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Combined tension–shear fracture analysis of V-notches with end holes

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Abstract

In the present research, two stress-based failure criteria were proposed to predict brittle fracture in components containing V-notches with end holes (VO-notches) under mixed mode I/II loading. The first criterion, called VO-MTS, was an extension of the maximum tangential stress (MTS) criterion, and the second one, called VO-MS, was developed based on the mean stress (MS) failure concept. Two different groups of critical distances were utilized in the predictions. The first group was equal to the critical distances for sharp crack, and the second one was computed by using the mode I fracture test results on notched specimens. To verify the criteria, the theoretical fracture curves were compared with numerous experimental results gathered from 108 new brittle fracture tests performed on the Brazilian disk specimens weakened by central V-notches with end holes and made of PMMA under mode I and mixed mode I/II loadings. It was found that both the criteria provide very good predictions to the experimental results for different mode mixity ratios. Also, found in this research was that the curves are almost independent of the critical distance groups, meaning that one can simply utilize the critical distances of sharp crack in both the VO-MTS and the VO-MS criteria without requiring performing mode I VO-notch fracture experiments.

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Abbreviations

CTSN:

Compact–tension–shear–notch

CZM:

Cohesive zone model

d:

Total slit length in the VO-BD specimen

D:

Diameter of the VO-BD specimen

d c :

Critical distance of the MS criterion for sharp cracks

d c,vo :

Critical distance of the MS criterion for VO-notch

E :

Young’s modulus

ERNFT:

Effective relative notch fracture toughness

FE:

Finite element

FFM:

Finite fracture mechanics

FNR:

Fictitious notch radius

\({K_{\rm eff}^{\rm vo}}\) :

Effective relative notch fracture toughness

K IC :

Plane-strain fracture toughness of material

\({K_{\rm I}^{\rm vo}}\) :

Mode I notch stress intensity factor

\({K_{\rm II}^{\rm vo}}\) :

Mode II notch stress intensity factor

\({K_{\rm Ic}^{\rm vo}}\) :

Mode I notch fracture toughness

MTS:

Maximum tangential stress

MS:

Mean stress

NDT:

Nondestructive testing

NFM:

Notch fracture mechanics

NFT:

Notch fracture toughness

NSIF:

Notch stress intensity factor

P:

Compressive load applied to the specimen

PMMA:

Polymethyl-methacrylate

PS:

Point stress

r c :

Critical distance of the MTS criterion for sharp cracks

r c,vo :

Critical distance of the MTS criterion for VO-notch

SED:

Strain energy density

TPB:

Three-point bending

UNBD:

U-notched Brazilian disk

VO-BD:

Brazilian disk specimen containing V-notch with end hole

VO-notch:

V-notch with end hole

β :

Loading angle in the VO-BD specimen

β II :

Mode II loading angle in the VO-BD specimen

λ i :

Eigenvalues

ϕ i :

Auxiliary parameters in the stress field around the VO-notch

\({\nu}\) :

Poisson’s ratio

ρ :

Notch radius

\({\sigma_{\rm u}}\) :

Ultimate tensile strength of material

\({\sigma_{{\rm r}\theta}}\) :

In-plane shear stress

\({\sigma_{\theta \theta}}\) :

Tangential stress

\({\overline{\sigma_{\theta \theta}}}\) :

Average value of the tangential stress over the critical distance

\({\sigma_{\rm c}}\) :

Critical stress

\({\theta_0}\) :

Fracture initiation angle for the MTS criterion

\({\theta_{0{\rm II}}}\) :

Fracture initiation angle for the MTS criterion under mode II loading

\({\overline{\theta_0}}\) :

Fracture initiation angle for the MS criterion

\({\overline{\theta_{0{\rm II}}}}\) :

Fracture initiation angle for the MS criterion under mode II loading

References

  1. Yuan H.: Singular stress fields at V-notch tips in elastoplastic pressure-sensitive materials. Acta Mech. 118, 151–170 (1996)

    Article  MATH  Google Scholar 

  2. Dornowski W., Perzyna P.: Localized fracture phenomena in thermo-visco-plastic flow processes under cyclic dynamic loadings. Acta Mech. 155, 233–255 (2002)

    Article  MATH  Google Scholar 

  3. Bian L., Cheng Y., Taheri F.: Elasto-plastic analysis of critical fracture stress and fatigue fracture prediction. Acta Mech. 225(11), 3059–3072 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bhat S., Narayanan S.: A computational model and experimental validation of shielding and amplifying effects at a crack tip near perpendicular strength-mismatched interfaces. Acta Mech. 216, 259–279 (2011)

    Article  MATH  Google Scholar 

  5. Lauterbach B., Gross D.: The role of nucleation and growth of micro cracks in brittle solids under compression: a numerical study. Acta Mech. 159, 199–211 (2002)

    Article  MATH  Google Scholar 

  6. Shindo Y., Narita F., Sato T.: Analysis of mode II interlaminar fracture and damage behavior in end notched flexure testing of GFRP woven laminates at cryogenic temperatures. Acta Mech. 187, 231–240 (2006)

    Article  MATH  Google Scholar 

  7. Guo J.H., Lu Z.X., Feng. X.: The fracture behavior of multiple cracks emanating from a circular hole in piezoelectric materials. Acta Mech. 215, 119–134 (2010)

    Article  MATH  Google Scholar 

  8. Brighenti R., Carpinteri A., Spagnoli A.: Influence of material microvoids and heterogeneities on fatigue crack propagation. Acta Mech. 225, 3123–3135 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gogotsi G.A.: Fracture toughness of ceramics and ceramic composites. Ceram. Int. 29, 777–784 (2003)

    Article  Google Scholar 

  10. Knesl Z.: A criterion of V-notch stability. Int. J. Fract. 48, 79–83 (1991)

    Article  Google Scholar 

  11. Nui L.S., Chehimi C., Pluvinage G.: Stress field near a large blunted tip V-notch and application of the concept of the critical notch stress intensity factor (NSIF) to the fracture toughness of very brittle materials. Eng. Fract. Mech. 49, 325–335 (1994)

    Article  Google Scholar 

  12. Seweryn A.: Brittle fracture criterion for structures with sharp notches. Eng. Fract. Mech. 47, 673–681 (1994)

    Article  Google Scholar 

  13. Strandberg M.: Fracture at V-notches with contained plasticity. Eng. Fract. Mech. 69, 403–415 (2002)

    Article  Google Scholar 

  14. Lazzarin P., Zambardi R.: A finite-volume-energy based approach to predict the static and fatigue behavior of components with sharp V-shaped notches. Int. J. Fract. 112, 275–298 (2001)

    Article  Google Scholar 

  15. Lazzarin P., Lassen T., Livieri P.: A notch stress intensity approach applied to fatigue life predictions of welded joints with different local toe geometry. Fat. Fract. Eng. Mater. Struct. 26, 49–58 (2003)

    Article  Google Scholar 

  16. Lazzarin P., Berto F., Radaj D.: Fatigue-relevant stress field parameters of welded lap joints: Pointed slit tip compared with keyhole notch. Fat. Fract. Eng. Mater. Struct. 32, 713–735 (2009)

    Article  Google Scholar 

  17. Yosibash Z., Bussiba A., Gilad I.: Failure criteria for brittle elastic materials. Int. J. Fract. 125, 307–333 (2004)

    Article  MATH  Google Scholar 

  18. Berto F., Lazzarin P., Yates J.R.: Multiaxial fatigue of V-notched steel specimens: a non-conventional application of the local energy method. Fat. Fract. Eng. Mater. Struct. 34, 921–943 (2011)

    Article  Google Scholar 

  19. Gomez F.J., Elices M.: A fracture criterion for sharp V-notched samples. Int. J. Fract. 123, 163–175 (2003)

    Article  Google Scholar 

  20. Gomez F.J., Elices M., Valiente A.: Cracking in PMMA containing U-shaped notches. Fat. Fract. Eng. Mater. Struct. 23, 795–803 (2000)

    Article  Google Scholar 

  21. Carpinteri A., Cornetti P., Pugno N., Sapora A., Taylor D.: A finite fracture mechanics approach to structures with sharp V-notches. Eng. Fract. Mech. 75(7), 1736–1752 (2008)

    Article  Google Scholar 

  22. Livieri P.: A new path independent integral applied to notched components under mode I loadings. Int. J. Fract. 123, 107–125 (2003)

    Article  Google Scholar 

  23. Matvienko Y.G., Morozov E.M.: Calculation of the energy J-integral for bodies with notches and cracks. Int. J. Fract. 125, 249–261 (2004)

    Article  MATH  Google Scholar 

  24. Berto F., Lazzarin P.: Relationships between J-integral and the strain energy evaluated in a finite volume surrounding the tip of sharp and blunt V-notches. Int. J. Solids Struct. 44, 4621–4645 (2007)

    Article  MATH  Google Scholar 

  25. Livieri P.: Use of J-integral to predict static failures in sharp V-notches and rounded U-notches. Eng. Fract. Mech. 75, 1779–1793 (2008)

    Article  Google Scholar 

  26. Barati E., Alizadeh Y., Aghazadeh J., Berto F.: Some new practical equations for rapid calculation of J-integral in plates weakened by U-notches under bending. Mater. Design 31, 2964–2971 (2009)

    Article  Google Scholar 

  27. Barati E., Alizadeh Y.: A numerical method for evaluation of J-integral in plates made of functionally graded materials with sharp and blunt V-notches. Fat. Fract. Eng. Mater. Struct. 34, 1041–1052 (2011)

    Article  Google Scholar 

  28. Becker T.H., Mostafavi M., Tait R.B., Marrow T.J.: An approach to calculate the J-integral by digital image correlation displacement field measurement. Fat. Fract. Eng. Mater. Struct. 35, 971–984 (2012)

    Article  Google Scholar 

  29. Ayatollahi M.R., Torabi A.R.: Brittle fracture in rounded-tip V-shaped notches. Mater. Design 31, 60–67 (2010)

    Article  Google Scholar 

  30. Ayatollahi M.R., Torabi A.R.: Tensile fracture in notched polycrystalline graphite specimens. Carbon 48, 2255–2265 (2010)

    Article  Google Scholar 

  31. Torabi A.R.: Estimation of tensile load-bearing capacity of ductile metallic materials weakened by a V-notch: The equivalent material concept. Mater. Sci. Eng. A 536, 249–255 (2012)

    Article  Google Scholar 

  32. Torabi A.R.: Fracture assessment of U-notched graphite plates under tension. Int. J. Fract. 181, 285–292 (2013)

    Article  Google Scholar 

  33. Torabi A.R.: On the use of the Equivalent Material Concept to predict tensile load-bearing capacity of ductile steel bolts containing V-shaped threads. Eng. Fract. Mech. 97, 136–147 (2013)

    Article  Google Scholar 

  34. Torabi A.R.: Ultimate bending strength evaluation of U-notched ductile steel samples under large-scale yielding conditions. Int. J. Fract. 180, 261–268 (2013)

    Article  Google Scholar 

  35. Torabi A.R., Fakoor M., Pirhadi E.: Tensile fracture in coarse-grained polycrystalline graphite weakened by a U-shaped notch. Eng. Fract. Mech. 111, 77–85 (2013)

    Article  Google Scholar 

  36. Torabi A.R., Berto F.: Fracture assessment of blunt V-notched graphite specimens by means of the strain energy density. Strength Mater. 45, 635–647 (2013)

    Article  Google Scholar 

  37. Torabi, A.R., Berto, F.: Notch fracture toughness evaluation for a brittle graphite material. Mater. Perform. Charact. 3(3), 1–16 (2014)

  38. Ayatollahi M.R., Torabi A.R.: Investigation of mixed mode brittle fracture in rounded-tip V-notched components. Eng. Fract. Mech. 77, 3087–3104 (2010)

    Article  Google Scholar 

  39. Ayatollahi M.R., Torabi A.R.: Failure assessment of notched polycrystalline graphite under tensile-shear loading. Mater. Sci. Eng. A 528, 5685–5695 (2011)

    Article  Google Scholar 

  40. Ayatollahi M.R., Torabi A.R.: Experimental verification of RV-MTS model for fracture in soda-lime glass weakened by a V-notch. J. Mech. Sci. Tech. 25, 2529–2534 (2011)

    Article  Google Scholar 

  41. Ayatollahi M.R., Torabi A.R., Azizi P.: Experimental and theoretical assessment of brittle fracture in engineering components containing a sharp V-notch. Exp. Mech. 51, 919–932 (2011)

    Article  Google Scholar 

  42. Ayatollahi M.R., Torabi A.R.: A criterion for brittle fracture in U-notched components under mixed mode loading. Eng. Fract. Mech. 76, 1883–1896 (2009)

    Article  Google Scholar 

  43. Torabi A.R.: Sudden fracture from U-notches in fine-grained isostatic graphite under mixed mode I/II loading. Int. J. Fract. 181, 309–316 (2013)

    Article  Google Scholar 

  44. Torabi A.R., Fakoor M., Pirhadi E.: Fracture analysis of U-notched disc-type graphite specimens under mixed mode loading. Int. J. Solids. Struct. 51, 1287–1298 (2014)

    Article  Google Scholar 

  45. Susmel L., Taylor D.: The theory of critical distances to predict static strength of notched brittle components subjected to mixed-mode loading. Eng. Fract. Mech. 75, 534–550 (2008)

    Article  Google Scholar 

  46. Susmel L., Taylor D.: On the use of the theory of critical distances to predict static failures in ductile metallic materials containing different geometrical features. Eng. Fract. Mech. 75, 4410–4421 (2008)

    Article  Google Scholar 

  47. Susmel L., Taylor D.: The theory of Critical Distances to estimate lifetime of notched components subjected to variable amplitude uniaxial fatigue loading. Int. J. Fatigue 33, 900–911 (2011)

    Article  Google Scholar 

  48. Yosibash Z., Priel E., Leguillon D.: A failure criterion for brittle elastic materials under mixed-mode loading. Int. J. Fract. 141, 291–312 (2006)

    Article  MATH  Google Scholar 

  49. Chen D., Ozaki S.: Investigation of failure criteria for a sharp notch. Int. J. Fract. 152, 63–74 (2008)

    Article  MATH  Google Scholar 

  50. Priel E., Yosibash Z., Leguillon D.: Failure initiation at a blunt V-notch tip under mixed mode loading. Int. J. Fract. 149, 143–173 (2008)

    Article  MATH  Google Scholar 

  51. Tovo R., Livieri P., Benvenuti E.: An implicit gradient type of static failure criterion for mixed-mode loading. Int. J. Fract. 141, 497–511 (2006)

    Article  MATH  Google Scholar 

  52. Gomez F., Elices M., Berto F., Lazzarin P.: Local strain energy to assess the static failure of U-notches in plates under mixed mode loading. Int. J. Fract. 145, 29–45 (2007)

    Article  MATH  Google Scholar 

  53. Gomez F., Elices M., Berto F., Lazzarin P.: A generalized notch stress intensity factor for U-notched components loaded under mixed mode. Eng. Fract. Mech. 75, 4819–4833 (2008)

    Article  Google Scholar 

  54. Gomez F., Elices M., Berto F., Lazzarin P.: Fracture of V-notched specimens under mixed mode (I+II) loading in brittle materials. Int. J. Fract. 159, 121–135 (2009)

    Article  Google Scholar 

  55. Gomez F., Elices M., Berto F., Lazzarin P.: Fracture of U-notched specimens under mixed mode: experimental results and numerical predictions. Eng. Fract. Mech. 76, 236–249 (2009)

    Article  Google Scholar 

  56. Berto F., Ayatollahi M.R.: Fracture assessment of Brazilian disc specimens weakened by blunt V-notched under mixed mode loading by means of local energy. Mater. Design 32, 2858–2869 (2010)

    Article  Google Scholar 

  57. Lazzarin P., Berto F., Elices M., Gomez J.: Brittle failures from U- and V-notches in mode I and mixed, I+II, mode: a synthesis based on the strain energy density averaged on finite-size volumes. Fat. Fract. Eng. Mater. Struct. 32, 671–684 (2009)

    Article  Google Scholar 

  58. Zappalorto M., Lazzarin P.: In-plane and out-of-plane stress field solutions for V-notches with end holes. Int. J. Fract. 168, 167–180 (2011)

    Article  MATH  Google Scholar 

  59. Lazzarin P., Zappalorto M., Berto F.: Generalized stress intensity factors for rounded notches in plates under in-plane shear loading. Int. J. Fract. 170, 123–144 (2011)

    Article  MATH  Google Scholar 

  60. Berto F., Zappalorto M.: Fictitious notch rounding concept applied to V-notches with end holes under mode I loading. Int. J. Fract. 171, 91–98 (2011)

    Article  MATH  Google Scholar 

  61. Lazzarin P., Berto F., Ayatollahi M.R.: Brittle failure of inclined key-hole notches in isostatic graphite under in-plane mixed mode loading. Fat. Fract. Eng. Mater. Struct. 36(9), 942–955 (2013)

    Article  Google Scholar 

  62. Kullmer G., Richard H.A.: Influence of the root radius of crack-like notches on the fracture load of brittle components. Arch. Appl. Mech. 76, 711–723 (2006)

    Article  MATH  Google Scholar 

  63. Torabi, A.R., Abedinasab, S.M.: Fracture study on key-hole notches under tension: two brittle fracture criteria and notch fracture toughness measurement by the disk test. Exp. Mech. (2014). doi:10.1007/s11340-014-9949-0

  64. Torabi A.R., Pirhadi E.: Stress-based criteria for brittle fracture in key-hole notches under mixed mode loading. Euro. J. Mech A/Solids. 49, 1–12 (2015)

    Article  Google Scholar 

  65. Torabi, A.R., Berto, F., Campagnolo, A.: Experimental and theoretical investigation of brittle fracture in key-hole notches under mixed mode I/II loading. Acta Mech. 226, 2313–2322 (2015)

  66. Berto F., Lazzarin P., Ayatollahi M.R.: Brittle fracture of sharp and blunt V-notches in isostatic graphite under pure compression loading. Carbon 63, 101–116 (2013)

    Article  Google Scholar 

  67. Torabi A.R., Ayatollahi M.R.: Compressive brittle fracture in V-notches with end holes. Euro. J. Mech A/Solids. 45, 32–40 (2014)

    Article  MathSciNet  Google Scholar 

  68. Torabi, A.R., Amininejad, SH.: Brittle fracture in V-notches with end holes. Int. J. Damage Mech. (2014). doi:10.1177/1056789514538293

  69. Torabi, A.R., Amininejad, SH.: Fracture assessment of VO-notches under mode II loading: experiments and theories. Theor. Appl. Fract. Mech. (2004). doi:10.1016/j.tafmec.2014.10.010

  70. Torabi A.R., Berto F., Campagnolo A.: Local strain energy density to predict mode II brittle fracture in Brazilian disk specimens weakened by V-notches with end holes. Mater. Design 69, 22–29 (2015)

    Article  Google Scholar 

  71. Novozhilov V.V.: On necessary and sufficient criterion of brittle fracture. Prikladnaja Matematica i Mechanika 33, 212–222 (1969)

    Google Scholar 

  72. Berto F., Lazzarin P., Radaj D.: Fictitious notch rounding concept applied to sharp V-notches: Evaluation of the microstructural support factor for different failure hypotheses. Part I: basic stress equations. Eng. Fract. Mech. 75(10), 3060–3072 (2008)

    Article  Google Scholar 

  73. Berto F., Lazzarin P., Radaj D.: Fictitious notch rounding concept applied to sharp V-notches: Evaluation of the microstructural support factor for different failure hypotheses. Part II: microstructural support analysis. Eng. Fract. Mech. 76(9), 1151–1175 (2009)

    Article  Google Scholar 

  74. Berto F., Lazzarin P., Radaj D.: Fictitious notch rounding concept applied to V-notches with root holes subjected to in-plane shear loading. Eng. Fract. Mech. 79, 281–294 (2012)

    Article  Google Scholar 

  75. Radaj D., Lazzarin P., Berto F.: Generalised Neuber concept of fictitious notch rounding. Int. J. Fatigue 51, 105–115 (2013)

    Article  Google Scholar 

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Authors and Affiliations

  1. Fracture Research Laboratory, Faculty of New Science and Technologies, University of Tehran, P.O. Box 13741-4395, Tehran, Iran

    Rouzbeh Riazi, A. R. Torabi, SH. Amininejad & M. H. Sabour

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  1. Rouzbeh Riazi
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  2. A. R. Torabi
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  3. SH. Amininejad
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Riazi, R., Torabi, A.R., Amininejad, S. et al. Combined tension–shear fracture analysis of V-notches with end holes. Acta Mech 226, 3717–3736 (2015). https://doi.org/10.1007/s00707-015-1404-5

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  • DOI: https://doi.org/10.1007/s00707-015-1404-5

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