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


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.


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

List of symbols


crack depth for cracked specimens and notch depth for notched ones


thickness of the specimen


loading position


cohesive zone model


Young modulus


generalised Young modulus

f( )

dimensionless function of the V-notch angle


cohesive strength


Lazzarin–Tovo mode I universal tensor function


Williams mode I universal tensor function


cohesive fracture energy


Lazzarin–Tovo mode II universal tensor function


Williams mode II universal tensor function


material toughness


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


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


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


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


notch stress intensity under mixed mode

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

critical notch stress intensity under mode I


characteristic length


support span


normal vector to the integration boundary


rupture load


notch root radius


polar coordinate


length magnitude


SED critical length


strain energy density


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


size of the specimen


averaged strain energy density


critical strain energy


critical crack opening displacement



notch angle


material coefficient


mode I strain energy density over total SED


polar coordinate


initial fracture angle


mode I eigenvalue


mode II eigenvalue


stress tensor


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


maximum principal stress


principal stress at the notch tip


tensile strength


Poisson’s ratio


distance from the notch edge


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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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