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

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4

equations, they obtained two equations dependent on the material’s properties, that is Lame’s constant λ and shear modulus G , on the shear factor k = 0 . 907, on the mean in-plane stress resultant N = ( N xx + N yy ) / 2, and on w .

N − w = 0 ,

∇ 2

h 2 3 k 2

λ + G 3 λ + 2 G ∇

1 2 kG

λ + 2 G 2 k λ G

λ 3 λ + 2 G

2 w − w =

N

(2)

where the stress resultants are simply the stress components integrated along the thickness. Exploiting of the Fourier Transform and the Wiener-Hopf technique near the crack tip, they derived the generalized plane strain stress resultants in the following form:

h K √ 2 π r

N i j =

g i j ( θ ) i , j = 1 , 2 ,

(3)

where K is the stress intensity factor in Mode I loading, and g i j ( θ ) are the classical angular functions of plane elastic crack problems. From Eq. (3), by means of N = ( N xx + N yy ) / 2, and from the second equation of Eq. (2), the harmonic equation ∇ 2 w = 0 for the out-of-plane displacement ahead the crack tip surface is obtained. Kotousov and Lew (2006) extended the previous approach to the analysis of the 3D stress field ahead of a V-notch, as suming a shear factor k = 1 and introducing an Airy’s stress function Φ , which automatically satysfies the equilibrium equations. For the detailed formulation see Kotousov and Lew (2006); Lazzarin and Zappalorto (2012). Applying the free-free boundary conditions along the edges of the V-notch, two eigenvalue problems have been deduced, symmetric and skew-symmetric, respectively. A recent new frame has been developed by Lazzarin and Zappalorto (2012), who converted any three-dimensional notch problem into a bi-harmonic and harmonic problem. The main assumption is again the Kane and Mindlin’s hy pothesis of generalized plane strain ( ε zz uniform along the thickness) on the displacement field of Eq. (1). It can be deduced that the normal strains ε ii (no repetition rule), and γ xy are not a function of z . Consequently, by means of the stress-strain relationships, also the correspondent stress components are independent on z . The only stress and strain components to be dependent on z are τ xz , τ yz , and γ xz , γ yz , respectively. The equilibrium in z-direction gives rise to the harmonic equation on w :

∇ 2 w = 0 .

(4)

As for the 2D problem, the Airy’s stress function Φ ( x , y ) must satisfies the bi-harmonic equation, as reported in detail in Lazzarin and Zappalorto (2012), that is:

∇ 4 Φ = 0 .

(5)

Hence, any 3D elastostatic notch problem, under the hypothesis of Eq. (1), can be entirely represented by Eq. (4) and (5), which must be satisfied simultaneously. Considering V-notch problems, Eq. (5) and (4) are satisfied by the functions Φ ( r , θ ) and w ( r , θ ), for in-plane (William’s solution Williams (1952)) and out-of-plane shear (Seweryn and Molski (1996)) loading, respectively: