PSI - Issue 18

Johannes Scheel et al. / Procedia Structural Integrity 18 (2019) 268–273 J. Scheel et al. / Structural Integrity Procedia 00 (2019) 000–000

270

3

∆ a − r

u + i

r

x 2

n

φ

x 1

D

C

B

A

( ∆ a − r , φ = π )

( r = L / 2 , φ = 0)

u − i

∆ a , L

a

Fig. 1. Virtual crack growth ∆ a and denominations for the crack closure integral

calculating the energy release rate for an infinitely small crack extension ∆ a . The index notation is used for tensor operations, implying summation over repeated indices. The energy release rate and the SIF are related as

K 2

2 I +

8 µ

1 + κ

III

K 2

G =

(2)

II + K

,

4 µ

where µ and κ are elastic constants. For the sake of convenience, symmetry with regard to the crack plane is assumed, i. e. u + i = − u − i . (3) Assuming mixed mode I / II-loading ( K III = 0) and inserting Eq. (2) in Eq. (1) yields

+ 2 ( ∆ a − r , π ) dr   .

8 µ (1 + κ ) ∆ a   ∆ a

+ 1 ( ∆ a − r , π ) dr + ∆ a

K 2

2 I = lim ∆ a → 0

II + K

σ 12 ( r , 0) u

σ 22 ( r , 0) u

(4)

The first and second integral in Eq. (4) can be directly associated with the applied loading mode, so that the SIF can be calculated independently, i. e. K 2 I = lim ∆ a → 0 8 µ (1 + κ ) ∆ a ∆ a σ 22 ( r , 0) u + 2 ( ∆ a − r , π ) dr , K 2 II = lim ∆ a → 0 8 µ (1 + κ ) ∆ a ∆ a σ 12 ( r , 0) u + 1 ( ∆ a − r , π ) dr . (5) For the stresses the near tip solutions on the ligament

K I √ 2 π r

K II √ 2 π r

σ 22 ( r , 0) =

σ 12 ( r , 0) =

(6)

,

,

are inserted in Eq. (5) yielding

8 µ (1 + κ ) ∆ a ∆ a

8 µ (1 + κ ) ∆ a ∆ a

u +

2 ( ∆ a − r , π ) √ 2 π r

u +

1 ( ∆ a − r , π ) √ 2 π r

K I = lim ∆ a → 0

K II = lim ∆ a → 0

dr ,

) dr ,

(7)

leaving only the displacement jumps on the crack faces divided by √ r to integrate. The displacement jumps are described using interpolation functions based on the Williams series [Williams (1957)]. As one possible approach three eigenfunctions are used so that the displacements are

√ r 3 a

rb 1 + ra 2 + √ r

3 ,

3 .

2 µ

2 µ − √

√ ra 1 − rb 2 −

1 + κ

1 + κ

3 b

u 2 ( r , π ) =

u 1 ( r , π ) =

(8)

The parameters a i , b i are calculated from the displacements on the crack faces of the FE analysis, choosing appropriate nodes and solving the system of algebraic equations. Assuming mode I loading conditions further simplifies the