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Characterization of crack tip stress fields in test specimens using mode mixity parameters

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Abstract

The aim of this study is to represent the combined effect of mode mixity, specimen geometry and relative crack length on the \(T\)-stress, elastic–plastic stress fields, integration constant \(I_{n}\), angle of initial crack extension, and the plastic stress intensity factor. The analytical and numerical results are obtained for the complete range of mixed modes of loading between mode I and mode II. For comparison purposes, the reference fields for plane mixed-mode problems governing the asymptotic behavior of the stresses and strains at the crack tip are developed in a power law elastic–plastic material. For the common experimental fracture mechanics specimen geometries considered, the numerical constant of the plastic stress field \(I_{n}\) and the \(T\)-stress distributions are obtained as a function of the dimensionless crack length and mode mixity. A method is also suggested for calculating the plastic stress intensity factor for any mixed-mode I/II loading based on the \(T\)-stress and power law solutions. It is further demonstrated that in both plane stress and the plane strain, the plastic stress intensity factor can be used to characterize the crack tip stress fields for a variety of specimen geometries and different mixed-mode loading. The applicability of the plastic stress intensity factor to analysis of the in-plane and out-of-plane constraint effect is also discussed.

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References

  • Anderson TL (2005) Fracture mechanics, 3rd edn. CRC Press, Boca Raton

    Google Scholar 

  • ANSYS Mechanical APDL Theory Reference (2012) Release 14.5// ANSYS, Inc. Southpointe. 275 Technology Drive. CanonBurg, PA

  • Aoki S, Kishimoto K, Yoshida T, Sakata M (1987) A finite element study of the near crack tip deformation of a ductile material under mixed mode loading. J Mech Phys Solids 35:431–455

    Article  Google Scholar 

  • Aoki S, Kishimoto K, Yoshida T, Sakata M, Richard H-A (1990) Elastic–plastic fracture behavior of an aluminum alloy under mixed mode loading. J Mech Phys Solids 38:195–214

    Article  Google Scholar 

  • Arun Roy Y, Narasimhan R (1997) J-dominance in mixed mode ductile fracture specimens. Int J Fract 88:259–279

    Article  Google Scholar 

  • Arun Roy Y, Narasimhan R (1999a) A finite element investigation of the effect of crack tip constraint on hole growth under mode I and mixed mode loading. Int J Solids Struct 36:1427–1447

    Article  Google Scholar 

  • Arun Roy Y, Narasimhan R (1999b) Constraint effects on ductile fracture processes near a notch tip under mixed-mode loading. Eng Fract Mech 62:511–534

    Article  Google Scholar 

  • Arun Roy Y, Narasimhan R, Arora PR (1999) An experimental investigation of constraint effects on mixed mode fracture initiation in a ductile aluminium alloy. Acta Mater 47:1587–1596

    Article  Google Scholar 

  • Ayatollahi MR, Pavier MJ, Smith DJ (1998) Determination of T-stress from finite element analysis for mode I and mixed mode I/II loading. Int J Fract 91:283–298

    Article  Google Scholar 

  • Ayatollahi MR, Sedighiani K (2012) A T-stress controlled specimen for mixed mode fracture experiments on brittle materials. Eur J Mech A/Solids 36:83–93

    Article  Google Scholar 

  • Betegon C, Hancock J (1991) Two-parameter characterization of elastic–plastic crack-tip fields. ASME J Appl Mech 58:104–113

    Article  Google Scholar 

  • Budiansky B, Rice JR (1973) Conservation laws and energy release rates. ASME J Appl Mech 40:201–203

    Article  Google Scholar 

  • Chao YJ, Yang S, Sutton NA (1994) On the fracture of solids characterized by one or two parameters: theory and practice. J Mech Phys Solids 42:629–647

    Article  Google Scholar 

  • Chrysakis AC (1987) Improvement of the max \(\sigma \)r, max \(\tau \)r\(\theta \) and max \(\sigma \)1, criteria for mixed mode fracture. Eng Fract Mech 26:651–656

    Article  Google Scholar 

  • Dhirendra VK, Narasimhan R (1998) Mixed-mode steady-state crack growth in elastic–plastic solids. Eng Fract Mech 59:543–559

    Article  Google Scholar 

  • Dolgorukov VA (1988) Elastic–plastic problem for determination of singular stress–strain-state at inclined crack tip under plane stress conditions. VINITI 4340–V88, pp 1–21 (in Russian)

  • Dong P, Pan J (1990a) Plane-strain mixed mode near-tip fields in elastic perfectly plastic solids under small-scale yielding conditions. Int J Fract 45:243–262

    Article  Google Scholar 

  • Dong P, Pan J (1990b) Plane-stress mixed mode near-tip fields in elastic perfectly plastic solids. Eng Fract Mech 37: 43–57

    Google Scholar 

  • Eftis J, Subramonian N (1978) The inclined crack under biaxial load. Eng Fract Mech 10:43–67

    Article  Google Scholar 

  • Erdogan F, Sih GS (1963) On crack extension in plates under plane loading and transverse shear. J Basic Eng Trans ASME 85:519–527

    Article  Google Scholar 

  • Faleskog J, Gao X, Shih CF (1998) Cell model for nonlinear fracture analysis—I. Micromechanics calibration. Int J Fract 89:355–373

    Article  Google Scholar 

  • Gonzales-Albuixech VF, Giner E, Fernandez-Saez J, Fernandez- Canteli A (2011) Influence of the t33-stress on the 3-D stress state around corner cracks in an elastic plate. Eng Fract Mech 78:412–427

    Google Scholar 

  • Guo W (1993) Elastoplastic three dimensional crack border field-II. Asymptotic solution for the field. Eng Fract Mech 46:105–113

    Google Scholar 

  • Gurson AL (1977) Continuum theory of ductile rupture by void nucleation and growth: part 1—yield criteria and flow rules for porous ductile media. J Eng Mater Technol 99:1–16

    Google Scholar 

  • Hebel J, Hohe J, Friedmann V, Siegele D (2007) Experimental and numerical analysis of in-plane and out-of-plane crack tip constraint characterization by secondary fracture parameters. Int J Fract 146:173–188

    Article  Google Scholar 

  • Henry BS, Luxmoore AR (1997) The stress triaxiality constraint and the Q-value as ductile fracture parameter. Eng Fract Mech 55:375–390

    Article  Google Scholar 

  • Hussain MA, Pu SL, Underwood J (1974) Strain energy release rate for a crack under combined mode I and mode II. Fract Anal ASTM Spec Tech Publ 560:2–28

    Google Scholar 

  • Hutchinson JW (1968a) Singular behaviour at the end of a tensile crack in a hardening material. J Mech Phys Solids 16: 13–31

    Google Scholar 

  • Hutchinson JW (1968b) Plastic stress and strain fields at a crack tip. J Mech Phys Solids 16:337–347

    Article  Google Scholar 

  • Kim JH, Paulino GH (2007) On fracture criteria for mixed-mode crack propagation in functionally graded materials. Mech Adv Mater Struct 14:227–244

    Article  Google Scholar 

  • Li Y, Wang Z (1986) High-order asymptotic field of tensile plane-strain nonlinear crack problems. Sci China Math 29(9):941–955

    Google Scholar 

  • Loghin A, Joseph P (2001) Asymptitic solutions for mixed mode loading of cracks and wedged in power law hardening materials. Eng Fract Mech 68:1511–1534

    Google Scholar 

  • Loghin A, Joseph P (2003) Mixed mode fracture in power law hardening materials near Mode I. Int J Fract 123:81–106

    Google Scholar 

  • Maiti SK, Smith RA (1983) Theoretical and experimental studies on the extension of cracks subjected to concentrated loading near their faces to compare the criteria for mixed mode brittle fracture. J Mech Phys Solids 31:389–403

    Article  Google Scholar 

  • Maiti SK, Smith RA (1984) Criteria for brittle fracture in biaxial tension. Eng Fract Mech 19:793–804

    Article  Google Scholar 

  • Matvienko YG, Shlyannikov VN, Boychenko NV (2013) In-plane and out-of-plane constraint parameters along a three-dimensional crack-front stress field under creep loading. Fatigue Fract Engng Mater Struct 36:14–24

    Google Scholar 

  • McClintock FA (1968) A criterion for ductile fracture by growth of holes. J Appl Mech 35:324–334

    Google Scholar 

  • McMeeking RM (1977) Finite deformation analysis of crack-tip opening in elastic–plastic materials and implications for fracture. J Mech Phys Solids 25:357–381

    Article  Google Scholar 

  • Muskhelishvili NI (1963) Some basic problems of the mathematical theory of elasticity, 4th edn. Noordhoff, Groningen

  • Nikishkov GP (1995) An algorithm and computer program for the three-term asymptotic expansion of elastic-plastic crack tip stress and displacement fields. Eng Fract Mech 50:65–83

    Google Scholar 

  • Nikishkov GP, Bruckner-Foit A, Munz D (1995) Calculation of the second fracture parameter for finite cracked bodies using a three-term elastic-plastic asymptotic expansion. Eng Fract Mech 52:685–701

    Google Scholar 

  • O’Dowd N (1995) Applications of two parameter approaches in elastic-plastic fracture mechanics. Eng Fract Mech 52:445–465

    Google Scholar 

  • O’Dowd N, Shih C (1991) Family of crack-tip fields characterized by a triaxiality parameter-I. Structure of fields. J Mech Phys Solids 39:989–1015

    Google Scholar 

  • Rice JR (1968) A path independent integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech 35:379–386

    Article  Google Scholar 

  • Rice JR (1974) Limitations to the small scale yielding approximation for crack tip plasticity. J Mech Phys Solids 22:17–26

    Article  Google Scholar 

  • Rice JR, Rosengren GF (1968) Plane strain deformation near a crack tip in a power-law hardening material. J Mech Phys Solids 16:1–12

    Article  Google Scholar 

  • Rice JR, Tracey DM (1969) On the ductile enlargement of voids in triaxial stress fields. J Mech Phys Solids 17:201–217

    Article  Google Scholar 

  • Richard HA (1985) Bruchvorhersagen bei berlagerter normal- und schubbean-spruchung von rissen. VDI Forscungsheft N 631, VDI-Verlag, Dusseldorf

  • Richard HA, Benitz K (1983) A loading device for the creation of mixed mode in fracture mechanics. Int J Fract 22:R55–R58

    Article  Google Scholar 

  • Sharma SM, Aravas N (1991) Determination of higher-order terms in asymptotic elastoplastic crack tip solutions. J Mech Phys Solids 39:1043–1072

    Article  Google Scholar 

  • Shih CF (1974) Small-scale yielding analysis of mixed mode plane-strain crack problems. Fract Anal ASTM Spec Tech Publ 560:187–210

    Google Scholar 

  • Shlyannikov VN, Dolgorukov VA (1988) Analysis of the crack propagation under biaxial cyclic load taking into accounts their orientation. In: Gzoboly E (ed) Proceedings of 7th European conference on fracture, Budapest, pp 1095–1103

  • Shlyannikov VN (1991) Crack paths in aluminum alloys under biaxial cyclic loading. Strength Mater N6:42–47

    Google Scholar 

  • Shlyannikov VN (1999) Mixed-mode static and fatigue crack growth in central notched and compact tension shear specimens. Mixed Mode Crack Behav ASTM Spec Tech Publ 1359:279–294

    Article  Google Scholar 

  • Shlyannikov VN (2003) Elastic–plastic mixed-mode fracture criteria and parameters. Springer, Berlin

    Book  Google Scholar 

  • Shlyannikov VN, Ilchenko BV, Boychenko NV (2009) Biaxial loading effect on higher-order crack tip parameters. Fatigue Fract Mech ASTM Spec Tech Publ 1508 36:609–640

    Google Scholar 

  • Shlyannikov VN, Boychenko NV, Tartygasheva AM (2011) In-plane and out-of-plane crack-tip constraint effects under biaxial nonlinear deformation. Eng Fract Mech 78:1771–1783

    Article  Google Scholar 

  • Shlyannikov VN (2013) T-stress for crack paths in test specimens subject to mixed mode loading. Eng Fract Mech 99:3–18

    Google Scholar 

  • Sih GC (1974) Strain energy density factor applied to mixed mode crack problems. Int J Fract 10:305–321

    Google Scholar 

  • Smith DJ, Ayatollahi MR, Pavier MJ (2001) The role of T-stress in brittle fracture for linear elastic materials under mixed-mode loading. Fatigue Fract Eng Mater Struct 24:137–150

    Google Scholar 

  • Smith EW, Pascoe KJ (1983) The behaviour of fatigue cracks subject to applied biaxial stress: a review of experimental evidence. Fatigue Eng Mater Struct 6:201–224

    Article  Google Scholar 

  • Theocaris PS, Philippidis TP (1987) The T-criterion for ductile fractures in HRR plastic singular fields. Int J Fract 35:21–37

    Google Scholar 

  • Tvergaard V (1982) Ductile fracture by cavity nucleation between larger voids. J Mech Phys Solids 30:265–286

    Article  Google Scholar 

  • Tvergaard V, Hutchinson JW (1992) The relation between crack growth resistance and fracture process parameters in elastic–plastic solids. J Mech Phys Solids 40:1377–1397

    Article  Google Scholar 

  • Vladimirov VI (1984) A physical nature of metals fracture. Metallurgiya Press, Moscow

    Google Scholar 

  • Williams ML (1957) On the stress distribution at the base of stationary crack. J Appl Mech 24:111–114

    Google Scholar 

  • Xiang M, Yu Z, Guo W (2011) Characterization of three-dimensional crack border fields in creeping solids. Int J Solids Struct 48:2695–2705

    Google Scholar 

  • Yang S, Chao Y, Sutton M (1993) Higher order asymptotic crack tip fields in a power-law hardening material. Eng Fract Mech 45:1–20

    Article  Google Scholar 

  • Zhao J, Guo W, She C (2007) The in-plane and out-of-plane stress constraint factors and K–T–T\(_{{\rm z}}\) description of stress field near border of semi-elliptical surface crack. Int J Fatigue 29:435–443

    Article  Google Scholar 

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Acknowledgments

The authors gratefully acknowledge the financial support of the Russian Foundation for Basic Research under the Project 13-08-00813.

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Correspondence to Valery Shlyannikov.

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Shlyannikov, V., Tumanov, A. Characterization of crack tip stress fields in test specimens using mode mixity parameters. Int J Fract 185, 49–76 (2014). https://doi.org/10.1007/s10704-013-9898-0

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  • DOI: https://doi.org/10.1007/s10704-013-9898-0

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