PSI - Issue 68

D. Linn et al. / Procedia Structural Integrity 68 (2025) 1259–1265 D. Linn, W. Becker / Structural Integrity Procedia 00 (2024) 000–000

1264

6

Δ w

FEM New approach

FEM

New approach

Δ u

Fig. 6. Crack opening displacements (left) and incremental energy release rate (right) for a patch on a substrate under thermal loading.

displacement functions of the uncracked section. 14 continuity conditions are derived from the variation at x = ∆ a , where the variations of the functions of the cracked and uncracked regions are considered to be equal ( . . . ) δ u (1) i = ( . . . ) δ u (1 , c) i , ( . . . ) δ w (1) i = ( . . . ) δ w (1 , c) i , with : i = 0 . . . 3 , ( . . . ) δ u (2) j = ( . . . ) δ u (2 , c) j , ( . . . ) δ w (2) j = ( . . . ) δ w (2 , c) j , with : j = 1 . . . 3 , (15) The other 16 continuity conditions result from the claim that the displacement functions are equal at the transition between the sections u (1) i = u (1 , c) i , w (1) i = w (1 , c) i , with : i = 0 . . . 3 , u (1) 0 = u (2 , c) 0 , w (1) 0 = w (2 , c) 0 , u (2) j = u (2 , c) j , w (2) j = w (2 , c) j , with : j = 1 . . . 3 . (16) From these 60 conditions now, the unknown constants can be determined and consequently the displacement field can be described as well as the strain and stress field within the layers. For the example of a patch on a substrate under a temperature change of ∆ T = − 100 K (see model in Figure 3 (left) and material parameters in Table 1), the crack opening displacements ( ∆ u = u (1 , c) 0 − u (2 , c) 0 and ∆ w = w (1 , c) 0 − w (2 , c) 0 ) for a crack length of ∆ a = 0 . 1 mm are now shown in Figure 6 (left), again in comparison with results of the FEM. Qualitatively, the plots of both displacements of the new displacement approach agree reasonably well with those of theFEM.

6.2. Finite Fracture Mechanics

Within the framework of Finite Fracture Mechanics, a debonding crack of finite length will form instantaneously when a coupled stress and energy criterion are satisfied simultaneously. Here, for this purpose, a quadratic stress criterion

σ 2 z σ 2 c

τ 2

xz

f ( σ z ,τ xz ,σ c ,τ c ) =

1 ,

(17)

c ≥

+

τ 2

and a linear energy criterion g G I , G II , G Ic , G IIc = G

I ( ∆ a ) G Ic

II ( ∆ a ) G IIc ≥

+ G

1 ,

(18)

are used. The stress criterion uses the interlaminar stresses σ z and τ xz at the interface of both layers of the uncracked configuration that are already discussed. The energy criterion uses the incremental energy release rate of mode I and II, G I and G II , which can be calculated from the crack opening displacements ( ∆ u and ∆ w ) and the interlaminar stresses ( σ z and τ xz ) G I = 1 2 ∆ a ∆ a 0 σ z w (1 , c) 0 − w (2 , c) 0 d s ; G II = 1 2 ∆ a ∆ a 0 τ xz u (1 , c) 0 − u (2 , c) 0 d s . (19)