Abstract
In the present work the J-integral (indicated here as JVρ because two parallel flanks are not present) was calculated by using, along the free border, the exact analytical stress distribution for the ellipse and the asymptotic one for parabolic notches. The material was assumed as homogeneous isotropic and linear elastic. First, for an ellipse under remote tensile loading, the expression of JVρ has been analytically calculated on the basis of Inglis’ equations. The equations have been used to prove that, in terms of J-integral, the crack is the limit case of an equivalent elliptic notch. Furthermore, by distinguishing the symmetric and skew-symmetric terms, the well-known Stress Intensity Factors (SIF) of mode I and II for a crack in a wide plate under tension are obtained by adding a limiting condition. Second, by means of Creager–Paris’ equations, JVρ has been analytically calculated for a parabolic notch of assigned tip notch radius ρ. The asymptotic value of JVρ and the relationship between the peak stress and the relative SIF are the same as the ellipse. Finally, as an engineering application, we provide an accurate formula for the evaluation of the Notch Stress Intensity Factors of a crack, mainly subjected to tensile stress, from the peak stress of the equivalent ellipse under the same loading.
Similar content being viewed by others
References
Berto F and Lazzarin P (2007). Relationships between J-integral and the strain energy evaluated in a finite volume surrounding the tip of sharp and blunt V-notches. Int J Solids Struct. 44: 4621–4645
Chen YZ (1988). Evaluation of K2 value from the solution of notch problem. Int J Fract 38: R61–R64
Chiang CR (1990). Evaluation of stress intensity factors from solution of the corresponding notch problems. Int J Fract 42: R61–R63
Creager M and Paris P (1967). Elastic field equations for blunt cracks with reference to stress corrosion cracking. Int J Fract 3: 247–252
Filippi S, Lazzarin P and Tovo R (2002). Developments of some explicit formulas useful to describe elastic stress fields ahead of notches in plates. Int J Solids Struct 39(17): 4543–4565
Hasebe N and Kutanda Y (1978). Calculation of stress intensity factors from stress concentration factor. Eng Fract Mech 10: 215–221
Inglis CE (1913). Stresses in a plate due to the presence of cracks and sharp corners. Trans Inst Naval Architects 55: 219–230
Irwin GR (1958). Fracture in Hanbuck der physik, VI, Elasticity and plasticity. Springer-Verlag, Berlin, 551–590
Livieri P (2003). A new path independent integral applied to notched components under mode I loadings. Int J Fract 123: 107–125
Livieri P (2008). Use of J -integral to predict static failures in sharp V-notches and rounded U-notches. Eng Fract Mech. 75: 1779–1793
Matvienko YG and Morozov EM (2004). Calculation of the energy J-integral for bodies with notches and cracks. Int J Fract 125: 249–261
Murakami Y (1987) Stess intensity factors handbook 2. Pergsamon Press
Rice JR (1968). A path independent integral and the approximate analysis of strain concentration by notches and cracks. ASME-J Appl Mech 35: 379–386
Rigby RH and Aliabadi MH (1998). Decomposition of the mixed-mode J-integral-revisited. Int J Solids Struct 35(17): 2073–2099
Shin CS (1986). A discussion on various estimations of elastic stress distributions and stress concentration factors for sharp edge notches. Int J Fatigue 8: 235–237
Shin CS, Man KC and Wang CM (1994). A practical method to estimate the stress concentration of notches. Int J Fatigue 16: 242–256
Sih GC, Paris PC and Erdogan F (1962). Crack-tip, stress-intensity factors for plane extension and plate bending problems. J Appl Mech Trans ASME 29: 306–312
Sih, GG, Liebowitz H (1968) Mathematical theories of brittle fracture. In: Liebowitz H (ed) Fracture-An Advanced Treatise. Vol. II, Mathematical Fundamentals. Academic Press
Tada H, Paris CP and Irwin GR (2000). The stress analysis of cracks handbook, 3rd edn. ASME, New York
Timoshenko SP, Goodier JN (1970) Theory of elasticity, 3rd edn. McGraw-Hill
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Livieri, P., Segala, F. Analytical evaluation of J-integral for elliptical and parabolic notches under mode I and mode II loading. Int J Fract 148, 57–71 (2007). https://doi.org/10.1007/s10704-008-9178-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10704-008-9178-6