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Basics of Fracture Mechanics

  • Meinhard KunaEmail author
Chapter
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 201)

Abstract

The theoretical foundations of fracture mechanics will be presented in this chapter. The main focus lies on the description of the available continuum-mechanical solutions for cracks. On the basis of stress and deformation situations determined this way, suitable parameters, which clearly describe the loading states during fractures, are then selected. These loading and fracture parameters shape the foundation for the formulation of fracture criteria. With their help, the behavior of cracks can be quantitatively evaluated. These usually closed mathematical solutions are the preconditions to being able to calculate the sizes of cracks with numerical methods later on. Naturally, the experimental test methods of fracture mechanics used to evaluate the material parameters are based on the understanding of the loading situation as well.

Notes

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References

  1. 1.
    Muskhelishvili NI (1971) Einige Grundaufgaben zur mathematischen Elastizitätstheorie. Fachbuchverlag LeipzigGoogle Scholar
  2. 2.
    Williams ML (1957) On the stress distribution at the base of a stationary crack. J Appl Mech 24:109–114MathSciNetzbMATHGoogle Scholar
  3. 3.
    Gross B, Srawley JE (1965) Stress-intensity factors for single-edge-notch specimens in bending or combined bending and tension by boundary collocation of a stress function. Technical Report NASA Technical Note D-2603, NASA, Lewis Research CenterGoogle Scholar
  4. 4.
    Xiao QZ, Karihaloo BL (2002) Coefficients of the crack tip asymptotic field for a standard compact tension specimen. Int J Fract 118:1–15CrossRefGoogle Scholar
  5. 5.
    Sneddon LN, Lowengrub M (1969) Crack problems in the classical theory of elasticity. Wiley, New YorkzbMATHGoogle Scholar
  6. 6.
    Fabrikant VI (2001) Application of potential theory in mechanics: a selection of new results. Kluwer Academic Publishers, DordrechtGoogle Scholar
  7. 7.
    Hartranft RJ, Sih GC (1969) The use of eigenfunction expansions in the general solution of three-dimensional crack problems. J Math Mech 19(2):123–138MathSciNetzbMATHGoogle Scholar
  8. 8.
    Hartranft RJ, Sih GC (1977) Stress singularity for a crack with an arbitrary curved crack front. Eng Fract Mech 9:705–718CrossRefGoogle Scholar
  9. 9.
    Sih GC (1973) Methods of analysis and solutions of crack problems. Mechanics of fracture, vol. 1. Noordhoff International Publisher, LeydenGoogle Scholar
  10. 10.
    Tamuz V, Romalis N, Petrova V (2000) Fracture of solids with microdefects. Nova Science Publishers, New YorkGoogle Scholar
  11. 11.
    Erdogan F, Gupta GD, Cook TS (1973) Numerical solution of singular integral equations. In: Sih GC (ed) Methods of analysis and solutions of crack problems. Mechanics of fracture, vol. 1. Noordhoff, Leyden, pp. 368–425Google Scholar
  12. 12.
    Kassir MK, Sih GC (1975) Three dimensional crack problems. Mechanics of fracture, vol. 2. Noordhoff Internatioanl Publisher, LeydenGoogle Scholar
  13. 13.
    Murakami Y (1987) Stress intensity factors handbook, vol. 1–5. Pergamon Press, OxfordGoogle Scholar
  14. 14.
    Rooke DP, Cartwright DJ (1976) Compendium of stress intensity factors. Her Majesty’s Stationary Office, LondonGoogle Scholar
  15. 15.
    Tada H, Paris P, Irwin G (1985) The stress analysis of cracks handbook, 2nd edn. Paris Production Inc., St. LouisGoogle Scholar
  16. 16.
    Theilig H, Nickel J (1987) Spannungsintensitätsfaktoren. Fachbuchverlag, LeipzigGoogle Scholar
  17. 17.
    Irwin GR (1957) Analysis of stresses and strains near the end of a crack traversing a plate. J Appl Mech 24:361–364Google Scholar
  18. 18.
    ISO12135 (2002) Metallic materials—Unified method of test for the quasitatic fracture toughness. Technical Report, International Organization for Standardization, GenfGoogle Scholar
  19. 19.
    ESIS P2 (1992) Procedure for determining the fracture behaviour of materials. Technical Report, European Structural Integrity SocietyGoogle Scholar
  20. 20.
    ASTM-E 1820 (2007) Standard test method for measurement of fracture toughness. Technical Report, American Society for Testing and Materials, West ConshohockenGoogle Scholar
  21. 21.
    Blumenauer H, Pusch G (1993) Technische Bruchmechanik, 3rd edn. Deutscher Verlag für Grundstoffindustrie, LeipzigGoogle Scholar
  22. 22.
    Griffith AA (1921) The phenomena of rupture and flow in solids. Philos Trans Ser A 221: 163–198Google Scholar
  23. 23.
    Irwin GR (1958) Fracture. In: Flügge S (ed) Handbuch der Physik. Band 6, Engineering fracture mechanics, Springer, Berlin, pp 551–590Google Scholar
  24. 24.
    Cherepanov G (1967) Rasprostranenie trechin v sploshnoi srede (about crack advance in the continuum). Prikladnaja Matematika i Mekhanica 31:478–488Google Scholar
  25. 25.
    Rice J (1968) A path independent integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech 35:379–386CrossRefGoogle Scholar
  26. 26.
    Altenbach H, Altenbach J, Rikards R (1996) Einführung in die Mechanik der Laminatwerkstoffe. Deutscher Verlag für Grundstoffindustrie, StuttgartGoogle Scholar
  27. 27.
    Lekhnitskii SG (1981) Theory of elasticity of an anisotropic body. Mir Publisher, MoscowzbMATHGoogle Scholar
  28. 28.
    Stroh AN (1962) Steady state problems on anisotropic elasticity. J Math Phys 41:77–103MathSciNetzbMATHGoogle Scholar
  29. 29.
    Sih GC, Paris PC, Irwin GR (1965) On cracks in rectilinear anisotropic bodies. Int J Fract Mech 1:189–203Google Scholar
  30. 30.
    Comninou M (1990) An overview of interface cracks. Eng Fract Mech 37:197–208CrossRefGoogle Scholar
  31. 31.
    Rice JR, Suo Z, Wang JS (1990) Mechanics and thermodynamics of brittle interface failure in bimaterial systems. In: Rühle M, Evans A, Ashby M, Hirth J (eds) Metal-Ceramic Interfaces. Pergamon Press, Oxford, pp 269–294Google Scholar
  32. 32.
    Rice JR (1988) Elastic fracture mechanics concepts for interfacial cracks. Trans ASME 55:98–103Google Scholar
  33. 33.
    Banks-Sills L, Ashkenazi D (2000) A note on fracture criteria for interface fracture. Int J Fract 103:177–188CrossRefGoogle Scholar
  34. 34.
    Qu J, Bassani JL (1993) Interfacial fracture mechanics for anisotropic bimaterials. J Appl Mech 60:422–431CrossRefzbMATHGoogle Scholar
  35. 35.
    Suo Z (1990) Singularities, interfaces and cracks in dissimilar anisotropic media. Proc Roy Soc London A 427:331–358MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Beom HG, Atluri SN (1995) Dependence of stress on elastic constants in an anisotropic bimaterial under plane deformation; and the interfacial crack. Comput Mech 16:106–113CrossRefzbMATHGoogle Scholar
  37. 37.
    Williams ML (1961) The bending stress distribution at the base of a stationary crack. J Appl Mech 28:78–82MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Sih GC, Paris PC, Erdogan F (1962) Crack-tip stress-intensity factors for plane extension and plate bending problems. J Appl Mech 29:306–312Google Scholar
  39. 39.
    Hartranft RJ, Sih GC (1968) Effect of plate thickness on the bending stress distribution around through cracks. J Math Phys 47:276–291zbMATHGoogle Scholar
  40. 40.
    Hui CY, Zehnder AT (1993) A theory for the fracture of thin plates subjected to bending and twisting moments. Int J Fract 61:211–229CrossRefGoogle Scholar
  41. 41.
    Rice JR (1972) Some remarks on elastic crack-tip stress fields. Int J Solid Struct 8:751–758CrossRefzbMATHGoogle Scholar
  42. 42.
    Stein E, Barthold FJ (1996) Werkstoffe - Elastizitätstheorie. In: Mehlhorn G (ed) Der Ingenieurbau, vol 4. Verlag Ernst & Sohn, Berlin, pp 165–425Google Scholar
  43. 43.
    Chen YZ (1989) Weight function technique in a more general case. Eng Fract Mech 33:983–986Google Scholar
  44. 44.
    Bueckner HF (1970) A novel principle for the computation of stress intensity factors. Zeitschrift für Angewandte Mathematik und Mechanik 50:529–546MathSciNetzbMATHGoogle Scholar
  45. 45.
    Paris PC, McMeeking RM, Tada H (1976) The weight function method for determining stress intensity factors. Cracks and Fracture, STP 601, American Society for Testing of Materials pp 471–489Google Scholar
  46. 46.
    Bueckner HF (1987) Weight functions and fundamental fields for the penny-shaped and the half-plane crack in three-space. Int J Solid Struct 23(1):57–93CrossRefzbMATHGoogle Scholar
  47. 47.
    Bueckner HF (1989) Observations of weight functions. Eng Anal Bound Elem 6:3–18CrossRefGoogle Scholar
  48. 48.
    Fett T, Munz D (1997) Stress intensity factors and weight functions. Computational Mechanics Publications, Southampton, BostonGoogle Scholar
  49. 49.
    Bortmann Y, Banks-Sills L (1983) An extended weight function method for mixed-mode elastic crack analysis. J Appl Mech 50:907–909CrossRefGoogle Scholar
  50. 50.
    Gao H, Rice JR (1987) Somewhat circular tensile cracks. Int J Fract 33:155–174MathSciNetCrossRefGoogle Scholar
  51. 51.
    Rice JR (1985) First order variation in elastic fields due to variation in location of a planar crack front. J Appl Mech 52:571–579CrossRefzbMATHGoogle Scholar
  52. 52.
    Qin QH (2001) Fracture mechanics of piezoelectric materials. WIT Press, Southampton, BostonGoogle Scholar
  53. 53.
    Kuna M (2006) Finite element analyses of cracks in piezoelectric structures: a survey. Arch Appl Mech 76:725–745CrossRefzbMATHGoogle Scholar
  54. 54.
    Sähn S, Göldner H (1993) Bruch- und Beurteilungskriterien in der Festigkeitslehre. Fachbuchverlag, Leipzig-KölnGoogle Scholar
  55. 55.
    Irwin GR (1956) Onset of fast crack propagation in high strength steel and aluminum alloys. In: Sagamore research conference proceedings, vol. 2, pp 289–305Google Scholar
  56. 56.
    BS 5447 (1974) Methods of testing for plane-strain fracture toughness (\(K_{Ic}\)) of metallic materials. Technical Report, British StandardsGoogle Scholar
  57. 57.
    Dugdale D (1960) Yielding of steel sheets containing slits. J Mech Phys Solid 8:100–104CrossRefGoogle Scholar
  58. 58.
    Wells AA (1961) Unstable crack propagation in metals: cleavage and fast fracture. In: Proceedings of the crack propagation symposium, vol. 1, Paper 84, Cranfield, UKGoogle Scholar
  59. 59.
    Burdekin FM, Stone DEW (1966) The crack opening displacement approach to fracture mechanics in yielding materials. J Strain Anal 1:145–153CrossRefGoogle Scholar
  60. 60.
    Schwalbe KH (1980) Bruchmechanik metallischer Werkstoffe. Carl Hanser Verlag, MünchenGoogle Scholar
  61. 61.
    ASTM-E 1290–93 (1993) Fracture toughness measurement crack-tip opening displacement (CTOD). Technical Report, American Society for Testing and Materials, PhiladelphiaGoogle Scholar
  62. 62.
    BS 5762 (1979) Methods for crack opening displacements (COD) testing. Technical Report, British StandardsGoogle Scholar
  63. 63.
    Harrison RPEA (1976) Assessments of the integrity of structures containing defects. CEGB-Report R/H.R6Google Scholar
  64. 64.
    Milne I, Ainsworth RA, Dowling AR, Stewart AT (1991) Assessments of the integrity of structures containing defects. British Energy-Report, R6-Revision 3Google Scholar
  65. 65.
    Zerbst U, Wiesner C, Kocak M, Hodulak L (1999) Sintap: Entwurf einer vereinheitlichten europäischen fehlerbewertungsprozedur - eine einführung. Technical Report, GKSS-Forschungszentrum, GeesthachtGoogle Scholar
  66. 66.
    Kocak M, Ainsworth RA, Dowling AR, Stewart AT (2008) FITNET Fitness-for-Service, Fracture-Fatigue-Creep-Corrosion, vol. I+II. GKSS Research CentreGoogle Scholar
  67. 67.
    Gross D, Seelig T (2001) Bruchmechanik: Mit einer Einführung in die Mikromechanik. Springer Verlag, BerlinGoogle Scholar
  68. 68.
    Shih CF (1973) Elastic-plastic analysis of combined mode crack problems. Ph.D. thesis, Harvard University Cambridge, MassachusettsGoogle Scholar
  69. 69.
    Hutchinson JW (1968) Plastic-stress and strain fields at a crack tip. J Mech Phys Solid 16:337–347Google Scholar
  70. 70.
    Hutchinson JW (1968) Singular behavior at the end of a tensile crack tip in a hardening material. J Mech Phys Solid 16:13–31CrossRefzbMATHGoogle Scholar
  71. 71.
    Rice JR, Rosengren GF (1968) Plain strain deformation near a crack tip in a power-law hardening material. J Mech Phys Solid 16:1–12CrossRefzbMATHGoogle Scholar
  72. 72.
    Uhlmann W, Knésl Z, Kuna M, Bilek Z (1976) Approximate representation of elastic-plastic small scale yielding solution for crack problems. Int J Fract 12:507–509Google Scholar
  73. 73.
    Larsson SG, Carlsson AJ (1973) Influence of non-singular stress terms and specimen geometry on small-scale yielding at track tips in elastic-plastic materials. J Mech Phys Solid 21:263–277CrossRefGoogle Scholar
  74. 74.
    Issler L, Ruoß H, Häfele P (2003) Festigkeitslehre - Grundlagen. Springer, BerlinGoogle Scholar
  75. 75.
    McMeeking RM, Parks DM (1979) On criteria for \(J\)-dominance of crack-tip fields in large-scale yielding. In: Landes JD, Begley JA, Clarke GA (eds) Elastic-plastic fracture, vol. ASTM STP 668, ASTM, pp. 175–194Google Scholar
  76. 76.
    Yuan H (2002) Numerical assessments of cracks in elastic-plastic materials. Springer, BerlinCrossRefzbMATHGoogle Scholar
  77. 77.
    Fett T (1998) A compendium of T-stress solutions. Technical Report FZKA 6057, Forschungszentrum Karlsruhe, Technik und UmweltGoogle Scholar
  78. 78.
    Sherry AH, France CC, Goldthorpe MR (1995) Compendium of T-stress solutions for two and three dimensional cracked geometries. Fatigue Fract Eng Mater Struct 18:141–155CrossRefGoogle Scholar
  79. 79.
    Rice JR (1974) Limitations to the small-scale yielding approximation for crack tip plasticity. J Mech Phys Solid 22:17–26CrossRefGoogle Scholar
  80. 80.
    Betegón C, Hancock JW (1991) Two-parameter characterization of elastic-plastic crack-tip fields. J Appl Mech 58:104–110CrossRefGoogle Scholar
  81. 81.
    Sharma SM, Aravas N (1991) Determination of higher-order terms in asymptotic crack tip solutions. J Mech Phys Solid 39(8):1043–1072CrossRefzbMATHGoogle Scholar
  82. 82.
    Yang S, Chao YJ, Sutton MA (1993) Higher order asymptotic fields in a power-law hardening material. Eng Fract Mech 45:1–20CrossRefGoogle Scholar
  83. 83.
    Nikishkov GP (1993) Three-term elastic-plastic asymptotic expansion for the description of the near-tip stress field. Technical Report, University of Karlsruhe, Institute for Reliability and Failure AnalysisGoogle Scholar
  84. 84.
    O’Dowd NP, Shih CF (1991) Family of crack-tip fields characterized by a triaxiality parameter: I structure of fields. J Mech Phys Solid 39:989–1015CrossRefGoogle Scholar
  85. 85.
    O’Dowd NP, Shih CF (1992) Family of crack-tip fields characterized by a triaxiality parameter: II fracture applications. J Mech Phys Solid 40:939–963CrossRefGoogle Scholar
  86. 86.
    Dodds RHJ, Tang M, Anderson TL (1994) Effects of prior ductile tearing on cleavage fracture toughness in the transition region. In: Kirk M, Bakker A (eds) Constraint effects in fracture—theory and applications, vol. ASTM STP 1244. ASTMGoogle Scholar
  87. 87.
    Dodds RH Jr, Shih CF, Anderson TL (1993) Continuum and micromechanics treatment of constraint in fracture. Int J Fract 64(2):101–133Google Scholar
  88. 88.
    Kirk MT, Koppenhoefer KC, Shih CF (1993) Effect of constraint on specimen dimensions needed to obtain structurally relevant toughness measures. Constraint effects in fracture, ASTM STP 1171, American Society for Testing and Materials, pp 79–103Google Scholar
  89. 89.
    Brocks W, Schmitt W (1994) The second parameter in J-R curves: constraint or triaxiality? Second symposium on constraint effects, ASTM STP 1244. In: Kirk MT, Bakker A (eds), American society for testing and materialsGoogle Scholar
  90. 90.
    Kumar V, German MD, Shih CF (1981) An engineering approach for elastic-plastic fracture analysis. EPRI-Report NP-1931Google Scholar
  91. 91.
    Begley JA, Landes JD (1972) The J-integral as a fracture criterion. ASTM STP 514, American Society of Testing and Materials, pp 1–20Google Scholar
  92. 92.
    Rice JR, Paris PC, Merkle JG (1973) Some further results of J-integral analysis and estimates. ASTM STP 536, American society of testing and materials, pp 231–245Google Scholar
  93. 93.
    Hutchinson JW, Paris PC (1979) Stability analysis of \(J\)-controlled crack growth. In: Landes J, Begley J, Clarke G (eds) Elastic-plastic fracture, vol. ASTM STP 668, ASTM, pp 37–64Google Scholar
  94. 94.
    Shih CF, deLorenzi HG, Andrews WR (1979) Studies on crack initiation and stable crack growth. Elastic-Plastic Fracture, pp 65–120Google Scholar
  95. 95.
    Kussmaul K, Roos E, Föhl J (1997) Forschungsvorhaben Komponentensicherheit (FKS). Ein wesentlicher Beitrag zur Komponentensicherheit. In: Kussmaul K (ed) 23. MPA Seminar Sicherheit und Verfügbarkeit in der Anlagentechnik, vol. 1, MPA, Stuttgart, pp 1–20Google Scholar
  96. 96.
    Kordisch H, Sommer E, Schmitt W (1989) The influence of triaxiality on stable crack growth. Nucl Eng Des 112:27–35CrossRefGoogle Scholar
  97. 97.
    Slepyan LI (1974) Growing crack during plane deformation of an elastic-plastic body. Mekh. Tverdogo Tela 9:57–67Google Scholar
  98. 98.
    Drugan WJ, Rice JR, Sham TL (1982) Asymptotic analysis of growing plane strain tensile cracks in elastic-ideally plastic solids. J Mech Phys Solid 30:447–473CrossRefzbMATHGoogle Scholar
  99. 99.
    Castañeda PP (1987) Asymptotic fields in steady crack growth with linear strain-hardening. J Mech Phys Solid 35:227–268Google Scholar
  100. 100.
    Turner CE (1990) A re-assessment of ductile tearing resistance. In: Firrao D (ed) Fracture behaviour and design of materials and structures, vol. 2, EMAS, Warley, pp. 933–949, 951–968Google Scholar
  101. 101.
    Memhard D, Brocks W, Fricke S (1993) Characterization of ductile tearing resistance by energy dissipation rate. Fatigue Fract Eng Mater Struct 16:1109–1124CrossRefGoogle Scholar
  102. 102.
    Cotterell B, Atkins AG (1996) A review of the \(J\) and \(I\) integrals and their implications for crack growth resistance and toughness in ductile fracture. Int J Fract 81:357–372CrossRefGoogle Scholar
  103. 103.
    Belytschko TKLW, Moran B (2001) Nonlinear finite elements for continua and structures. Wiley, New YorkGoogle Scholar
  104. 104.
    Moran B, Shih CF (1987) Crack tip and associated domain integrals from momentum and energy balance. Eng Fract Mech 27:615–642CrossRefGoogle Scholar
  105. 105.
    Nguyen QS (1991) An energetic analysis of elastic-plastic fracture. In: Blauel JG, Schwalbe KH (eds) Defect assessment in components—fundamentals and applications. Mechanical Engineering Publications, London, pp 75–85Google Scholar
  106. 106.
    Yuan H, Brocks W (1991) On the \(J\)-integral concept for elastic-plastic crack extension. Nucl Eng Des 131:157–173CrossRefGoogle Scholar
  107. 107.
    Paris P, Erdogan F (1963) A critical analysis of crack propagation laws. J Basic Eng 85:528–534CrossRefGoogle Scholar
  108. 108.
    Erdogan F, Ratwani M (1970) Fatigue and fracture of cylindrical shells containing a circumferential crack. Int J Fract Mech 6:379–392Google Scholar
  109. 109.
    ESA (2000) ESACRACK User’s manual. European Space Research and Technology Centre (ESTEC)Google Scholar
  110. 110.
    Newman JC (1981) A crack-closure model for predicting fatigue crack growth under aircraft spectrum loading. In: Chang JB, Hudson CM (eds) Methods and models for predicting fatigue crack growth under random loading. ASTM STP 748, pp 53–84Google Scholar
  111. 111.
    Rice JR (1967) Mechanics of crack tip deformation and extension by fatigue. In: Fatigue crack propagation, vol. ASTM STP 415, American Society for Testing and Materials, London, pp 247–309Google Scholar
  112. 112.
    Wheeler OE (1972) Spectrum loading and crack growth. J Basic Eng 94:181–186CrossRefGoogle Scholar
  113. 113.
    Elber W (1970) Fatigue crack closure under cyclic tension. Eng Fract Mech 2:37–45CrossRefGoogle Scholar
  114. 114.
    Suresh S, Ritchie RO (1984) Propagation of short fatigue cracks. Int Metall Rev 29:445–476Google Scholar
  115. 115.
    Westermann-Friedrich A, Zenner (Verfasser) H (1999) Zählverfahren zur Bildung von Kollektiven aus Zeitfunktionen - Vergleich der verschiedenen Verfahren und Beispiele. Technical Report, Forschungsvereinigung Antriebstechnik. FVA-Merkblatt Nr. 0/14Google Scholar
  116. 116.
    Dominguez J (1994) Fatigue crack growth under variable amplitude loading. In: Carpinteri A (ed) Handbook of fatigue crack propagation in metallic structures, Elsevier Science, pp 955–997Google Scholar
  117. 117.
    NASA (2000) Fatigue crack growth computer program "NASGRO" version 3.0. JSC-22267B, Johnson Space Center, TexasGoogle Scholar
  118. 118.
    de Koning AU (1981) A simple crack closure model for prediction of fatigue crack growth rates under variable-amplitude loading. In: Fracture mechanics, vol. ASTM STP 743, American Society for Testing and Materials, pp 63–85Google Scholar
  119. 119.
    Padmadinata UH (1990) Investigation of crack-closure prediction models for fatigue in aluminum sheet under flight-simulation loading. Ph.D. Thesis, Delft University of TechnologyGoogle Scholar
  120. 120.
    Führing H, Seeger T (1979) Dugdale crack closure analysis of fatigue cracks under constant amplitude loading. Eng Fract Mech 11:99–122CrossRefGoogle Scholar
  121. 121.
    Schijve J (2001) Fatigue of structures and materials. Kluwer Academic Publisher, DordrechtGoogle Scholar
  122. 122.
    Schijve J (1979) Four lectures on fatigue crack growth. Eng Fract Mech 11:176–221Google Scholar
  123. 123.
    Miller KJ, de los Rios ER (1992) Short fatigue cracks, vol. ESIS 13. Mechanical Engineering Publications, LondonGoogle Scholar
  124. 124.
    Ravichandran KS, Ritchie RO, Murakami Y (1999) Small fatigue cracks. mechanics, mechanisms and applications. Elsevier, AmsterdamGoogle Scholar
  125. 125.
    Richard HA (1985) Bruchvorhersagen bei überlagerter Normal- und Schubbeanspruchung von Rissen. VDI-Forschungsheft 631Google Scholar
  126. 126.
    Erdogan F, Sih GE (1963) On the crack extension in plates under plane loading and transverse shear. J Basic Eng 85:519–527CrossRefGoogle Scholar
  127. 127.
    Hussain MA, Pu SL, Underwood J (1974) Strain energy release rate for a crack under combined Mode I and Mode II. ASTM STP 560:2–28Google Scholar
  128. 128.
    Ichikawa M, Tanaka S (1982) A critical analysis of the relationship between the energy release rate and the stress intensity factors for non-coplanar crack extension under combined mode loading. Int J Fract 18:19–28CrossRefGoogle Scholar
  129. 129.
    Lo KK (1978) Analysis of branched cracks. J Appl Mech 45:797–802CrossRefzbMATHGoogle Scholar
  130. 130.
    Nuismer RJ (1975) An energy release rate criterion for mixed mode fracture. Int J Fract 11:245–250CrossRefGoogle Scholar
  131. 131.
    Sih GC (1974) Strain energy density factor applied to mixed mode crack problems. Int J Fract 10:305–321CrossRefGoogle Scholar
  132. 132.
    Radaj D, Heib M (1978) Energy density fracture criteria for cracks under mixed mode loading. Materialprüfung 20:256–62Google Scholar
  133. 133.
    Richard HA, Fulland M, Sander M (2005) Theoretical crack path prediction. Fatigue Fract Eng Mater Struct 28:3–12CrossRefGoogle Scholar
  134. 134.
    Pook LP (2002) Crack paths. WIT Press, BostonGoogle Scholar
  135. 135.
    Schöllmann M, Richard HA, Kullmer G, Fulland M (2002) A new criterion for the prediction of crack development in multiaxially loaded structures. Int J Fract 117:129–141CrossRefGoogle Scholar
  136. 136.
    Richard HA, Kuna M (1990) Theoretical and experimental study of superimposed fracture modes I. II and III. Eng Fract Mech 35(6):949–960CrossRefGoogle Scholar
  137. 137.
    Chen CS, Wawrzynek PA, Ingraffea AR (1997) Methodology for fatigue crack growth and residual strength prediction with applications to aircraft fuselages. Comput Mech 19:527–532CrossRefGoogle Scholar
  138. 138.
    Sumi Y, Nemat-Nasser S, Keer LM (1985) On crack path instability in a finite body. Eng Fract Mech 22:759–771CrossRefGoogle Scholar
  139. 139.
    Cotterell B, Rice JR (1980) Slightly curved or kinked cracks. Int J Fract 16:155–169CrossRefGoogle Scholar
  140. 140.
    Freund LB (1998) Dynamic fracture mechanics. Cambridge University Press, CambridgeGoogle Scholar
  141. 141.
    Ravi-Chandar K (2003) Dynamic fracture. In: Milne I, Ritchie RO, Karihaloo B (eds) Comprehensive structural integrity—fundamental theories and mechanisms of failure, vol 2. Elsevier, Oxford, pp 285–361CrossRefGoogle Scholar
  142. 142.
    Miannay DP (2001) Time-dependent fracture mechanics. Springer, New YorkCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Institute für Mechanik und FluiddynamikTU Bergakademie FreibergFreibergGermany

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