PSI - Issue 2_A

J. Hein et al. / Procedia Structural Integrity 2 (2016) 2462–2254

2252

J. Hein, M. Kuna / Structural Integrity Procedia 00 (2016) 000–000

7

J increases mostly in surface-near regions until the maximum value max( J ) = 16 . 23 N / mm is reached at ϕ = 0 and t = t ∗ = 20 s. Afterwards, J decreases at the surface immediately and for ϕ > π/ 6 at a later time.

4

t [s]

f p

0

x

0 a

W

3

J

J

J

N mm

0

π/ 6

π/ 2

0

π/ 2

ϕ

ϕ

contours

(a) J -integral along crack front at di ff erent times.

(b) J max of 13 contours.

(c)

Path-independence

( ϕ = 0 → ϕ = π/ 2). Fig. 6: Quadratic FGM f p = 0 − 4 %: (a) Transient J -integral ; (b) J max ; (c) Path-independence of J -integral at time step of J max . By means of this example, the path-independence of the extended J -integral is demonstrated. This is evident from the coincident lines along crack front obtained for all contours of various radii r ∈ [0 . 0036 , 0 . 1852] a in Fig. 6b or from the horizontal lines in Fig. 6c. Hereby, J max is the curve J ( ϕ, t = t ∗ ) along the crack front at time t ∗ , at which the maximum value max( J ) occurs. The influence of material gradation expressed by concentration of pore forming agents is now studied in detail. Figures 7a and 7b show results for quadratic variations of f p between 0% and 4% together with reference values obtained for homogeneous materials with these concentrations (blue lines). In an analogous manner, the concentration f p is varied between 4 % and 12% in figures 7c and 7d.

— t ∗ = 22 s, max( J ) = 16 . 79 N / mm — t ∗ = 20 s, max( J ) = 16 . 23 N / mm - - t ∗ = 21 s, max( J ) = 15 . 60 N / mm - - t ∗ = 28 s, max( J ) = 11 . 26 N / mm - - t ∗ = 29 s, max( J ) = 11 . 60 N / mm — t ∗ = 30 s, max( J ) = 11 . 63 N / mm

- - t ∗ = 28 s, max( J ) = 11 . 26 N / mm — t ∗ = 28 s, max( J ) = 11 . 06 N / mm - - t ∗ = 28 s, max( J ) = 11 . 07 N / mm · · · t ∗ = 36 s, max( J ) = 10 . 48 N / mm - - t ∗ = 37 s, max( J ) = 10 . 80 N / mm — t ∗ = 36 s, max( J ) = 10 . 39 N / mm

4

12

J max N

mm

J max N

J max / J max ( f p = 4 %)

J max / J max ( f p = 4 %)

mm

f p

f p

0

4

0 a

W

0 a

W

x

x

3

3

π/ 2

π/ 2

π/ 2

π/ 2

ϕ

ϕ

ϕ

ϕ

(a) J max for gradations in f p = [0 , 4] %.

(b) J max / J max ( f p = 4 %).

(c) J max for gradations in f p = [4 , 12] %.

(d) J max / J max ( f p = 4 %).

Fig. 7: Comparing J max of gradations: (a) and (b) f p = [0 , 4] % ; (c) and (d) f p = [4 , 12] %.

The absolute values of J -integral along the crack front are depicted in figures 7a and 7c for that time t ∗ , where max( J ) is reached. One can see from Fig. 7a that the highest and lowest curves result from the smallest and largest