Size-Dependent Flexural Analysis of Thick Microplates Using Consistent Couple Stress Theory
Abstract
1. Introduction
2. Solution Procedure Based on Couple Stress Theory
2.1. Governing Equations
2.2. Analytical Solution
2.2.1. System of Equations
2.2.2. Determination of Coefficients, aij
- In Equations (A1), (A11) and (A24), the same coefficients of can be obtained. Thus, one can obtain .
- In Equations (A4), (A14) and (A21), the same coefficients of can be obtained. Thus, one can obtain .
- In Equations (A3), (A13) and (A26), although the coefficients of are different, these equations are identical.
- In Equations (A6), (A16) and (A23), although the coefficients of are different, these equations are identical.
- In Equations (A2), (A12) and (A25), the same coefficients of can be obtained. Thus, one can obtain .
- In Equations (A5), (A15) and (A22), the same coefficients of can be obtained. Thus, one can obtain .
- In Equations (A8), (A18) and (A30), although the coefficients of are different, these equations are identical.
- In Equations (A10), (A20) and (A28), the same coefficients of can be obtained.
- In Equations (A9), (A19) and (A27), the same coefficients of can be obtained.
- Solving the equations, the coefficients are determined as follows:
2.2.3. Boundary Conditions
- Bottom surface: .
- Top surface: .
3. Classical Plate Theory
4. Numerical Results and Discussion
5. Conclusions
- 1-
- The material length-scale parameter has a significant influence on the bending behavior of the microplate and cannot be neglected.
- 2-
- The microplate is more flexible and experiences higher stress levels compared to classical elasticity predictions.
- 3-
- The absolute values of displacements decrease as the thickness of the microplate increases.
- 4-
- For a constant h/l ratio, the displacements calculated using couple stress theory are greater than those predicted by classical theory.
- 5-
- As the thickness of the microplate increases, the absolute values of stresses decrease.
- 6-
- Unlike classical theories, and are not identical.
- 7-
- Compared to classical elasticity theory, the out-of-plane stresses attain larger values, and these components increase further as the thickness decreases.
- 8-
- An increase in thickness results in a reduction in the absolute value of .
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A
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Elasticity Theory | Classical Plate Theory | |||||
---|---|---|---|---|---|---|
9.61 × 10−7 | 9.61 × 10−7 | 8.19 × 10−6 | 4.46 × 104 | 4.46 × 104 | 1.97 × 10−9 | |
1.21 × 10−7 | 1.21 × 10−7 | 5.31 × 10−7 | 4.46 × 104 | 4.46 × 104 | 4.91 × 10−10 |
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Shaban, M.; Minaeii, S.; Kalhori, H. Size-Dependent Flexural Analysis of Thick Microplates Using Consistent Couple Stress Theory. J. Compos. Sci. 2025, 9, 142. https://doi.org/10.3390/jcs9030142
Shaban M, Minaeii S, Kalhori H. Size-Dependent Flexural Analysis of Thick Microplates Using Consistent Couple Stress Theory. Journal of Composites Science. 2025; 9(3):142. https://doi.org/10.3390/jcs9030142
Chicago/Turabian StyleShaban, Mahdi, Saeid Minaeii, and Hamed Kalhori. 2025. "Size-Dependent Flexural Analysis of Thick Microplates Using Consistent Couple Stress Theory" Journal of Composites Science 9, no. 3: 142. https://doi.org/10.3390/jcs9030142
APA StyleShaban, M., Minaeii, S., & Kalhori, H. (2025). Size-Dependent Flexural Analysis of Thick Microplates Using Consistent Couple Stress Theory. Journal of Composites Science, 9(3), 142. https://doi.org/10.3390/jcs9030142