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

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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.

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Abbreviations

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

f t :

cohesive strength

g :

Lazzarin–Tovo mode I universal tensor function

g * :

Williams mode I universal tensor function

G F :

cohesive fracture energy

h :

Lazzarin–Tovo mode II universal tensor function

h * :

Williams mode II universal tensor function

K IC :

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

K V,R :

notch stress intensity under mixed mode

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

critical notch stress intensity under mode I

l ch :

characteristic length

m :

support span

n j :

normal vector to the integration boundary

P :

rupture load

R :

notch root radius

r :

polar coordinate

r 0 :

length magnitude

R c :

SED critical length

SED:

strain energy density

u i :

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

W c :

critical strain energy

w c :

critical crack opening displacement

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

σ ij :

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

σ max :

maximum principal stress

σ tip :

principal stress at the notch tip

σ u :

tensile strength

ν :

Poisson’s ratio

ξ :

distance from the notch edge

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Authors and Affiliations

  1. Department of Materials Science, Universidad Politécnica de Madrid, E.T.S. Ingenieros de Caminos, 28040, Madrid, Spain

    F. J. Gómez & M. Elices

  2. Department of Management and Engineering, University of Padova, Stradella S. Nicola 3, 36100, Vicenza, Italy

    F. Berto & P. Lazzarin

Authors
  1. F. J. Gómez
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  2. M. Elices
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  3. F. Berto
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  4. P. Lazzarin
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Correspondence to M. Elices.

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Gómez, F.J., Elices, M., Berto, F. et al. Fracture of V-notched specimens under mixed mode (I + II) loading in brittle materials. Int J Fract 159, 121–135 (2009). https://doi.org/10.1007/s10704-009-9387-7

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  • DOI: https://doi.org/10.1007/s10704-009-9387-7

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