PSI - Issue 13

Jan Klusák et al. / Procedia Structural Integrity 13 (2018) 1261–1266 Author name / Structural Integrity Procedia 00 (2018) 000–000

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3

material. Note that the definition of the stability of notches means determination of conditions under which a crack initiates from the notch tip. The classic fracture mechanics approach of comparison of the stress intensity factor K I with its critical value K Icrit is generalized to the following relation: (2) The stability condition can be understood in the following way. A crack is not initiated at the notch tip if the value H 1 is lower than its critical value H 1,crit . The value H 1 (σ appl ) follows from a numerical solution and depends mainly on the level of external loading and on the global geometry. Its critical value H 1,crit depends on the critical material characteristic K Ic or K Ith and has to be deduced with the help of the controlling variable L , see Knésl et al. (2007). The controlling variable L needs to have a clear and identical physical meaning in the case of assessing both a crack in homogeneous material and a sharp or bi-material notch. With respect to particularities of the notches the average value of the strain energy density factor Σ has been chosen as the controlling variable L . 3. The average strain energy density factor criterion In the linear elastic fracture mechanics of cracks, Sih’s strain energy density factor (SEDF) criterion can be used to predict crack propagation conditions (Sih (1973)). This approach can also be applied for the same purpose on other cases of stress concentrations (Sih (1991)). In the case of a crack the direction of the extreme (minimum) of the SEDF is independent of radial distance. General cases however show the radial distance dependence. Therefore, a mean value of the SEDF over a distance d , which is a distance related to fracture mechanism or material microstructure, is used. In Klusák and Knésl (2010) they applied mean value of the SEDF to assess the stability of bi-material notches. No instances of the SEDF criterion employment with consideration of not only singular but also higher non-singular terms are found in the literature to the best of authors’ knowledge. As known, some cases of V-notches and bi-material notches are characterized by rather weak singularities in comparison to crack problems (e.g. notches with opening angle greater than 120°). In such cases, the singular terms may describe stress field precisely only on distances smaller than distance d related to fracture mechanism or material microstructure. Because of that, the SEDF criteria without consideration of higher order terms may give either overconservative or underestimated failure load prediction. In order to mitigate such discrepancy, the SEDF can be calculated using n singular and non-singular terms. The definition of the SEDF is (Sih (1973)): t 1 1,c appl I,cr rit i ( , ...) ( , ...) K H H  

d

0 d r W r V        d

(3)

where the d W /d V represents strain energy density (strain energy per volume). When considering the brittle fracture the crack initiates when the SEDF reaches its critical value Σ c (4), which is material parameter determined in relation to the fracture toughness of given material K Ic, m . Further  m is shear modulus of material m and k m depends on Poisson’s ratio ν m . For crack problems:

1 2 for plane strain 1 for plane stress 1 m   

       

2 Ic,

k K

4 m m m 

, where

(4)

 

k

m

m

c,

m m

In the case of a bi-material notch, the SEDF will be determined for both material regions and the interface, thus for index m = 1, 2, interface. As discussed above, because of the dependence of the direction of SEDF extreme on radial distance, its mean value m  over specific distance d is determined as:

1

d

0 d     m

(5)

d m r

We consider the formula for calculation of the SEDF for plane problems: