PSI - Issue 2_B

Per Ståhle et al. / Procedia Structural Integrity 2 (2016) 589–596

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4 P.Ståhle et al. / Structural Integrity Procedia 00 (2016) 000–000 In the present study only quasi-static evolution is considered. The body may be in a thermodynamic state but the mechanical state is assumed to be static. By putting the mobility parameter L u i � ∂ u i /∂ t implying that, δ F el δ u i = 0 ⇒ 2 μ� i j , j + λ� j j , i − 2(3 λ + 2 μ ) � s ( ψ ) , i = 0 . (15) The governing equation for the evolution of the displacements, u i , is as follows

1 1 − 2 ν

1 + ν 1 − 2 ν

2 ) ψ

u j , ji =

u i , j j +

� v (1 − ψ

(16)

, i ,

which is identical to σ i j , j = 0 for a material with an isotropic stress free volumetric expansion � v . With the length unit g b / p and the time unit ( L ψ p ) − form

1 used in Eqs. (12) and (16), the governing equations take the

ψ ,αα =

1 4 ˆ � v ˆ σ ββ + ψ (1 − ψ

∂ψ ∂ ˆ t −

2 ) ,

(17)

(1 + ν ) 2 1 − 2 ν

2(1 + ν ) 1 − 2 ν

2 ) ψ

ˆ u α,ββ +

ˆ u β,αβ = 2

ˆ � v (1 − ψ

(18)

,α

where the following scalings are made

x i , ˆ u i =

ˆ t = L ψ p t , ˆ x i =

p g b

E g b

u i .

(19)

and consequently

ˆ � αα =

ˆ � v =

E p � i j ,

E p � v ,

1 √ Ep

ˆ σ αα =

(20)

σ i j , .

The index notation for partial derivatives using greek letters α , and β denotes that the partial derivatives are taken with respect to the non-dimensional coordinates, e.g. ( ) ,α denotes ∂ ( ) /∂ ˆ x α , where α = 1, 2 or 3.

3. Solutions for Plane and Shallow Wavy Interfaces

In the absence of mechanical load an analytical solution is given for a straight edge by Ginzburg and Landau (1950). Closed form solutions that include the e ff ect of an external homogeneous stress field are easily obtained as is shown in the following section. The modelled geometry is a strip with the dimensions of | x 1 | ≤ h / 2 and | x 2 | ≤ 3 h / 2, where h is height of the studied geometry. The interface is parallel with the x 1 axis. and is initially placed at x 2 = 0 . In the following subsections constant displacements along the edges at | x 1 | = h / 2, respectively constant stress along the edges at | x 2 | = 3 h / 2 are applied. In both cases the applied load creates stress and strain fields that are only dependent on the x 2 coordinate and time. 3.1. Uniaxial stress field applied in the x 2 direction, ψ = f ( ˆ x 2 , ˆ t ) Here the solution for an edge moving under steady state conditions in a uniaxial stress field, under plane stress condition, is derived. The boundary conditions are σ 22 ( ± 3 h / 2) = σ o and σ 12 ( ± 3 h / 2) = 0 , (21) where the stress σ o is constant. Remaining boundaries are traction free. The phase boundary conditions are