PSI - Issue 3

F. Berto et al. / Procedia Structural Integrity 3 (2017) 135–143 F. Berto et al. / Structural Integrity Procedia 00 (2017) 000–000

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Lazzarin, 2014). According to this volume-based criterion, the failure occurs when the mean value of the strain energy density W over a control volume with a well-defined radius R 0 is equal to a critical value W C , which does not depend on the notch sharpness. The critical value and the radius of the control volume (which becomes an area in bi-dimensional problems) are dependent on the material (Berto and Lazzarin, 2014).

Table 1. Fatigue results from uncoated and coated (HDG) welded specimens. UNCOATED SPECIMENS

Δ

Δ

COATED SPECIMENS

W  [N.mm/mm 3 ]

W  [N.mm/mm 3 ]

N [cycles] 494000 1079000 4800000 85000 436500 978200 96820 905500 1125546 3800000 1500000 4500000 4000000 101200 195000 250000 1940000 42000 115000

N [cycles] 168750 81500 181484 445750 572333 5000000 803000 523983 804960 556990 645140 45000 5000000 173000 205616

σ [MPa]

σ [MPa]

260 320 260 220 180 140 160 160 140 140 160 320 120 220 220

0.5692 0.8622 0.5692 0.4075 0.2728 0.1650 0.2155 0.2155 0.1650 0.1650 0.2155 0.8622 0.1212 0.4075 0.4075

140 120 100 260 140 120 220 120 110 100 110 110 110 260 170 170 110 320 220

0.1650 0.1212 0.0842 0.5692 0.1650 0.1212 0.4075 0.1212 0.1019 0.0842 0.1019 0.1019 0.1019 0.5692 0.2433 0.2433 0.1019 0.8622 0.4075

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The SED approach was formalized and applied first to sharp, zero radius, V-notches (Lazzarin and Zambardi, 2001), considering bi-dimensional problems (plane stress or plane strain hypothesis). The volume over which the strain energy density is averaged is then a circular area Ω of radius R 0 centred at the notch tip, symmetric with respect to the notch bisector (Fig. 4), and the stress distributions are those by Williams (Williams, 1952), written according to Lazzarin and Tovo formulation (Lazzarin and Tovo, 1998). Dealing with sharp V-notches the strain energy density averaged over the area Ω turns out to be:

2

2

e K e K   

  

(1)

W





1

1

2

2



 



1

1



1 



2 

E R 

E R 

0

0

Where E is the Young’s modulus of the material, λ 1 and λ 2 are Williams’ eigenvalues (Williams, 1952), e 1 and e 2 are two parameters dependent on the notch opening angle 2 α and on the hypothesis of plane strain or plane stress considered. Those parameters are listed in Table 1 as a function of the notch opening angle 2 α , for a value of the Poisson’s ratio ν = 0.3 and plane strain hypothesis. K 1 and K 2 are the Notch Stress Intensity Factors (NSIFs) according to Gross and Mendelson (Gross and Mendelson, 1972):









1



1 

2 lim 

 ,   r  ,   r  r

0

K

r





 

 

1

0

r



(2)







1



2 

2 lim 

0

K

r



 



 

2

0

r



The SED approach was then extended to blunt U- and V-notches (Lazzarin et al., 2009; Lazzarin and Berto, 2005), by means of the expressions obtained by Filippi et al. (Filippi et al., 2002) for the stress fields ahead of blunt notches, and to the case of multiaxial loading (Lazzarin et al., 2008), by adding the contribution of mode III.