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Fracture of V-notched specimens under mixed mode (I + II) loading in brittle materials

  • F. J. Gómez
  • M. Elices
  • F. Berto
  • P. Lazzarin
Original Paper

Abstract

The purpose of this research is threefold. First, to provide experimental results of fracture loads for V-notched beams loaded under mixed mode. Second, to check the suitability of fracture criteria based on the cohesive zone model and strain energy density when applied to those samples. And, third, to suggest a very simple fracture criterion, based on the dominance of the local mode I, for notched samples (with different V-notch angles and notch root radii) loaded under mixed (I + II) mode. This proposal unifies predictions for the experimental results obtained under mode I and mixed mode loading. To this end, 36 fracture tests on V-notched beams were performed and reported: three V-notched angles were investigated (90°, 60°, 30°, four different loadings (mixed modes I and II) were selected and three samples were tested for each configuration.

Keywords

Notched components Mixed mode loading Rupture Cohesive crack Strain energy density Notch angle 

List of symbols

a

crack depth for cracked specimens and notch depth for notched ones

B

thickness of the specimen

b

loading position

CZM

cohesive zone model

E

Young modulus

E

generalised Young modulus

f( )

dimensionless function of the V-notch angle

ft

cohesive strength

g

Lazzarin–Tovo mode I universal tensor function

g*

Williams mode I universal tensor function

GF

cohesive fracture energy

h

Lazzarin–Tovo mode II universal tensor function

h*

Williams mode II universal tensor function

KIC

material toughness

\({K_{1}^{V}}\)

mode 1 notch stress intensity factor of a sharp V-notch

\({K_{2}^{V}}\)

mode 2 notch stress intensity factor of a sharp V-notch

\({K_{1}^{V,R}}\)

mode 1 notch stress intensity factor of a blunt V-notch

\({K_{2}^{V,R}}\)

mode 2 notch stress intensity factor of a blunt V-notch

KV,R

notch stress intensity under mixed mode

\({K_{\rm IC}^{V,R}}\)

critical notch stress intensity under mode I

lch

characteristic length

m

support span

nj

normal vector to the integration boundary

P

rupture load

R

notch root radius

r

polar coordinate

r0

length magnitude

Rc

SED critical length

SED

strain energy density

ui

i component of the displacement field

\({\hat{u}_i ^{\rm I}}\)

auxiliary displacement field in mode I

\({\hat{u}_i ^{\rm II}}\)

auxiliary displacement field in mode II

W

size of the specimen

\({\bar{W}}\)

averaged strain energy density

Wc

critical strain energy

wc

critical crack opening displacement

Greek

α

notch angle

β

material coefficient

χ

mode I strain energy density over total SED

θ

polar coordinate

φ

initial fracture angle

λI

mode I eigenvalue

λII

mode II eigenvalue

σ

stress tensor

σij

ij component of the stress tensor

\({\hat{\sigma}_{ij} ^{\rm I}}\)

stress tensor of the auxiliary field in mode I

\({\hat{\sigma}_{ij} ^{\rm II}}\)

stress tensor of the auxiliary field in mode I

σmax

maximum principal stress

σtip

principal stress at the notch tip

σu

tensile strength

ν

Poisson’s ratio

ξ

distance from the notch edge

References

  1. Atkinson C, Bastero JM, Martínez-Esnaola JM (1988) Stress analysis in sharp angular notches using auxiliary fields. Eng Fract Mech 31: 637–646CrossRefGoogle Scholar
  2. Atzori B, Lazzarin P (2001) Notch sensitivity and defect sensitivity: two sides of the same medal. Int J Fract 107(1): L3–L8CrossRefGoogle Scholar
  3. Atzori B, Lazzarin P, Meneghetti G (2003) Fracture mechanics and notch sensitivity. Fatigue Fract Eng Mater Struct 26: 257–267CrossRefGoogle Scholar
  4. Atzori B, Lazzarin P, Meneghetti G (2005) Unified treatment of fatigue limit of components weakened by notches and defects subjected to prevailing mode I stresses. Int J Fract 133: 61–87CrossRefGoogle Scholar
  5. Ayatollahi MR, Aliha MRM (2009) Analysis of a new specimen for mixed mode fracture tests on brittle materials. Eng Fract Mech 76: 1563–1573CrossRefGoogle Scholar
  6. Bažant ZP, Planas J (1998) Fracture and size effect in concrete and other quasibrittle materials. CRC Press, Boca Raton and LondonGoogle Scholar
  7. Berto F, Lazzarin P, Gómez FJ, Elices M (2007) Fracture assessment of U-notches under mixed mode loading: two procedures based on the equivalent local mode I concept. Int J Fract 148: 415–433CrossRefGoogle Scholar
  8. Carpenter WC (1984) A collocation procedure for determining fracture mechanics parameters at a corner. Int J Fract 24: 255–266CrossRefGoogle Scholar
  9. Carpinteri A (1987) Stress singularity and generalised fracture toughness at the vertex of re-entrant corners. Eng Fract Mech 26: 143–155CrossRefGoogle Scholar
  10. Chen DH, Ozaki S (2008) Investigation of failure criteria for a sharp notch. Int J Fract 152: 63–74CrossRefGoogle Scholar
  11. Dini D, Hills D (2004) Asymptotic characterisation of nearly sharp notch root stress fields. Int J Fract 130: 651–666CrossRefGoogle Scholar
  12. Elices M, Guinea GV, Gómez FJ, Planas J (2002) The cohesive zone model: advantages, limitations and challenges. Eng Fract Mech 69: 137–163CrossRefGoogle Scholar
  13. Erdogan F, Sih GC (1963) On the crack extension in plates under plane loading and transverse shear. J Basic Eng Trans ASME 85d: 519–525Google Scholar
  14. Filippi S, Lazzarin P, Tovo R (2002) Developments of some explicit formulas useful to describe elastic stress fields ahead of notches in plates. Int J Solids Struct 39: 4543–4565zbMATHCrossRefGoogle Scholar
  15. Gogotsi GA (2003) Fracture toughness of ceramics and ceramic composites. Ceram Int 7: 777–884CrossRefGoogle Scholar
  16. Gómez FJ, Elices M (2003) Fracture of components with V-shaped notches. Eng Fract Mech 70: 1913–1927CrossRefGoogle Scholar
  17. Gómez FJ, Elices M (2003) A fracture criterion for sharp V-notched samples. Int J Fract 123: 163–175CrossRefGoogle Scholar
  18. Gómez FJ, Elices M (2004) A fracture criterion for blunted V-notched samples. Int J Fract 127: 239–264CrossRefGoogle Scholar
  19. Gómez FJ, Elices M, Valiente A (2000) Cracking in PMMA containing U-shaped notches. Fatigue Fract Eng Mat Struct 23: 795–803CrossRefGoogle Scholar
  20. Gómez FJ, Elices M, Planas J (2005) The cohesive crack concept: application to PMMA at −60° C. Eng Fract Mech 72: 1268–1285CrossRefGoogle Scholar
  21. Gómez FJ, Elices M, Berto F, Lazzarin P (2007) Local strain energy to asses the static failure of U-notches in plates under mixed mode loading. Int J Fract 145: 29–45CrossRefGoogle Scholar
  22. Gómez FJ, Elices M, Berto F, Lazzarin P (2008) A generalizad notch stress intensity factor for U-notched components loaded under mixed mode. Eng Fract Mech 75: 4819–4833CrossRefGoogle Scholar
  23. Gómez FJ, Elices M, Berto F, Lazzarin P (2009) Fracture of U-notched specimens under mixed mode experimental results and numerical predictions. Eng Fract Mech 76: 236–249CrossRefGoogle Scholar
  24. Gross R, Mendelson A (1972) Plane elastostatic analysis of V-notched plates. Int J Fract Mech 8: 267–276CrossRefGoogle Scholar
  25. Knésl Z (1991) A criterion of V-notch stability. Int J Fract 48: R79–R83CrossRefGoogle Scholar
  26. Kullmer G, Richard HA (2006) Influence of the root radius of crack-like notches on the fracture load of brittle components. Arch Appl Mech 76: 711–723zbMATHCrossRefGoogle Scholar
  27. Lazzarin P, Berto F (2005) Some expressions for the strain energy in a finite volume surrounding the root of blunt V-notches. Int J Fract 135: 161–185CrossRefGoogle Scholar
  28. Lazzarin P, Berto F (2005) From Neuber’s elementary volume to Kitagawa and Atzori’s diagrams: an interpretation based on local energy. Int J Fract 135: L33–L38CrossRefGoogle Scholar
  29. Lazzarin P, Berto F (2008) Control volumes and strain energy density under small and large scale yielding due to tension and torsion loading. Fatigue Fract Eng Mater Struct 31: 95–107Google Scholar
  30. Lazzarin P, Tovo R (1996) A unified approach to the evaluation of linear elastic stress fields in the neighbourhood of cracks and notches. Int J Fract 78: 3–19CrossRefGoogle Scholar
  31. Leguillon D, Yosibash Z (2003) Crack onset at a V-notch influence of the notch tip radius. Int J Fract 122: 1–21CrossRefGoogle Scholar
  32. Lazzarin P, Zambardi R (2001) A finite-volume-energy based approach to predict the static and fatigue behaviour of components with sharp V-shaped notches. Int J Fract 112: 275–298CrossRefGoogle Scholar
  33. Leguillon D, Quesada D, Putot C, Martin E (2007) Prediction of crack initiation at blunt notches and cavities—size effects. Eng Fract Mech 74: 2420–2436CrossRefGoogle Scholar
  34. Livieri P, Lazzarin P (2005) Fatigue strength of steel and aluminium welded joints based on generalised stress intensity factors and local strain energy values. Int J Fract 133: 247–278CrossRefGoogle Scholar
  35. Neuber H (1958) Theory of notch stresses. Springer, BerlinGoogle Scholar
  36. Nui LS, Chehimi C, Pluvinage G (1994) 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: 325–335CrossRefGoogle Scholar
  37. Papadopoulos GA, Paniridis PI (1988) Crack initiation from blunt notches under biaxial loading. Eng Fract Mech 31(1): 65–78CrossRefGoogle Scholar
  38. Planas J (2009) A note on pseudo-cohesive behavior in quasi-bidimensional brittle fracture, engineering failure analysis. doi: 10.1016/j.engfailanal.2009.04.021
  39. Planas J, Elices M (1992) Asymptotic analysis of a cohesive crack: 1. Theoretical background. Int J Fract 55: 153–177CrossRefADSGoogle Scholar
  40. Planas J, Elices M (1993) Asymptotic analysis of a cohesive crack: 2. Influence of the softening curve. Int J Fract 64: 221–237CrossRefADSGoogle Scholar
  41. Planas J, Sancho JM (2007) Computational orientated finite elements. COFE. Internal report. JP0501. Departamento de Ciencia de los Materiales. Universidad Politécnica de MadridGoogle Scholar
  42. Priel E, Bussiba A, Gilad I, Yosibash Z (2007) Mixed mode failure criteria for brittle elastic V-notched structures. Int J Fract 144: 247–265CrossRefGoogle Scholar
  43. Priel E, Yosibash Z, Leguillon D (2008) Failure initiation of a blunt V-notch tip Ander mixed mode loading. Int J Fract 149: 143–173CrossRefGoogle Scholar
  44. Sancho JM, Planas J, Cendón DA, Reyes E, Gálvez JC (2007) An embedded cohesive crack model for finite element analysis of concrete fracture. Eng Fract Mech 74: 75–86CrossRefGoogle Scholar
  45. Schleicher F (1926) Der Spannungszustand an der Fliessgrenze (Plastizitätsbedingung). Zeitschrift für angewandte Mathematik und Mechanik 6(3): 199–216zbMATHGoogle Scholar
  46. Seweryn A (1994) Brittle fracture criterion for structures with sharp notches. Eng Fract Mech 47: 673–681CrossRefGoogle Scholar
  47. Seweryn A, Lucaszewicz A (2002) Verification of brittle fracture criteria for elements with V-shaped notches. Eng Fract Mech 69: 1487–1510CrossRefGoogle Scholar
  48. Seweryn A, Mróz Z (1995) A non-local stress failure condition for structural elements under multiaxial loading. Eng Fract Mech 51: 955–973CrossRefGoogle Scholar
  49. Strandberg M (2002) Fracture at V-notches with container plasticity. Eng Fract Mech 69: 403–415CrossRefGoogle Scholar
  50. Taylor D (2004) Predicting the fracture strength of ceramic materials using the theory of critical distances. Eng Fract Mech 71: 2407–2416CrossRefGoogle Scholar
  51. Williams ML (1952) Stress singularities resulting from various boundary conditions in angular corners of plates in extension. J Appl Mech 19: 526–528Google Scholar
  52. Yosibash Z, Bussiba Ar, Gilad I (2004) Failure criteria for brittle elastic materials. Int J Fract 125: 307–333CrossRefGoogle Scholar
  53. Yosibash Z, Priel E, Leguillon D (2006) A failure criterion for brittle elastic materials under mixed-mode loading. Int J Fract 141: 291–312CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • F. J. Gómez
    • 1
  • M. Elices
    • 1
  • F. Berto
    • 2
  • P. Lazzarin
    • 2
  1. 1.Department of Materials ScienceUniversidad Politécnica de MadridMadridSpain
  2. 2.Department of Management and EngineeringUniversity of PadovaVicenzaItaly

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