Preface because of its inherent generality: in proceeding to more complex syste one needs only to modify existing terms, or add new ones, in the expression for the total energy of the crack system. All soundly-based fracture theories derive either directly from the griffith concept or from some alternative concept with underlying equivalence, such as Irwin,s stress-intensity factor In attempting to construct an integrated picture of fracture one becomes aware of widely diverse perspectives on brittle cracks. Most traditional is the perspective of the engineer, who sees cracks in terms of a slit continuum, treating the tip and its surrounds as a singular(black box) zone. At the opposite end of the spectrum is the crack-tip'enclave perspective of the physicist-chemist, who defines the processes of discrete bond rupture in terms of intersurface force functions. Both viewpoints are valuable: the first gives us general parameters such as mechanical-energy release rate G and stress-intensity factor K for quantifying themotive for of extraneous variables like applied loads, specime geometry, environmental concentration, etc. the second provides us with a basis for describing the fundamental structure of atomically sharp cracks and thereby defining laws of extension. And now we must add a relatively new perspective, that of the materials scientist, who seeks to incorporate discrete dissipative elements into ceramic microstructures in order to overcome the intrinsic brittleness. It is at this level that the concept of shielding emerges, in the form of an intervening dissipative zone which screens the crack-tip enclave from the external applied loads. Innovations in microstructural shielding processes hold the key to the next generation of strong and tough brittle materials As with any attempt to tie these disparate perspectives into a cohesive description, it is inevitable that conflicts in notation will arise. In seeking compromise I have leant toward materials terminology. Among the more conspicuous symbols is the Griffith c rather than the solid mechanics a for crack size. Also notable are the symbols for toughness, R and T, in place of the engineering parameters Gr and Kg; the former serve to emphasise that the intrinsic resistance to crack propagation is an equilibrium materia property, ultimately expressible as an integral of a constitutive stress- displacement relation without reference to fracture at all The layout of the book follows a loose progression from scientific fundamentals at one end to engineering design at the other. Historical and conceptual foundations are laid in chapter l, with a review of the energy balance concept and flaw hypothesis of Griffith. Chapters 2 and 3 develop a theoretical description of crack propagation in terms of continu fracture mechanics, with an emphasis on equilibrium configurations
Preface xi because of its inherent generality: in proceeding to more complex systems one needs only to modify existing terms, or add new ones, in the expression for the total energy of the crack system. All soundly-based fracture theories derive either directly from the Griffith concept or from some alternative concept with underlying equivalence, such as Irwin's stress-intensity factor. In attempting to construct an integrated picture of fracture, one becomes aware of widely diverse perspectives on brittle cracks. Most traditional is the 'global' perspective of the engineer, who sees cracks in terms of a slit continuum, treating the tip and its surrounds as a singular (black box) zone. At the opposite end of the spectrum is the crack-tip 'enclave' perspective of the physicist-chemist, who defines the processes of discrete bond rupture in terms of intersurface force functions. Both viewpoints are valuable: the first gives us general parameters such as mechanical-energyrelease rate G and stress-intensity factor Kfor quantifying the 'motive' for fracture in terms of extraneous variables like applied loads, specimen geometry, environmental concentration, etc.; the second provides us with a basis for describing the fundamental structure of atomically sharp cracks and thereby defining laws of extension. And now we must add a relatively new perspective, that of the materials scientist, who seeks to incorporate discrete dissipative elements into ceramic microstructures in order to overcome the intrinsic brittleness. It is at this level that the concept of shielding emerges, in the form of an intervening dissipative zone which screens the crack-tip enclave from the external applied loads. Innovations in microstructural shielding processes hold the key to the next generation of strong and tough brittle materials. As with any attempt to tie these disparate perspectives into a cohesive description, it is inevitable that conflicts in notation will arise. In seeking compromise I have leant toward materials terminology. Among the more conspicuous symbols is the Griffith c rather than the solid mechanics a for crack size. Also notable are the symbols for toughness, R and T, in place of the engineering parameters GR and KR; the former serve to emphasise that the intrinsic resistance to crack propagation is an equilibrium material property, ultimately expressible as an integral of a constitutive stressdisplacement relation without reference to fracture at all. The layout of the book follows a loose progression from scientific fundamentals at one end to engineering design at the other. Historical and conceptual foundations are laid in chapter 1, with a review of the energybalance concept and flaw hypothesis of Griffith. Chapters 2 and 3 develop a theoretical description of crack propagation in terms of continuum fracture mechanics, with an emphasis on equilibrium configurations
Presa Chapters 4 and 5 extend these considerations to moving cracks, dynamic (fast )and kinetic(slow), with special attention in the latter case to environmental chemistry. In chapter 6 we analyse crack-tip processes at the atomic level, again with provision to include chemistry in the fundamental crack laws. Chapter 7 considers the influence of micro- structure on the fracture mechanics, with accent on some of the promising shielding mechanisms that are emerging in the toughness description. One of the most powerful and widespread methodologies for evaluating ceramic materials, indentation fracture, is surveyed in chapter 8. In chapter 9 we deal with the issue of flaws and crack initiation. Finally, in chapter 10. strength and reliability are addressed An understanding of fracture mechanics is best obtained by con centrating on basic principles rather than on factual information. Conse- quently, our attention to ' materials like homogeneous glass and polycrystalline alumina should be seen as essential groundwork for ultimate extension to more complex engineering materials. That phil osophy extends to the literature citations. We have not sought to provide an extensive reference list, but rather a selective bibliography. It is a hope that, in an age where the published word is fast becoming a lost forum of communication, the reader will be persuaded to consult the open literature Many colleagues and students have contributed greatly to this venture Special mention is due to Rodney wilshaw, former co-author and old friend, with whom the first edition was conceived and produced. Soon after publication of that earlier version Rod turned from academic endeavours to a life on the land. He gracefully withdrew his name from the cover of this edition. His spirit is nevertheless still to be found in the ensuing pages. Other major contributors over the years include: S.J. Bennison, L M Braun, S.J. Burns, H. M. Chan, P, Chantikul, R. F Cook, T. P. Dabbs, F C. Frank, E R. Fuller, B J Hockey, R.G. Horn, S. Lathabai, Y.W Mai. D. B. Marshall. n. P. Padture. D. h. Roach. Rodelj e. sinclair M.V. Swain, R. M. Thomson, K.-T. Wan and S m. wiederhorn. I also k R w. Cahn for his encouragement to embark on this second edition, and his perseverance during its completion. Finally, to my wife Valerie, my heartfelt appreciation for enduring it all Brian lawn
xii Preface Chapters 4 and 5 extend these considerations to moving cracks, dynamic ('fast') and kinetic ('slow'), with special attention in the latter case to environmental chemistry. In chapter 6 we analyse crack-tip processes at the atomic level, again with provision to include chemistry in the fundamental crack laws. Chapter 7 considers the influence of microstructure on the fracture mechanics, with accent on some of the promising shielding mechanisms that are emerging in the toughness description. One of the most powerful and widespread methodologies for evaluating ceramic materials, indentation fracture, is surveyed in chapter 8. In chapter 9 we deal with the issue of flaws and crack initiation. Finally, in chapter 10, strength and reliability are addressed. An understanding of fracture mechanics is best obtained by concentrating on basic principles rather than on factual information. Consequently, our attention to 'model' materials like homogeneous glass and polycrystalline alumina should be seen as essential groundwork for ultimate extension to more complex engineering materials. That philosophy extends to the literature citations. We have not sought to provide an extensive reference list, but rather a selective bibliography. It is a hope that, in an age where the published word is fast becoming a lost forum of communication, the reader will be persuaded to consult the open literature. Many colleagues and students have contributed greatly to this venture. Special mention is due to Rodney Wilshaw, former co-author and old friend, with whom the first edition was conceived and produced. Soon after publication of that earlier version Rod turned from academic endeavours to a life on the land. He gracefully withdrew his name from the cover of this edition. His spirit is nevertheless still to be found in the ensuing pages. Other major contributors over the years include: S. J. Bennison, L. M. Braun, S. J. Burns, H. M. Chan, P. Chantikul, R. F. Cook, T. P. Dabbs, F. C. Frank, E. R. Fuller, B. J. Hockey, R. G. Horn, S. Lathabai, Y.-W. Mai, D. B. Marshall, N. P. Padture, D. H. Roach, J. Rodel, J. E. Sinclair, M. V. Swain, R. M. Thomson, K.-T. Wan and S. M. Wiederhorn. I also thank R. W. Cahn for his encouragement to embark on this second edition, and his perseverance during its completion. Finally, to my wife Valerie, my heartfelt appreciation for enduring it all. Brian Lawn
Glossary of symbols and abbreviations SI units are used throughout, with the following prefixes k kilo H micro nano T tera p pIco 10-12 Symbols(with units) a inclusion or pore radius(um); characteristic contact critical contact size (um) A cross-sectional area(mm2); Auerbach constant minor axis in Inglis elliptical cavity (um); magnitude of Burgers vector(nm) bo lattice spacing(nm) characteristic crack size(um) cb crack size at branching(um) critical crack size (um) flaw size(um) cr crack size at failure(um) crack size at pop-in(um) crack size at activated failure(um
Glossary of symbols and abbreviations k M G T kilo mega giga tera 103 106 109 1012 SI units are used throughout, with the following prefixes: m milli 10-3 u micro 10~6 n nano 10~9 p pico 10-12 f femto 10~15 a atto 10~18 Symbols (with units) a inclusion or pore radius (um); characteristic contact radius (urn) ac critical contact size (um) a0 atomic spacing (nm) A cross-sectional area (mm2 ); Auerbach constant b minor axis in Inglis elliptical cavity (urn); magnitude of Burgers vector (nm) b0 lattice spacing (nm) c characteristic crack size (um) cB crack size at branching (um) cc critical crack size (um) cf flaw size (um) cv crack size at failure (um) c{ crack size at pop-in (um) cM crack size at activated failure (um)
Glossary of symbols and abbreviations starter crack(notch) size(mm) C crack area(um) d beam thickness(mm); characteristic spacing between Youn E E, plane stress; E/(I-v2), plane strain(GPa) nit length)(N m") Fr force on stretched atomic bond (nN) Fo lattice-modified force(nN) △F f angular function in crack-tip displacement field f lar function in crack-tip stress field o net crack-extension force, or'motive(J") rgy-release rate (J m- 2) G global mechanical-energy-release rate (J m 2) G mechanical-energy-release rate (J m 2) Gr G in material with shielding (J m 2) G, crack-tip enclave mechanical-energy-release rate (J m-2) Gu shielding-zone mechanical-energy- release rate(J m 2) cohesion-zone mechanical-energy- release rate( m) h cantilever-beam crack-opening displacement (um) h Planck constant (6.6256x 10-4 J s) H indentation hardness(GPa) J Rice line integral (J m 2) k elastic coefficient for Hertzian contact Boltzmann constant(1.3805 x 10-JK-) net K-field at singular tip(MPa m/ 2) K stress-intensity factor (MPa m2) K global stress-intensity factor (MPa m) Kn stress-intensity factor at crack branching(MPa m/) Kc critical stress-intensity factor(MPa m) Kr residual stress-intensity factor(MPa m 2) Kg Ka in material with shielding(MPa m) K shielding-zone stress-intensity factor (MPa m) K& crack-tip enclave stress-intensity factor(MPa m 2)
xiv Glossary of symbols and abbreviations c0 starter crack (notch) size (mm) C crack area (um2 ) d beam thickness (mm); characteristic spacing between microstructural elements (urn) E Young's modulus (GPa) E' E, plane stress; E/(l — v2 ), plane strain (GPa) F line force (force per unit length) (N m"1 ) FB force on stretched atomic bond (nN) Fn lattice-modified force (nN) AF activation free energy (aJ molec"1 ) ft angular function in crack-tip displacement field ftj angular function in crack-tip stress field p net crack-extension force, or 'motive ' (J m~2 ) G mechanical-energy-release rate (J m~2 ) GA global mechanical-energy-release rate (J m~2 ) Gc critical mechanical-energy-release rate (J m~2 ) GR GA in material with shielding (J m~2 ) G* crack-tip enclave mechanical-energy-release rate (J m~2 ) G^ shielding-zone mechanical-energy-release rate (J m~2 ) Go cohesion-zone mechanical-energy-release rate (J m~2 ) h cantilever-beam crack-opening displacement (jam) h Planck constant (6.6256 x 10"34 J s) H indentation hardness (GPa) / Rice line integral (J m~2 ) k elastic coefBcient for Hertzian contact k Boltzmann constant (1.3805 x 1(T23 J Kr1 ) / net AT-field at singular tip (MPa m1/2) K stress-intensity factor (MPa m1/2) KA global stress-intensity factor (MPa m1/2) KB stress-intensity factor at crack branching (MPa m1/2) Kc critical stress-intensity factor (MPa m1/2) KR residual stress-intensity factor (MPa m1/2) KR KA in material with shielding (MPa m1/2) K^ shielding-zone stress-intensity factor (MPa m1/2) K* crack-tip enclave stress-intensity factor (MPa m1/2)
Glossary of symbols and abbreviations K, cohesion-zone stress-intensity factor(MPa m) KI Ku, Ku mode I, Il, IlI stress-intensity factors(MPa m) I beam span in flexure specimen(mm); grain size (um) Ic critical grain size for spontaneous microcracking (um) L bridging zone length(mm); specimen dimension(mm) molecular mass(10-kg) crack velocity power-law exponent; number of atoms in lattice-crack chair Pc critical bridging stress(MPa) pe critical fibre pullout stress(MPa) po fibre debonding stress(MPa) P nvironmental gas pressure( kPa) theoretical cohesive stress(GPa) cohesive surface stress at crack interface(GPa) microstructural shielding tractions at crack interface (MPa) Po mean contact pressure(MPa applied point load, contact load (N) Pc itical contact load P+, P applied load extremes for lattice trapping(N) P probability of failure e heat input () radial crack-tip coordinate(um); fibre or sphere radius ck-resistance energy per unit area(J m-2) R-curve resistance-curve Re crack-resistance energy in interactive environment (J m-2) Ru. microstructural shielding component of resistance energy Ro crack-resistance energy in a vacuum(J"?) R steady-state crack-resistance energy (J m-2) R,R crack-resistance trapping range (J m-2) R quasi-equilibrium crack-resistance energy (J m-2) arc length(m s entropy (JK-)
Glossary of symbols and abbreviations xv Ko cohesion-zone stress-intensity factor (MPa m1/2) Tj, Kn, KUI mode I, II, III stress-intensity factors (MPa m1/2) / beam span in flexure specimen (mm); grain size (um) /c critical grain size for spontaneous microcracking (um) L bridging zone length (mm); specimen dimension (mm) m molecular mass (10~27 kg) n crack velocity power-law exponent; number of atoms in lattice-crack chain />c critical bridging stress (MPa) pi critical fibre pullout stress (MPa) p B fibre debonding stress (MPa) /7E environmental gas pressure (kPa) pTh theoretical cohesive stress (GPa) py cohesive surface stress at crack interface (GPa) p^ microstructural shielding tractions at crack interface (MPa) p0 mean contact pressure (MPa) P applied point load, contact load (N) Pc critical contact load (N) P+, P_ applied load extremes for lattice trapping (N) P probability of failure Q heat input (J) r radial crack-tip coordinate (um); fibre or sphere radius (um) R crack-resistance energy per unit area (J m~2 ) 7^-curve resistance-curve RF crack-resistance energy in interactive environment (J m~2 ) R^ microstructural shielding component of resistance energy (Jm-2 ) Ro crack-resistance energy in a vacuum (J m~2 ) RK steady-state crack-resistance energy (J m~2 ) R+ , R~ crack-resistance trapping range (J m"2 ) R' quasi-equilibrium crack-resistance energy (J m"2 ) s arc length (m) S entropy (J Kr1 )