vi Glossary of symbols and abbreviations fp time to failure(lifetime)(s) T toughness(MPa m/2) T-curve toughness-curve T toughness in interactive environment(MPa m) (MPa m 2) To cuum(MP/ 2) T steady-state toughness(MPa m) absolute temperature(K) traction vector in J-integral(MPa) t(um); load displacement(um) displacement vector(um) placement vector (um) uz crack-opening displacement at edge of traction zone(um) u, crack-opening displacement in cohesion zone(nm) U system internal energy () Ua energy of applied loading system J Uw cohesion energy of molecule A-A (fD) Uan energy of terminal bond A-B-(f) Ub energy of stretched cohesive bond(f) UBB cohesion energy of bond-B-B(f) elastic strain energy () U, U initial, final energy states(J) Uk kinetic energy () Us surface energy of crack area () AU on energy (J m) strain energy density in J-integral (J m - crack velocity(m") U longitudinal wave velocity(km s") terminal velocity(km s) UI, UnI, Uum velocities in regions I, I, Il volume fraction
xvi Glossary of symbols and abbreviations t time (s) t¥ time to failure (lifetime) (s) T toughness (MPa m1/2) /"-curve toughness-curve TE toughness in interactive environment (MPa m1/2) 7^ microstructural shielding component of toughness (MPa m1/2) To toughness in a vacuum (MP m1/2) Tx steady-state toughness (MPa m1/2) T absolute temperature (K) ZT traction vector in J-integral (MPa) u crack-opening displacement (um); load-point displacement (jam) u displacement vector (urn) ut component of displacement vector (um) wz crack-opening displacement at edge of traction zone (um) uy crack-opening displacement in cohesion zone (nm) U system internal energy (J) UA energy of applied loading system (J) UAA cohesion energy of molecule A-A (fJ) UAB energy of terminal bond A-B - (f J) UB energy of stretched cohesive bond (fJ) UBB cohesion energy of bond -B-B - (f J) UB elastic strain energy (J) Ui9 Ut initial, final energy states (J) UK kinetic energy (J) UM mechanical energy (J) Us surface energy of crack area (J) A UAd adsorption energy (J m~2 ) °U strain energy density in /-integral (J rrr3 ) v crack velocity (m s"1 ) v} longitudinal wave velocity (km s"1 ) vT terminal velocity (km s' 1 ) v i> ^n» r m velocities in regions I, II, III Vf volume fraction
Glossary of symbols and abbreviations XVII specimen width(mm) critical width of frontal-zone wake(um) rk of adhesion(J m-2) h same, for crack growth through healed interface (J m-2) w same, for crack growth through virgin solid( m-2) ab work to separate unlike bodies a-B in a vacuum(J m 2) bodies B-B in work to separate like bodies B-B in environment Em-2) Cartesian coordinates for crack system(m) X crack-interface coordinate measured from crack tip(mm) crack-interface coordinate at edge of traction shielding zone (mm) specimen geometry edge correction factor; activation area (nm2molec- ); lattice spring constant(nN nm"); thermal (K-) contact geometry coefficient b gas pressure coefficient in crack velocity equation; lattice pring constant (nn nm- ) normalised radial coordinate of contact crack initiation y surface or interface energy per unit area(J m intrinsic(inert )surface energy of solid body b m) interfacial energy for body b in environmental medium E (mJ m grain boundary energy (mJ m) fault energy for interface healed in environment(mJ m-2) cophase boundary energy(mJ m") r Gibbs surface excess(nm-2) ttice-trapping modulation factor in cohesion energy (Jm-2) d Barenblatt crack-opening displacement(nm) e bridge rupture strain a constrained microcrack-zone dilational strain &T constrained transformation-zone dilational strain
Glossary of symbols and abbreviations xvii w specimen width (mm) wc critical width of frontal-zone wake (jam) W Dupre work of adhesion (J irT2 ) h W same, for crack growth through healed interface (J m~2 ) V W same, for crack growth through virgin solid (J m~2 ) WAB work to separate unlike bodies A-B in a vacuum (J m~2 ) WBB work to separate like bodies B-B in a vacuum (J m~2 ) ^BEB work to separate like bodies B-B in environment E (J m~2 ) x, y, z Cartesian coordinates for crack system (m) X crack-interface coordinate measured from crack tip (mm) Xz crack-interface coordinate at edge of traction shielding zone (mm) a specimen geometry edge correction factor; activation area (nm2 molec"1 ); lattice spring constant (nN nm"1 ); thermal expansion coefficient (K"1 ) a0 contact geometry coefficient fi gas pressure coefficient in crack velocity equation; lattice spring constant (nN nm"1 ); normalised radial coordinate of contact crack initiation y surface or interface energy per unit area (J m~2 ) yB intrinsic (' inert') surface energy of solid body B (J m~2 ) yBF interfacial energy for body B in environmental medium E (mJ m"2 ) yGB grain boundary energy (mJ m~2 ) yhE fault energy for interface healed in environment (mJ m~2 ) yIB interphase boundary energy (mJ m~2 ) F Gibbs surface excess (nm"2 ) F B lattice-trapping modulation factor in cohesion energy (J m-2 ) S Barenblatt crack-opening displacement (nm) s strain £B bridge rupture strain e M constrained microcrack-zone dilational strain e T constrained transformation-zone dilational strain
Glossary of symbols and abbreviations rupture strain for plastic bridge dilational strain in frontal-zone shielding field s kink coordinate(nm) B,o polar coordinates for crack system g fractional surface adsorption coverage k Knudsen attenuation factor for free molecular flow 2 elastic compliance (m N-); Barenblatt zone length(nm a entropy production rate( Js") critical range for stress cutoff at edge of closure zone(um) s critical cutoff range for bridge disengagement (um s critical cutoff range for fibre pullout(um) a stress(MPa) pplied uniform stress(MPa) ac stress at tip of elliptical cavity(GPa) a critical activation stress for microcracking(MPa) ac critical activation stress for transformation(MPa) σ yield stress(MPa) σ failure stress(MPa omponent of stress tensor(MPa) G activated failure stress(MPa) gr proof stress(MPa at Hertzian contact circle(M stress in frontal-zone shielding field Weibull scaling stress(MPa)
xviii Glossary of symbols and abbreviations e Y rupture strain for plastic bridge e^ dilational strain in frontal-zone shielding field C kink coordinate (nm) n order of chemical interaction 0, (/> polar coordinates for crack system 9 fractional surface adsorption coverage K Knudsen attenuation factor for free molecular flow X elastic compliance (m N"1 ); Barenblatt zone length (nm) A entropy production rate (J s"1 ) fi friction coefficient; shear modulus (GPa) v Poisson's ratio v0 lattice frequency (Hz) £ critical range for stress cutoff at edge of closure zone (jam) £B critical cutoff range for bridge disengagement (urn) £ p critical cutoff range for fibre pullout (um) p tip radius of elliptical cavity (nm); density (kg irr3 ); radial coordinate (m) o stress (MPa) < TA applied uniform stress (MPa) GC stress at tip of elliptical cavity (GPa) 0-J? critical activation stress for dislocation motion (MPa) G™ critical activation stress for microcracking (MPa) crj critical activation stress for transformation (MPa) <7c yield stress (MPa) GF failure stress (MPa ) cTj component of stress tensor (MPa ) <7j inert strength (MPa) GM activated failure stress (MPa) aF proof stress (MPa) < j R residual stress (MPa) as surface stress (MPa) aT tensile stress at Hertzian contact circle (MPa) <7TS thermal shock stress (MPa) G^ dilational stress in frontal-zone shielding field G0 Weibull scaling stress (MPa)
t interfacial friction stress(MPa) Φ indenter half- angle x indentation residual-contact coefficient y crack-geometry factor Abbreviat T compact tension specimen DCB er beam specimen DT double-torsion specimen NDE non-destructive evaluation PSZ partially stabilised zirconia SEnb single-edge notched beam specimen TEM transmission electron microscope
XIX T interfacial friction stress (MPa) O indenter half-angle X indentation residual-contact coefficient y/ crack-geometry factor Abbreviations CT compact tension specimen DCB double-cantilever beam specimen DT double-torsion specimen NDE non-destructive evaluation PSZ partially stabilised zirconia SENB single-edge notched beam specimen TEM transmission electron microscope
The Griffith concept Most materials show a tendency to fracture when stressed beyond some critical level. This fact was appreciated well enough by nineteenth century structural engineers, and to them it must have seemed reasonable suppose strength to be a material property. After all, it had long been established that the stress response of materials within the elastic limit could be specified completely in terms of characteristic elastic constants. Thus arose the premise of acritical applied stress, and this provided the basis of the first theories of fracture The idea of a well-defined stress limit was(and remains) particularly attractive in engineering design; one simply had to ensure that the maximum stress level in a given structural component did not exceed this limit. However, as knowledge from structural failures accumulated the universal validity of the critical applied stress thesis became more suspect The fracture strength of a given material was not, in general, highly reproducible, in the more brittle materials fluctuating by as much as an order of magnitude. Changes in test conditions, e. g. temperature, chemical environment, load rate, etc, resulted in further, systematic variations in strengths. Moreover, different material types appeared to fracture in radically different ways: for instance, glasses behaved elastically up to the critical point, there to fail suddenly under the action of a tensile stress component, while many metallic solids deformed extensively by plastic flow prior to rupture under shear. The existing theories were simply ncapable of accounting for such disparity in fracture behaviour. This, then, was the state of the subject in the first years of the present entury. It is easy to see now, in retrospect, that the inadequacy of the critical stress criterion lay in its empirical nature: for the notion that a solid should break at a characteristic stress level, however intuitively appealing, is not based on sound physical principles. There was a need to take a closer
1 The Griffith concept Most materials show a tendency to fracture when stressed beyond some critical level. This fact was appreciated well enough by nineteenth century structural engineers, and to them it must have seemed reasonable to suppose strength to be a material property. After all, it had long been established that the stress response of materials within the elastic limit could be specified completely in terms of characteristic elastic constants. Thus arose the premise of a 'critical applied stress', and this provided the basis of the first theories of fracture. The idea of a well-defined stress limit was (and remains) particularly attractive in engineering design; one simply had to ensure that the maximum stress level in a given structural component did not exceed this limit. However, as knowledge from structural failures accumulated, the universal validity of the critical applied stress thesis became more suspect. The fracture strength of a given material was not, in general, highly reproducible, in the more brittle materials fluctuating by as much as an order of magnitude. Changes in test conditions, e.g. temperature, chemical environment, load rate, etc., resulted in further, systematic variations in strengths. Moreover, different material types appeared to fracture in radically different ways: for instance, glasses behaved elastically up to the critical point, there to fail suddenly under the action of a tensile stress component, while many metallic solids deformed extensively by plastic flow prior to rupture under shear. The existing theories were simply incapable of accounting for such disparity in fracture behaviour. This, then, was the state of the subject in the first years of the present century. It is easy to see now, in retrospect, that the inadequacy of the critical stress criterion lay in its empirical nature: for the notion that a solid should break at a characteristic stress level, however intuitively appealing, is not based on sound physical principles. There was a need to take a closer 1