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.
Similar content being viewed by others
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
References
Atkinson C, Bastero JM, Martínez-Esnaola JM (1988) Stress analysis in sharp angular notches using auxiliary fields. Eng Fract Mech 31: 637–646
Atzori B, Lazzarin P (2001) Notch sensitivity and defect sensitivity: two sides of the same medal. Int J Fract 107(1): L3–L8
Atzori B, Lazzarin P, Meneghetti G (2003) Fracture mechanics and notch sensitivity. Fatigue Fract Eng Mater Struct 26: 257–267
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: 61–87
Ayatollahi MR, Aliha MRM (2009) Analysis of a new specimen for mixed mode fracture tests on brittle materials. Eng Fract Mech 76: 1563–1573
Bažant ZP, Planas J (1998) Fracture and size effect in concrete and other quasibrittle materials. CRC Press, Boca Raton and London
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: 415–433
Carpenter WC (1984) A collocation procedure for determining fracture mechanics parameters at a corner. Int J Fract 24: 255–266
Carpinteri A (1987) Stress singularity and generalised fracture toughness at the vertex of re-entrant corners. Eng Fract Mech 26: 143–155
Chen DH, Ozaki S (2008) Investigation of failure criteria for a sharp notch. Int J Fract 152: 63–74
Dini D, Hills D (2004) Asymptotic characterisation of nearly sharp notch root stress fields. Int J Fract 130: 651–666
Elices M, Guinea GV, Gómez FJ, Planas J (2002) The cohesive zone model: advantages, limitations and challenges. Eng Fract Mech 69: 137–163
Erdogan F, Sih GC (1963) On the crack extension in plates under plane loading and transverse shear. J Basic Eng Trans ASME 85d: 519–525
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: 4543–4565
Gogotsi GA (2003) Fracture toughness of ceramics and ceramic composites. Ceram Int 7: 777–884
Gómez FJ, Elices M (2003) Fracture of components with V-shaped notches. Eng Fract Mech 70: 1913–1927
Gómez FJ, Elices M (2003) A fracture criterion for sharp V-notched samples. Int J Fract 123: 163–175
Gómez FJ, Elices M (2004) A fracture criterion for blunted V-notched samples. Int J Fract 127: 239–264
Gómez FJ, Elices M, Valiente A (2000) Cracking in PMMA containing U-shaped notches. Fatigue Fract Eng Mat Struct 23: 795–803
Gómez FJ, Elices M, Planas J (2005) The cohesive crack concept: application to PMMA at −60° C. Eng Fract Mech 72: 1268–1285
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: 29–45
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: 4819–4833
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: 236–249
Gross R, Mendelson A (1972) Plane elastostatic analysis of V-notched plates. Int J Fract Mech 8: 267–276
Knésl Z (1991) A criterion of V-notch stability. Int J Fract 48: R79–R83
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: 711–723
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: 161–185
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–L38
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: 95–107
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: 3–19
Leguillon D, Yosibash Z (2003) Crack onset at a V-notch influence of the notch tip radius. Int J Fract 122: 1–21
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: 275–298
Leguillon D, Quesada D, Putot C, Martin E (2007) Prediction of crack initiation at blunt notches and cavities—size effects. Eng Fract Mech 74: 2420–2436
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: 247–278
Neuber H (1958) Theory of notch stresses. Springer, Berlin
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: 325–335
Papadopoulos GA, Paniridis PI (1988) Crack initiation from blunt notches under biaxial loading. Eng Fract Mech 31(1): 65–78
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
Planas J, Elices M (1992) Asymptotic analysis of a cohesive crack: 1. Theoretical background. Int J Fract 55: 153–177
Planas J, Elices M (1993) Asymptotic analysis of a cohesive crack: 2. Influence of the softening curve. Int J Fract 64: 221–237
Planas J, Sancho JM (2007) Computational orientated finite elements. COFE. Internal report. JP0501. Departamento de Ciencia de los Materiales. Universidad Politécnica de Madrid
Priel E, Bussiba A, Gilad I, Yosibash Z (2007) Mixed mode failure criteria for brittle elastic V-notched structures. Int J Fract 144: 247–265
Priel E, Yosibash Z, Leguillon D (2008) Failure initiation of a blunt V-notch tip Ander mixed mode loading. Int J Fract 149: 143–173
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: 75–86
Schleicher F (1926) Der Spannungszustand an der Fliessgrenze (Plastizitätsbedingung). Zeitschrift für angewandte Mathematik und Mechanik 6(3): 199–216
Seweryn A (1994) Brittle fracture criterion for structures with sharp notches. Eng Fract Mech 47: 673–681
Seweryn A, Lucaszewicz A (2002) Verification of brittle fracture criteria for elements with V-shaped notches. Eng Fract Mech 69: 1487–1510
Seweryn A, Mróz Z (1995) A non-local stress failure condition for structural elements under multiaxial loading. Eng Fract Mech 51: 955–973
Strandberg M (2002) Fracture at V-notches with container plasticity. Eng Fract Mech 69: 403–415
Taylor D (2004) Predicting the fracture strength of ceramic materials using the theory of critical distances. Eng Fract Mech 71: 2407–2416
Williams ML (1952) Stress singularities resulting from various boundary conditions in angular corners of plates in extension. J Appl Mech 19: 526–528
Yosibash Z, Bussiba Ar, Gilad I (2004) Failure criteria for brittle elastic materials. Int J Fract 125: 307–333
Yosibash Z, Priel E, Leguillon D (2006) A failure criterion for brittle elastic materials under mixed-mode loading. Int J Fract 141: 291–312
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10704-009-9387-7