PSI - Issue 77

Alireza Shadmani et al. / Procedia Structural Integrity 77 (2026) 221–228

225

Shadmani et al. / Structural Integrity Procedia 00 (2026) 000–000

5

(a)

(b)

Fig. 3: (a) Finite element model geometry of the PU coating and (b) mesh distribution

2.2. Continuum damage mechanics

In order to represent the material degradation, continuum damage mechanics (CDM) establishes a damage variable, represented as D , demonstrating the micro-damage state of the material. In the case of fatigue, the material loses its strength with an increase of repeated cycles as the damage accumulates due to growth of micro-damages in cyclic loading conditions. To explain the damage evolution due to fatigue loading conditions, a damage evolution law is proposed in Pandey et al. (2023) which is given as:

dD dN

M (1 − e α ) α

D R 0 . 5 β v

∆ σ β e α

(4)

=

where, M , α , and β are material constants which are obtained from the S-N curve and α is the material constant resulting from damage vs. the number of cycles curve. The value of M and β , and α are presented in Pandey et al. (2023), where the value of M and β are 8 . 47 × 10 − 16 and6 . 6, respectively, and α is equal to 6 . 1. Although the damage evolution model is originally developed for metals Pandey et al. (2019), it has been applied to polymers (Pandey et al. (2023); Kuthe et al. (2025)). However, further experimental validation is necessary to confirm its applicability to the PU coating under high-speed droplet impacts, specifically, to determine the model parameters that allow accurate characterization of damage evolution. This damage evolution model is based on the assumption that the damage evolution is driven by the maximum principal stress, ∆ σ , and the stress triaxiality function, R v , is used to account for the e ff ect of multiaxial stress states, concerning the fact that the material is subjected to a complex stress state due to the transient pressure profile. R v denotes the stress triaxiality function expressed as:

2 3 (1 + ν ) + 3(1 − 2 ν )(

σ max , H σ max , eq

) 2

R v =

(5)

where, σ max , H represents maximum hydrostatic stress, and ν is Poisson’s ratio. For uniaxial loading, the value of R v becomes unity. The component is considered to fail as damage approaches unity. Therefore, the fatigue life, i.e., N = N f , can be identified after substituting D = 1. The relation between damage variable D and fraction of life ( N N f )