PSI - Issue 13

Ondřej Krepl et al. / Procedia Structural Integrity 13 (2018) 1279 – 1284 Ond ř ej Krepl & Jan Klusák / Structural Integrity Procedia 00 (2018) 000–000

1282

4

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

       

where

depending on Poisson’s ratio ν m of the material m .

k

m

m m

Finally based on the SEDF approach, we determine the generalized fracture toughness:

k

2

m

(4)

1c, H K  m

m n n

Ic,

k l        k l

1

d

1 k l k    



( ) 

k l  

U

klm

1 1

Note that all the critical values H 1 c, 1 , H 1 c, 2 and H 1c , interface should be evaluated for calculated corresponding angles of crack initiation  0,1 ,  0,2 and  0,interface respectively. Once the critical fracture toughness values are known, in order to assess stability, the generalized stability condition as stated in equation (2) is used. 4. Numerical example Three point bending specimen with sharp material inclusion is modeled with geometry as shown in the Fig. 2. The geometric parameters are L = 76.2 mm, h = 17.8 mm and thickness b = 12.7 mm. The problem is solved in 2D by the finite element analysis (FEA), plane strain state is assumed. The inclusion opening angle is 2α = 90° and the inclusion depth to the specimen height ratio is a/h = 0.2. The specimen is loaded with F = 100 N. The material part 1 (inclusion) consists of PMMA and the material part 2 (matrix) of aluminum with material properties given in the Table 1. In the same table fracture tougnesses of inclusion, matrix and the interface are listed. The fracture parameters of the interface are taken from experimental analysis for bi-material notch of PMMA/W10/Aluminum in Krishnan and Xu (2011). The problem is examined by the use of multi-parameter SEDF criterion with specific distance chosen to d = 1 mm. The global minimum occurs in PMMA as shown in Fig. 3, whereas solution by two singular terms and solution by two singular and two non-singular terms gives identical critical angle of  0,1 = 0°. Local minimum of SEDF occurs in aluminum and again both approaches give identical critical angle  0,2 = 179.9°. Both of the critical angles correspond to the symmetry of the problem. Nevertheless, there is a difference in the values of resulting generalized fracture toughnesses H 1C,m , see the right side of the Table 1. This is in accordance to the apparent differences in m  as shown in Fig. 3 which is calculated by either singular (yellow makers) or singular and non-singular terms (cyan markers). Please note that the eigenfunction f rrkm (  ) which is an input for the augmented function U klm (  ) is discontinuous at the interface. To calculate H 1c , interface the radial function of PMMA material region is used. The lowest critical value of generalized fracture toughness is found in the interface  0,interface = 45° or  0,interface = -45° therefore it is the expected crack initiation direction.

Fig. 2. Three point bending specimen model with sharp material inclusion.