# Mesostructural Model for the Fatigue Analysis of Open-Cell Metal Foams
## 3 Proposed Model
This criterion is founded on the idea that a material's mechanical qualities deteriorate over time. Three failure criteria are considered: the [PERSON] et al. criterion [14], the peak compression stress reduction criterion [15], and the peak tensile stress reduction criterion [15]. The [PERSON] et al. [14] criterion established that the ratio between the prepeak tensile and prepeak compressive slope \((R=\frac{HCN}{HTN})\) is initially close to one. As the number of cycles increases, this ratio also increases, and failure is reached when \(R=1.5\). Figure 3a shows a schematic representation of the [PERSON] et al. [14] criterion. The peak compression and peak tensile stress reduction criteria [15] consider that the mechanical properties of the material decrease over its lifetime. These two criteria are defined as follows: failure is considered to have occurred when the tensile stress reaches 80% of its maximum value or when the compressive stress reaches 90% of its maximum value. Figure 3b shows a schematic representation of the peak compression and peak tensile stress reduction criteria [15]. Implementation of the proposed model: the proposed model is divided as follows:  
Figure 3: Schematic representation of the proposed failure criteria: (**a**) [PERSON] et al. criterion [14] and (**b**) peak compression and peak tensile stress reduction criteria [15]. Step 1: determine the stresses applied to each ligament. The finite element model is created following the basic unit cell, as shown in Figure 2a, and it is replicated in three dimensions, as shown in Figure 2b. The stresses applied to each ligament are calculated for a constant strain amplitude \((\epsilon)\). The model allows for iterating \((i)\) for each ligament \((j)\) to determine the stresses applied to each ligament \((\sigma_{ij})\). Step 2: determine the fatigue life of each ligament. A fatigue analysis is conducted for each ligament using the Weibull fatigue model (Equation (1)). The stresses applied individually are known from Step 1. As a result of the Weibull fatigue analysis, the base material S-N curve is obtained; thus, the number of cycles to failure for each stress can be determined \((n_{ij})\). Step 3: identify the first ligaments to fail, remove these ligaments from the finite element model, and record the number of cycles. Since the number of cycles to failure for each ligament is known from Step 2, it is possible to determine which ligament is going to be the first to fail. The number of cycles to failure \((N_{i})\) is given by:  
\[N_{i}=N_{i-1}+\min(n_{ij}) \tag{3}\]  
where \(n_{ij}\) is the number of cycles to failure at iteration \(i\) of ligament \(j\). Note that the number of cycles to failure of iteration 0 is 0 (\(N_{i=0}=0\)). At this point, it is necessary to check the failure criteria to determine if the fatigue life of the foam has been reached. If the failure criteria have been satisfied, the analysis is finished. If the failure criteria have not yet been satisfied, then the analysis continues to Step 4. Step 4: recalculate the stresses \((\sigma_{ij})\) of the remaining ligaments with the new geometric configuration in the finite element model. After new stresses are determined, the algorithm returns to Step 2; therefore, the cycles of failure for each ligament are determined and recorded as \(n_{ij}\). Step 5: apply the new stresses to each remaining ligament and use the equivalent number of cycle model to determine the number of cycles remaining until each ligament's fatigue life. The applied stresses and corresponding fatigue lives were determined for each ligament in Step 4. However, the determined number of cycles to failure must be restated according to the applied cycles. The history of previously applied cycles is considered using the equivalent number of cycles model.