# Mesostructural Model for the Fatigue Analysis of Open-Cell Metal Foams
## 3 Proposed Model  
The philosophy behind the proposed model is to determine the fatigue life of the foam by analyzing the fatigue life of its ligaments. Various models are employed to develop the proposed model: a finite element model, a fatigue analysis model, an equivalent number of cycles model, and a failure criterion model. Next, each model is described. Computational methods: the proposed model considers a basic tetrakaidekahead until cell composed of linear ligaments forming a three-dimensional structure, as shown in Figure 2a. Each three-dimensional structure conforms to the 3-D arrangement for modeling the metal foam, as indicated in Figure 2b. Then, the stress in each ligament is evaluated. The material properties used in the computational analysis (Duocel(r) foam, ERG Aerospace Corporation, Oakland, CA, USA) were derived from experimental data obtained from open-cell aluminum foams with 40 pores per inch (ppi), as detailed in our previous study. The base material for the foam is 6061 aluminum alloy, which exhibits a relative density of 6-8%. The Young's modulus, Poisson's ratio, and other relevant mechanical properties are measured in both compression and tension tests, ensuring accurate representation in the simulations. These properties are input into the computational models to simulate the realistic behavior of the foam under cyclic loading conditions. The boundary conditions in the computational models were carefully chosen to reflect the experimental setup. The models employ fixed boundary conditions at the base of the foam structure to simulate a constrained scenario, while the top surface is subjected to a cyclic load. The load is applied as a fully reversed strain-controlled cycle with amplitudes ranging from 0.003 to 0.0125, mimicking the experimental fatigue tests. The cyclic load is uniformly distributed across the surface, ensuring consistent stress and strain distribution throughout the foam structure. This approach allows us to replicate the stress-strain behavior observed in physical tests, providing a reliable basis for evaluating the fatigue life of the foam under varying strain amplitudes. Fatigue analysis model: the proposed model considers a fatigue analysis using the Weibull fatigue model, which is based on a probabilistic approach and has been used and  
Figure 1: Metal foams: (**a**) open-cell foam and (**b**) closed-cell foam. Figure 2: (**a**) Basic unit cell and (**b**) 3-dimensional arrangement for modeling the metal foam. calibrated by [PERSON] et al. [23] and [PERSON] et al. [24; 25; 26]. The Weibull fatigue model can be expressed as follows:  
\[N^{*}=\exp\biggl{\{}B+\frac{\lambda+\delta(-\log(1-p))^{1/\beta}}{\log(\epsilon_ {a}^{*})-C}\biggr{\}} \tag{1}\]  
where \(N^{*}\) is the number of cycles to failure, \(p\) is the probability of failure, \(\epsilon_{a}\) is the strain amplitude, \(B\) and \(C\) are the threshold parameters, while \(\beta\), \(\lambda\), and \(\delta\) are Weibull parameters. The detailed derivation of the Weibull model can be seen in [23; 24; 25; 26]. Equivalent number of cycles model: estimating the number of cycles to failure becomes a variable amplitude fatigue analysis because the ligaments that have reached failure must be removed and, therefore, the stress recalculated. In this research, we use the equivalent number of cycles model, which is obtained from the Weibull model detailed in [PERSON] et al. [27]. The equivalent number of cycles model can record the previous history of the loads applied to the material, and it is calculated as follows:  
\[N_{\text{eq}}^{*}=\exp\biggl{\{}B+\frac{(\log(n_{1})-B)(\log(\sigma_{1})-C)}{ \log(\sigma_{2})-C}\biggr{\}} \tag{2}\]  
where \(N_{\text{eq}}^{*}\) corresponds to the equivalent number of cycles applied at \(\sigma_{2}\) that generate the same damage level as \(n_{1}\) cycles applied at \(\sigma_{1}\). Failure criterion model: this study considers a failure criterion model related to calculating the fatigue life of foam.