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A Novel Algorithm in Stochastic Chopped Carbon Fiber Composite Structure-A Study of RVE Size Effect and Homogenization Response of Directional Modulus

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Abstract

Low volume fraction (~ 3%) chopped fiber reinforced resin matrix composites have both structural toughness and strength. In this work, bisphenol F epoxy resin was used as the matrix, which has low viscosity and high strength, so it is not easy to make fiber agglomeration. The aim of this study is to explore the size effect and the response of directional modulus to the dispersion angle in the homogenization process with low volume fraction (~ 1.24%) chopped fiber reinforced resin based RVE structure. In this work, a novel mesoscale chopped fiber filled resin matrix random field algorithm named spatial direction sensitive Hotelling expansion (SDSHE) is proposed to study the change of composite properties caused by the random distribution of chopped carbon fiber (CCF) unit cells (UCs) in the representative volume element (RVE) structure. Meanwhile, several influencing factors to the size effect in these RVEs structure are also studied. A post-processing scheme of subdomain in RVE structure based on Gauss theorem is proposed for the first time. Namely, the volume integral of the continuously differentiable vector field (displacement, force, et al.) in the subdomain calculated by finite element method is extracted, which is used to calculate the average strain and stress in three-dimensional structure. The results shows that the SDHSE algorithm matches well with low volume fraction chopped carbon fiber reinforced composites, and based on the generated RVEs, the decay length t is 0-21 μm, the average directive modulus also appear respective sensitivities with rotation angles.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China, (No. 51561010). Jiangxi Province Key Research and Development Projects of China (No. 20192BBE50033, 20202BBEL53031, 20161BBH80062), Major Scientific Research Project of Jiangxi Academy of Sciences, China (No. 2020-YZD-2).

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Authors and Affiliations

  1. Institute of Applied Physics, Jiangxi Academy of Sciences, NanChang, 330012, China

    Jin Liu, Yuan Fu, Yun Chen, Jiang Jiang & You Liang Zhang

  2. School of Environmental Science and Engineering, Tianjin University, 300072, Tianjin, China

    Yuan Fu & Yin Xu

  3. Songshan Lake Materials Laboratory, Guangdong, 523808, Dongguan, China

    Yuan Fu & Shang Lin

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Appendix. Related algorithms of this work

Appendix. Related algorithms of this work

Algorithm 1 Space Direction-sensitive Hotelling expansion

Input: The RVE length L, Vccf, and Fiber length limits (Llower < Lccf < Lhigher,)

Output: The position Si and random orientation Rij (x, y, z, δ) of the fibers

1: Create a cubic matrix with size of L × L × L and oriented along the axes x’, y’ and z’ of the global Cartesian coordinate system of RVE x’ y’ z’;

2: Set one of matrix corners as the origin of the coordinate system x’ y’ z’;

3: Generate a given Vccf = vf (or fiber number Nf) of fibers within the UC by algorithm 2;

4: repeat:

5: Generate a new candidate fiber i with position Si with (xi, yi, zi) by Eq. (9) and a random length Llower < Li < Lhigher with Weibull distribution by algorithm 2;

6: for each generated fiber i do:

7: Assign a random fluctuation < \(\theta_{i}^{rd}(x,y,z,\delta)\)> with Eq. (16)

8: If the fiber i collides or intersects with the RVE surfaces then:

9: Create fiber i’s periodic image(s) ip using the method of algorithm 3;

10: end if

11: Check for penetration of fiber i with other fibers;

12: If fiber i without any penetration then:

13: Create the fiber and record Si and random fluctuation < \(\theta_{i}^{rd}(x,y,z,\delta)\)>;

14: else If fiber i penetrate with fiber j (i > j) then:

15: readjustment < \(\theta_{i}^{rd}(x,y,z,\delta)\)> and recheck penetration of fiber i;

16: If step 12 repeated over 5 times and penetration check failed then:

17: Delete Si and < \(\theta_{i}^{rd}(x,y,z,\delta)\)> and return to step 5;

18: until fiber i with no penetration;

19: end if

20: end if

21: Initialize the vf = vcurrent, Fiber number Nf = Ncurrent;

22: end for

23: until (vcurrent ≥ Vccf), Record Nf = N, and extract RVE model

Algorithm 2 Generate a given UC with wavy fiber and interphase

Input: Fiber radius dccf and average length with a random length Llower < Li < Lhigher, matrix length Lm/Li ∈ (0,1), crimp amplitude A, crimp period length ω, and radius of interphase di

Output: stochastic UCs

1: Initialize the vf = 0 and fiber number Nf = 0;

2: repeat:

3:for the fiber number i do:

4: Generate a candidate wavy fiber i with stochastic by a spatial random field H (x, y, z, δ) with Eq. 9. The length of fiber i submit to Weibull distribution, the waviness, thickness and interphase also generated according to input parameters;

5: Record the origin of UC coordinates Si with coordinate value (xi, yi, zi);

6: Create a cubic matrix of UC structure with size of (A + di) × (A + di) × (Li + Lm) and oriented along the axes xi in UC coordinate system. And set the point (xi-Lm/2, yi + (A + di)/2, zi + (A + di)/2) as matrix corners of UC;

7: Calculate the current Fredholm integral equation according Eq. 10 and 11 get current value of λi, and φi;

8: Update Eq. 9 with λi, and φi and get spatial random field H (x, y, z, δ) for fiber i + 1 and update current vf = vi, Nf = Ni;

9: end for

10: until (vf ≥ Vccf)

Algorithm 3 Periodic boundary conditions

Given a position Sf, length Lf of fiber f and length L of the cubic RVE, periodic images number Np of fiber f depends on the RVE surfaces number Ns that collide or intersect with fiber

f, and the periodic filling algorithm is given as follows:

(i) Np = 1 if and only if -Lf/2 < Sf,i < Lf/2 or L-Lf/2 ≤ Sf,i ≤ L + Lf/2 with i ∈ {1,2,3};

(ii) Np = 3 if and only if (-Lf/2 < Sf,i < Lf/2 or L-Lf/2 ≤ Sf,i ≤ L + Lf/2) and (-Lf/2 < Sf,j < Lf/2 or L-Lf/2 ≤ Sf,j ≤ L + Lf/2) with i ∈ {1,2}, j ∈ {2,3} and i ≠ j;

(iii) Np = 7 if and only if (-Lf/2 < Sf,i < Lf/2 or L-Lf/2 ≤ Sf,i ≤ L + Lf/2) and (-Lf/2 < Sf,j < Lf/2 or L-Lf/2 ≤ Sf,j ≤ L + Lf/2) and (-Lf/2 < Sf,k < Lf/2 or L-Lf/2 ≤ Sf,k ≤ L + Lf/2) with i = 1, j = 2, and k = 3

The position Sf,p of the periodic images of the fiber f are determined by equation:

Sf,p = Sf-vf,p with p ∈ {1,2,…,Np}

Where vf,p is shifted vector of the periodic image p regarding the fiber f. The components of vector vf,p takes the value of {-L,0,L}. Denote the vectors V1 = (-L,0,0), V2 = (0,-L,0), V3 = (0,0,-L), V4 = (L,0,0), V5 = (0,L,0), V6 = (0,0,L), and the components of the vector vf,p can be determined as:

(i) For the case of Np = 1

(1) if and only if -Lf/2 < Sf,i ≤ Lf/2: vf,p=1 = Vi with i ∈ {1,2,3};

(2) if and only if L-Lf/2 ≤ Sf,i < L + Lf/2:vf,p=1 = Vi+3 with i ∈ {1,2,3}

(ii) For the case of Np = 3

(1) if and only if -Lf/2 < Sf,i ≤ Lf/2 and -Lf/2 < Sf,j ≤ Lf/2:vf,p=1 = Vi,vf,p=2 = Vj and vf,p=3 = Vi + Vj with i ∈ {1,2}, j ∈ {2,3} and i ≠ j;

(2) if and only if L-Lf/2 ≤ Sf,i < L + Lf/2 and -Lf/2 < Sf,j ≤ Lf/2:vf,p=1 = Vi+3,vf,p=2 = Vj and vf,p=3 = Vi+3 + Vj with i ∈ {1,2}, j ∈ {2,3} and i ≠ j;

(3) if and only if -Lf/2 < Sf,i ≤ Lf/2 and L-Lf/2 ≤ Sf,j < L + Lf/2:vf,p=1 = Vi,vf,p=2 = Vj+3 and vf,p=3 = Vi + Vj+3 with i ∈ {1,2}, j ∈ {2,3} and i ≠ j;

(4) if and only if L-Lf/2 ≤ Sf,i < L + Lf/2 and L-Lf/2 ≤ Sf,j < L + Lf/2:vf,p=1 = Vi+3 and vf,p=2 = Vj+3 and vf,p=3 = Vi+3 + Vj+3 with i ∈ {1,2}, j ∈ {2,3} and i ≠ j

(iii) For the case of Np = 7

(1) if and only if -Lf/2 < Sf,i ≤ Lf/2 and -Lf/2 < Sf,j ≤ Lf/2 and -Lf/2 < Sf,k ≤ Lf/2:vf,p=1 = Vi,vf,p=2 = Vj, vf,p=3 = Vk, vf,p=4 = Vi + Vj, vf,p=5 = Vi + Vk, vf,p=6 = Vj + Vk and vf,p=7 = Vi + Vj + Vk with i = 1, j = 2 and k = 3;

(2) if and only if L-Lf/2 ≤ Sf,i < L + Lf/2 and -Lf/2 < Sf,j ≤ Lf/2 and -Lf/2 < Sf,k ≤ Lf/2:vf,p=1 = Vi+3,vf,p=2 = Vj, vf,p=3 = Vk, vf,p=4 = Vi+3 + Vj, vf,p=5 = Vi+3 + Vk, vf,p=6 = Vj + Vk and vf,p=7 = Vi+3 + Vj + Vk with i = 1, j = 2 and k = 3;

(3) if and only if -Lf/2 < Sf,i ≤ Lf/2 and L-Lf/2 ≤ Sf,j < L + Lf/2 and -Lf/2 < Sf,k ≤ Lf/2:vf,p=1 = Vi,vf,p=2 = Vj+3, vf,p=3 = Vk, vf,p=4 = Vi + Vj+3, vf,p=5 = Vi + Vk, vf,p=6 = Vj+3 + Vk and vf,p=7 = Vi + Vj+3 + Vk with i = 1, j = 2 and k = 3;

(4) if and only if -Lf/2 < Sf,i ≤ Lf/2 and -Lf/2 < Sf,j ≤ Lf/2 and L-Lf/2 ≤ Sf,k < L + Lf/2:vf,p=1 = Vi,vf,p=2 = Vj, vf,p=3 = Vk+3, vf,p=4 = Vi + Vj, vf,p=5 = Vi + Vk+3, vf,p=6 = Vj + Vk+3 and vf,p=7 = Vi + Vj + Vk+3 with i = 1, j = 2 and k = 3;

(5) if and only if L-Lf/2 ≤ Sf,i < L + Lf/2 and L-Lf/2 ≤ Sf,j < L + Lf/2 and -Lf/2 < Sf,k ≤ Lf/2:vf,p=1 = Vi+3, vf,p=2 = Vj+3, vf,p=3 = Vk, vf,p=4 = Vi+3 + Vj+3, vf,p=5 = Vi+3 + Vk, vf,p=6 = Vj+3 + Vk and vf,p=7 = Vi+3 + Vj+3 + Vk with i = 1, j = 2 and k = 3;

(6) if and only if L-Lf/2 ≤ Sf,i < L + Lf/2 and -Lf/2 < Sf,j ≤ Lf/2 and L-Lf/2 ≤ Sf,k < L + Lf/2:vf,p=1 = Vi+3,vf,p=2 = Vj, vf,p=3 = Vk+3, vf,p=4 = Vi+3 + Vj, vf,p=5 = Vi+3 + Vk+3, vf,p=6 = Vj + Vk+3 and vf,p=7 = Vi+3 + Vj + Vk+3 with i = 1, j = 2 and k = 3;

(7) if and only if -Lf/2 < Sf,i ≤ Lf/2 and L-Lf/2 ≤ Sf,j < L + Lf/2 and L-Lf/2 ≤ Sf,k < L + Lf/2:vf,p=1 = Vi,vf,p=2 = Vj+3, vf,p=3 = Vk+3, vf,p=4 = Vi + Vj+3, vf,p=5 = Vi + Vk+3, vf,p=6 = Vj+3 + Vk+3 and vf,p=7 = Vi + Vj+3 + Vk+3 with i = 1, j = 2 and k = 3;

(8) if and only if L-Lf/2 ≤ Sf,i < L + Lf/2 and L-Lf/2 ≤ Sf,j < L + Lf/2 and L-Lf/2 ≤ Sf,k < L + Lf/2:vf,p=1 = Vi+3, vf,p=2 = Vj+3, vf,p=3 = Vk+3, vf,p=4 = Vi+3 + Vj+3, vf,p=5 = Vi+3 + Vk+3, vf,p=6 = Vj+3 + Vk+3 and vf,p=7 = Vi+3 + Vj+3 + Vk+3 with i = 1, j = 2 and k = 3

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Liu, J., Fu, Y., Chen, Y. et al. A Novel Algorithm in Stochastic Chopped Carbon Fiber Composite Structure-A Study of RVE Size Effect and Homogenization Response of Directional Modulus. Appl Compos Mater 29, 565–595 (2022). https://doi.org/10.1007/s10443-021-09947-2

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