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On the use of the extended finite element and incremental methods in brittle fracture assessment of key-hole notched polystyrene specimens under mixed mode I/II loading with negative mode I contributions

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Abstract

The aim of the present study is to assess the suitability of the extended finite element method (XFEM) combined with the cohesive zone model (CZM) and also the incremental method together with the maximum tangential stress (MTS) criterion in predicting the fracture load and crack trajectory of key-hole notched brittle components subjected to mixed mode I/II loading with negative mode I contributions. For this purpose, a total number of 63 fracture test results, reported recently in the literature on the key-hole notched Brazilian disk (Key-BD) specimens made of the general-purpose polystyrene (GPPS) under mixed mode I/II loading with negative mode I contributions, are first collected. Then, the experimentally obtained fracture loads of the tested GPPS specimens are theoretically predicted by means of XFEM combined with CZM. Additionally, the crack trajectory in the tested Key-BD specimens is predicted by using both XFEM combined with CZM and the incremental method combined with MTS criterion. Finally, it is shown that both the fracture load and the crack trajectory could successfully be predicted by means of the two proposed methods for different notch geometries.

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Abbreviations

ASED:

Averaged strain energy density

ASED-EFC:

Averaged strain energy density based on the equivalent factor concept

\(b_{i}\) :

Gradient vector of the shape function associated with node i

CTSN:

Compact-tension-shear-notched

CZM:

Cohesive zone model

D :

Fourth-order elastic moduli tensor

E :

Young’s modulus

ES :

The element size applied to the notch border

\(f(\,)\) :

Softening function

\(G_\mathrm{f}\) :

Specific fracture energy

FIA:

Fracture initiation angle

FVSD:

Flattened V-notched semi-disk

GPPS:

General-purpose polystyrene

\(H(\,)\) :

Heaviside function

Key-MS:

Key-hole notch mean stress

Key-MTS:

Key-hole notch maximum tangential stress

\(K_{\mathrm{I}}\) :

Mode I stress intensity factor

\(K_{\mathrm{II}}\) :

Mode II stress intensity factor

\(K_{\mathrm{Ic}}\) :

Plane strain fracture toughness

Key-BD:

Key-hole notched Brazilian disk

LEFM:

Linear elastic fracture mechanics

LCC:

Load-carrying capacity

\(l_\mathrm{ch}\) :

Characteristic length

\(L_{1}\) :

Total slit length in the Key-BD specimen

\(L_{2}\) :

Diameter of the Key-BD specimen

MS:

Mean stress

MTS:

Maximum tangential stress

n :

Unitary vector normal to the maximum principal stress

NFM:

Notch fracture mechanics

\(N_{i}(\,)\) :

Shape function associated with node i

PS:

Point stress

RNL:

Relative notch length

SED:

Strain energy density

SIF:

Stress intensity factor

t :

Traction vector

T :

Cohesive traction

u():

Displacements field

\(u_{i}\) :

Nodal displacements of node i

VSC:

V-notched stepped cottage

w :

Crack opening vector

\({\tilde{w}} \) :

Equivalent crack opening

XFEM:

Extended finite element method

\(\beta \) :

Loading angle in the Key-BD specimen

\({\beta }_{\mathrm{II}}\) :

Loading angle corresponding to pure mode \(\mathrm{I}\mathrm{I}\) loading

\(\delta \) :

Virtual crack opening displacement

\(\nu \) :

Poisson’s ratio

\(\rho \) :

Notch tip radius

\({\sigma }_{\mathrm{u}} \) :

Ultimate tensile strength

\({\sigma }_{\vartheta \vartheta } \) :

Tangential stress

\(\vartheta _{{0,}{\mathrm{Exp.}}}\) :

Fracture initiation angle obtained from the experiment

\(\vartheta _{{0,}{\mathrm{Key-MTS}}}\) :

Fracture initiation angle obtained from the key-hole notch maximum tangential stress criterion

\(\vartheta _{{0,}{\mathrm{XFEM}}}\) :

Fracture initiation angle obtained from the extended finite element method

References

  1. Livieri, P.: A new path independent integral applied to notched components under mode I loadings. Int. J. Fract. 123, 107–25 (2003)

    Article  Google Scholar 

  2. Matvienko, Y.G., Morozov, E.M.: Calculation of the energy J-integral for bodies with notches and cracks. Int. J. Fract. 125, 249–61 (2004)

    Article  MATH  Google Scholar 

  3. 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, 4621–45 (2007)

    Article  MATH  Google Scholar 

  4. 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. Fatigue Fract. Eng. Mater. Struct. 35, 971–84 (2012)

    Article  Google Scholar 

  5. 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, 275–98 (2001)

    Article  Google Scholar 

  6. Lazzarin, P., Berto, F.: Some expressions for the strain energy in a finite volume surrounding the root of blunt V-notches. Int. J. Fract. 135, 161–85 (2005)

    Article  MATH  Google Scholar 

  7. Gomez, F.J., Elices, M., Berto, F., Lazzarin, P.: A generalised notch stress intensity factor for U-notched components loaded under mixed mode. Eng. Fract. Mech. 75, 4819–33 (2008)

    Article  Google Scholar 

  8. Torabi, A.R., Majidi, H.R., Ayatollahi, M.R.: Fracture prediction of key-hole notched graphite plates by using the strain energy density based on the equivalent factor concept. J. Struct. Fluid Mech. 7(2), 1–15 (2017)

    Google Scholar 

  9. Ayatollahi, M.R., Torabi, A.R.: Brittle fracture in rounded-tip V-shaped notches. Mater. Des. 31, 60–7 (2010)

    Article  Google Scholar 

  10. Ayatollahi, M.R., Torabi, A.R.: Investigation of mixed mode brittle fracture in rounded-tip V-notched components. Eng. Fract. Mech. 77, 3087–104 (2010)

    Article  Google Scholar 

  11. Ayatollahi, M.R., Torabi, A.R.: Tensile fracture in notched polycrystalline graphite specimens. Carbon 48(8), 2255–65 (2010)

    Article  Google Scholar 

  12. 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, 919–32 (2011)

    Article  Google Scholar 

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

    Article  Google Scholar 

  14. Gomez, F.J., Elices, M.: A fracture criterion for sharp V-notched samples. Int. J. Fract. 123, 163–75 (2003)

    Article  MATH  Google Scholar 

  15. Gomez, F.J., Guinea, G.V., Elices, M.: Failure criteria for linear elastic materials with U-notches. Int. J. Fract. 141, 99–113 (2006)

    Article  MATH  Google Scholar 

  16. Cendon, D., Torabi, A.R., Elices, M.: Fracture assessment of graphite V-notched and U-notched specimens by using the cohesive crack model. Fatigue Fract. Eng. Mater. Struct. 38, 563–73 (2015)

    Article  Google Scholar 

  17. Dugdale, D.S.: Yielding of steel sheets containing slits. J. Mech. Phys. Solids. 8, 100–4 (1960)

    Article  Google Scholar 

  18. Barenblatt, G.I.: The mathematical theory of equilibrium cracks in brittle fracture. Adv. Appl. Mech. 7, 55–129 (1962)

    Article  MathSciNet  Google Scholar 

  19. Hillerborg, A., Modeer, M., Petersson, P.E.: Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem. Concr. Res. 6, 773–81 (1976)

    Article  Google Scholar 

  20. Wells, G., Sluys, L.: A new method for modelling cohesive cracks using finite elements. Int. J. Numer. Methods Eng. 50, 2667–82 (2001)

    Article  MATH  Google Scholar 

  21. Moes, N., Belytschko, T.: Extended finite element method for cohesive crack growth. Eng. Fract. Mech. 69, 813–33 (2002)

    Article  Google Scholar 

  22. Mariani, S., Perego, U.: Extended finite element method for quasi-brittle fracture. Int. J. Numer. Methods Eng. 58, 103–26 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Meschke, G., Dumstorff, P.: Energy-based modeling of cohesive and cohesionless cracks via X-FEM. Comput. Methods Appl. Mech. Eng. 196, 2338–57 (2007)

    Article  MATH  Google Scholar 

  24. Cox, J.V.: An extended finite element method with analytical enrichment for cohesive crack modeling. Int. J. Numer. Methods Eng. 78, 48–83 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Giner, E., Sukumar, N., Tarancon, J., Fuenmayor, F.: An Abaqus implementation of the extended finite element method. Eng. Fract. Mech. 76, 347–68 (2009)

    Article  Google Scholar 

  26. Lewicki, D.G., Spievak, L.E., Wawrzynek, P.A., Ingraffea, A.R., Handschuh, R.F.: Consideration of moving tooth load in gear crack propagation predictions. J. Mech. Des, Trans ASME. Am. Soc. Mech. Eng. 123(1), 118–124 (2001)

    Google Scholar 

  27. Spievak, L.E., Wawrzynek, P.A., Ingraffea, A.R., Lewicki, D.G.: Simulating fatigue crack growth in spiral bevel gears. Eng. Fract. Mech. 68, 53–76 (2001)

    Article  Google Scholar 

  28. Ural, A., Wawrzynek, P.A., Ingraffea, A.R., Lewicki, D.G.: Simulating fatigue crack growth in spiral bevel gears using computational fracture mechanics. In: ASME Conference Proceedings 195-9 (2003)

  29. Seidenfuss, M., Samal, M., Roos, E.: On critical assessment of the use of local and nonlocal damage models for prediction of ductile crack growth and crack path in various loading and boundary conditions. Int. J. Solids Struct. 48, 3365–81 (2011)

    Article  Google Scholar 

  30. Areias, P., Rabczuk, T.: Finite strain fracture of plates and shells with configurational forces and edge rotations. Int. J. Numer. Methods Eng. 94, 1099–122 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Areias, P., Rabczuk, T., Dias-da-Costa, D.: Element-wise fracture algorithm based on rotation of edges. Eng. Fract. Mech. 110, 113–37 (2013)

    Article  MATH  Google Scholar 

  32. Areias, P., Rabczuk, T., Camanho, P.: Finite strain fracture of 2D problems with injected anisotropic softening elements. Theor. Appl. Fract. Mech. 72, 50–63 (2014)

    Article  Google Scholar 

  33. Areias, P., Msekh, M., Rabczuk, T.: Damage and fracture algorithm using the screened Poisson equation and local remeshing. Eng. Fract. Mech. 158, 116–43 (2016)

    Article  Google Scholar 

  34. Rabczuk, T., Belytschko, T.: Cracking particles: a simplified meshfree method for arbitrary evolving cracks. Int. J. Numer. Methods Eng. 61, 2316–43 (2004)

    Article  MATH  Google Scholar 

  35. Rabczuk, T., Zi, G., Bordas, S., Nguyen-Xuan, H.: A geometrically non-linear three-dimensional cohesive crack method for reinforced concrete structures. Eng. Fract. Mech. 75, 4740–58 (2008)

    Article  Google Scholar 

  36. Rabczuk, T., Zi, G., Gerstenberger, A., Wall, W.A.: A new crack tip element for the phantom-node method with arbitrary cohesive cracks. Int. J. Numer. Methods Eng. 75, 577–99 (2008)

    Article  MATH  Google Scholar 

  37. Rabczuk, T., Bordas, S., Zi, G.: On three-dimensional modelling of crack growth using partition of unity methods. Comput. Struct. 88, 1391–411 (2010)

    Article  Google Scholar 

  38. Rabczuk, T., Zi, G., Bordas, S., Nguyen-Xuan, H.: A simple and robust three-dimensional cracking-particle method without enrichment. Comput. Methods Appl. Mech. Eng. 199, 2437–55 (2010)

    Article  MATH  Google Scholar 

  39. Ren, H., Zhuang, X., Cai, Y., Rabczuk, T.: Dual-horizon peridynamics. Int. J. Numer. Methods Eng. 108, 1451–76 (2016)

    Article  MathSciNet  Google Scholar 

  40. Ren, H., Zhuang, X., Rabczuk, T.: Dual-horizon peridynamics: a stable solution to varying horizons. Comput. Methods Appl. Mech. Eng. 318, 762–82 (2017)

    Article  MathSciNet  Google Scholar 

  41. Ayatollahi, M.R., Torabi, A.R., Majidi, H.R.: Brittle fracture in key-hole notched polymer specimens under combined compressive-shear loading. Amirkabir J. Mech. Eng. (2017, Accepted Manuscript)

  42. Torabi, A.R., Majidi, H.R., Ayatollahi, M.R.: Brittle failure of key-hole notches under mixed mode I/II loading with negative mode I contributions. Eng. Fract. Mech. 168, 51–72 (2016)

    Article  Google Scholar 

  43. Lazzarin, P., Berto, F., Ayatollahi, M.R.: Brittle failure of inclined key-hole notches in isostatic graphite under in-plane mixed mode loading. Fatigue Fract. Eng. Mater. Struct. 36, 942–55 (2013)

    Article  Google Scholar 

  44. Torabi, A.R.: Closed-form expressions of mode I apparent notch fracture toughness for key-hole notches. J. Strain Anal. Eng. Des. 49, 583–91 (2014)

    Article  Google Scholar 

  45. 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. 55(2), 393–401 (2015)

  46. Torabi, A.R., Abedinasab, S.M.: Mode II notch fracture toughness measurement for key-hole notches by the disk test. J. Strain Anal. Eng. Des. 50, 264–75 (2015)

    Article  Google Scholar 

  47. Torabi, A.R., Abedinasab, S.M.: Brittle fracture in key-hole notches under mixed mode loading: experimental study and theoretical predictions. Eng. Fract. Mech. 134, 35–53 (2015)

    Article  Google Scholar 

  48. Torabi, A.R., Campagnolo, A., Berto, F.: Experimental and theoretical investigation of brittle fracture in key-hole notches under mixed mode I/II loading. Acta. Mech. 226(7), 2313–23 (2015)

    Article  Google Scholar 

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

    Article  Google Scholar 

  50. 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, 711–23 (2006)

    Article  MATH  Google Scholar 

  51. Berto, F., Lazzarin, P., Ayatollahi, M.R.: Brittle fracture of sharp and blunt V-notches in isostatic graphite under pure compression loading. Carbon 63, 101–16 (2013)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  53. Torabi, A.R., Firoozabadi, M., Ayatollahi, M.R.: Brittle fracture analysis of blunt V-notches under compression. Int. J. Solids Struct. 67, 219–230 (2015)

    Article  Google Scholar 

  54. Ayatollahi, M.R., Torabi, A.R., Firoozabadi, M.: Theoretical and experimental investigation of brittle fracture in V-notched PMMA specimens under compressive loading. Eng. Fract. Mech. 135, 187–205 (2015)

    Article  Google Scholar 

  55. Ayatollahi, M.R., Torabi, A.R., Rahimi, A.S.: Brittle fracture assessment of engineering components in the presence of notches: a review. Fatigue Fract. Eng. Mater. Struct. 39(3), 267–291 (2015)

    Article  Google Scholar 

  56. Belytschko, T., Black, T.: Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Methods Eng. 45, 601–20 (1999)

    Article  MATH  Google Scholar 

  57. Irwin, G.R.: Analysis of stresses and strains near the end of a crack traversing a plate. J. Appl. Mech. 24, 361–4 (1957)

    Google Scholar 

  58. Erdogan, F., Sih, G.: On the crack extension in plates under plane loading and transverse shear. J. Basic Eng. 85, 519–27 (1963)

    Article  Google Scholar 

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Authors and Affiliations

  1. Fatigue and Fracture Research Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran, 16846, Iran

    H. R. Majidi & M. R. Ayatollahi

  2. Fracture Research Laboratory, Faculty of New Sciences and Technologies, University of Tehran, P.O. Box 14395-1561, Tehran, Iran

    A. R. Torabi

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  1. H. R. Majidi
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Correspondence to M. R. Ayatollahi.

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Majidi, H.R., Ayatollahi, M.R. & Torabi, A.R. On the use of the extended finite element and incremental methods in brittle fracture assessment of key-hole notched polystyrene specimens under mixed mode I/II loading with negative mode I contributions. Arch Appl Mech 88, 587–612 (2018). https://doi.org/10.1007/s00419-017-1329-7

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