## 4 Discussion
For example, some researchers have used finite element methods to analyze the effects of different thicknesses, embedding positions, and curvatures of carbon plates on foot biomechanics [40; 41]. However, the material properties of the carbon plates used were isotropic, and the impact of these factors on the FLBS of the shoe sole was not further analyzed. This study validates the method of increasing stiffness by stacking additional carbon plates, as proposed in previous research, through finite element simulations. Unlike earlier FE simulations of carbon plate running shoes, this study employs a more realistic approach by using carbon fiber composite materials and laminate structures during micro-scale modeling, further simulating the mechanical properties of the carbon plates. This study also finds that when the thickness is consistent, the stiffness increases slightly with the increase in the curvature radius, and this phenomenon is observed in both carbon plate and 'plate-sole' systems. Previous research has shown that changes in carbon plate curvature affect the biomechanics of the ankle and metatarsophalangeal joints, and an appropriate curvature radius can reduce the net energy loss at the metatarsophalangeal joint [42]. [PERSON] [41] used finite element methods to investigate the effects of different thicknesses of flat and curved carbon plates on the stress and strain of foot and metatarsal soft tissues. It was found that carbon plates with a larger forefoot curvature (i.e., smaller curvature radius) can reduce peak pressure in the forefoot. Compared to flat carbon plates, when wearing running shoes with forefoot-curved carbon plates, the thicker midsole beneath the metatarsophalangeal joint may play a significant role in cushioning impact and absorbing energy during running [43]. The present research shows that carbon plates with a smaller curvature radius exhibit lower FLBS, and for the human foot, this lower FLBS helps reduce the peak pressure in the forefoot. This may be due to changes in the carbon plate's bending structure and the distribution of midsole material as the curvature radius increases. Regarding the relationship between the stiffness of the carbon plate and the 'plate-sole' system, this study finds that regardless of changes in thickness or curvature radius, the stiffness of the 'plate-sole' is significantly lower than that of the carbon plate when the plate is embedded into the sole. However, changes in thickness or curvature radius affect the extent of stiffness reduction. Specifically, when the thickness remains unchanged, increasing the curvature radius slightly reduces the rate of stiffness decrease in the 'plate-sole'. In contrast, when the curvature radius remains constant, increasing the carbon plate thickness leads to a progressively larger decrease in the stiffness of the 'plate-sole'. In other words, for the same thickness, the larger the curvature radius, the smaller the stiffness reduction in the 'plate-sole'; conversely, for the same curvature radius, the thicker the carbon plate, the greater the reduction in the stiffness of the 'plate-sole'. This suggests that the overall response of the 'plate-sole' is not a simple sum of the individual contributions from the shoe and the carbon plate. Instead, the shoe material modifies the overall stiffness of the system, leading to a noticeable reduction that increases significantly with the carbon plate thickness. This non-additive response highlights the significant role of the shoe material in altering the mechanical properties of the carbon plate, which is a crucial consideration for optimizing shoe design. This study has several limitations. Firstly, the model assumes linear elastic material behavior, but under large deformations, materials may exhibit nonlinear behaviors such as plasticity or viscoelasticity. Additionally, time-dependent effects, such as viscoelasticity, were not considered in this study, but they could become significant under different loading rates or longer durations.