PSI - Issue 68

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

1261

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

3

The first integral describes the strain energy of the first layer and the second integral the strain energy of the second layer. In case of applied mechanical loads these are taken into account by a corresponding external potential Π ext . In the next step, Hooke’s law

   

    ε ( k )

   

    ε

    σ x σ y σ z τ xz

    ( k )

x − α th k ∆ T − α th k ∆ T ( k ) z − α th k ∆ T γ ( k ) xz

1 − ν k ν k

0 0

ν k

E k 2(1 + ν k )(1 − 2 ν k )

ν k 1 − ν k ν k

, with : k = 1 , 2 ,

(5)

=

ν k 1 − ν k 0

ν k

1 − 2 ν k 2

0 0 0

the kinematic relations ε ( k ) x = δ u ( k ) δ x , ε ( k )

δ w ( k ) δ z

δ u ( k ) δ z

δ w ( k ) δ x

( k ) z =

, γ ( k )

y = 0 , ε

, with : k = 1 , 2 ,

(6)

xz =

+

and the displacement approach are inserted, so that the total potential energy is only dependent on functions of x . To return to the principle of minimum total potential energy, it states that the potential energy must become minimal. This results in δ Π= 0. If this variation is carried out for all unknown displacement functions, a non-homogeneous second order di ff erential equation system and the corresponding boundary conditions are obtained. To solve this system of 14 ordinary di ff erential equations, it is transformed into a first-order system of the kind A ϕ + B ϕ ′ = b , (7) where the array ϕ consists of the 14 unknown displacement functions dependent on x and their derivatives. The solution of this non-homogeneous di ff erential equation system can be represented as the sum of a homogeneous solution ϕ h and a particular solution ϕ p ϕ = ϕ h + ϕ p . (8) Using the method of undetermined coe ffi cients provides the following particular solution (9) For the homogeneous solution, an eigenvalue problem must be solved. Using an exponential solution approach ϕ h = ve µ x leads to an eigenvalue system, so that the eigenvalues µ i are determined by setting the determinant of the coe ffi cient matrix equal zero det  − B − 1 A − µ i E  = 0 , (10) where E denotes the unit matrix. The eigenvectors v i are calculated by  − B − 1 A − µ i E  v i = 0 . (11) The general solution with the 28 unknown constants C 1 ... 28 then is given by ϕ = C 1 v 1 e µ 1 x + C 2 v 2 e µ 2 x + . . . + C 28 v 28 e µ 28 x + b . (12) Using the 28 boundary conditions at the left and right edge of the model these unknown constants can be determined and the deformation field as well as the strain and stress field can be fully described. ϕ p = A − 1 b .

4. Finite Element Model

In order to compare and validate the analytical model, the interlaminar stresses are compared using a numerical model created with the Finite Element Method in Simulia Abaqus CAE (see Figure 2). The stress singularity is expected at the edge at the interface between both layers. Therefore, the mesh is finest in this region with elements with an edge length of 0.005 mm (where the layer thicknesses are t 1 = 1 mm and t 2 = 4 mm). Towards the other edges, the mesh becomes coarser. The model makes use of quadratic basis functions (CPE8 elements) and has about 1.1 million degrees of freedom.