PSI - Issue 17

W. Reheman et al. / Procedia Structural Integrity 17 (2019) 850–856 W Reheman et al. / Structural Integrity Procedia 00 (2019) 000–000

853

4

required to maintain a homogeneous stress state in the respective half-space. The attempt is to calculate the growth rate of sinusoidal perturbations given by

d w d ξ

= ak cos k ξ ,

(5)

w ( ξ ) = a sin k ξ and

where k is the wave number. Thus, the force contribution from the traction d F applied on the lower part B, at x 1 = ξ becomes d F = ( σ ∞ − 1 2 E s )d w = ( σ ∞ − 1 2 E s ) d w d ξ d ξ = ( σ ∞ − 1 2 E s ) ak cos x 1 k d ξ , (6)

cf. Fig. (3c). Transformation from polar to Cartesian coordinates the following gives σ 11 = σ ρρ cos 2 ϕ, σ 22 = σ ρρ sin 2 ϕ, and σ 12 = − σ ρρ sin 2 ϕ .

(7)

By using Eqs. (4), (6), (7) and that cos ϕ = ( x 1 − ξ ) /ρ , sin ϕ = − x 2 /ρ and ρ = ( x 1 − ξ ) 2 + x 2 stresses are obtained after integration along the entire interface,

2 , the following total

π 2 akI 11 , σ 22 = − ( σ ∞ −

E s ) 1 −

1 2

1 2

1 2

π 2 akI 22 , and σ 12 = − ( σ ∞ −

π 2

E s )

E s )

akI 12 ,

σ 11 = ( σ ∞ −

(8)

where

I 11 = I 22 = I 12 = − −∞ −∞ + ∞ + ∞

cos( k ξ ) ( x 1 − ξ ) 3

x 2 k sin x

π 2 (2 + x 2 k ) e

ρ 4 d ξ =

1 k ,

2 2

( x 1 − ξ ) x

x 2 k sin x

(9)

π 2 x 2 ke

cos( k ξ )

d ξ = −

1 k ,

ρ 4

+ ∞

2 x

cos( k ξ ) ( x 1 − ξ )

x 2 k cos x

π 2 (1 + x 2 k ) e

d ξ = −

1 k .

2

ρ 4

−∞

The calculations requires integration by parts. The relations ∞ −∞ t cos t

t 2 + q 2 d t = 0 and

∞ −∞

t sin t t 2 + q 2 d t = π e −|

q | become useful.

3. Gibbs’ Free Energy Density

Both the elastic energy density, W , and the required interfacial energy, γ , increase with the wave number, k . The growth rate of the waviness of an interface that is sinusoidal positioned at x 2 = a sin( kx 1 ) is controlled by the avail ability of free energy. In the absence of no other energy resources than U and γ that seems reasonable to consider, the rate of growth of the waviness is assumed to be proportional to the excess of free energy due to a change of the wave amplitude. With the proportionality factor, L , being a non-negative constant the following is obtained, ∂ a ∂ t = L ∂ ∂ a ( W − γ ) , (10) in accordance with Landau and Lifshitz (1935).

3.1. Free elastic energy

The remote normal stress, σ ∞ ± E s / (1 − ν 2 ), plus or minus depending in the specific half-plane, is assumed to dominate everywhere. Due to the wavy interface with a small amplitude a 1 / k , there will be a small perturbation of the stresses and strains of the order of ka . These stresses are significant only in the vicinity of the interface. The assumed plane stress, i.e., σ 13 = σ 23 = σ 33 = 0, reduces the involved stress components to those only in the x 1 - x 2 plane. The elastic energy density per unit volume for plane stress is defined by

1 2 E

12 ,

σ i j i j 2

σ 2

2 22 − 2 ν σ 11 σ 22 + (1 + ν ) σ 2

W =

(11)

11 + σ

=