The shadow optical method of caustics - crack tip stress intensity factors

TWI Bulletin, May/June 1996

The shadow optical method of caustics for determining crack tip stress intensity factors is seldom applied, either at TWI or elsewhere. Mike Ellis reviews the method and a number of specific applications for this technique.

The word 'caustic' has two definitions in common use: to describe a substance that burns or corrodes organic tissue and to describe the nature of a person's humour, ie sarcastic or biting. However, a third, and often forgotten definition, is that a caustic is formed by the intersection of rays reflected or refracted from a curved surface. When applied to fracture mechanics, the optical method of caustics opens up a number of opportunities for characterising crack tip stress intensity factors.

In linear elastic fracture mechanics, the stress intensity factor (K) is still the most widely used parameter for characterisation of the stress field around the tip of a crack. Knowledge of K, together with relevant material properties, enables estimates to be made of the resistance to failure ( eg by fatigue, brittle fracture, etc) of a cracked component or structure.

The value of K may be determined in a variety of ways. The conventional linear elastic fracture mechanics approach requires knowledge of details such as crack length, component geometry and the applied load in order to evaluate K. In contrast, if the shadow optical method of caustics is used, K can be obtained directly from the cracked component without knowledge of these parameters. Therefore, this approach is useful in assessing cracked bodies and crack configurations which are difficult to study, both practically and analytically. In the shadow optical method of caustics it is the reflected (or transmitted) light from a curved surface associated with the crack tip which is used to optically characterise the stress intensity, K _opt.

The shadow optical method of caustics should be used to analyse cracks in real structures such as pressure vessels, oil rig nodes and welded joints. At present, TWI is pursuing the possibility of using this technique.

Caustic technique

K _opt = MD ^5/2     [1]

where M = Kz _o m ^3/2 d and D = r _o mF (see Fig.3) where D is the transverse diameter of the caustic, f is a numerical constant (= 3.17), K is a constant which incorporates the opto-elastic constant (Poisson's ratio divided by Young's modulus) and f, d is the specimen thickness, and m is the magnification factor of the caustic relative to the optical set-up ( Fig.2) where m = Z ₁ /Z ₁ +Z ₀ . ^[1,2]

A typical optical arrangement is shown in Fig.2. Laser light (A) is used for illumination, producing a light beam which is partially collimated by an optical collimator (B and C). The light beam striking the specimen is reflected onto a bi-convex lens (E) and collected by the camera (F). As the laser beam emitted by the collimator is not necessarily parallel (the light being divergent), the actual size of the recorded caustic in the camera is not the same as that on the specimen. The appropriate equations of all magnification factors and their correction are given elsewhere. ^[2]

As an example of the validity of the application of the shadow optical method of caustics, K was measured on a cracked specimen for which it could also be calculated accurately, this was a disc-shaped specimen made from epoxy resin. ^[3] K for this geometry is given by the following equation.

Where P is the applied load, F(a/W) is the stress intensity factor which relates to specimen geometry, B is the specimen thickness, W is the specimen width and a is the crack length.

A fatigue crack was grown in the specimen using a servo-hydraulic testing machine to a/W between 0.45 and 0.55. These cracked discs were subsequently polished to lµm diamond paste giving an optically smooth surface. They were then placed back in the load frame and the optical arrangement shown in Fig.2 was placed around the epoxy specimens.

Static loads were applied, in small increments, to the discs and the resulting caustic shadow ( Fig.4) around the crack tip was recorded. The resulting values of K _opt ( Fig.5), calculated using Equation 1, are compared with the K _mech ( Fig.6). The excellent correlation between K _opt and K _mech for monotonic loading of an epoxy resin disc is thus highlighted ( Fig.7).

As mentioned earlier, the major benefit associated with the shadow optical method of caustics is that values of the crack tip stress intensity factor can be determined without any knowledge of the applied load (P) or crack length (a). For this reason, provided that optically smooth surfaces are present, K can be evaluated in real structures.

General fracture mechanics

A literature survey was carried out, to list some of the uses of the shadow optical method of caustics:

It has been applied in the study of mixed mode loading of cracks ( ie opening and shear) in polymers and metals. [4,5] High speed photography allied to the shadow optical method of caustics has been used to evaluate stress intensity factors under dynamic loading in pre-cracked bend bars. [6] A study of the behaviour of propagating and subsequently arresting cracks under impact loading has also been undertaken, [7] and the impact toughness response of structural steels has been investigated by observing the dynamic stress intensity factor against time. [8] The method has also been used to predict values of J integral in polycarbonate and Al alloy specimens. [9]

Fatigue

A prediction of fatigue crack growth has been achieved by using caustics to indicate the presence of a plastic zone of a notch up to a compressive overload. [10] This size of plastic zone was comparable with that determined by subsequent fatigue crack propagation. Plasticity-introduced crack tip closure has been studied using caustics in 7xxx series Al alloys. [11,12] A fatigue crack was induced by cyclic compression so that the crack was free from any closure-related deformation. At low levels of Δ K, the shadow optical method was found to underestimate the extent to which plasticity induces non-closure. However, as Δ K is increased, the value of K opt was seen to increase with increasing crack length, and, when all the load was removed, a residual caustic was seen due to plastic stretching of the surrounding area. Further work showed that crack closure and related shielding mechanisms could be studied successfully by applying the shadow optical method. [12]

Stress corrosion

The shadow optical method of caustics has been used to characterise the stress corrosion cracking (SCC) behaviour of Al alloys. [2,13,14] Development of SCC was assessed using caustics to characterise mechanically the crack tip in 7xxx alloy. Crack tip solution chemistries were monitored during SCC. At high values of K the caustic shadow diameter was seen to increase as SCC was initiated.

Concluding remarks

The shadow optical method of caustics is, in principle, a versatile experimental method for measuring crack tip stress intensity factors, the equipment is portable (in a hand-held case) and can be purchased at relatively low cost (less than £1K). The fact that K opt can be determined without detailed knowledge of the crack length and applied load means that difficult analytical situations can be successfully studied. Coupled with computer-based stress analysis methods, such as finite element analysis, the method can also be used to verify calculated values of K.

Acknowledgements

The author would like to thank Alcan International Ltd for their kind permission to publish results of the work carried out on the epoxy resin.

References

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