# Generalised stress intensity factors for rounded notches in plates under in-plane shear loading

- 366 Downloads
- 32 Citations

## Abstract

The Notch Stress Intensity Factors (NSIFs) quantify the intensities of the asymptotic linear elastic stress distributions of sharp (zero radius) V-shaped notches. When the notch tip radius is different from zero, the singular sharp-notch field diverges from the rounded-notch solution in the close neighborhood of the notch tip. Nevertheless the NSIFs might continue to be parameters governing fracture if the notch root radius is small enough. Otherwise they can be seen simply as stress field parameters useful in quantifying the stress distributions ahead of the specific notch. Taking advantage of some analytical formulations which are able to describe stress distributions ahead of parabolic, hyperbolic and V-shaped notches with end holes, the paper discusses the form and the significance of the NSIFs with reference to in-plane shear loading, considering explicitly the role played by the notch opening angle and the notch tip radius. These parameters quantify the stress redistribution due to the root radius with respect to the sharp notch case to which they should naturally tend for decreasing values of the notch radius.

## Keywords

In-plane shear Mode II Blunt V-notches Notch stress intensity factors Elasticity## References

- Atzori B, Lazzarin P (2001) Notch sensitivity and defect sensitivity: two sides of the same medal. Int J Fract 107: L3–L8CrossRefGoogle Scholar
- Atzori B, Lazzarin P, Meneghetti G (2003) Fracture mechanics and notch sensitivity. Fatigue Fract Eng Mater Struct 26: 257–267CrossRefGoogle Scholar
- Atzori B, Lazzarin P, Tovo R (1999) Stress field parameters to predict the fatigue strength of notched components. J Strain Anal 34: 437–453CrossRefGoogle Scholar
- Atzori B, Lazzarin P, Tovo R (1999) From the local stress approach to fracture mechanics: a comprehensive evaluation of the fatigue strength of welded joints. Fatigue Fract Eng Mater Struct 22: 369–382CrossRefGoogle Scholar
- 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
- Ayatollahi MR, Dehghany M (2010) On T-stresses near V-notches. Int J Fract 165: 121–126CrossRefGoogle Scholar
- Ayatollahi MR, Torabi AR (2010) Investigation of mixed mode brittle failure in rounded-tip V-notches components. Eng Fract Mech 2010(77): 3087–3104CrossRefGoogle Scholar
- Ayatollahi MR, Torabi AR (2011) Failure assessment of notched polycrystalline graphite under tensile-shear loading. Mater Sci Eng. doi: 10.1016/j.msea.20121.04.066 (in press)
- Ayatollahi MR, Nejati M (2011) Determination of NSIFs and coefficients of higher order terms for sharp notches using finite element method. Int J Mech Sci. doi: 10.1016/j.ijmecsci.2010.12.005 (in press)
- Benthem JP (1987) Stresses in the region of rounded corners. Int J Solids Struct 23: 239–252CrossRefGoogle Scholar
- Berto F, Lazzarin P, Gomez 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
- Berto F, Lazzarin P, Harding S, Kotousov A (2011) Out-of-plane singular stress fields in V-notched plates and welded lap joints induced by in-plane shear load conditions. Fatigue Fract Eng Mater Struct 34: 291–304CrossRefGoogle Scholar
- Boukharouba T, Tamine T, Nui L, Chehimi C, Pluvinage G (1995) The use of notch stress intensity factor as a fatigue crack initiation parameter. Eng Fract Mech 52: 503–512CrossRefGoogle Scholar
- Carpinteri A, Cornetti P, Pugno N, Sapora A (2010) On the most dangerous V-notch. Int J Solids Struct 47: 887–893CrossRefGoogle Scholar
- Chen DH (1995) Stress intensity factors for V-notched strip under tension or in-plane bending. Int J Fract 70: 81–97CrossRefGoogle Scholar
- Cheng YZ (1988) Evaluation of K
_{2}values from the solution of notch problem. Int J Fract 38: R61–64Google Scholar - Creager M, Paris PC (1967) Elastic field equations for blunt cracks with reference to stress corrosion cracking. Int J Fract Mech 3: 247–252Google Scholar
- Dini D, Hills D (2004) Asymptotic characterisation of nearly-sharp notch root stress fields. Int J Fract 130: 651–666CrossRefGoogle Scholar
- Dunn ML, Suwito W, Cunningham S (1997a) Fracture initiation at sharp notches: correlation using critical stress intensities. Int J Solids Struct 34: 3873–3883CrossRefGoogle Scholar
- Dunn ML, Suwito W, Cunningham S, May CW (1997b) Fracture initiation at sharp notches under mode I, mode II, and mild mixed mode loading. Int J Fract 84: 367–381CrossRefGoogle Scholar
- 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: 4543–4565CrossRefGoogle Scholar
- Glinka G (1985) Calculation of inelastic notch-tip strain-stress histories under cyclic loading. Eng Fract Mech 22: 839–854CrossRefGoogle Scholar
- Gogotsi GA (2003) Fracture toughness of ceramics and ceramic composites. Ceram Int 7: 777–884CrossRefGoogle Scholar
- Gómez FJ, Elices M (2003) A fracture criterion for sharp V-notched samples. Int J Fract 123: 163–175CrossRefGoogle Scholar
- Gómez FJ, Elices M (2004) A fracture criterion for blunted V-notched samples. Int J Fract 127: 239–264CrossRefGoogle Scholar
- Gómez FJ, Guinea GV, Elices M (2006) Failure criteria for linear elastic materials with U-notches. Int J Fract 141: 99–113CrossRefGoogle Scholar
- Gómez FJ, Elices M (2006) Fracture loads for ceramic samples with rounded notches. Eng Fract Mech 73: 880–894CrossRefGoogle Scholar
- Gómez FJ, Elices M, Berto F, Lazzarin P (2007) Local strain energy density to assess the static failure of U-shaped notches in plates under mixed mode loading. Int J Fract 145: 29–45CrossRefGoogle Scholar
- Gómez FJ, Elices M, Berto F, Lazzarin P (2008) A generalised notch stress intensity factor for U-notched components under mixed mode. Eng Fract Mech 75: 4819–4833CrossRefGoogle Scholar
- Gómez FJ, Elices M, Berto F, Lazzarin P (2009) Fracture of V-notched specimens under mixed mode (I+II) loading in brittle materials. Int J Fract 159: 121–135CrossRefGoogle Scholar
- Gross R, Mendelson A (1972) Plane elastostatic analysis of V-notched plates. Int J Fract Mech 8: 267–276CrossRefGoogle Scholar
- Harding S, Kotousov A, Lazzarin P, Berto F (2010) Transverse singular effects in V-shaped notches stressed in mode II. Int J Fract 164: 1–14CrossRefGoogle Scholar
- Irwin GR (1958). Fracture. In: Handbuch der Physik 6. Springer, Berlin, pp 551–590Google Scholar
- Knésl Z (1991) A criterion of V-notch stability. Int J Fract 48: R79–R83CrossRefGoogle Scholar
- Kotousov A, Lew TL (2006) Stress singularities resulting from various boundary conditions in angular corners of plates of arbitrary thickness in extension. Int J Solids Struct 43: 5100–5109CrossRefGoogle Scholar
- Kotousov A (2007) Fracture in plates of finite thickness. Int J Solids Struct 44: 8259–8273CrossRefGoogle Scholar
- Kotousov A, Lazzarin P, Berto F, Harding S (2010) Effect of the thickness on elastic deformation and quasi-brittle fracture of plate components. Eng Fract Mech 77: 1665–1681CrossRefGoogle Scholar
- Lazzarin P, Tovo R (1996) A unified approach to the evaluation of linear elastic stress fields in the neighborhood of cracks and notches. Int J Fract 78: 3–19CrossRefGoogle Scholar
- Lazzarin P, Tovo R (1998) A notch intensity approach to the stress analysis of Welds. Fatigue Fract Eng Mater Struct 21: 1089–1104CrossRefGoogle Scholar
- Lazzarin P, Filippi S (2006) A generalised stress intensity factor to be applied to rounded V-shaped notches. Int J Solids Struct 43: 2461–2478CrossRefGoogle Scholar
- Lazzarin P, Zappalorto M, Yates JR (2007) Analytical study of stress distributions due to semi-elliptic notches in shafts under torsion loading. Int J Eng Sci 45: 308–328CrossRefGoogle Scholar
- Leguillon D, Yosibash Z (2003) Crack onset at a V-notch. Influence of the notch tip radius. Int J Fract 122: 1–21CrossRefGoogle Scholar
- 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
- Muskhelishvili NI (1977) Some basic problems of the mathematical theory of elasticity, 4th edn. Noordhoof International, LeydenGoogle Scholar
- Nakamura T, Parks DM (1989) Antisymmetrical 3-D stress field near the crack front of a thin elastic plate. Int J Solids Struct 25: 1411–1426CrossRefGoogle Scholar
- Neuber H (1958) Kerbspannungslehre 2nd edn. Springer-Verlag, BerlinGoogle Scholar
- Noda N, Takase Y (2003) Generalized stress intensity factors for V-shaped notch in a round bar under torsion, tension and bending. Eng Fract Mech 70: 1447–1466CrossRefGoogle Scholar
- 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
- Pook LP (2000) Finite element analysis of corner point displacements and stress intensity factors for narrow notches in square sheets and plates. Fatigue Fract Eng Mater Struct 23: 979–992CrossRefGoogle Scholar
- 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
- Radaj D, Zhang S (1993) On the relations between notch stress and crack stress intensity in plane shear and mixed mode loading. Eng Fract Mech 44: 691–704CrossRefGoogle Scholar
- Savruk MP, Kazberuk A (2006) A Relationship between the stress intensity and stress concentration factors for sharp and rounded notches. Mater Sci 42: 725–738CrossRefGoogle Scholar
- Savruk MP, Kazberuk A (2010) A two-dimensional fracture mechanics problems for solids with sharp and rounded V-notches. Int J Fract 161: 79–95CrossRefGoogle Scholar
- Seweryn A, Molski K (1996) Elastic stress singularities and corresponding generalized stress intensity factors for angular corners under various boundary condition. Eng Fract Mech 55: 529–556CrossRefGoogle Scholar
- Sih GC, Liebowitz H (1968) Mathematical theories of brittle fracture. In fracture, vol II. Mathematical fundamentals. Academic Press, London, UKGoogle Scholar
- Strandberg M (1999) A numerical study of the elastic stress field arising from sharp and blunt V-notches in a SENT-specimen. Int J Fract 100: 329–342CrossRefGoogle Scholar
- Strandberg M (2001) Upper bounds for the notch intensity factor for some geometries and their use in general interpolation formulae. Eng Fract Mech 68: 577–585CrossRefGoogle Scholar
- Verreman Y, Nie B (1996) Early development of fatigue cracking at manual fillet welds. Fatigue Fract Eng Mater Struct 19: 669–681CrossRefGoogle Scholar
- Westergaard HM (1939) Bearing pressures and cracks. J Appl Mech 6: A49–A53Google Scholar
- Williams ML (1952) Stress singularities resulting from various boundary conditions in angular corners of plate in extension. J Appl Mech 19: 526–528Google Scholar
- Zappalorto M, Lazzarin P, Yates JR (2008) Elastic stress distributions resulting from hyperbolic and parabolic notches in round shafts under torsion and uniform antiplane shear loadings. Int J Solids Struct 45: 4879–4901CrossRefGoogle Scholar
- Zappalorto M, Lazzarin P, Berto F (2009) Elastic notch stress intensity factors for sharply V-notched rounded bars under torsion. Eng Fract Mech 76: 439–453CrossRefGoogle Scholar
- Zappalorto M, Lazzarin P, Filippi S (2010) Stress field equations for U and blunt V-shaped notches in axisymmetric shafts under torsion. Int J Fract 164: 253–269CrossRefGoogle Scholar
- Zappalorto M, Lazzarin P (2011) In-plane and out-of-plane stress field solutions for V-notches with end holes. Int J Fract 168: 167–180CrossRefGoogle Scholar
- Zhao Z, Hahn HG (1992) Determining the SIF of a V-notch from the results of a mixed-mode crack. Eng Fract Mech 43: 511–518CrossRefGoogle Scholar