热环境中功能梯度圆板的非线性动力响应分析

研究了热环境中功能梯度圆板在横向简谐激励作用下的非线性动力响应和动应力问题。针对陶瓷-金属功能梯度圆板, 考虑几何非线性、材料物理性质参数随温度变化及材料组分沿厚度方向按幂律分布的情况, 应用虚功原理给出了热载荷与横向简谐载荷共同作用下的非线性振动偏微分方程。在固支无滑动的边界条件下, 利用伽辽金法得到了达芬型非线性强迫振动方程。通过数值算例, 给出了关于体积分数指数的分岔图, 相图、Poincare映射等响应图以及动应力变化规律图, 讨论了材料体积分数指数和温度场对功能梯度圆板非线性动力响应的影响。结果表明: 热环境中功能梯度圆板随体积分数指数的变化可使系统出现周期响应、倍周期响应和混沌响应。功能梯度圆板中心处动应力在系统发生分岔或出现混沌响应时出现大幅变化, 而且在混沌响应时具有不可预测性。 

Nonlinear dynamic response of circular functionally graded plate in thermal environment

The nonlinear dynamic responses and the dynamic stress of a circular plate in thermal environment were studied. The effect of geometric nonlinearity and temperature-dependent material properties were both taken into account. The material properties of the functionally graded plate were assumed to vary continuously through the thickness according to a power law distribution of the volume fraction of the constituents. Using the principle of virtual work, the nonlinear partial differential equations of functionally graded plate subjected to transverse harmonic excitation and thermal loads were derived. For the circular plate with clamped immovable edge, the Duffing nonlinear forced vibration equation was deduced by using Galerkin method. Through the numerical example given in the paper, the bifurcation diagram for material's volume fraction index, phasepotrait, Poincare map and the dynamic stress variation were plotted. Besides, the influences of materials volume fraction index and thermal loads on the nonlinear dynamic response of functionally graded plate were discussed. The results show that periodic, multiplier periodic and chaotic motions exist for the functionally graded plate with the change of the volume fraction index. The dynamic stress at the center of the circular plate varies sharply when the system appears bifurcation or chaos and becomes unpredictable when the system appears chaotic motions.