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

1260

2

l

l

E 1 , 1 , th1 E 2 , 2 , th2

Δ T

Δ T

z

z

t 1

t 1

layer 1 layer 2

layer 1 layer 2

x

x

t 2

t 2

Fig. 1. A two-layer system of a patch on a substrate under thermal loading (left) and reduction to the overlap length (right).

many works, for example by Hell et al. (Hell et al. (2014)), by Stein et al. (Stein et al. (2015)) or by Mendoza-Navarro et al. (Mendoza-Navarro et al. (2013)). For this criterion, the stresses along the interface and the incremental energy release rate for a potential crack are used for evaluation. In order to determine these quantities a closed-form analyti cal approach is developed. The quality of the prediction is evaluated by comparative Finite Element Analysis and the presented approach is confirmed as very useful for practical application.

2. Mechanical situation

A two-layer system of a patch with length l and thickness t 1 on a substrate (thickness t 2 ) is considered under thermal loading ∆ T (see Figure 1, left). Both layers are modeled with linear elastic isotropic material behavior with Young’s modulus E 1 , E 2 , Poisson’s ratio ν 1 , ν 2 and thermal expansion coe ffi cient α th1 , α th2 . Plane-strain behavior in y -direction, i.e. out of the drawing plane, is also assumed. Now the idea is to reduce the analysis to the overlap length as a generic configuration (see Figure 1, right). The advantage of this generic configuration is that other load cases can be modeled as well, simply by changing the boundary conditions. The goal is to get an approximate closed-form analytical description of the displacements and the strain and stress field within the layers. To describe the displacement field a higher-order displacement approach is chosen, namely a second order dis placement approach, which is extended with an additional term. When the situation is analyzed, it is clear that at the free edges between the two layers, stress singularities occur. These singularities are taken into account in these additional terms by using a respective displacement exponent λ . For the configuration of an aluminum patch on a steel layer an exponent of λ = 0 . 7 is used. The displacements u (1) in longitudinal direction and w (1) in thickness direction within the patch (layer 1) are described as follows u (1) ( x , z ) = u (1) 0 ( x ) + z u (1) 1 ( x ) + z 2 u (1) 2 ( x ) + z λ u (1) 3 ( x ) , w (1) ( x , z ) = w (1) 0 ( x ) + zw (1) 1 ( x ) + z 2 w (1) 2 ( x ) + z λ w (1) 3 ( x ) . (1) Analogously, the displacements u (2) and w (2) within the substrate (layer 2) are described as u (2) ( x , z ) = u (1) 0 ( x ) − z u (2) 1 ( x ) + ( − z ) 2 u (2) 2 ( x ) + ( − z ) λ u (2) 3 ( x ) , w (2) ( x , z ) = w (1) 0 ( x ) − zw (2) 1 ( x ) + ( − z ) 2 w (2) 2 ( x ) + ( − z ) λ w (2) 3 ( x ) , (2) where the functions u (1) 0 and w (1) 0 are the same z -independent functions as in (1) in order to ensure continuity along the interface. In the following, the goal is to determine the unknown functions u (1) 0 ( x ), u (1) 1 ( x ), u (1) 2 ( x ), u (1) 3 ( x ), u (2) 1 ( x ), u (2) 2 ( x ), u (2) 3 ( x ), w (1) 0 ( x ), w (1) 1 ( x ), w (1) 2 ( x ), w (1) 3 ( x ), w (2) 1 ( x ), w (2) 2 ( x ), w (2) 3 ( x ) dependent on x . This is done with the help of the minimum total energy principle, which postulates that the total energy Π , which is the sum of the internal Π int and external potential Π ext , must become minimal Π=Π int +Π ext = min . (3) The internal potential can be calculated from the strain energy as follows 3. Analytical model

(1) x − α th1 ∆ T + σ

(1) z − α th1 ∆ T + τ (2) xz γ

l 0 σ

(1) x ε

(1) z ε

1 2 0 − t 2

t 1

(1) y ( − α th1 ∆ T ) + σ

(1) xz γ

(1) xz d z

σ

Π int =

0

(4)

(2) x ε

(2) z ε

(2) x − α th2 ∆ T + σ

(2) z − α th2 ∆ T + τ

(2) xz d z d x .

+

(2) y ( − α th2 ∆ T ) + σ