## 1. Introduction

_{2}emissions. The Environmental Protection Agency ($EPA$) defines 82.5 g/km for passenger vehicles manufactured before 2026, while in the European Union the limit is 95 g/km until 2025; this limit is expected to decrease by 30% in 2030 and become zero in 2035 [3,4].

## 3. Fatigue in Composites

$${S}_{R}={S}_{max}-{S}_{min}$$

$${S}_{a}={\displaystyle \frac{{S}_{R}}{2}}={\displaystyle \frac{{S}_{max}-{S}_{min}}{2}}$$

_{m}) is expressed by the mean value of the maximum and minimum loads:

$${S}_{m}={\displaystyle \frac{{S}_{max}+{S}_{min}}{2}}$$

_{n}), which depends on the mechanical behavior of the matrix and the reinforcement [69]. It is expressed as follows:

$${D}_{n}={\displaystyle \frac{{E}_{o}-{E}_{n}}{{E}_{o}-{E}_{f}}}$$

$${R}_{II}^{r}={R}_{u}^{s}-\sum _{i=1}^{m}\left({R}_{II}^{s}-{\sigma}_{max}\right){\displaystyle \frac{\Delta {n}_{i}}{{N}_{f}{\sigma}_{max}}}$$

$$E\left(n\right)={\displaystyle \frac{{\sigma}_{max}}{\epsilon \left(n\right)}}={\displaystyle \frac{q{\sigma}_{ult}}{\epsilon \left(n\right)}}$$

$$D\left({n}_{i}\right)={\left({\displaystyle \frac{{n}_{i}+{n}_{i,i-1}^{\frac{log{\sigma}_{i}}{log{\sigma}_{i-1}}}}{{N}_{i}}}\right)}^{{\displaystyle \frac{{q}_{w}(1-R)}{{p}_{w}}}}$$

$${n}_{i,i-1}={N}_{i}{\left({\displaystyle \frac{sin{q}_{i-1}{X}_{i-1}cos({q}_{i-1}-{p}_{i-1})}{sin{q}_{i-1}cos({q}_{i-1}{X}_{i-1}-{p}_{i-1}}}\right)}^{{n}_{i,i-1}}$$

where ${X}_{i-1}={n}_{i-1}/{N}_{i-1}$, and ${p}_{w}$ and ${q}_{w}$ are weighted parameters defined by

$${p}_{w}={\displaystyle \frac{{\sigma}_{min}}{{\sigma}_{max}}}\left(\sqrt{{p}_{i-1}+{p}_{i}}\right)$$

$${q}_{w}=\sqrt{{q}_{i-1}+{q}_{i}}$$

$$\frac{{\sigma}_{11}^{2}}{{X}_{r}^{2}}}+{\displaystyle \frac{{\tau}_{12}^{2}}{{S}_{r}}}=1$$

_{r}), the failure of the matrix can be expressed as follows [73]:

$$\frac{{\sigma}_{22}^{2}}{{Y}_{r}^{2}}}+{\displaystyle \frac{{\tau}_{12}^{2}}{{S}_{r}}}=1$$

$${\epsilon}_{12}^{*}={C}_{12}^{*}{\sigma}_{12}$$

$${\sigma}_{12}^{*}={S}_{12}^{*}{\sigma}_{12}$$

where ${\epsilon}_{12}^{*}$ is the transformation strain or eigenstrain, and ${\sigma}_{12}^{*}$ is the uniform stress. ${C}_{12}^{*}$ and ${S}_{12}^{*}$ are the effective stiffness and compliance tensor, respectively, described as follows:

$${S}_{ij}^{*}=\left({V}_{m}{S}_{ik}^{m}+{V}_{k}{S}_{ip}^{f}{H}_{pk}\right){\left({V}_{m}{I}_{kj}+{V}_{f}{H}_{kj}\right)}^{-1}$$

$${C}_{ij}^{*}=\left({V}_{m}{C}_{ik}^{m}+{V}_{f}{C}_{ip}^{f}{T}_{pk}\right){\left({V}_{m}{I}_{kj}+{V}_{f}{T}_{kj}\right)}^{-1}$$

where ${V}_{f}$ and ${V}_{m}$ are the volume fractions of the fibers and matrix, respectively. ${I}_{ij}$ is the second-order unit tensor. ${T}_{ij}$ and ${H}_{ij}$ can be obtained [74]

$${T}_{ij}={\left[{I}_{ij}+{K}_{ik}{S}_{kp}^{m}({C}_{pj}^{f}-{C}_{pj}^{m})\right]}^{-1}$$

$${H}_{ij}={C}_{ik}^{f}{T}_{kp}{S}_{pj}^{m}={C}_{ik}^{f}{\left[{I}_{kq}+{K}_{kl}{S}_{lp}^{m}({C}_{pq}^{f}-{C}_{pq}^{m})\right]}^{-1}{S}_{qj}^{m}$$

$$\frac{{\sigma}_{1}}{{X}_{t}}}-{\nu}_{12}{\displaystyle \frac{{\sigma}_{2}}{{X}_{t}}}+{\nu}_{f12}{\displaystyle \frac{{E}_{1}}{{E}_{2}}}\xb7{m}_{\sigma f}{\displaystyle \frac{{\sigma}_{2}}{{X}_{t}}}=1$$

and for ${\sigma}_{1}<0$

$$\left|{\displaystyle \frac{{\sigma}_{1}}{{X}_{t}}}-{\nu}_{12}{\displaystyle \frac{{\sigma}_{2}}{{X}_{t}}}+{\nu}_{f12}{\displaystyle \frac{{E}_{1}}{{E}_{2}}}\xb7{m}_{\sigma f}{\displaystyle \frac{{\sigma}_{2}}{{X}_{t}}}\right|+{\left(10{\gamma}_{12}\right)}^{2}=1$$

where X is the strength of the lamina under tension (t) and compression (c), respectively. The subscript f denotes the fiber properties [75,76].

## 4. Neural Networks in Fatigue Life Prediction

$${I}_{i}=\sum _{j=1}^{n}{x}_{j}{w}_{ij}$$

$${s}_{i}=\sum _{j=1}^{n}{x}_{j}{w}_{ij}-{\theta}_{i}$$

$${y}_{i}=f\left({u}_{i}\right)=f\left(\sum _{j=1}^{n}{x}_{j}{w}_{ij}-{\theta}_{i}\right)$$

where f is the activation function. Although the most commonly used activation function is the S form, expressed mathematically by the sigmoid function, the ramp and step functions can also be used [106].

$$f\left(u\right)={\displaystyle \frac{1}{1+{e}^{-u}}}$$

The connection between layers is not direct between the input and output; there are hidden layers with different configurations depending on the main target.

$$\delta k=({t}_{k}-{O}_{k}){O}_{k}(1-{O}_{k})$$

where ${O}_{k}(1-{O}_{k})$ is the derivative of the trigger function. The weights (${\mathsf{\omega}}_{j,k}$) are functions of the error at node k times the activation node (j), including the learning rate (${l}_{r}$), and the change in the weight ($\Delta {W}_{j,k}$) is defined by

$$\Delta {W}_{j,k}={l}_{r}{\delta}_{k}{X}_{k}$$

$$\Delta {W}_{j,k}={l}_{r}{\delta}_{k}{X}_{k}+\Delta {W}_{j,k}^{n-1}\mu $$

$$E={\displaystyle \frac{1}{2}}\sum \left(\sum {\left({t}_{k}-Ok\right)}^{2}\right)$$

$$RMSE=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^{N}}{\left({t}_{i}-t{d}_{i}\right)}^{2}}$$

$$y\left(t\right)=f\left[y(t-1),\dots ,y(t-{n}_{y},u(t-1),\dots ,u(t-{n}_{u}+e\left(t\right)\right]$$

where ${n}_{y}$ and ${u}_{n}$ are the output and the past input, respectively. The delay units ($z-1$) are used to predict the next point on the input layer. The output data are computed using the bipolar sigmoid function (${f}_{1}$), which is expressed by

$${y}_{p}\left(t\right)={f}_{1}\left(\sum _{i=1}^{K}I{w}_{i}{u}_{i}+\sum _{j=1}^{K}I{w}_{o}{y}_{j}+{b}_{i}\right)$$

The application of artificial intelligence helps predict nonlinear behaviors, such as fatigue life in composite materials. The degradation of materials generates the need to implement numerical processes that allow updating of the behavior of the material. The topology and characteristics of the neural network will depend on the available parameters that can be used as input to predict fatigue life. It is possible to include material characteristics as inputs, which can include chemical composition, and also mechanical properties such as Young’s modulus, yield stress, UTS, and strain at fracture. Fatigue can also be predicted directly by the load amplitudes and their characteristics, such as maximum effort, minimum effort, and relationship between efforts. Likewise, numerical models can be included that help the network predict durability; this can be done through nested networks including damage models. By using the results of experimental tests, the results are validated by having a low error percentage.

## 5. Conclusions

The battery housings of electric cars meet different conditions to protect the batteries from vibrations, impacts, and humidity. This could cause an unwanted battery reaction, but at the same time the housings protect against electromagnetic fields and in the case of fire retard the spread of flames. These loads generate accumulated damage during operation, so predicting the durability of the components is essential to prevent failures throughout the useful life of the vehicle and the battery.

Mechanical fatigue analysis is essential for predicting component failure during operation, particularly for safety-critical systems. While systems or vehicles may continue functioning until a failed component is replaced, the failure of safety systems can lead to significant risks, including damage to the mechanical components and vehicle occupants or pedestrians. Achieving sustainable mobility requires integrating advanced composite materials due to their superior weight-to-strength ratio, making them ideal for reducing vehicle weight while maintaining structural integrity. However, the long-term durability of these materials, especially in electric vehicle battery housings, remains underexplored.

Fatigue prediction in battery housings is crucial for ensuring long-term performance and preventing failures that may compromise the battery system’s operation and protective capabilities. Although AI is increasingly used to predict mechanical behavior, challenges remain, particularly in extrapolating predictions over extended lifespans and different load cases. These challenges necessitate extensive experimental testing to account for diverse driving conditions, environmental influences, and vehicle characteristics. Furthermore, AI models are data-intensive, requiring large datasets for training and validation, which presents an additional limitation.

Environmental factors and load changes are significant sources of variability in mechanical fatigue. As a component’s strength changes, the load it experiences can shift from a biaxial to a multiaxial stress state, particularly in electric vehicle battery housings. These loads come from various sources, including battery weight, vehicle acceleration forces, thermal loads, road impacts, and environmental factors, contributing to cumulative damage.

Neural networks have shown promise in improving the reliability of fatigue life predictions. However, accurate modeling must incorporate detailed information on material properties, load conditions, and manufacturing processes. Given their complexity, composite materials require precise models that reflect their anisotropic behavior and failure mechanisms, including matrix failure, reinforcement failure, and delamination. Dynamic models that update material properties based on real-time data are crucial to reducing variability and improving prediction accuracy.

The numerical model to be used to predict mechanical fatigue depends on the amount of information available or relevant to analyze the physical phenomenon. Although there are machine learning techniques that can predict phenomena more accurately, they eventually fall into overfitting because they require a large number of data. Sometimes networks such as the feedforward are better when there are few data, as is the case of mechanical properties such as the life curve (S–N), or few results for the training of the network.

The integration of AI in fatigue life prediction provides an opportunity to optimize the performance of lightweight components. However, AI models must be informed by the underlying physics of the materials rather than being treated as “black boxes”. By integrating AI with a physics-based approach, the accuracy of fatigue life predictions can be enhanced, leading to the development of lighter, more efficient components that extend vehicle range, reduce environmental impact, and allow for the recycling of components into systems with lower mechanical strength requirements.

The use of composite materials for the manufacture of battery housing for electric cars allows the development of innovative concepts to achieve the objective of reducing weight. However, an area of opportunity lies in the different recycling processes. Due to their nature, composite materials are difficult to recycle; it is important to consider the material recovery stage at the end of their useful life. Mechanical, chemical, and thermal methods can be implemented to separate the fibers from the matrix. In some cases, the matrix material can be incinerated for energy generation and at the same time the reinforcing fiber can be recovered. However, some of these processes can emit harmful gases, negatively impacting the environment.

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Moises Jimenez-Martinez www.mdpi.com