基于Mindlin横剪变形理论的功能梯度板热屈曲分析

Thermal buckling analysis of functionally graded plates based on Mindlin transverse shear deformable theory

Mindlin板理论对挠度和转角采用各自独立的场函数以反映一阶横向剪切变形,具有简明的表达式,适于建立功能梯度板的热屈曲分析模型。本文假设功能梯度材料沿板厚方向的分布为幂函数,采用混合定律和Mori-Tanaka方法计算功能梯度板的均质化等效力学性能。基于Mindlin板理论和von Karman应变-位移关系导出功能梯度板的非线性静力平衡方程,采用3结点三角形MIN3单元建立功能梯度板热屈曲的有限元模型,并分析了典型功能梯度板的热屈曲稳定性和热后屈曲变形。陶瓷-金属功能梯度板的数值计算结果表明:材料分布幂指数越大,即组份中陶瓷体积含量越少、金属体积含量越多,则陶瓷-金属功能梯度板的屈曲温度越低,且热后屈曲变形越大。这与陶瓷的弹性模量比金属的弹性模量大,但金属的热膨胀系数比陶瓷高有关;固支功能梯度板的热屈曲变形幅值比简支功能梯度板的热屈曲变形幅值低,但偏差量随着材料分布幂指数的增大略微降低。

The Mindlin plate theory uses two separate field functions for deflection and rotation, so the formulations in Mindlin plate are simple, furthermore, the first-order transverse shear deformation is considered in this theory. So the Mindlin plate theory is suitable for establishing the functionally graded(FG) plate model for thermal buckling analysis. In this paper, the power law distribution of constituent materials along the plate thickness is assumed, the effective homogeneous properties are calculated by the rule of mixture and Mori-Tanaka method, the Mindlin plate theory and von Karman strain-displacement relations are utilized to deduce the nonlinear static equilibrium equations of FG plates, and a three-node triangular MIN3 element is adopted to establish the finite element model for thermal buckling analysis of FG plates. The thermal buckling stability and post-buckling deflection of ceramic-metal FG plates are analyzed using this model, and numerical results show that with the increase of the power law exponent of material gradation, the volume fraction of ceramics decreases while the volume fraction of metals increases, and then the critical buckling temperature decreases and the thermal post-buckling deflection increases. This is partly due to the fact that the higher elastic modulus and lower thermal expansion coefficient of ceramics than that of metals. The thermal post-buckling deflection of clamped FG plate is lower than that of simply supported FG plate. However, the effect of boundary conditions on the thermal buckling deflection slightly decreases with the increase of the power law exponent of material gradation.