Effect of second non‐singular term of mode I near the tip of a V‐notched crack

Abstract

A B S T R A C TThe effect of a second non-singular term of mode I in the vicinity of the tip of a V-notched crack is discussed. The undetermined eigenvector coefficients of singular and non-singular terms of eigenfunction expansion were calculated by the numerical analysis method using the reciprocal work contour integral method (RWCIM) based on Betti's reciprocal work theorem and finite-element analysis. This study demonstrates that the second non-singular term of mode I can have a significant effect on the size and shape of plastic zones as an additional fracture mechanics parameter along with the conventional parameter K.Keywords RWCIM; T-stress; V-notched crack; V-stress. N O M E N C L A T U R Ea = half crack length or half V-notched crack length A R , A I = eigenvector coefficients related to mode I and mode II C = closed contour C i = inner contour C o = outer contour F I , F II = dimensionless stress intensity factors F V = dimensionless V-stress K n I , K n II = notch stress intensity factors for the mode I and mode II r, θ = polar coordinate of general point P T = T-stress V = V-stress α = half-wedge angle β = half-notch angle γ = bisector angle of notch μ, ν = shear modulus and Poisson's ratio ϕ, η = eigenvalues related to mode I and mode II σ o = tensile yield stress I N T R O D U C T I O NIt is well known that a linear elastic body with a sharp notch shape has stress singularities at the tip of the Vnotched crack. The crack propagation may commence from the tip of V-notched crack due to these singular stresses. Therefore, researchers aimed at establishing methods for evaluating the stress intensity factors representing the singular fields near the tip of the V-notched crack as a fracture mechanics parameter have made rapid progress.For a V-notched crack, Williams 1 presented an eigenvalue formulation based on the notch angle by using eigenfunction expansion. The stress intensity factors related to the orders of stress singularity were investigated by Gross and Mendelson 2 using a boundary collocation method and by Lin and Tong 3 who employed special hybrid elements. For a homogeneous material and two dissimilar isotropic materials with a notch shape, the reciprocal work contour integral method (RWCIM) used by Stern et al. 4 was employed by Carpenter 5-7 to calculate the undetermined eigenvector coefficients associated with each eigenvalue. Zhao and Hahn 8 and Chen 9 also obtained the stress intensity factors under mixed mode 346

Determination of NSIFs and coefficients of higher order terms for sharp notches using finite element method

“…They showed that when the control volume surrounding the pointed slit tip is large enough, the stress intensity factors together with the T-stress are not able to characterize efficiently the stress field in the control volume, and hence the higher order terms must be taken into account. Meanwhile, Kim and Cho [19] suggested that even for a non-zero notch angle (g a0) the size and shape of the plastic zone developing around the notch tip depends significantly on the second term of mode I in Eq. (1).…”

“…They showed that the least-squares method can be used to calculate NSIFs accurately if more than two stress terms are included. Kim and Cho [19] applied the reciprocal work contour integral method based on Betti's reciprocal work theorem and finite element analysis to calculate the coefficients of singular and non-singular terms of notch eigenfunction expansion. In order to calculate NSIFs from coarse meshes, Lazzarin et al [47] also suggested the use of the Strain Energy Density (SED) averaged over a control volume embracing the notch tip.…”

Application of artificial neural network for predicting crack growth direction in multiple cracks geometry

“…The crack growth direction for two edge cracks on one side with initial conditions a 1 /a 2 = 1, H/a ≥ 1 and S/a = 0 is found to be less than 10 • . According to MTS [4,19], SED [15][16][17][18], DET [36] criterion, the crack extension angle  0 remains less than 10 • for K I /K II >10. This indicates that crack grows mostly under the influence of mode I and the effect of mode II stress intensity factor is about 1/10th of mode I and hence the effect of mode II is negligible as compared to mode I.…”

“…The T-stress is considered in some studies as an auxiliary parameter for increasing the accuracy of the results for KI. Kim and Cho [15], for instance, showed that this non-singular term has noticeable effects on the size and shape of plastic zone near the notch tip. Ayatollahi and colleagues demonstrated that the first non-singular term have considerable contributions to the stress components around the notch tip and also on the fracture resistance of notched components under mode I loading [16][17][18].…”

On T-Stresses Near V-Notches

“…The only notable work is related to Kim et al (2009) who studied the effect of first non-singular term of mode I on the size and shape of plastic zone around a notch tip. Since cracks are special cases of sharp notches, one can expect that at least for very small notch opening angles, the second term of Williams' series expansions has significant effect on the fracture behavior of notched components.…”