. As a result the stress intensity (K_I) normally increases as the crack (a) grows.

Because of this, most specimen designs produce a rising R-curve. This means that when the load is increased to the point where K_I=K_Ic and the crack starts to grow, the stress intensity (K_I) at the crack tip will increase and produce instant complete fracture. The load versus extension plot will be a straight line and only a single assessment of K_Ic will be possible.

Chevron Notch

The chevron notched crack is a powerful contribution to create a falling R-curve. Because of the triangular shape of the ligament (= the part that will crack) the width increases with crack growth. Because the stress is distributed over less crack width, K_I is proportionally greater.

As the crack grows, K_I will decrease at first because the influence of the increasing width is stronger than that of the crack length. Only beyond a_m the R-curve rises again.

Comparison of a straight and a chevron notched specimen

The plot below shows four increasing levels of applied load on a chevron notched specimen. With the first (lower) two levels K_I is less than K_Ic and nothing happens; the crack length stays at a₀.  At the third level the crack has grown up to the intersection of this curve with the material's K_Ic.

As the load continues to increase to the top curve the crack growth should be stable, following the K_Ic curve, up to a crack length of a_m where the R-curve has its minimum. In this case it is possible to assess the fracture toughness (K_Ic) from a₀ to a_m, and it can be verified if the specimen is large enough for the toughness of the material, i.e. if the R-curve of the material is flat enough.

When the crack grows beyond a_m instant complete fracture occurs because from this point on, the R-curve rises.

With many ceramic materials there may be little doubt if these are brittle enough for the specimen size. In such cases, with a chevron notch only the force at fracture is needed to calculate K_Ic (at a_m).

It will be clear that a chevron notch does not require any more pre-cracking than the slightly more complex sawing part.

Specimen Compliance

As the crack grows in a controlled manner, the load extension plot will bend. The length of the crack can be inferred when the mathematical relationship between compliance (C), the crack length (a), and other dimensions of the specimen is known.

The compliance can be assessed by partially unloading during the test as shown with the solid parts of the blue lines. The compliance is the angle of such a line. With a perfectly brittle material, all these lines point to the origin (dotted) as in the left plot.

As more plasticity occurs, the unloading lines converge less to the origin. The right plot shows this rather exaggerated (this much plasticity is likely to render the test invalid).

The ASTM suggests two methods to compensate for an acceptable amount of (tip) plasticity. With only a small amount of plastic deformation, an iterative mathematical procedure is recommended. With larger amounts of plasticity, the ASTM advices the secant method. At the same time it seems to be that also with only little plasticity the secant methods gives better results than the more complex mathematical approach.

The secant methods is not very complex. The idea is shown in the right plot. The compliance is calculated as if the unloading lines do pass through the origin, as with the black dotted line. Of course a line from the origin to anywhere on the plot can be used to assess the 'corrected' compliance.

Note that the secant method also does not require actual unloading.

Reference

Anderson, TL (1991); Fracture Mechanics : fundamentals and applications; CRC Press, Inc.; ISBN 0-8493-4277-5.