Size-dependent thermoelastic initially stressed micro-beam due to a varying temperature in the light of the modified cou
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APPLIED MATHEMATICS AND MECHANICS (ENGLISH EDITION) https://doi.org/10.1007/s10483-020-2676-5
Size-dependent thermoelastic initially stressed micro-beam due to a varying temperature in the light of the modified couple stress theory∗ A. E. ABOUELREGAL1,2,† 1. Department of Mathematics, College of Science and Arts, Jouf University, Al-Qurayyat 77423, Saudi Arabia; 2. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt (Received Mar. 24, 2020 / Revised Aug. 3, 2020) Abstract The bending of the Euler-Bernoulli micro-beam has been extensively modeled based on the modified couple stress (MCS) theory. Although many models have been incorporated into the literature, there is still room for introducing an improved model in this context. In this work, we investigate the thermoelastic vibration of a micro-beam exposed to a varying temperature due to the application of the initial stress employing the MCS theory and generalized thermoelasticity. The MCS theory is used to investigate the material length scale effects. Using the Laplace transform, the temperature, deflection, displacement, flexure moment, and stress field variables of the micro-beam are derived. The effects of the temperature pulse and couple stress on the field distributions of the micro-beam are obtained numerically and graphically introduced. The numerical results indicate that the temperature pulse and couple stress have a significant effect on all field variables. Key words thermoelectricity, micro-beam, initial stress, temperature pulse, modified couple stress (MCS) theory Chinese Library Classification O321, O327, O343 2010 Mathematics Subject Classification 74H45, 70J10, 74B10
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Introduction
More recently, small scale and nano-scale structures have been used in several design devices such as nuclear drive and switches. A few models are developed for small scale and nanostructures, including strain gradient and couple stress. The modified couple stress (MCS) theory can be seen as an exceptional example of the strain gradient theory. The MCS theory considers the revolution as a variable in determining curvature, while the strain inclination theory considers the strain to be variable in curvature pronounce[1–4] . The classical elasticity theory is not suitable to capture the effect of the size of the microstructure. In addition, it is appropriate to study the material behavior on a large scale. As ∗ Citation: ABOUELREGAL, A. E. Size-dependent thermoelastic initially stressed micro-beam due to a varying temperature in the light of the modified couple stress theory. Applied Mathematics and Mechanics (English Edition), 41(12), 1805–1820 (2020) https://doi.org/10.1007/s10483-020-2676-5 † Corresponding author, E-mail: [email protected] c The Author(s) 2020
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the size of the study decreases, the accuracy of the classical theory diminishes and therefore its expectation of the material behavior on the micro- and nano-scales does not match with experimental results. It is observed that the explanation behind this
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