## Posts Tagged ‘fracture mechanics’

### On Griffith’s criterion for brittle fracture

November 10, 2008

In a paper published in 1920 [1], A A Griffith laid the foundations of fracture mechanics with his criterion for brittle fracture: the wiki page on Griffith as well as this and this short notes by Zhigang Suo have more information on the man and his achievements. In this blog post, which is part of my monthly Classics in Materials Science series, I would like to discuss Griffith’s paper (with quotes from the paper itself — as usual); by the way, the wiki page on fracture mechanics also explains the idea behind Griffith’s work in a lucid manner, and links to several wonderful online resources.

The idea

Griffith noticed that polished surfaces and materials with smaller sized cracks have enhanced strength against rupture as opposed to materials with scratched surfaces and larger sized cracks. These observations were not satisfactorily explained by the then existing hypotheses of rupture in materials. So, at the very beginning, Griffith introduces his alternate criterion, which is solely based on energy minimization:

According to the well-known “theorem of minimum energy,” the equilibrium state of an elastic body, deformed by specified surface forces, is such that the potential energy of the whole system is a minimum. The new criterion of rupture is obtained by adding to this theorem the statement that the equilibrium position, if equilibrium is possible, must be one in which rupture of the solid has occurred, if the system can pass from the unbroken to the broken condition by a process involving a continuous decrease in potential energy.

Thus, Griffith’s criterion is nothing but a necessary, thermodynamic criterion. In modern parlance, if the elastic energy due to the presence of a crack can be relieved by opening the crack up, the system will do so — provided the cost paid to open it up (in terms of the creation of newer surfaces) is more than compensated by the elastic energy gain.

This idea is rather general, and, as Griffith himself notes,

Up to this point the theory is quite general, no assumption having been introduced regarding the isotropy or homogeneity of the substance, or the linearity of its stress strain relations.

Griffith goes on to consider two specific cases — the case of a cracked plate and that of the strength of thin fibres.

The cracked plate and thin fibres — theory and experiments

The first illustration that Griffith considers is as lens shaped crack that runs through the length thickness (Thanks Rajesh for the careful reading!) of the specimen; and the plate is under tensile load such that the tensile axis is perpendicular to the crack surface — as shown in the figure below.Given this scenario, now, it is just a question of calculating the strain energy and the surface energy. Griffith does both and obtains the well known expression for the stress $\sigma_f$ at which this material would rupture as $\sigma_f = \sqrt{\frac{2 \gamma Y}{\pi c}}$, where $\gamma$ is the surface energy and Y is the Young’s modulus. [Note that the elastic energy expressions in the paper are incorrect -- as Griffith himself notes at the end of the paper].

Griffith went on to do experiments on thin tubes and spherical bulbs of hard English to verify his expression; he measured the surface tension in these glasses, and then, after introducing scratches on them, ruptured them by the application of stress. Knowing the elastic modulus of the glasses and the scratch size, it was then easy to calculate the $\sigma_f$ and compare it with the experimentally observed values. The match was quite good — save a systematic error; and, Griffith attributes the error to stress intensity at the crack tips using the following observation, which I liked a lot (the emphasis and the paranthetical remarks are mine):

… It must be regarded as improbable that the error in the estimated surface tension is large enough to account for this difference [10% below the theoretical value], as this view would render necessary a somewhat unlikely deviation from the linear law [of variation of surface tension of glass with temperature].

A more probable explanation is to be obtained from an estimate of the maximum stress in the cracks. An upper limit to the magnitude of the radius of curvature at the ends of the crack was obtained by inspection of the interference colours shown there. Near the ends a faint brownish tint was observed, and this gradually died out as the end was approached, until finally nothing at all was visible. It was inferred that the width of the cracks at the ends was not greater than one-quarter of the shortest wavelength of visible light, or about $4 \times 10^{-6}$ inch. Hence, $\rho$ [radius of the crack tip] could not be greater than $2 \times 10^{-6}$ inch.

Griffiths ends the section on cracked plates with a discussion, in which, he succinctly sums up:

The general conclusion may be drawn that the weakness of isotropic solids, as ordinarily met with, is due to the presence of discontinuities, or flaws as they may be more correctly called, whose ruling dimensions are large compared with molecular distances. The effective strength of technical materials might be increased 10 or 20 times at least if these flaws could be eliminated.

Having noted that the elimination of flaws would lead to larger strengths, it was only natural for Griffith to test the idea and add support to the validity of his hypothesis; this he did with his experiments on thin fibres:

… very small solids of given form, e.g., wires or fibers, might be expected to be stronger than large ones, as there must in such cases be some additional restriction on the size of the flaws. In the limit, in fact, a fibre consisting of a single line of molecules must possess the theoretical molecular tensile strength. …

This conclusion has been verified experimentally for the glass used in the previous tests, strengths of the same order as the theoretical tenacity having been observed.

The rest

Apart from these discussions mentioned above which have become part and parcel of every materials scientists basic knowledge base, Griffith goes on to discuss many other things in the later sections of the paper — molecular theory, limitations of elastic theory and the application of his theory to liquids. They are very interesting as they are speculative (at least at that time). For example, here is Griffith on the validity of the continuum assumption for small scale systems, which, anybody who works on nanomaterials today will appreciate:

It is a fundamental assumption of the mathematical theory that it is legitimate to replace summation of the molecular forces by integration. In general this can only be true if the smallest material dimension, involved in the calculations, is large compared with the unit of structure of the substance.

The impact of the paper

The scientific study of fracture begain with Griffith’s work on cracks in brittle solids like glass …

– J D Eshelhy [2]

In real life, crystals deform at stresses which are much lower than those predicated by theories; this discrepancy is what gave birth to the theory, and later the experimental observation and confirmation, of dislocations. Similarly, in real life, materials rupture at stresses much lower than that predicted by theories; by introducing the idea of flaws, Griffith removed this inconsistency and showed a way of quantifying things.  In doing so, Griffith gave rise to the field of study, now known as fracture mechanics. His elegant and simple arguments have now become part of every basic materials engineering course and more detailed and varied  studies based on his work continue to be pursued (like this study on the beginning of earthquakes, for example) — which makes his paper a must-read. Finally, the last, and probably the best, reason to pursue this classic is perhaps also its clarity and the pleasure it gives on perusal. Have fun!

Note:

• Thanks to my colleague Rajesh (discussions with whom prompted me to pick Griffith for this post), I understand that a volume of Transactions of the American Society for Metals has re-printed Griffith’s paper — probably with annotations [3]; unfortunately, I am not able to locate the volume; but it might be well worth the effort; by the way, if you manage to do, send me a note or leave a comment below.
• As an historical aside, G I Taylor, one of the proponents of the dislocation theory is also the one who communicated Griffith’s paper; in addition, Taylor and Griffith also worked together on stress estimation; however, Griffith’s work on fracture predates that of Taylor on dislocations by more than a decade; on the other hand, while Taylor’s work lead to vigorous activity in the study of dislocations, Griffith’s was not followed up for quite some time — till after second world war or so.

References:

[1] A A Griffith, The phenomena of rupture and flow in solids, Philosophical Transactions of the Royal Society of London, Series A, 221,  163-198, 1920.

[2] J D Eshelby, Fracture Mechanics, Science Progress, 59, 161-179, 1971.

[3] Metallurgical classics, Transactions of the American Society for Metals, 61, 871-906 (1968).

Update – Nov 14, 2008: Though I vaguely remembered the story about a fire accident involving the experiments of Griffith with glass rods, I could not locate a reference earlier. Today, I found it in Gordon’s The New Science of Strong Materials (the entire section on Griffith and glass fracture is well worth your time):

I never knew Griffith himself but Sir Ben Lockspeiser, who acted as Griffith’s assistant at this time, told me something about the circumstances under which the work was done. In those days research workers were expected to earn their money by being practical, and in the case of materials they were expected to confine their experiments to proper engineering materials like wood and steel. Griffith wanted a much simpler experimental material than wood or steel and one which would have an uncomplicated brittle fracture, for these reasons he chose glass as what is now called a ‘model’ material. In those days models were all very well in the wind tunnel for aerodynamic experiments but, damn it, who ever heard of a model material?

These things being so, Griffith and Lockspeiser took care not to bring the details of their experiments too much to the notice of the authorities. The experiments, however, involved drawing fibres and blowing bubbles of molten glass and one day, after the work has been going on for some months, Lockspeiser went home leaving the gas torch used foe melting the glass still burning. After the enquiry into the resulting fire, Griffith and Lockspeiser were commanded to cease wasting their time. Griffith was transferred to other work and became very famous engine designer. The feeling about glass died hard. Many years later, about 1943, I introduced a distinguished Air Marshal to one of the first of the airborne glass-fibre radomes, a biggish thing intended to be hung under a Lancaster bomber. ‘Whats’s it made of?’ ‘Glass sir.’ ‘GLASS! — GLASS! I won’t have you putting glass on any of my bloody aeroplanes, blast you!’ The turnover of the fibreglass industry passed the £ 100,000,000  mark about 1959 I believe.