Abstract

This study proposes a computationally efficient approach about the dynamic response of a homogeneous, transversely isotropic, thermoelastic micro-beam resonator subjected to time-dependent thermal loading. Due to the shortcomings of power law distributions in fractional derivatives, the usage of some other forms of derivatives with few other kernel functions are proposed. With this motivation, the heat transport equation is defined in an integral form of a common derivative on a slipping interval by incorporating the three-phase-lag memory-dependent heat transfer which is modeled based on Euler-Bernoulli beam theory. The beam is assumed to be clamped-clamped conditions on its axial ends. Employing the Laplace transform technique, the analytical solutions have been obtained. Further, incorporating residual calculus, the inversion of the transformed solutions are performed. The analytical expressions for the deflection has been computed numerically for a Silicon micro-beam using Maple software. The new approach is able to portray material heterogeneities and the non-locality of the new kernel allows the full describing of the memory within structure and media with different scales, which cannot be described by classical fractional derivative and also that of the memory-dependent derivative.

摘要

本研究提出了一种计算有效的方法，该方法可以计算均匀、横向各向同性、热弹性微梁谐振器在时间相关热载荷下的动态响应。由于分数导数中幂律分布的缺点，提出了使用一些其他形式的导数，而其他核函数很少。有了这个动机，通过结合基于 Euler-Bernoulli 梁理论建模的三相滞后记忆相关传热，将传热方程定义为滑动区间的共同导数的积分形式。假设梁在其轴向端处于夹紧-夹紧状态。采用拉普拉斯变换技术，得到了解析解。此外，结合残差演算，执行转换后的解决方案的反演。已使用 Maple 软件对硅微梁的偏转解析表达式进行了数值计算。新方法能够描述材料的异质性，并且新内核的非局部性允许完整描述不同尺度的结构和介质内的记忆，这是经典分数导数和记忆相关导数无法描述的.