A New Approach of Mathematical Analysis of Structure of Graphene as a Potential Material for Composites
Abstract
1. Introduction
2. Assumptions for Modeling the Graphene Structure
3. Basic Model of Graphene Cell (Calculations of Strains)
4. Numerical Model of Graphene Band
5. The Strain in y-Direction
6. Results
7. Strains in Both Directions
8. Summary
Author Contributions
Funding
Conflicts of Interest
Nomenclature
Dimension between atoms bond | |
Length of graphene bar | |
Diameter of graphene bar | |
Thickness of single graphene sheet for one layer | |
Length of segment | |
Tension force for armchair edge (x-direction) | |
Tension force for zigzag edge (y-direction) | |
Angle between graphene bar | |
Increment of elongation of segment caused by tension | |
Increment of elongation of bar | |
Total elongation caused by bending | |
Young’s modulus of segment of graphene sheet | |
Young’s modulus of graphene bar | |
Moment of inertia of graphene bar | |
Cross-section area | |
Total increment of elongation due to bending and tension | |
Length of graphene segment | |
Poisson’s ratio for graphene bar | |
Poisson’s ratio for graphene cut-out for x-direction | |
Poisson’s ratio for graphene cut-out for y-direction | |
Strain of segment in x-direction caused by tension and bending in x-direction | |
Strain of segment in y-direction caused by tension and bending in x-direction | |
Strain in x-direction of alternative segment in x-direction | |
Strain of segment in y-direction caused by tension and bending in y-direction | |
Strain of segment in x-direction caused by tension and bending in y-direction | |
Strain in y-direction of alternative segment in y-direction | |
νxy | Poisson’s ratio of graphene calculated for two perpendicular forces |
kxy | Ratio of forces |
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Number of Calculation Variant | Direction of Tension | l (Å) | a (Å) | h (Å) | d (Å) | Y (Tpa) | (-) | EGr (TPa) (Equation (10)) | (-) (Equation (12)) (Equation (17)) | (-) FEM |
---|---|---|---|---|---|---|---|---|---|---|
1 | x | 1.42 | 2.46 | 0.44 | 0.44 | 1.15 | 0.3 | 40.95 | 0.800 | 0.812 |
2 | x | 1.42 | 2.46 | 0.75 | 0.75 | 1.15 | 0.3 | 13.09 | 0.663 | 0.666 |
3 | x | 1.42 | 2.46 | 0.89 | 0.89 | 1.15 | 0.3 | 9.630 | 0.580 | 0.620 |
4 | x | 1.42 | 2.46 | 1.00 | 1.00 | 1.15 | 0.3 | 7.932 | 0.546 | 0.59 |
5 | x | 1.42 | 2.46 | 1.42 | 1.42 | 1.15 | 0.3 | 4.727 | 0.463 | 0.515 |
6 | x | 1.42 | 2.46 | 2.00 | 2.00 | 1.15 | 0.3 | 3.058 | 0.411 | 0.471 |
7 | x | 1.42 | 2.46 | 2.42 | 2.42 | 1.15 | 0.3 | 2.448 | 0.392 | 0.453 |
8 | y | 1.42 | 2.46 | 0.44 | 0.44 | 1.15 | 0.3 | 40.94 | 0.800 | 0.791 |
9 | y | 1.42 | 2.46 | 0.75 | 0.75 | 1.15 | 0.3 | 13.08 | 0.633 | 0.618 |
10 | y | 1.42 | 2.46 | 0.89 | 0.89 | 1.15 | 0.3 | 9.620 | 0.580 | 0.564 |
11 | y | 1.42 | 2.46 | 1.00 | 1.00 | 1.15 | 0.3 | 7.930 | 0.546 | 0.527 |
12 | y | 1.42 | 2.46 | 1.42 | 1.42 | 1.15 | 0.3 | 4.726 | 0.463 | 0.445 |
13 | y | 1.42 | 2.46 | 2.00 | 2.00 | 1.15 | 0.3 | 3.057 | 0.411 | 0.394 |
14 | y | 1.42 | 2.46 | 2.42 | 2.42 | 1.15 | 0.3 | 2.448 | 0.392 | 0.373 |
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Jaroniek, M.; Czechowski, L.; Kaczmarek, Ł.; Warga, T.; Kubiak, T. A New Approach of Mathematical Analysis of Structure of Graphene as a Potential Material for Composites. Materials 2019, 12, 3918. https://doi.org/10.3390/ma12233918
Jaroniek M, Czechowski L, Kaczmarek Ł, Warga T, Kubiak T. A New Approach of Mathematical Analysis of Structure of Graphene as a Potential Material for Composites. Materials. 2019; 12(23):3918. https://doi.org/10.3390/ma12233918
Chicago/Turabian StyleJaroniek, Mieczysław, Leszek Czechowski, Łukasz Kaczmarek, Tomasz Warga, and Tomasz Kubiak. 2019. "A New Approach of Mathematical Analysis of Structure of Graphene as a Potential Material for Composites" Materials 12, no. 23: 3918. https://doi.org/10.3390/ma12233918
APA StyleJaroniek, M., Czechowski, L., Kaczmarek, Ł., Warga, T., & Kubiak, T. (2019). A New Approach of Mathematical Analysis of Structure of Graphene as a Potential Material for Composites. Materials, 12(23), 3918. https://doi.org/10.3390/ma12233918