B. Cotterell Engineering Fracture Mechanics 69(2002)533-553 Brittle Ductile 5| v453 → Precracked 105 666X73 8 88 72 Fig. 2. Welded and notched wide-plate tests on I in thick low carbon steel plates [31]- glass was accurately predicted by Griffith's theory. However, size effects were not so obvious in metal specimens. It was the work of Orowan [22] that led to the generalisation of Griffith's work to less brittle materials. Orowan [22] studied the depth of plastic strain beneath cleavage facets in low carbon steel using X-ray scattering. Irwin [25] noted that the energy expended in this plastic straining could be estimated from Orowan's result. This fracture energy, yn, for low carbon steel around oC turns out to be roughly two thousands times the surface energy, ,s. Irwin concluded that Griffiths theory could be used if the plastic work were substituted for the surface energy [25]. Orowan presented the same idea a little later [32] However, it was Irwin who grasped the engineering significance of the extension of Griffith's work and went on to develop LEFM. One very interesting paper on the direct measurement of ,, is that of Wells [33] whe used a thermocouple to measure the plane temperature wave emanating from a fast propagating fracture from which the heat source and ,p could be calculated At first Irwins development of LEFM was in terms of energy. He defined the elastic energy released for a unit increase in crack area, the crack extension force, G,-[5]. A fracture would initiate when G reached a critical value Gc= 2y. Work on hot stretching of PMMA [34]led Kies, a collaborator of Irwin at NRL, t observe that the critical stress for a given crack size depended only on GcE, where e is the elastic modulus The response of the Boeing engineers, who had initiated the work on hot stretching, was to use(Gce) which they termed the fracture toughness, Kc, in recognition of Kies, as their fracture parameter [35]. Irwin [36, 37). using Westergaard's paper [38], related G to the stress field at the crack tip and introduced the stress intensity factor K=(GE)"- which was also named in honour of Kies. For any symmetrical geometry, K can be expressed as 2 Irwin used the term crack extension force in analogy to a force on a dislocation and chose a Gothic letter G in honour of Griffith which caused difficulties in reproducing, now fortunately a Roman G is acceptable
glass was accurately predicted by Griffith’s theory. However, size effects were not so obvious in metal specimens. It was the work of Orowan [22] that led to the generalisation of Griffith’s work to less brittle materials. Orowan [22] studied the depth of plastic strain beneath cleavage facets in low carbon steel using X-ray scattering. Irwin [25] noted that the energy expended in this plastic straining could be estimated from Orowan’s result. This fracture energy, cp, for low carbon steel around 0 C turns out to be roughly two thousands times the surface energy, cs. Irwin concluded that Griffith’s theory could be used if the plastic work were substituted for the surface energy [25]. Orowan presented the same idea a little later [32]. However, it was Irwin who grasped the engineering significance of the extension of Griffith’s work and went on to develop LEFM. One very interesting paper on the direct measurement of cp is that of Wells [33] who used a thermocouple to measure the plane temperature wave emanating from a fast propagating fracture from which the heat source and cp could be calculated. At first Irwin’s development of LEFM was in terms of energy. He defined the elastic energy released for a unit increase in crack area, the crack extension force, G, 2 [5]. Afracture would initiate when G reached a critical value Gc ¼ 2cp. Work on hot stretching of PMMA[34] led Kies, a collaborator of Irwin at NRL, to observe that the critical stress for a given crack size depended only on GcE, where E is the elastic modulus. The response of the Boeing engineers, who had initiated the work on hot stretching, was to use ðGcEÞ 1=2 , which they termed the fracture toughness, Kc, in recognition of Kies, as their fracture parameter [35]. Irwin [36,37], using Westergaard’s paper [38], related G to the stress field at the crack tip and introduced the stress intensity factor K ¼ ðGEÞ 1=2 which was also named in honour of Kies. For any symmetrical geometry, K can be expressed as 2 Irwin used the term crack extension force in analogy to a force on a dislocation and chose a Gothic letter G in honour of Griffith which caused some difficulties in reproducing, now fortunately a Roman G is acceptable. Fig. 2. Welded and notched wide-plate tests on 1 in. thick low carbon steel plates [31]. 538 B. Cotterell / Engineering Fracture Mechanics 69 (2002) 533–553
B. Cotterell Engineering Fracture Mechanics 69(2002 )533-553 K=a√mF(a/W) where a is a representative stress, a is the crack length, and F(a/w)is a function of the geometry LEFM is one of the most successful concepts of continuum mechanics and has gone on the to be applied 1. fatigue crack growth, 2. stress-corrosion cracking, 3. dynamic fracture mechanics, 4. creep-and visco-elastic fracture However, there is not space for a discussion of these topics in this paper. Apart from a few notable exceptions, such as Wells whom regularly visited Irwin in America, the de- velopment of fracture mechanics in Europe and America during the 40s and 50s was quite separate In the main researchers in Europe were interested in low strength steels where the problem was more of transition from ductile to brittle behaviour whereas in America there was more interest in high strength steels, which were used in rocket motor cases. Griffith's theory of fracture had little direct quantitative application to low strength steels whereas it had application to high strength steels and it is not surprising that the general development of LEFM should have occurred first in America. However, there some isolated developments in LEFM in Europe whose general significance was not seen at the time. One such paper is that by rivlin and Thomas [39]on the rupture of rubber. As in the case of high strength metals, the deformation of rubber is essentially elastic except for a small region at the crack tip. However, the extension of rubber is very large and the deformation is non-linear. Rivlin and Thomas [39] independently proposed an application of a generalised Griffith theory of fracture to rubber. They proposed that the critical condition for catastrophic rupture would be when the energy released by crack propagation became equal to a tearing energy, T, which is equivalent to Irwins Gc, but since the deformation is non-linear, Griffith's equations could not be used. Rubber is almost invariably tested under fixed grip conditions so that the energy released comes only from the strain energy stored. Rivlin and Thomas [39]used a graphical method of determining the strain energy release rate and showed that rupture occurred when this reached a critical value equal to the rupture energy, T. It is interesting to note that the first application of lefM in a structural code occurred in the australia Timber Engineering Code AS CA65 of 1972 [40]as a result of the work of Leicester in the CSIRo Division of Building Research [41] 5. The development of elasto-plastic fracture mechanics LEFM predicts infinite stress at the crack tip, so that obviously there must be an inner core where the elastic solution breaks down. This factor was early recognised by Irwin [38] who, by simple equilibrium arguments, estimated the size of the plastic zone, dp, at a crack tip in a material with yield strength of oy K where n= l for plane stress and n= 3 for plane strain. Provided the stress field outside of the plastic zone is dominated by the K-field, then LEFM can be applied, which means that dp must be small compared with the dimensions of the specimen. There is a regime where the plastic zone does have a significant effect on the outside stress field, but so not so great that it completely destroys the K-field. In this regime, Irwin [38]
K ¼ r ffiffiffiffiffi pa p F ða=W Þ ð1Þ where r is a representative stress, a is the crack length, and F ða=W Þ is a function of the geometry. LEFM is one of the most successful concepts of continuum mechanics and has gone on the to be applied to: 1. fatigue crack growth, 2. stress-corrosion cracking, 3. dynamic fracture mechanics, 4. creep- and visco-elastic fracture. However, there is not space for a discussion of these topics in this paper. Apart from a few notable exceptions, such as Wells whom regularly visited Irwin in America, the development of fracture mechanics in Europe and America during the 40s and 50s was quite separate. In the main researchers in Europe were interested in low strength steels where the problem was more of transition from ductile to brittle behaviour whereas in America there was more interest in high strength steels, which were used in rocket motor cases. Griffith’s theory of fracture had little direct quantitative application to low strength steels whereas it had application to high strength steels and it is not surprising that the general development of LEFM should have occurred first in America. However, there some isolated developments in LEFM in Europe whose general significance was not seen at the time. One such paper is that by Rivlin and Thomas [39] on the rupture of rubber. As in the case of high strength metals, the deformation of rubber is essentially elastic except for a small region at the crack tip. However, the extension of rubber is very large and the deformation is non-linear. Rivlin and Thomas [39] independently proposed an application of a generalised Griffith theory of fracture to rubber. They proposed that the critical condition for catastrophic rupture would be when the energy released by crack propagation became equal to a tearing energy, T, which is equivalent to Irwin’s Gc, but since the deformation is non-linear, Griffith’s equations could not be used. Rubber is almost invariably tested under fixed grip conditions so that the energy released comes only from the strain energy stored. Rivlin and Thomas [39] used a graphical method of determining the strain energy release rate and showed that rupture occurred when this reached a critical value equal to the rupture energy, T. It is interesting to note that the first application of LEFM in a structural code occurred in the Australian Timber Engineering Code AS CA65 of 1972 [40] as a result of the work of Leicester in the CSIRO Division of Building Research [41]. 5. The development of elasto-plastic fracture mechanics LEFM predicts infinite stress at the crack tip, so that obviously there must be an inner core where the elastic solution breaks down. This factor was early recognised by Irwin [38] who, by simple equilibrium arguments, estimated the size of the plastic zone, dp, at a crack tip in a material with yield strength of rY to be dp ¼ 1 np K rY 2 ð2Þ where n ¼ 1 for plane stress and n ¼ 3 for plane strain. Provided the stress field outside of the plastic zone is dominated by the K-field, then LEFM can be applied, which means that dp must be small compared with the dimensions of the specimen. There is a regime where the plastic zone does have a significant effect on the outside stress field, but so not so great that it completely destroys the K-field. In this regime, Irwin [38] B. Cotterell / Engineering Fracture Mechanics 69 (2002) 533–553 539