Chen Wenxing, Dai Shuyang, Zheng Baojuan
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China.
Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan 430072, China.
Micromachines (Basel). 2022 May 10;13(5):753. doi: 10.3390/mi13050753.
Elastic materials include metal plates, rubber, foam, airbags and so on, which have a good buffer effect, toughness and strong recovery ability. In this paper, the deformation and thermal diffusion of 2D and 3D thin plates are studied. Two models are established for the deformation of 2D thin plates. The bending deformation equation of rectangular and circular plates is derived, and the semi-analytical solution of the deflection function w(x,y) is found through the Fourier series approximation in the polar coordinate. The consistencies of the numerical solution and the theoretical solution are verified by numerical method. Then, we find that the factors affecting the deformation are related to the Young's modulus, load, plate length and deformation factor α of the material. In a separate temperature physics field, we establish a heat conduction model of 2D graphene film. Three numerical schemes of the transient heat conduction equation of FDM-FEM are given. In contrast, this paper uses the implicit Euler method to discrete the time term. Furthermore, we compared the difference between the adiabatic condition and the convection condition by the graphical method and the curve trend. The results show that the temperature near the adiabatic boundary is higher. Finally, we proposed a 3D dynamic thermal-mechanical coupling model (3D-DTMCM) that has been established. A laser heating monocrystalline silicon sheet with periodic motion formula is given. The temperature radiation of the laser heat source has Gaussian distribution characteristics. Our proposed model can dynamically determine Young's modulus with a variable temperature. The numerical results show that the higher the temperature is, the higher the strain energy density of the plate is. In addition, the deformation amplitude of the plates in the coupling field is larger than that in the single mechanical field. Finally, we also discussed the stress field distribution of mixed cracks under high temperature and high load. Our research provides theoretical support for the deformation of different plates, and also reflects the value of the coupled model in practical applications.
弹性材料包括金属板、橡胶、泡沫、安全气囊等，它们具有良好的缓冲效果、韧性和较强的恢复能力。本文研究了二维和三维薄板的变形及热扩散。针对二维薄板的变形建立了两个模型。推导了矩形板和圆形板的弯曲变形方程，并通过极坐标下的傅里叶级数近似求出了挠度函数(w(x,y))的半解析解。用数值方法验证了数值解与理论解的一致性。然后，发现影响变形的因素与材料的杨氏模量、载荷、板长和变形因子(\alpha)有关。在单独的温度物理场中，建立了二维石墨烯薄膜的热传导模型。给出了有限差分法-有限元法瞬态热传导方程的三种数值格式。相比之下，本文采用隐式欧拉方法对时间项进行离散。此外，通过图形法和曲线趋势比较了绝热条件和对流条件之间的差异。结果表明，绝热边界附近的温度较高。最后，提出了一个已建立的三维动态热-机械耦合模型（3D-DTMCM）。给出了具有周期性运动公式的激光加热单晶硅片。激光热源的温度辐射具有高斯分布特性。我们提出的模型可以动态地确定随温度变化的杨氏模量。数值结果表明，温度越高，板的应变能密度越高。此外，耦合场中板的变形幅度大于单机械场中的变形幅度。最后，还讨论了高温和高载荷下混合裂纹的应力场分布。我们的研究为不同板的变形提供了理论支持，也体现了耦合模型在实际应用中的价值。
