PSI - Issue 2_A

J. Hein et al. / Procedia Structural Integrity 2 (2016) 2462–2254 J. Hein, M. Kuna / Structural Integrity Procedia 00 (2016) 000–000

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4

whereby J represents an averaged J -value and the crack area is given by ∆ A = ∆ s ∆ l ( s )d s = ∆ a ∆ s l ( s )d s .

(10)

In the 3D case the domain around the crack front becomes a volume V , bounded by the closed surface S = S + S + + S − + S end − S C (Fig. 2).

crack front

S

x 2

C

S C

l(s)

S +

s

v m

Δ s

x 1

S -

Δ a

x 3

S end

crack surface

Fig. 2: Virtual displacement of the crack tip (compare Kuna (2013)).

For an e ffi cient FEM analysis, Shih et al. (1986) developed the so-called equivalent domain integral (EDI) for homo geneous materials. In case of FGM, the released energy equation (9) can now be generalized (see Kuna (2013) for details) as follows J ∆ A = lim r → 0 − S Q m j n j ∆ l m d S + S + S + + S − + S end Q m j n j ∆ l m d S . (11) Applying a smoothly varying function inside of V q m ( x ) = 0 on S , S end ∆ l m on S C , (12) the 3D domain J -integral at a certain crack front position s can be obtained in a way similar to the derivation of the domain integral in 2D case (compare Hein and Kuna (2014))

1 ∆ A

( − I 1 − I 2 + I 3 )

(13)

lim r → 0

J = J ( s ) =

with

V V

σ i j

∂ T ∂ x m

∂ x m

δ i j q m d V , (14)

I 1 = I 2 = I 3 =

∂ C i jkl ∂ x m

∂ C i jkl ∂ T

1 2

∂ ∆ T ( x i )

1 2

∂ T ∂ x m −

∂α ∂ T

∂α ∂ x m

m i j

m

m i j

m kl

∆ T ( x i ) + α ( x i , T ( x i ))

kl +

+

U δ mk − σ ik u i , m q m , k d V , Un m − t i u i , m q m d S .

(15)

S + + S −

(16)