Fracture mechanics is basically about scaling. Leonardo da Vinci (1452–1519) was the first to record some understanding of scaling in fracture, by designing strength tests on iron wires of different length. While we don’t know whether Leonardo actually ever realized such tests, which would have required a high degree of precision for his times, he described in great detail the method to follow. Notably, he paid the greatest attention to the wire length, neglecting at all the diameter. By this measure, we should take him as the first to have seriously posited the problem of scaling in fracture. Almost two centuries later, Galileo Galilei with his Discorsi e dimostrazioni matematiche intorno a due nuove scienze (Dialogues and mathematical proofs concerning two new sciences, 1638) was the first to give the correct scaling laws for bars under tension and bending. Size effects are very important in fracture problems, and Galileo saw that this effect placed a limit on the size of structures, either man-made or natural: Impossibil sarebbe fabbricar navilii, palazzi o templi vastissimi, […] non potrebbe la natura far alberi di smisurata grandezza, poiché i rami loro, gravati dal proprio peso, finalmente si fiaccherebbero. (It would be impossible to build ships, palaces, or temples of enormous size […] nor could nature produce trees of extraordinary size because their branches would break down under their own weight.) If you remember my previous write-up about Galileo’s mistake with his discussion of Dante Alighieri’s “roof of the Hell” in 1587, you will agree that fifty years later he had well learned his lesson.

One key point that ancient scholars were missing was the notion of elasticity. This is just one aspect of Isaac Newton’s principle of action and reaction, although apparently the Great Man didn’t notice. The man who instead noticed it was Robert Hooke, with the statement that he famously hid in an anagram, Ut tensio sic vis (the force is proportional to the elongation), for which he is therefore considered the father of the theory of elasticity. But still, even Hooke did not consider the shape and size of the body. It would take Thomas Young to reach a clear understanding of tensions. In 1807, Young had just been fired from the Royal Society because his work was considered too practical; fortunately, he was soon hired by the Bureau des Longitudes. In that same year he announced the law that bears his name, namely 𝝈=E𝜺 in modern terms, with 𝝈 the stress and 𝜺 the deformation, also introducing the quantity E that we call today Young’s modulus. Elasticity is key in the fracture of a material, be it brittle or ductile, since it is the atomically-based mechanism by which the body responds to the applied load. On the one hand, the applied load deforms the body, and in return the body modifies the load, redistributing the forces according to its anisotropic elastic moduli.

It took another whole century after Young, for the concept of stress concentration to become the cornerstone of materials mechanics. Alan Arnold Griffith was a young engineer at the Royal Aircraft Establishment during WWI, when he started thinking around a puzzling problem. At the time it was generally assumed that the strength of any material was about E/10, a tenth of its Young’s modulus; however it was well known that many materials would often fail at just a thousandth of this predicted value. Griffith hypothesized that the microcracks existing in any imperfect material lowered the overall strength, by concentrating the stress at crack tips and other defects. This concentration would allow the stress to reach the critical E/10 at the tip of some cracks, long before the material as a whole attained such stress level. In 1921 Griffith published his famous work The phenomena of rupture and flow in solids [Phil. Trans. Royal Soc. A 221, 163]. Basically, Griffith understood the brittle vs. ductile behavior of materials as an energy competition between bond breaking across fracture surfaces and dislocation plasticity. Materials like glass or ceramic that have strong atomic bonds, break with a fast, fragile crack; materials like copper or wax that allow easy nucleation and motion of crystal defects (e.g. dislocations) are less hard, but can sustain larger stresses by deforming before breaking, they are tough and ductile.

Liberty cargo ships, built in the United States during World War II primarily for carrying troops, are the best known example of the importance of stress concentration. Eighteen American shipyards built 2751 such ships between 1941 and 1945. They were British in conception but adapted by the United States, cheap and quick to build. However, the initial design was modified by the U.S. Maritime Commission to conform to American construction practices. The new design replaced much riveting, which accounted for one-third of the labor costs, with welding. But no attention was paid to how welding would affect the structure. As a result, many Liberty ships suffered disastrous hull and deck cracks. Ships could broke in half without warning, like the one shown in the photo enclosed below. The blame was initially attributed to inexperienced workers and new welding techniques to produce the large numbers of ships quickly. It was only later discovered that the problem lied in the embrittlement of steel, caused by the very low temperatures of the North Atlantic.The predominantly welded hull allowed cracks to run long distances, with the the welding sutures acting as stress concentrators that “guided” the crack along its path. Since then, understanding why a material is brittle or ductile and how it can turn from one behavior to the other as a function of various parameters, such as temperature, has become a fundamental question in materials science. 

In 1996-7, a two-year research program took place at the Institute for Theoretical Physics (now Kavli Institute) in UC Santa Barbara, on Modeling of Industrial Materials: Connecting Atomistic and Continuum Scales. Several scientists could alternate in residence, and every one or two weeks a 2-3 days workshop was held on some hot topic. In those days there was a large effort among the American materials science community, to bring more robust atomistic foundations into the realm of materials mechanics, and notably the microscopic study of fracture and dislocations, both in metallic and covalent materials. My main fixation at the time was a new method to obtain the so-called “gamma surface” that describes the stress profile of a dislocation in a crystal lattice in terms of the sliding of one half of the crystal with respect to the other. This idea dated back to a famous paper by Rudolf Peierls [“The size of a dislocation”, Proc. Phys. Soc. 52, 34 (1940)], and had been subsequently expanded by Nabarro, Eshelby and others.  To me such a picture appeared somewhat too simplistic. So, I was fighting to obtain a better definition. I brought my obsession at the Santa Barbara meetings, where I spent three terms among the large crowd gathering both bright-old-guns and young-shining-stars (I was neither one). I proposed my idea, mostly to the old guns, without meeting much interest. A few months after the Santa Barbara meeting, another fracture mechanics meeting took place in Washington. There I met Robb Thomson and Frank Nabarro, and I extended also to them my frustrating concerns about the gamma surface: Nabarro was little interested and seemed not to get my main point, while Thomson picked up the idea but found my calculation method unpractical, compared to the simplicity of the gamma surface concept. (For the specialists out there, the problem lies in the difficulty of obtaining the local stress components for each atom from density-functional theory.) Eventually, I published my model despite their criticism [Phys Rev Lett  79, 1309 (1997)], only to discover that a quite similar calculation had been already done a couple of years before by Richard Hoagland at Los Alamos. I was awfully disappointed, and started losing interest in materials mechanics.

However, I remember the Santa Barbara meeting also because I met there Huajian Gao, at the time still at Stanford as a young associate professor in mechanics, and we remained friends since then. He was and still is an extremely kind person, and an extremely bright scientist; he obtained his PhD at Harvard when he was just 25, under the guidance of Jim Rice. Huajian (who in the meantime moved from the young-shining-stars section to the bright-old-guns one, with professorships at Max Planck Stuttgart, Brown, and Nanyang Singapore) has recently published a very interesting work on the mechanical behavior of two-dimensional hexagonal boron nitride (h-BN). He and his coworkers showed that this material has an intrinsic toughening mechanism, thus contradicting its predicted reputation for brittleness [Nature 594, 57 (2021)]. Such unexpected behavior of h-BN defies Griffith’s description of fracture mechanics and material toughness. As with most material, cracks in 2D materials also form at sites of concentrated stress. Their peculiar structure, however, makes cracks propagate straight through, opening up the bonds between atoms like a zipper. This happens for example in graphene, the epitome of 2D materials: graphene is very strong since its bonds are hard to break, but this also makes it very brittle, and its fracture toughness is very low. h-BN is a dielectric two-dimensional material with high strength of about 100 gigapascals, and Young’s modulus of about 0.8 terapascals, very similar to the corresponding values of graphene. Therefore, according to Griffith it should be very much brittle. Contrary to expectations, however, the fracture toughness of single-crystal monolayer h-BN found by Gao and coworkers is one order of magnitude higher than the Griffith theory prediction, and notably much larger than what reported for graphene. They observed stable crack propagation in monolayer h-BN, with crack deflection and branching occurring repeatedly. This is attributed to asymmetric elastic properties at the crack tip, and to a phenomenon of “edge swapping” during crack propagation, which makes the cracks to branch. This effect intrinsically toughens the material, because branching means that additional energy is required to drive the crack further, even in the absence of dislocations that usually round off the cracks in ductile metals. The h-BN already known resistance to heat, stability to chemicals, and dielectric properties already make it ideal as both a supporting base and an insulating layer for placing between electronic components. Now, the recent Huajian’s discovery that h-BN is also exceptionally tough means that it could be used to add tear and break resistance to flexible electronics.

Our understanding of fracture mechanics has gone a long way, from Leonardo’s wires to the Liberty ships to hexagonal two-dimensional materials, and still is full of surprises.

The breaking of our Liberties

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