PSI - Issue 25

Marco Maurizi et al. / Procedia Structural Integrity 25 (2020) 268–281 M. Maurizi and F. Berto / Structural Integrity Procedia 00 (2019) 000–000

273

6

where

√ 2 π τ z θ ( r , 0) r

1 − λ 3 a

K N

III = lim r → 0

(13)

is the notch (generalized) stress intensity factor (NSIF). Based on the same definition, the Mode I and II NSIFs can be expressed as follows:

√ 2 π σ θθ ( r , 0) r √ 2 π τ r θ ( r , 0) r

K N

1 − λ 1

I = lim r → 0 II = lim r → 0

(14)

K N

1 − λ 2 ,

(15)

from which the in-plane stress fields around the notch tip, exploiting of the superposition principle (valid in LEFM), have the following representation in polar coordinates, as proposed by Gross and Mendelson (1972):

K N

K N II √

I √

r λ 1 − 1 ψ

r λ 2 − 1 η

i j ( θ, λ 1 , γ ) +

i j ( θ, λ 2 , γ ) ,

(16)

σ i j =

2 π

2 π

where ψ i j and η i j are functions not reported for the sake of brevity (for details Lazzarin and Tovo (1998)). The proposed analytical framework allows to determine the 3D singular stress field near the crack / notch tip as a function of the NSIFs, whenever the Kane and Mindlin’s hypothesis on the displacements is satisfied. Besides, it is worth to be noticed that τ rz and τ θ z are the only shear stress components to have a dependence (linear) on z, fundamental to predict the behavior of coupled modes along the thickness. Obviously, the crack case is recovered for γ = π .

3. General Considerations on 3D E ff ects

Several numerical, analytical and experimental studies (as reviewed by He et al. (2016)) have been conducted to understand the behavior of the 3D singular stress field in the proximity of a crack / notch tip surface. One of the basic results is the radius of influence of 3D e ff ects around the crack tip: they tend to disappear at a radial distance approximately equal to half of the plate thickness (particularly evident in Harding et al. (2010)), converging to a 2D plane stress condition after a distance roughly equivalent to the plate thickness (Nakamura and Parks (1989),Berto et al. (2011c),Berto et al. (2012),He et al. (2016)). It means that for bodies with in-plane dimensions comparable with the thickness (no plates), the 3D e ff ects spread almost all over the body and an asymptotic plane stress field is not possible; by definition of plates, the in-plane dimensions are greater than the thickness, i.e. plane stress conditions can be reached far away from the crack / notch front, giving the possibility to study cracks and notches locally. Indeed, in finite element (FE) simulations the 3D stress state zone is often considered encapsulated by the K-dominance zone (He et al. (2016)), implying that coupled fracture modes must be related to the far-field applied modes. SIFs (or NSIFs) and the strength of singularity (1 − λ ) are the main parameters considered to characterize the 3D singular stress field. The previous described 3D analytical frames predict solutions which di ff er from those of the classical 2D plane elasticity for the presence of along-the-thickness stress and displacement components (along the z-axis), dependent on the z coordinate. In terms of fracture parameters, they provide that only K N III ( z ) is a function of z , i.e. it changes along the plate thickness, whereas K N I and K N II are supposed to be independent of z. However, numerical results of cracked / notched specimens under Mode I and II, such as that of She and Guo (2007) for through-