Elastic Constitutive Relationship of Metallic Materials Containing Grain Shape
Abstract
:1. Introduction
2. Elastic Constitutive Relationship of Metallic Materials Containing Grain Shape Effect
2.1. Grain Shape Function and Shape Coefficient Expression
2.2. Elastic Constitutive Relation Considering Average Grain Shape
3. Parameterization and Experiment of Metal Material Grain Shape
3.1. Extraction of Grain Image
3.2. Multi-String Method for Grain Shape Parameterization
3.3. Experimental Study on Grain Shape Coefficient
4. Parametric Experimental Study on Grain Shape Evolution
5. Conclusions
- Polycrystalline metallic materials are composed of small grains, and their constitutive relations must be related to the characteristics of grains, e.g., the average shape and orientation distribution of grains. According to the principle of no difference in the material frame, we choose the ratio of the arbitrary radius value in the grain to the average radius value of the grain, and establish the GSF, which can be used to describe the average shape of the grain. The GSF can be expanded into infinite series on the basis of Wigner D function, and the expanded coefficient is defined as the grain shape coefficient .
- We discuss the shape coefficients of special grains with weak anisotropy, and obtain the expression of the shape coefficients. Considering the average grain shape effect, we study the elastic constitutive relation of metallic multi-grain materials, and derive the analytical formula of elastic constitutive relation containing the grain shape coefficient .
- By using the power transformation, we improve the linear contrast of the digital image of the grain in polycrystalline metal materials, adopt the open and close operations in mathematical morphology to smooth the image, and then carry out histogram equalization and filtering noise removal processing to obtain a more ideal grain boundary. The approximate boundary of the image is extracted by the Canny operator, and then the boundary image is linearly expanded and refined. Then, using the internal function of MATLAB, a single, complete grain with clear and accurate boundary is obtained, and a total of 15 grain images of three grains are extracted under five loading steps.
- We discuss the mathematical description method of grain shape, and propose the multi-chord method to segment grains to represent the grain shape. When the grain shape is particularly irregular, the seven chord method is more reasonable. Furthermore, we carry out the experimental research on the shape function and shape coefficient of grains, fit the shape function of irregular grains, and prove that the shape function of grains can better describe the shape of irregular grains.
- Using the digital image analysis method of grains (e.g., the grain boundary processing, grain image acquisition, and grain shape parameterization), we track the shape evolution of the target grains in the metal materials under stress, obtain the parameterized representation of grain deformation, and analyze the relationship between the metal materials’ micro deformation and the materials’ macro mechanical properties.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Chord Number | 1 Chord | 3 Chord | 5 Chord | 7 Chord | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Grain Image Number | H. 1 | W. 2 | S.R. 3 | H. 1 | W. 2 | S.R. 3 | H. 1 | W. 2 | S.R. 3 | H. 1 | W. 2 | S.R. 3 |
GRAB1 | 67 | 45 | 1.489 | 65.67 | 45.33 | 1.449 | 67.4 | 46.4 | 1.453 | 68.14 | 46.71 | 1.459 |
GRAB5 | 69 | 44 | 1.568 | 70 | 44.67 | 1.567 | 68.8 | 44.2 | 1.557 | 70.57 | 45.13 | 1.564 |
GRAB10 | 70 | 44 | 1.591 | 70.33 | 44 | 1.598 | 70.8 | 44 | 1.609 | 71.28 | 44.85 | 1.59 |
GRAB14 | 75 | 41 | 1.829 | 75.67 | 40.67 | 1.861 | 75.6 | 41 | 1.844 | 76.11 | 40.14 | 1.896 |
GRAB15 | 77 | 40 | 1.925 | 77 | 40.33 | 1.909 | 77 | 40.6 | 1.897 | 77.57 | 39.42 | 1.968 |
Chord Number | 1 Chord | 3 Chord | 5 Chord | 7 Chord | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Grain Image Number | H. 1 | W. 2 | S.R. 3 | H. 1 | W. 2 | S.R. 3 | H. 1 | W. 2 | S.R. 3 | H. 1 | W. 2 | S.R. 3 |
GRAB1 | 51 | 44 | 1.159 | 53 | 39.67 | 1.336 | 53.8 | 40.7 | 1.322 | 53.37 | 40.93 | 1.304 |
GRAB5 | 56 | 45 | 1.244 | 53 | 40 | 1.325 | 53.6 | 40.6 | 1.321 | 53.73 | 40.87 | 1.314 |
GRAB10 | 50 | 46 | 1.087 | 51 | 41 | 1.244 | 53.2 | 40.4 | 1.317 | 53.84 | 40.23 | 1.338 |
GRAB14 | 59 | 46 | 1.283 | 55.33 | 40.67 | 1.361 | 54.4 | 41.1 | 1.324 | 54.33 | 41.54 | 1.307 |
GRAB15 | 55 | 45 | 1.222 | 55.67 | 40.33 | 1.38 | 55.2 | 41.7 | 1.324 | 55.87 | 41.67 | 1.341 |
Chord Number | 1 Chord | 3 Chord | 5 Chord | 7 Chord | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Grain Image Number | H. 1 | W. 2 | S.R. 3 | H. 1 | W. 2 | S.R. 3 | H. 1 | W. 2 | S.R. 3 | H. 1 | W. 2 | S.R. 3 |
GRAB1 | 103 | 109 | 0.945 | 106.67 | 109.33 | 0.976 | 102.8 | 108.6 | 0.947 | 103.57 | 109.17 | 0.949 |
GRAB5 | 105 | 108 | 0.972 | 108 | 107.67 | 1.003 | 106.2 | 108.2 | 0.982 | 104.58 | 108.71 | 0.962 |
GRAB10 | 108 | 106 | 1.019 | 109.33 | 106 | 1.031 | 108.4 | 105.8 | 1.025 | 109.17 | 105.14 | 1.038 |
GRAB14 | 111 | 103 | 1.078 | 110.67 | 103.33 | 1.071 | 112.2 | 102.6 | 1.094 | 111.34 | 102.47 | 1.087 |
GRAB15 | 113 | 100 | 1.13 | 114.33 | 100 | 1.143 | 113 | 99.2 | 1.139 | 114.02 | 99.42 | 1.147 |
Point Number | X | Y | Point Number | X | Y |
---|---|---|---|---|---|
Transverse point 1 | −22 | 27 | Transverse point 8 | −1 | 27 |
Transverse point 2 | −27 | 19 | Transverse point 9 | 0 | 19 |
Transverse point 3 | −25 | 11 | Transverse point 10 | 25 | 11 |
Transverse point 4 | −20 | 1 | Transverse point 11 | 24 | 1 |
Transverse point 5 | −15 | −7 | Transverse point 12 | 18 | −7 |
Transverse point 6 | −13 | −15 | Transverse point 13 | 17 | −15 |
Transverse point 7 | −11 | −23 | Transverse point 14 | 18 | −23 |
Point Number | X | Y | Point Number | X | Y |
---|---|---|---|---|---|
Vertical point 1 | −20 | 28 | Vertical point 8 | −20 | −1 |
Vertical point 2 | −13 | 34 | Vertical point 9 | −13 | −18 |
Vertical point 3 | −6 | 33 | Vertical point 10 | −6 | −27 |
Vertical point 4 | 0 | 21 | Vertical point 11 | 0 | −30 |
Vertical point 5 | 7 | 19 | Vertical point 12 | 7 | −31 |
Vertical point 6 | 14 | 16 | Vertical point 13 | 14 | −31 |
Vertical point 7 | 21 | 13 | Vertical point 14 | 21 | −5 |
Load (N) | Shape Parameter | Load (N) | Shape Parameter | Load (N) | Shape Parameter |
---|---|---|---|---|---|
0 | 0.717802 | 1600 | 0.713215 | 2800 | 0.695813 |
200 | 0.721352 | 1800 | 0.720267 | 3000 | 0.67768 |
600 | 0.721482 | 2000 | 0.703568 | 3200 | 0.681628 |
1000 | 0.717047 | 2200 | 0.723909 | 3400 | 0.687682 |
1200 | 0.717419 | 2400 | 0.704046 | 3600 | 0.622458 |
1400 | 0.711743 | 2600 | 0.718436 | 3800 | 0.545502 |
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Lan, Z.; Shao, H.; Zhang, L.; Yan, H.; Huang, M.; Zhao, T. Elastic Constitutive Relationship of Metallic Materials Containing Grain Shape. Crystals 2022, 12, 1768. https://doi.org/10.3390/cryst12121768
Lan Z, Shao H, Zhang L, Yan H, Huang M, Zhao T. Elastic Constitutive Relationship of Metallic Materials Containing Grain Shape. Crystals. 2022; 12(12):1768. https://doi.org/10.3390/cryst12121768
Chicago/Turabian StyleLan, Zhiwen, Hanjie Shao, Lei Zhang, Hong Yan, Mojia Huang, and Tengfei Zhao. 2022. "Elastic Constitutive Relationship of Metallic Materials Containing Grain Shape" Crystals 12, no. 12: 1768. https://doi.org/10.3390/cryst12121768