The Griffith concept crack was observed to retreat and apparently heal', but re- insertion of the wedge revealed a perceptible reduction in cleavage strength. These resul imply the existence in the energy balance of dissipative elements 1.5 Molecular theory of strength Although Griffith formulated his criterion for fracture in terms of macroscopic thermodynamical quantities, he was aware that a complete description required an evaluation of events at the molecular level. He argued that the maximum stress at the tip of an equilibrium crack must correspond to the theoretical cohesive strength of the solid; that is, the largest possible stress level that the molecular structure can sustain by virtue of its intrinsic bond strength. Griffith accordingly estimated the theoretical strength of his glass from the stress-concentration formula (1. 4), inserting p a 0.5 nm(molecular dimensions), as areasonable tip radius for a crack growing by sequential bond rupture, together with his measured value c/= Op c0/ at instability(sect. 1. 3). The value obtained ac 23 GPa, is an appreciable fraction of Youngs modulus for glass representing a bond strain of some 0.3-0.4. Griffith appreciated that Hooke's law could hardly be assumed to hold at such strain levels, for the force-separation relationship for interatomic bonds surely becomes nonlinear immediately prior to rupture. Nor could (1.4), based on the continuum concept of matter, be relied upon to give accurate results on the molecular scale, with due allowance for these factors, Griffith concluded hat the limiting cohesive strain was probably in the vicinity of 0.1 By way of confirmation of his estimate, Griffith consulted the literature for values of the intrinsic pressureof solids(as determined for instance, from the heat of vaporisation or equation of state). Since both theoretical strength and intrinsic pressure essentially measure the mol cular cohesion, their magnitudes should be comparable, at least for nearly isotropic solids. Griffith determined this to be the case. He thus inferred that the theoretical strength should be a material constant, closely related to the energy of cohesive bonds, with a value of order E/10 for all solids Thus with both p and ac effectively predetermined by the molecular structure of the solid, the critical applied tension in the Inglis equation(1. 4) becomes dependent on the crack size. There is an implication here of an invariant crack-tip structure. The last obstacle to a basic fracture criterion (sect. 1. 1)is thereby removed
12 The Griffith concept crack was observed to retreat and apparently 'heal', but re-insertion of the wedge revealed a perceptible reduction in cleavage strength. These results imply the existence in the energy balance of dissipative elements. 1.5 Molecular theory of strength Although Griffith formulated his criterion for fracture in terms of macroscopic thermodynamical quantities, he was aware that a complete description required an evaluation of events at the molecular level. He argued that the maximum stress at the tip of an equilibrium crack must correspond to the theoretical cohesive strength of the solid; that is, the largest possible stress level that the molecular structure can sustain by virtue of its intrinsic bond strength. Griffith accordingly estimated the theoretical strength of his glass from the stress-concentration formula (1.4), inserting p « 0.5 nm (molecular dimensions), as a 'reasonable' tip radius for a crack growing by sequential bond rupture, together with his measured value crAc 1/2 = aF cj/2 at instability (sect. 1.3). The value obtained, ac « 23 GPa, is an appreciable fraction of Young's modulus for glass, representing a bond strain of some 0.3-0.4. Griffith appreciated that Hooke's law could hardly be assumed to hold at such strain levels, for the force-separation relationship for interatomic bonds surely becomes nonlinear immediately prior to rupture. Nor could (1.4), based on the continuum concept of matter, be relied upon to give accurate results on the molecular scale. With due allowance for these factors, Griffith concluded that the limiting cohesive strain was probably in the vicinity of 0.1. By way of confirmation of his estimate, Griffith consulted the literature for values of the ' intrinsic pressure' of solids (as determined, for instance, from the heat of vaporisation or equation of state). Since both the theoretical strength and intrinsic pressure essentially measure the molecular cohesion, their magnitudes should be comparable, at least for nearly isotropic solids. Griffith determined this to be the case. He thus inferred that the theoretical strength should be a material constant, closely related to the energy of cohesive bonds, with a value of order E/10 for all solids. Thus with both p and ac effectively predetermined by the molecular structure of the solid, the critical applied tension in the Inglis equation (1.4) becomes dependent on the crack size. There is an implication here of an invariant crack-tip structure. The last obstacle to a basic fracture criterion (sect. 1.1) is thereby removed
Griffith flaws 1.6 Griffith flaws The argument in the previous section gave an indication of the strength that could be achieved by an ideal solid, an ultimate target in the fabrication of strong solids. Griffith was intrigued by the fact that the strengths of real materials'fell well short of this level, typically by two orders of magnitude, despite great care in maintaining specimen perfection on an optical scale. a further discrepancy was also evident. If a solid were to fail at its theoretical strength the applied stress would reach a maximum at rupture, implying a zero elastic modulus at this point at such a rupture point a sudden release of stored elastic strain energy, equivalent ap- proximately to the heat of vaporisation, would be expected to manifest itself as an explosive separation of the constituent atoms. Again, real materials behaved differently, parting instead with relatively little kinetic energy on a more or less well-defined separation plane Griffith concluded that the typical brittle solid must contain a profusion of submicroscopic flaws, microcracks or other centres of heterogeneity too small to be detected by ordinary means. The'effective length, Co =Cp, of these so-called Griffith faws was calculated by inserting the tensile strength of the strongest as-received glass specimen tested (sect. 1.3), op 170 MPa, along with the previously measured values of E and ,, into critical condition(1. 11): this gave c, 2 um. We may deduce from(1.2) that a molecularly sharp microcrack of this length has a wall separation b≈0.05 ch is about one-tenth of the wavelength of and therefore barely on the limit of optical delectability. The theoretical stress-concentration factor(1. 4)is of order 100 in this instance, emphasis ower of even the most minute of flaws To test his flaw hypothesis Griffith ran a series of experiments on the strength of glass fibres. The fibres were drawn from the same glass as used in the previous tests(sect. 1.3), and were broken either in tension or in bending under a monotonically increasing dead weight. Well-prepare pristine fibres showed unusually high strengths, shattering in the explosi manner expected of ideal, flawless solids. However, on exposure to laboratory atmosphere all fibres declined steadily in strength, reaching after a few hours a steady state value more typical of ordinary glass specimens. Griffith next tested a large number of such ' fibres with diameters ranging from I mm down to 3 um, and found an apparent size effect; the thinner specimens showed a tendency to greater strength Arguing that a single chain of molecules must possess the theoretical
Griffith flaws 13 1.6 Griffith flaws The argument in the previous section gave an indication of the strength that could be achieved by an ideal solid, an ultimate target in the fabrication of strong solids. Griffith was intrigued by the fact that the strengths of ' real materials' fell well short of this level, typically by two orders of magnitude, despite great care in maintaining specimen perfection on an optical scale. A further discrepancy was also evident. If a solid were to fail at its theoretical strength the applied stress would reach a maximum at rupture, implying a zero elastic modulus at this point: at such a rupture point a sudden release of stored elastic strain energy, equivalent approximately to the heat of vaporisation, would be expected to manifest itself as an explosive separation of the constituent atoms. Again, real materials behaved differently, parting instead with relatively little kinetic energy on a more or less well-defined separation plane. Griffith concluded that the typical brittle solid must contain a profusion of submicroscopicyfaws, microcracks or other centres of heterogeneity too small to be detected by ordinary means. The 'effective length', c0 = ct, of these so-called 'Griffith flaws' was calculated by inserting the tensile strength of the strongest as-received glass specimen tested (sect. 1.3), crF = 170 MPa, along with the previously measured values of E and y, into the critical condition (1.11): this gave cl « 2 jam. We may deduce from (1.2) that a molecularly sharp microcrack of this length has a wall separation 2b ~ 0.05 um, which is about one-tenth of the wavelength of visible light and therefore barely on the limit of optical delectability. The theoretical stress-concentration factor (1.4) is of order 100 in this instance, emphasising the potential weakening power of even the most minute of flaws. To test his flaw hypothesis Griffith ran a series of experiments on the strength of glass fibres. The fibres were drawn from the same glass as used in the previous tests (sect. 1.3), and were broken either in tension or in bending under a monotonically increasing dead weight. Well-prepared, pristine fibres showed unusually high strengths, shattering in the explosive manner expected of ideal, flawless solids. However, on exposure to laboratory atmosphere all fibres declined steadily in strength, reaching after a few hours a ' steady state' value more typical of ordinary glass specimens. Griffith next tested a large number of such 'aged' fibres with diameters ranging from 1 mm down to 3 urn, and found an apparent size effect; the thinner specimens showed a tendency to greater strength. Arguing that a single chain of molecules must possess the theoretical
The griffith strength(since such a chain could hardly sustain a flaw), he extrapolated his data to molecular di to one-tenth of the elastic modulus. thus in the one series of tests griffith had demonstrated convincingly not only that sources of weakness exist in the average specimen, but also that these could be avoided if sufficient care d skill were to be exercised in preparation. The production of ultra-high strength optical fibres, in which freshly drawn glass filaments are coated with a protective resin, is a modern exploitation of this principle It remained only for Griffith to speculate on the genesis of these flaws. He actually rejected the possibility that the faws might be real microcracks, since the observed decrease in fibre strength with time would require the system energy to increase spontaneously by the amount of surface energy of the crack faces. He also rejected the possibility that the flaws might generate spontaneously by stress-assisted thermal fluctuations, regarding as highly improbable the synchronised rupture of a large number(say 10) of neighbouring bonds, except perhaps at temperatures close to the melting oint. Griffith considered that the most likely explanation lay in a highly localised rearrangement of molecules within the glass network, wit transformations from the metastable, amorphous state into a higher density, crystalline phase(devitrification). He envisaged sheet-like units with an associated internal field capable of nucleating full-scale fractures e later, Griffith's speculations on the origin and nature of flaws have largely been superseded. The basic notion of the faw as a source of weakness in a solid has, nevertheless, played a vital part in the historical development of the present-day theory of strength 1. 7 Further considerations With his energy-balance concept (pertaining to crack propagation) and flaw hypothesis(pertaining to crack initiation), Griffith had laid a solid foundation for a general theory of fracture In a second paper in 1924 he developed his ideas still further, giving explicit consideration to the effect of applied stress state on the critical fracture conditions, and discoursing on the factors which determine brittleness. With regard to stress state, Griffith extended his analysis of sect. 1.3 to the case of a biaxial applied stress field, in which the crack plane is subjected to both normal (tensile or compressive) stress and shear stress. Referring once more to the Inglis stress analysis of an elliptical cavity, he argued that the location of the local
14 The Griffith concept strength (since such a chain could hardly sustain a flaw), he extrapolated his data to molecular dimensions, and once again arrived at a value close to one-tenth of the elastic modulus. Thus in the one series of tests Griffith had demonstrated convincingly not only that sources of weakness exist in the average specimen, but also that these could be avoided if sufficient care and skill were to be exercised in preparation. The production of ultra-high strength optical fibres, in which freshly drawn glass filaments are coated with a protective resin, is a modern exploitation of this principle. It remained only for Griffith to speculate on the genesis of these flaws. He actually rejected the possibility that the flaws might be real microcracks, since the observed decrease in fibre strength with time would require the system energy to increase spontaneously by the amount of surface energy of the crack faces. He also rejected the possibility that the flaws might generate spontaneously by stress-assisted thermal fluctuations, regarding as highly improbable the synchronised rupture of a large number (say 108 ) of neighbouring bonds, except perhaps at temperatures close to the melting point. Griffith considered that the most likely explanation lay in a highly localised rearrangement of molecules within the glass network, with transformations from the metastable, amorphous state into a higher density, crystalline phase (devitrification). He envisaged sheet-like units with an associated internal field capable of nucleating full-scale fractures. As we shall see later, Griffith's speculations on the origin and nature of flaws have largely been superseded. The basic notion of the flaw as a source of weakness in a solid has, nevertheless, played a vital part in the historical development of the present-day theory of strength. 1.7 Further considerations With his energy-balance concept (pertaining to crack propagation) and flaw hypothesis (pertaining to crack initiation), Griffith had laid a solid foundation for a general theory of fracture. In a second paper in 1924 he developed his ideas still further, giving explicit consideration to the effect of applied stress state on the critical fracture conditions, and discoursing on the factors which determine brittleness. With regard to stress state, Griffith extended his analysis of sect. 1.3 to the case of a biaxial applied stress field, in which the crack plane is subjected to both normal (tensile or compressive) stress and shear stress. Referring once more to the Inglis stress analysis of an elliptical cavity, he argued that the location of the local
Further considerations tensile stress at the near-tip contour, hence the direction of crack extension will rotate away from the major axis of the ellipse as the shear component increases. Conclusions concerning the crack path and critical applied loading could then be drawn. A somewhat surprising result of the analysis is that the crack tip may develop high tensile stresses even when both principal stresses of the applied field are compressive, provided the principal stresses are unequal. This concept has been developed most strongly in rock mechanics, where compressive stress states are the norm As to the question of brittleness, Griffith could but touch on the complications that were apparent in the fracture of many different material types. In many structural steels, for instance, the incidence of plastic flow prior to or during rupture was known to have a profound effect on the strength, but there seemed no way of reconciling this essentially irreversible behaviour with the energy-balance model. It will be recalled that Griffith had based his original model on the notion of an 'ideally brittle solid in which the creation of new fracture surface by the conservative rupture of cohesive bonds constitutes the sole mode of mechanical energy absorption. Inreal materials, however, irreversible processes inevitably accompany crack growth, and a substantially greater amount of mech anical energy may be consumed in the process of separating the material Thus it was recognised that different materials might exhibit different degrees of brittleness. A theoretical understanding of this factor remained an important and difficult problem for future researchers What follows in the subsequent chapters is the logical extension of the theory of brittle fracture from the fundamental concepts expounded by Griffith
Further considerations 15 tensile stress at the near-tip contour, hence the direction of crack extension, will rotate away from the major axis of the ellipse as the shear component increases. Conclusions concerning the crack path and critical applied loading could then be drawn. A somewhat surprising result of the analysis is that the crack tip may develop high tensile stresses even when both principal stresses of the applied field are compressive, provided these principal stresses are unequal. This concept has been developed most strongly in rock mechanics, where compressive stress states are the norm. As to the question of brittleness, Griffith could but touch on the complications that were apparent in the fracture of many different material types. In many structural steels, for instance, the incidence of plastic flow prior to or during rupture was known to have a profound effect on the strength, but there seemed no way of reconciling this essentially irreversible behaviour with the energy-balance model. It will be recalled that Griffith had based his original model on the notion of an ' ideally' brittle solid in which the creation of new fracture surface by the conservative rupture of cohesive bonds constitutes the sole mode of mechanical energy absorption. In 'real materials', however, irreversible processes inevitably accompany crack growth, and a substantially greater amount of mechanical energy may be consumed in the process of separating the material. Thus it was recognised that different materials might exhibit different ' degrees of brittleness'. A theoretical understanding of this factor remained an important and difficult problem for future researchers. What follows in the subsequent chapters is the logical extension of the theory of brittle fracture from the fundamental concepts expounded by Griffith
Continuum aspects of crack propagation I: linear elastic crack-tip field The Griffith study usefully identifies two distinct stages in crack evolution initiation and propagation. Of these, initiation is by far the less amenable to systematic analysis, governed as it invariably is by complex(and often ill defined) local nucleation forces that describe the faw state. Accordingly, we defer investigation of crack initiation to chapter 9. A crack is deemed have entered the propagation stage when it has outgrown the zone of influence of its nucleating forces. The term 'propagation' is not necessarily to imply departure from an equilibrium state: indeed for the present w shall concern ourselves exclusively with equilibrium crack propagation Usually(although not always), a single well-developed crack, by relieving the stress field on neighbouring nucleation centres, propagates from a dominant faw'at the expense of its potential competitors. In the construction of experimental test specimens for studying propagation mechanics such a well-developed crack may be artificially induced, e. g. by machining a surface notch. This pervasive notion of a well-developed crack, taken in conjunction with the fundamental Griffith energy-balance concept, provides us with the starting point for a powerful analytical tool called fracture mechanics, the many facets of which will become manifest in the remaining chapters The formulation of fracture mechanics began with Irwin and his associates round about 1950. The impetus for the development of this discipline originally came from the increasing demand for more reliable safety criteria in engineering design. In more recent times there has been a growing trend toward a materials science'perspective, where fracture mechanics is used to provide insight into the fundamental processes of fracture themselves. at the microstructural and atomic levels. This trend has been especially evident in the current surge toward stronger and tougher ceramic materials. The Irwin formulation, couched in the
Continuum aspects of crack propagation I: linear elastic crack-tip field The Griffith study usefully identifies two distinct stages in crack evolution, initiation and propagation. Of these, initiation is by far the less amenable to systematic analysis, governed as it invariably is by complex (and often illdefined) local nucleation forces that describe the flaw state. Accordingly, we defer investigation of crack initiation to chapter 9. A crack is deemed to have entered the propagation stage when it has outgrown the zone of influence of its nucleating forces. The term ' propagation' is not necessarily to imply departure from an equilibrium state: indeed, for the present we shall concern ourselves exclusively with equilibrium crack propagation. Usually (although not always), a single' well-developed' crack, by relieving the stress field on neighbouring nucleation centres, propagates from a 'dominant flaw' at the expense of its potential competitors. In the construction of experimental test specimens for studying propagation mechanics such a well-developed crack may be artificially induced, e.g. by machining a surface notch. This pervasive notion of a well-developed crack, taken in conjunction with the fundamental Griffith energy-balance concept, provides us with the starting point for a powerful analytical tool called fracture mechanics, the many facets of which will become manifest in the remaining chapters. The formulation of fracture mechanics began with Irwin and his associates round about 1950. The impetus for the development of this discipline originally came from the increasing demand for more reliable safety criteria in engineering design. In more recent times there has been a growing trend toward a 'materials science' perspective, where fracture mechanics is used to provide insight into the fundamental processes of fracture themselves, at the microstructural and atomic levels. This trend has been especially evident in the current surge toward stronger and tougher ceramic materials. The Irwin formulation, couched in the 16