Micromechanical Modeling of Fiber-Reinforced Composites with Statistically Equivalent Random Fiber Distribution
2. Algorithm Development
- Consider a rectangular objective window with length a and width b, a random fiber is first created in the central area of the window with the coordinates of the center (x1, y1), as shown in Figure 2a. The diameter of fibers can be identical or variable, and, in this study, the values are drawn from the experimentally measured diameter distribution. The RVE size must be large enough to be representative in predicting the mechanical properties of the composite material. As from the study of Trias et al. , a minimum length and width of 50 times of the fiber radius is recommended.
- Following the realistic nearest neighbor distribution, a nearest neighbor distance is assigned to the newly generated fiber (for example, fiber #1 for the first cycle).
- A new fiber (fiber #2) is then generated around the current reference fiber (fiber #1). The position of the new fiber is calculated based on the inter-fiber distance d12 and the orientation angle θ12 (see Figure 2b), which are determined based on the probability equation and the distribution of nearest fiber distance and orientation angle.
- In the original NNA method, the newly generated fiber (fiber #2) is checked and if the inter-fiber distance between it and the reference fiber is less than the nearest neighbor distance of the reference fiber, it will be regenerated. This rule could result in the increase of the inter-fiber distance and a relatively low frequency of small inter-fiber distance distribution. To overcome this issue, a random number P with the value between 0 and 1 is generated and is associated with a probability selection rule. If P is less than a predefined threshold value P0 (0 < P0 < 0.3), the newly generated fiber will be kept even though it does not satisfy the nearest neighbor distance. P0 is taken as 0.15 in this work.
- Steps 2–4 are repeated to generate more fibers around the current reference fiber, until the maximum iteration numbers (usually between 3 and 5) are reached. The inter-fiber distances between each newly generated fiber and all the existing fibers need to be examined separately. For example, in Figure 2c, the inter-fiber distance between fiber #1 and fiber #4 should be larger than the nearest neighbor distance of fiber #1. However, the probability selection rule of step 4 always applies.
- Followed by the generation of each new fiber, the frequency of the nearest neighbor distance of fibers needs to be updated. The equation is expressed as
- The current cycle ends when there is no more new fiber generation around the current reference fiber. The reference fiber is then switched to a different fiber based on the fiber number, e.g., fiber #2 will be the reference fiber of the second cycle.
- The above process is repeated until the requested fiber volume ratio or maximum number of cycles is reached.
3. Statistical Characterization
3.1. Nearest Neighbor Distance
3.2. Nearest Neighbor Orientation
3.3. Ripley’s K Function
3.4. Radial Distribution Function
4. Prediction of Elastic Properties
Conflicts of Interest
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|Properties||Carbon Fiber T300||Epoxy Resin 914C|
|Longitudinal Young’s Modulus E1 (GPa)||230||4|
|Transverse Young’s Modulus E2 (GPa)||15||4|
|Longitudinal Shear Modulus G12 (GPa)||15||1.481|
|Transverse Shear Modulus G23 (GPa)||7||1.481|
|Major Poisson’s Ratio ν12||0.2||0.35|
|Transverse Poisson’s Ratio ν23||0.07||0.35|
|Properties||E1 (GPa)||E2 (GPa)||G12 (GPa)||G23 (GPa)||ν12||ν23|
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Wang, W.; Dai, Y.; Zhang, C.; Gao, X.; Zhao, M. Micromechanical Modeling of Fiber-Reinforced Composites with Statistically Equivalent Random Fiber Distribution. Materials 2016, 9, 624. https://doi.org/10.3390/ma9080624
Wang W, Dai Y, Zhang C, Gao X, Zhao M. Micromechanical Modeling of Fiber-Reinforced Composites with Statistically Equivalent Random Fiber Distribution. Materials. 2016; 9(8):624. https://doi.org/10.3390/ma9080624Chicago/Turabian Style
Wang, Wenzhi, Yonghui Dai, Chao Zhang, Xiaosheng Gao, and Meiying Zhao. 2016. "Micromechanical Modeling of Fiber-Reinforced Composites with Statistically Equivalent Random Fiber Distribution" Materials 9, no. 8: 624. https://doi.org/10.3390/ma9080624