PSI - Issue 26

Marco Maurizi et al. / Procedia Structural Integrity 26 (2020) 336–347

340

M. Maurizi and F. Berto / Structural Integrity Procedia 00 (2019) 000–000

5

where E is the Young’s modulus, λ 1 is the eigenvalue of the 2D stress singularity under mode 1, obtained from sin ( λ 1 2 γ ) + λ 1 sin (2 γ ) = 0 (Williams (1952)), χ 1 = − sin [(1 − λ 1 ) γ ] / sin [(1 + λ 1 ) γ ] (Lazzarin and Tovo (1998)), whereas f rr ( θ, λ 1 , χ 1 ) and f θθ ( θ, λ 1 , χ 1 ) are elementary trigonometric functions, which appear in the expressions for the stress components σ rr and σ θθ (Lazzarin and Tovo (1998)), respectively, reported for the sake of clarity in the Ap pendix (Eq. (A.1)-(A.2)). By imposing unitary plane stress NSIF ( K ∞ 1 = 1 MPa mm 1 − λ 1 ) in Eq. (1a)-(1b), numerical simulations (experiments) under mode 1 were performed varying the Poisson’s ratio in the range -0.3 to 0.45, assumed to be the direct cause of vertex singularities, for 30 ◦ and 60 ◦ of V-notch opening angle 2 α . Besides, a Young’s modulus of 200 GPa was arbitrarily adopted, without loss of generality. To provide some observations on the e ff ects of the 3D corner point singularity on the out-of-plane shear stress com ponent σ yz in the proximity of V-notches / cracks under anti-plane loading, a few numerical simulations were carried out by exploiting of the FE model (Fig. 3b), setting material constant and geometrical parameters to E = 200 GPa, ν = 0 . 3, and 2 α = 45 ◦ , respectively. By virtue of the anti-symmetry of the problem, the displacements u x and u y of the nodes at the mid-plane were fully constrained. Moreover, analogously to Eq. (1a)-(1b), the plane stress asymptotic anti-plane displacement field, reported in Eq. (2a)-(2c), was applied to the outer boundary of the cylindrical plate:

2 K ∞ 3 G √ 2 π

r λ 3 sin ( λ

3 θ )

(2a)

u z ( r , θ ) =

u r = 0 u θ = 0 ,

(2b) (2c)

where u z is the displacement component along z (coordinate system Fig. 2a), K ∞ 3 is the far-field mode 3 NSIF, imposed to be equal to 1 MPa mm 1 − λ 3 , while G is the shear modulus, which can be expressed due to the isotropy as G = E / 2(1 + ν ). The eigenvalue λ 3 for a sharp V-notch under anti-plane loading can be found as solution of the simple equation cos ( λ 3 γ ) = 0 (see Berto et al. (2013)). Besides, two orders of magnitude, i.e. 2 and 20 mm, for the plate thickness were tested to investigate possible e ff ects on the out-of-plane shear stress component distribution on the plate surface. Hypothesizing that the i j -th stress component, σ yy for mode 1 or σ yz for mode 3, around the notch front in a 3D problem can be expressed in cylindrical coordinates (Fig. 2a) as follows:

1 √ 2 π

K ( z , 2 α, ν ) r λ ( z , 2 α,ν ) − 1 f

σ i j ( r , θ, z ) =

i j ( θ, λ, 2 α ) ,

(3)

where the NSIF K ( z , 2 α, ν ) depends on the geometry, i.e. notch opening angle 2 α , on the type of loading and boundary conditions, and it is assumed it varies along the notch front ( z ) and with the Poisson’s ratio. The power-law exponent is hypothesized to change with z , 2 α and ν . The unknown function f i j performs the same role of the angular functions of Eq. (1a)-(1b). The singularity exponent, i.e. λ − 1, and the NSIF, i.e. K are hence su ffi cient to characterize the stress field, which is influenced near the corner point by the vertex singularity. The stress components, i.e. σ yy for mode 1 or σ yz for mode 3, were extracted along the bisector line from the notch tip up to a distance equal to R / 10, for di ff erent z -coordinates, i.e. planes z = constant. By using a log-log regression analysis, under the hypothesis of Eq. (3), λ − 1 was determined as the slope of the line log( σ i j ) versus log( r ), whereas the intensity factor by the following classic definition:

K ( z , ν ) = √ 2 π lim r → 0

1 − λ ,

σ i j r

(4)