University of LjubljanaFaculty of Mathematics and Physics

Department of Physics

Seminar

Material failure

Author: Aleksandar Popadic

Advisor: prof. dr. Primoz Ziherl

Ljubljana, 2013

Abstract

Materials fail by recurring rupture of atomic bonds at the atomic scale leading to the overall failureof the material. We discuss how materials fail and how biological materials achieve their extraordinarytoughness. We also look at how fracture mechanics can be applied to study of earthquakes and welook at supersonic rupture propagation in earthquakes.

Contents

1 History 1

2 Griffith criterion for rupture 3

3 Biological materials 6

4 Earthquakes and supersonic crack propagation 7

5 Conclusion 10

References 10

1 History

While fracture mechanics as we know it today was born in the early 20th century, many have studied thestrength of materials before [1]. Da Vinci measured the strength of iron wires and found that the strengthis inversely proportional to the length of the wire [2]. Da Vinci’s results already implied that flaws inmaterials control the strength of materials. The larger the specimen the greater is the probability offlaws in the material. Galilei also studied the strength of wires [2]. He studied the strength as a functionof thickness and thus provided the basis for the concept of stress defined as force per unit area.

Material failure has been a problem for as long as mankind has been building man-made structures.From the earliest times when people started to build it was necessary to have some information ofstrength of structural materials. No doubt the Egyptians had some empirical rules for material strengthwhen building pyramids. It was a trial and error approach then and only the successful designs stoodthe test of time. Romans supposedly tested their bridges by forcing the engineers to stand underneathwhile chariots drove over the bridge [1]. This would act as an incentive for good design and it wouldweed out bad engineers by a kind of Darwinian natural selection.

Prior to the industrial revolution there was a limited choice of materials and brick and mortarwere primarily used. Brick and mortar are relatively brittle and unreliable for carrying tensile loads,consequently pre-industrial revolution structures were designed to be loaded in compression. This isthe reason they used arch shapes in their designs. Although arch shapes are aesthetically pleasing, theprimary purpose was more pragmatic. Figure 1 shows the arches of the ancient Roman aqueduct Pont duGard. By comparing that structure to modern structures we can see that today much lighter structuresare built. In the Roman days they did not know how to select a proper shape and size and usually tooksemicircular shapes of relatively small span compared to newer structures.

With the industrial revolution came mass production of iron and steel. The new materials allowedfor designs that carried tensile stresses. Steel, however, brought new problems. Occasionally a steelstructure would fail unexpectedly at stress levels well below the anticipated tensile strength [1]. To avoidfailure, designers typically overdesigned their designs. Nowadays the practice of overdesigning is stillin use, but today, with all the complex structures we make, more can go wrong and better tools areneeded. We can not afford a plane crashing because of a structural failure for example. This is why goodunderstanding of material failure is important.

In 1920 Griffith gave a quantitative connection between fracture stress and flaw size [1]. Griffith’smodel correctly predicted the relationship between strength and flaw size in glass, an ideally brittlematerial. It failed to predict the connection in metals, however. Griffith’s work stayed largely unnoticeduntil after the World War II, when Irwin extended the model making it applicable to metals [4].

During the World War II USA was supplying ships to Britain and the German navy was sinking theships faster than they could build them [1]. They then came up with a new revolutionary procedure forfabricating ships. The new ships, which became known as the Liberty ships, had an all-welded hull, asopposed to the riveted construction of traditional ship designs. The Liberty ship program was a greatsuccess, until one of the ships broke completely in half in 1943. Subsequent failures followed. Figure 2shows the T2 tanker Schenectady broken almost completely in two. When the ship broke it was securelymoored at the fitting dock and the water was calm. The fracture happened without a warning and couldbe heard at least a mile away [5]. The failures of Liberty ships were studied in great detail by a fracture

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Figure 1: Pont du Gard built in the 1st century AD in ancient Rome. On it we can clearly see thesemicircular arches typical for structures of that time. We can also recognize the small arch span andconsequently a much heavier structure compared to newer structures [3].

mechanics research group at the Naval Research Laboratory led by Irwin and it is there, where the fieldof science now known as fracture mechanics was born [1].

Figure 2: The tanker Schenectady broken almost completely in two. When the ship broke it was securelymoored at the dock and the water was calm [5].

A number of successful early applications of fracture mechanics followed and the field has beendeveloping ever since [4]. Numerous extensions to nonlinear material behaviour have been made andrecent advancements in computer technology have encouraged microstructural models connecting thelocal material behaviour and the global material failure [1].

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2 Griffith criterion for rupture

Fracture can be viewed as dissipation of elastic energy into breaking of atomic bonds. Material fractureswhen sufficient stress and work are applied to break the bonds at the atomic level. Here we have widelydiverse length scales. Storing of elastic energy is a process associated with length scales of the macroscopicspecimen and the breaking of the atomic bonds happens at the atomic level. How a material behavesunder stress is essentially governed by the interatomic interactions.

Figure 3 shows the potential and force between the atoms with equilibrium at x0. The bond energy

Figure 3: Potential energy and the force between two atoms [1].

is equal to the applied work needed to break the bond:

Wb =

∞∫x0

F dx. (1)

By multiplying the above expression with number of bonds per unit area we get the applied work perunit area needed to break the bonds

wb =

∞∫x0

σ dx, (2)

where σ now represents stress. Let us approximate the force between the atoms with one half of theperiod of a sine (Fig. 3)

σ ∼ σc sin(x0 +

π

λ(x− x0)

). (3)

σc is the critical stress needed to break the bonds, x0 the equilibrium distance between the atoms and λis the length of half of the period of the sine wave and it represents the range of the interatomic potential.

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Then the applied work needed to break the bonds is

wb =

x0+λ∫x0

σc sin(x0 +

π

λ(x− x0)

)dx = 2σc

λ

π. (4)

When a material fractures two new surfaces are formed, so the applied work goes into formation of thesetwo new surfaces and it must be equal to twice the surface energy

σcλ

π= γs (5)

For small displacements ∆x = x− x0 the material is linear elastic, so we linearise Eq. (3)

σ ∼ σcπ

λ∆x = E

∆x

x0, E = σc

πx0λ, (6)

where E is the Young’s modulus. Putting this into Eq. (5) we obtain for the critical stress:

σc =

√Eγsx0

. (7)

In reality the stresses needed to fracture a material are typically three to four orders of magnitudelower [4]. As already mentioned, da Vinci’s experiments implied that the size dependent strength ofmaterials is due to microscopic flaws in materials. The flaws must lower the strength of the material byincreasing the stress locally around the flaw. It is common knowledge that notches or cracks are veryhelpful in breaking materials. For instance when cutting glass or ceramic tiles, a fine scratch is madeon the surface to weaken the material around the scratch. The material can then be broken in twoby application of force. Inglis, whose work Griffith based his work on, studied stress in a plate due topresence of cracks and sharp corners [6]. Consider an elliptical hole in a plate nowhere near the edges ofthe plate with stress applied perpendicular to the major axis of the ellipse as is shown on Fig. 4. Thenthe stress at the tip of the major axis (point A) is

σA = σ

(1 +

2a

b

), (8)

where 2a and 2b are the major and minor axes of the ellipse. Inglis also showed that this is a goodapproximation for a crack that is not elliptical except at the tip.

Figure 4: An elliptical hole in a plate with stress applied perpendicular to the major axis [6].

Inglis’s result further predicts that stress is infinite at the tip of an infinitely sharp crack. This causedconcern since no material is capable of withstanding such a load and would fail upon application of aninfinitesimally small load. This led Griffith to develop a theory based on energy rather than stress [1].Griffith applied the first law of thermodynamics to fracture mechanics. When a system goes from anonequilibrium state to an equilibrium state there is a net decrease in energy. For crack propagation

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this means that a crack can grow only if this would cause the energy to decrease. Critical conditions arethen found by demanding no change in net energy when a crack grows.

Consider a semi-infinite crack, where a crack reaches half way into the material and stress is appliedperpendicularly to the crack length, as is shown in Fig. 5. Far enough away into the uncracked material(area 1) the material is strained and there is no stress concentration due to crack tip. The elastic energyper unit area stored in area 1 is then w1 = σ2ξ/2E, where ξ is the width of the strip. Similarly thematerial far away from the crack tip into the cracked material (area 2) is completely relaxed and thereis no stress concentration, therefore the elastic energy per unit area in area 2 is w2 = 0. As the cracktip propagates by the length a, the length of the area of strained material decreases by a and the lengthof the relaxed material increases by the same amount. In other words the rate of change of energy withcrack area is the difference between the elastic energies per unit area of relaxed and strained material.At critical conditions this energy is equal to the surface energy of the two newly formed surfaces, whichgives the critical stress condition

σc =

√4Eγsξ

. (9)

Figure 5: A semi-infinite crack with a crack reaching halfway into the material [7]

The energy per unit area released with propagation of crack is also called energy release rate and isdenoted with G. The energy balance can then be writen

G = 2γs, (10)

which is often referred to as the Griffith criterion [7]. This condition is applicable only to ideally brittlematerials. Griffith succeeded in predicting fracture strength of glass, but failed to predict fracturestrength of metals. Griffith criterion severely underestimates fracture strength of metals. Irwin latermodified Griffith criterion by recognizing that formation of new surface is not the only possible energydissipation mechanism [4]. Crack tip propagation may be associated with amorphization at the crack tip,crack surface reconstructions or lattice reorganization mechanisms [7]. In these situations the Griffithcriterion must be modified. The criterion should include another dissipation term γdiss yielding

G = 2γs + γdiss. (11)

Thus the critical stress reads

σc =

√2(2γs + γdiss)E

ξ. (12)

Additional dissipation mechanisms increase the strength of the material and many materials engi-neering approaches to increase material strength are based on this concept of of introducing additionaldissipation mechanisms to prevent material failure [7]. For comparison, the energy release rate for glass,where the surface energy term dominates, is G ∼ 2 J/m2 and for steel, where the plastic effects dominate,G ∼ 1000 J/m2.

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3 Biological materials

Biological protein materials are all around us. They are a critical building block of life, forming adiverse group of materials. These materials play a crucial role on the biological function of all cells andtissues within organisms. Among these materials are also materials whose primary function is to provideprotection and support for the body, such as bone, spider silk, wood and nacre among others. Biologicalmaterials have a highly hierarchical design which has evolved over billions of years of natural selection.It is believed that their exceptional mechanical properties come from functional adaptation of structureat all levels of hierarchy [8].

Because of the complexity introduced by the hierarchical design the method of failure in such materialsis not fully understood. Better understanding of failure of biological materials could provide us withbetter therapeutic treatments. It may also lead to new methods of development of materials withbottom up approach, where we design materials from the atomic scale up.

Bone

Here we will take a closer look at the hierarchical structure of bone and failure mechanisms at differentlength scales. The hierarchical structure of bone is shown in Fig. 6. At the smallest scale we havetropocollagen protein molecules, each built from three polypeptides arranged in a triple helix. Staggeredarrays of tropocollagen molecules then form collagen fibrils, the smallest building block of bone. Tinycrystals of hydroxyapatite assemble in the gaps between collagen fibrils and the fibrils become mineralizedas the bone tissue grows and matures. Several collagen fibrils, each linked by an organic phase, form fibrilarrays. Each array makes up a single collagen fibre and several fibres arrange into randomly orientedparallel, tilted or woven bundles in bone and these bundles are organized into a lamellar structure. Thismicrostructure forms distinct mesoscale structural arrangements, such as compact or spongy bone [9].

When discussing toughening mechanisms in bone we separate them into two distinct groups. Weseparate them into intrinsic toughening mechanisms ahead of the crack tip and extrinsic tougheningmechanisms mainly behind the crack tip [10]. Intrinsic toughening is effective against both the initia-tion of a crack and its propagation. It is primarily related to plasticity and its origins tend to be onsubmicrometer length scales. Intrinsic toughening is the primary source of fracture resistance in ductilematerials, such as metals. Extrinsic toughening on the other hand is effective only against propagationof fractures, it has no effect on crack initiation. It acts primarily on micrometer length scales and above.Extrinsic toughening is the primary source and in many cases the only source of toughening in brittlematerials. Where in human made materials one type of toughening typically excludes the other, naturehas been successful in incorporating both types of toughening in materials.

In Fig. 6 different dissipation mechanisms in bone are shown. On the smallest length scale we havestretching and unwinding of individual collagen molecules. On a slightly larger length scale, intermolec-ular sliding and breaking of weak and strong bonds between tropocollagen molecules in collagen fibrilsis present. These sliding motions provide the basis for large plastic strains without catastrophic brittlefailure [10]. The mineralized collagen fibrils as the smallest building block of bone are of particular sig-nificance. The presence of hydroxyapatite is critical to the stiffness of bone since hydroxyapatite has overan order of magnitude higher elastic modulus than collagen. At somewhat coarser levels, the toughnessof bone has been tied to the presence of sacrificial bonds and hidden length. At several length scalesfrom sub-micrometer to tens of micrometers, the process of microcracking is present.

Microcracking is not only a process of plastic deformation, it is also essential for the developement ofextrinsic toughening processes. In cortical bone the path of least microstructural resistance is along thecement lines, which are the hyper-mineralized interfaces between the bone matrix and the osteons [11].These regions are therefore preferential sites for microcracks to form. In transverse crack orientation(along the length of the bone), the cement-line microcracks are oriented roughly perpendicular to thecrack path, where they act as barriers. The microcracks blunt any growing cracks by deflecting andtwisting the crack path. This generates an extremely rough crack. In the longitudinal crack orientationthe toughening mechanism is quite different. Now the cement-line microcracks are roughly parallel tothe growing crack and thus form ahead of and parallel to the main crack tip. This leads to formation ofuncracked regions along the crack, which bridge the crack and carry load that would otherwise be usedto propagate the crack. There are other crack bridging modes at smaller dimensions. One prevalentform of bridging is that due to fibrils which span many of the microcracks that form in bone.

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Macro

Mac

ro

Nano

Micro

Micro

Amino acids~1 nm

Tropocollagen~300 nm

Mineralizedcollagen f brils

~1 mm

Fibril arrays~10 mm

Fibre patterns~50 mm

Osteons andHaversian canals

~100 mm

Bone tissue~50 cm Compact

bone

Spongy bone

Structure Mechanisma b

Osteon

Crack def ection and twist Hidden length(sacrif cial bonds)

Collagen f breUncracked-ligament bridging

Collagen-f bril bridging

Collagen f bres

Bridge

Fibrillar sliding

Constrained microcracking

Microcracks Molecular uncoiling

Tropocollagen

Microcracking

Mineralizedcollagen f bril

Mic

ro

Nano

Tropocollagenmolecule

Hydroxyapatitecrystal

Extrinsic > 1µm Intrinsic < 1µmAhead ofcrack tip

Behindcrack tip

i

i

i

i

i

i

l

Figure 6: a Hierarchical structure of bone from nano scale in the bottom to macroscopic bone in thetop. b Dissipation mechanisms in bone. Extrinsic toughening mechanisms acting primarily behind thecrack tip are shown in the left column and intrinsic toughening mechanisms, which are primarily aheadof the crack tip, are shown in the right column.

As we can clearly see, there are many different energy dissipation mechanisms in bone, which providefor bone toughness.

4 Earthquakes and supersonic crack propagation

Fracture mechanics is also applicable in the study of earthquakes. Two tectonic plates are pressed againsteach other and subject to shear tension, while elastic energy is stored in the system over the course ofmany years as tectonic plates plates deform under tension [7]. The earthquake occurs when the elasticenergy is released and the two tectonic plates slide against each other. Elastic waves emitted duringthis process may lead to strong vibrations of ground on the surface, which may lead to severe damageson structures and injuries. Good understanding of dynamics of earthquakes is important for betterprediction of future events and the damage they may cause.

Figure 7 shows an analysis of the 1999 earthquake in Kocaeli, Turkey. In Fig. 7a the path of theearthquake can be seen. The earthquake started in Kocaeli (marked with a star) and spread almostlinearly along the weak plane in earth’s crust, where the two tectonic plates meet, towards Izmit Bayand Eften lake. This earthquake is historically significant because this is the first earthquake in whichsupersonic propagation was observed, a phenomenon which has been proposed earlier based on theoreticalstudies [7]. The graph in Fig. 7b shows the distance of the earthquake advancement with time fromwhich the speed can be read. After a short propagation with subsonic speed (blue line) a sudden jumpin speed follows and the earthquakes propagates with supersonic speed (red line).

The phenomenon has been studied further since the first observation and it was reproduced in labo-ratories with the so called laboratory earthquakes [12]. Figure 8 shows the setup of such an experiment.In the laboratory earthquakes the earth’s crust is scaled down and modelled by two photoelastic plates.

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(a) (b)

Figure 7: (a) The path of the 1999 earthquake in Kocaeli in Turkey. The earthquake starts in Kocaeli(marked with a star) and spreads almost linearly along the weak plane in earth’s crust towards IzmitBay and Eften lake. (b) Position of the earthquake with time. After a short subsonic propagation (blueline) a sudden jump in speed follows and the earthquake propagates supersonically (red line) [7].

The two photoelastic plates, representing the two tectonic plates, are put together so that the interfacebetween them is inclined by an angle. The pressure is then applied to the plates from the top and bottomso that they fell compressive and shear force at the interface. The rupture is initiated by in the middleof the interface with an exploding wire and the experiment is recoded by illuminating the photoelasticplates with a collimated laser beam from one end and capturing it with a high speed camera from theother. These experiments have suggested that supersonic rupture propagation is possible through the socalled mother-daughter mechanism where a secondary rupture (daughter) is born ahead of the primaryrupture (mother). This process can be seen in Fig. 9. In Fig. A the rupture speed is subsonic and itcan be seen that it is approximately the speed of shear waves. In Fig. B a secondary rupture is bornahead of the primary one. In Fig. C the two ruptures join and the leading edge now propagates atsupersonic speed. Figure D shows the rupture tip position with time on which the sudden jump in speedand position (birth of the secondary rupture) is more clearly seen.

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Figure 8: A Setup of a laboratory earthquake experiment with photoelastic plates, representing thetectonic, plates in the middle. B Picture of the experiment. C Close up of the interface between thephotoelastic plates at the moment of rupture initiation [12].

Figure 9: Visualization of subsonic to supersonic fracture transition in the photoelastic plate. A - thespeed is subsonic and approximately the speed of shear waves. B - a secondary rupture is born ahead ofthe primary one. C - the rupture propagates supersonically. D - plot of rupture tip position with time.The birth of secondary rupture and the jump in speed can be clearly seen at time ∼30 µs [12].

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5 Conclusion

In this seminar we looked at how materials fail and we took a closer look at failure mechanisms inbone. Good understanding of material failure mechanisms is important for engineering of new materialsand for biological materials it may lead to better therapeutic treatments. We also briefly looked atsupersonic rupture propagation in earthquakes. In rapidly propagating cracks the Griffith criterion hasto be modified to take into account the kinetic energy [13]. The rapidly moving cracks are studied indynamic fracture mechanics [14]. The field of fracture mechanics is a rapidly advancing field. Herewe only talked about linear elastic materials, fracture mechanics, however, has also been extended tononlinear elastic materials [1]. In the recent time the molecular dynamics approach has been gaininginterest, providing for much insight into fracture mechanisms.

References

[1] T. L. Anderson, Fracture mechanics: fundamentals and applications, 3rd edition (CRC Press, BocaRaton, 2004).

[2] S. P. Timoshenko, History of strength of materials (Dover Publications, New York, 1983).

[3] Pont du Gard by Emanuele on Flickr, CC-BY-SA-2.0.

[4] F. Erdogan, Int. J. Solids Struct. 37, 171 (2000).

[5] Wikipedia, SS Schenectady, http://en.wikipedia.org/wiki/SS_Schenectady (visited on26/03/2013).

[6] C. E. Inglis, Trans. Inst. Naval Architects 55, 219 (1913).

[7] M. J. Buehler, Rev. Mod. Phys. 82, 1459 (2010).

[8] P. Fratzl and R. Weinkamer, Prog. Mater. Sci. 52, 1263 (2007).

[9] R. O. Ritchie, Nat. Mater. 10, 817 (2011).

[10] R. O. Ritchie, M. J. Buehler and P. Hansma, Phys. Today 62, 41 (2009).

[11] M. E. Launey, M. J. Buehler and R. O. Ritchie, Ann. Rev. Mater. Res. 40, 25 (2010).

[12] K. Xia1, A. J. Rosakis and H. Kanamori, Science 303, 1859 (2004).

[13] A. J. Rosakis, O. Samudrala and D. Coker, Science 284, 1337 (1999).

[14] L. B. Freund, Dynamic fracture mechanics (Cambridge University Press, Cambridge, 1998).

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