PSI - Issue 75

Thomas Constant et al. / Procedia Structural Integrity 75 (2025) 660–676 Author name / Structural Integrity Procedia 00 (2025) 000–000

663

4

with M ∈ R n dof × n dof , C ∈ R n dof × n dof , K ∈ R n dof × n dof are respectively the mass, damping and sti ff ness matrices and u ( x A , t ) ∈ R n dof is the displacement vector. f ( x A , t ) ∈ R n dof corresponds to the force vector applied to the system. We here assume that it can be described from a reduced number of load parameters gathered in x A . At that stage, the motion of the system can be expressed in the basis of the real eigenmodes Φ of the system:

ω 2 M

Φ = K Φ

(4)

Using a truncated modal basis up to the m -th mode (with m ≤ n dof ), the displacement vector is approximated as:

m i = 1

u ( x A , t ) ≈

q i ( x A , t ) ϕ i ,

(5)

where q i ( x A , t ) are the time-dependent modal coordinates, and ϕ i are the corresponding eigenmodes. The truncated modal basis is denoted as Φ = [ ϕ 1 ϕ 2 · · · ϕ m ]. This basis diagonalizes the mass M and sti ff ness matrices K , see equation 4. Under the assumption of proportional damping—or, alternatively, in the case of low damping with well separated modal frequencies, or under the Caughey damping assumption—the damping matrix is also diagonalized in the modal space. Finally, the system of equations of motion 3 reduces to m decoupled equations, each of which can be solved independently.

f i ( x A , t ) m i

¨ q i ( x A , t ) + 2 ξ i ω i ˙ q i ( x A , t ) + ω 2

i q i ( x A , t ) =

(6)

Equation 6 is expressed for the i -th mode, with f i ( x A , t ) = ϕ T

i f ( x A , t ) the generalized force of the i -th mode, ω 2 i

the i -th eigenvalue associated with ϕ i , ξ i the i -th generalized damping factor, and m i the i -th generalized mass. It should also be noted that one can define a truncated modal stress matrix Ψ = [ ψ 1 ψ 2 · · · ψ m ], where each column represents the modal stress vector associated with a corresponding mode shape. Finally, the stress values of the system are gathered into a modal stress vector, by linearly combining the individual modal stress components.

m i = 1

q i ( x A , t ) ψ i ,

σ ( x A , t ) ≈

(7)

The latter can be rewritten a of temporal stress tensors σ k ( x A , t ) defined at each node k ∈N c of a critical location of the system, see Equation 1.

2.2. Mathematical description of model m B

In structural durability, fatigue refers to the phenomenon of material or component degradation resulting from cyclic loading that remains below the material’s yield strength. In the scope of this research, the initiation of a single crack at any critical location within the system indicates the fatigue failure of the entire system. Fatigue failure can manifest either in LCF (low cycle fatigue) or HCF (high cycle fatigue). HCF typically occurs if small elastic or plastic strains exist over a high number of cycles. Conversely, LCF is distinguished by significant plastic strains, resulting in failure after a comparatively small number of cycles (see Salifu and Olubambi (2024)). In uniaxial fatigue, the high-cycle fatigue (HCF) regime can be, on one hand, modeled using the Basquin’s power law ∆ ϵ e 2 = σ ′ f E (2 N f ) b ,which relates the elastic strain amplitude to fatigue life. On the other hand, a power-law such as the one proposed by Co ffi n