PSI - Issue 17

W. Reheman et al. / Procedia Structural Integrity 17 (2019) 850–856

854

W Reheman et al. / Structural Integrity Procedia 00 (2019) 000–000

5

where the tensor notation for summation over equal indices is used. The average elastic energy density, W o , for a flat interface is

1 2 E

E s

2

1 2

σ 2

(12)

W o =

∞ −

.

Insertion of Eqs. (8), (9) gives

2 (1 −

2 .

1 2 E

1 2

π 2

π 2

π 2

π 2 akI 22 + (1 + ν )(

π 2

2 + (

2 − 2 ν (1 −

E s )

(13)

W =

akI 11 )

akI 22 )

akI 11 )

akI 12 )

( σ ∞ −

Terms independent of the perturbation amplitude a does not contribute to the growth rate as it is readily seen in Eq. (10). Other terms only containing a single integral I 11 , I 22 or I 12 become proportional to either cos x 1 k or sin x 1 k and vanish therefore, as an average over each period length 2 π/ k , which leave us with

ak ) 2 I 2

12 .

1 2

1 2 E

π 2

2 (

2 22 − 2 ν I 11 I 22 + (1 + ν ) I 2

W =

E s )

( σ ∞ −

11 + I

(14)

Integration over the period 0 ≤ x 1 ≤ 2 π/ k and the entire lower half-plane gives an average per period k 2 π 2 π/ k 0 I 2 11 d x 1 = π 2 8 (2 + x 2 k ) 2 e 2 x 2 k , k 2 π 2 π/ k 0 I 2 22 d x 1 = π 2 8 x 2 2 k 2 e 2 x 2 k , k 2 π 2 π/ k 0 I 11 I 22 d x 1 = 0 and k 2 π 2 π/ k 0 I 2 12 d x 1 = π 2 8 (1 + x 2 k ) 2 e 2 x 2 k . Integration over the entire lower half-plane finally gives

(15)

+ ∞ −∞

+ ∞ −∞

+ ∞ −∞

k 2 π

k 2 π

k 2 π

2 π/ k

2 π/ k

2 π/ k

5 π 2 32

π 2 32 k

π 2 32 k

I 2 11 d x 1 d x 2 =

I 2 22 d x 1 d x 2 =

I 2 12 d x 1 d x 2 =

, and

. (16)

,

0

0

0

Thus, the elastic strain energy area density is summarised to W = π 4 4 7 + ν E ( σ ∞ − 1 2 E s ) 2 a 2 k .

(17)

3.2. Interfacial energy density

A wavy interface that is sinusoidally positioned at w = a sin( x 1 k ) have an area average calculated for a period length 2 π/ k ,

k 2 π

2 π/ k 0

o 1 + (d w / d x 1 )

a 2 k 2 2

cos 2 x

3 ,

2 + (d w ) 2 = A

2 = A

A = A o

(d x 1 )

o (1 +

1 k ) + O ( ak )

(18)

where A o is the area of the original flat interface. For small values of d w / d x 1 1 the following change of interfacial energy is obtained

2

2 π/ k

a 2 k 2

1 4

cos 2 ( kx

2 k 2 ) ,

γ = γ o A / A o = γ o + γ o

1 )d x 1 = γ o (1 +

γ o a

(19)

0

where γ o is the flat interface energy which is assumed to be constant per unit of true area. The waviness causes an increase of the interface energy γ as it appears on a structural length scale.

4. Discussion and Conclusions

The growth rate according to Eq.(10) with the insertion of Eqs. (17) and (19) gives ∂ a ∂ t = L ∂ ∂ a ( W − γ ) = L a k π 4 64 7 + ν E σ 2 ∞ + ( 1 2 E s ) 2 − 1 2 γ o k .

(20)