PSI - Issue 68

D. Linn et al. / Procedia Structural Integrity 68 (2025) 1259–1265

1263

D. Linn, W. Becker / Structural Integrity Procedia 00 (2024) 000–000

5

Aluminum l = 25 mm Steel

F = 100 N

l 0 = 25 mm

l = 25 mm

u l1 = 0.5 mm

z

z

t 2 = 2 mm t 1 = 2 mm

t 2 = 2 mm t 1 = 2 mm

Steel Steel

x

x

F = 100 N

FEM

New approach FEM

New approach

Fig. 4. Interlaminar stresses along the interface between both layers for a Double Cantilever Beam (left) and a Single Lap Joint (right).

l

Δ T

Δ T

Δ a

z

t 1

layer 1 layer 2

1

2

x

t 2

Fig. 5. Patch on a substrate with interface crack of length ∆ a (left) and model with combination of two regions (right).

6.1. Refinement of analytical model

In order to represent a potential debonding behavior with the analytical model, the model must be extended. For this purpose, it is combined by two sections – the cracked section and the uncracked region (see Figure 5, right). For these two sections, di ff erent displacement approaches are used. For the uncracked section the displacement approach which has been used so far ((1),(2)) is still applied. For the cracked region the displacement approaches are extended in a way that at the interface ( z = 0) a separation can be taken into account u (1 , crack) ( x , z ) = u (1 , c) 0 ( x ) + z u (1 , c) 1 ( x ) + z 2 u (1 , c) 2 ( x ) + z λ u (1 , c) 3 ( x ) , w (1 , crack) ( x , z ) = w (1 , c) 0 ( x ) + zw (1 , c) 1 ( x ) + z 2 w (1 , c) 2 ( x ) + z λ w (1 , c) 3 ( x ) , (13) u (2 , crack) ( x , z ) = u (2 , c) 0 ( x ) − z u (2 , c) 1 ( x ) + ( − z ) 2 u (2 , c) 2 ( x ) + ( − z ) λ u (2 , c) 3 ( x ) , w (2 , crack) ( x , z ) = w (2 , c) 0 ( x ) − zw (2 , c) 1 ( x ) + ( − z ) 2 w (2 , c) 2 ( x ) + ( − z ) λ w (2 , c) 3 ( x ) . (14) The further procedure is now analogous to the uncracked configuration. For the identification of the 30 unknown functions depending on x , the minimum total potential energy principle is used again. As the configuration is only subject to thermal loads, the total potential is only expressed by the internal potential, which is composed of two portions. The first portion is over the cracked section from 0 to ∆ a and the second one is over the uncracked region ∆ a to l . The variation is performed analogously for the unknown functions and again leads to a non-homogeneous di ff erential equation system of second order and the corresponding boundary conditions. The system of di ff erential equations is solved in the same way as before, the only di ff erence is in the determination of the free constants C 1 ... 60 . For these constants 30 boundary conditions at the model edges and 30 continuity conditions at x =∆ a are used. At the left model edge ( x = 0) the boundary conditions result from the variations of the 16 displacement functions of the approach of the cracked section and at the right model edge ( x = l ) they result from the variations of the 14