# The Response of Nanobeams with Temperature-Dependent Properties Using State-Space Method via Modified Couple Stress Theory

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## Abstract

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## 1. Introduction

## 2. Mathematical Modeling for Nonlocal Beam Theory

## 3. Formulation of the Problem

## 4. Analytical Solution

## 5. Laplace Transform Technique

## 6. Mathematical Method of State-Space Approach

## 7. Laplace Transforms Inversion

## 8. Numerical Results

## 9. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

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**Figure 2.**The physical field variables versus $x$ for different values of nonlocal parameter $\xi $.

**Figure 3.**The physical field variables versus $x$ when thermal conductivity $K$ is temperature-dependent.

**Figure 4.**The physical field variables versus $x$ for different values of small scale parameter $l$.

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**MDPI and ACS Style**

Abouelregal, A.E.; Marin, M.
The Response of Nanobeams with Temperature-Dependent Properties Using State-Space Method via Modified Couple Stress Theory. *Symmetry* **2020**, *12*, 1276.
https://doi.org/10.3390/sym12081276

**AMA Style**

Abouelregal AE, Marin M.
The Response of Nanobeams with Temperature-Dependent Properties Using State-Space Method via Modified Couple Stress Theory. *Symmetry*. 2020; 12(8):1276.
https://doi.org/10.3390/sym12081276

**Chicago/Turabian Style**

Abouelregal, Ahmed E., and Marin Marin.
2020. "The Response of Nanobeams with Temperature-Dependent Properties Using State-Space Method via Modified Couple Stress Theory" *Symmetry* 12, no. 8: 1276.
https://doi.org/10.3390/sym12081276