Snap-through Buckling Analysis of P-FGM Shallow Spherical Shells under Thermomechanical Loads
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摘要: 基于经典壳体理论和Sanders非线性应变-位移关系，导出了幂律型功能梯度材料(P-FGM)扁球壳在热-机械荷载作用下的几何非线性常微分控制方程。推导过程考虑了沿厚度存在一维热传导温度场和法向均布荷载作用。采用打靶法求解了由控制方程和固定夹紧边界条件构成的两点边值问题。得到了FGM扁球壳的一些典型的屈曲平衡路径和双稳态构形。对热-机械荷载作用的FGM扁球壳的跳跃屈曲行为进行了参数影响分析。结果表明：温度上升时，球壳上临界荷载显著增加、下临界荷载变化不明显。梯度指数增加时，球壳上、下临界荷载均显著减小。组分材料模量增加时，球壳上、下临界荷载均显著增加。当底圆半径和厚度给定时，随壳体中面曲率半径增加，球壳上、下临界荷载均显著增加。当中面曲率半径和厚度给定时，随底圆半径增加，球壳下临界荷载显著减小，上临界荷载几乎不变。
基于经典壳体理论和Sanders非线性应变-位移关系，导出了幂律型功能梯度材料(P-FGM)扁球壳在热-机械荷载作用下的几何非线性常微分控制方程。推导过程考虑了沿厚度存在一维热传导温度场和法向均布荷载作用。采用打靶法求解了由控制方程和固定夹紧边界条件构成的两点边值问题。得到了FGM扁球壳的一些典型的屈曲平衡路径和双稳态构形。对热-机械荷载作用的FGM扁球壳的跳跃屈曲行为进行了参数影响分析。结果表明：温度上升时，球壳上临界荷载显著增加、下临界荷载变化不明显。梯度指数增加时，球壳上、下临界荷载均显著减小。组分材料模量增加时，球壳上、下临界荷载均显著增加。当底圆半径和厚度给定时，随壳体中面曲率半径增加，球壳上、下临界荷载均显著增加。当中面曲率半径和厚度给定时，随底圆半径增加，球壳下临界荷载显著减小，上临界荷载几乎不变。
Abstract: Based on the classical shell theory and Sanders nonlinear strain-displacement relationship, the geometric nonlinear ordinary differential governing equations of power-law functionally graded material (P-FGM) shallow spherical shells under thermal mechanical loads are derived. One-dimensional heat conduction temperature field along the thickness and normal uniformly distributed load are considered in the derivation. The two-point boundary value problem composed of the governing equations and clamped boundary condition is solved by the shooting method. Some typical buckling equilibrium paths and bistable configurations of FGM shallow spherical shells are obtained. The influence of parameters on the snap-through buckling behavior of FGM shallow spherical shells under thermal mechanical load is analyzed. The results show that when the temperature rises, the upper critical load of the shells increases significantly and the lower critical load does not change obviously. When gradient index increases, the upper and lower critical loads of the shells decrease significantly. When the constituent material modulus increases, the upper and lower critical loads of the shells increase significantly. When bottom circle radius and thickness are given, the upper and lower critical loads of the shells increase significantly with the decrease of the radius of curvature of the shells middle surface. When middle surface curvature radius and thickness of the shells are given, with the increase of the radius of the bottom circle, the lower critical load of the spherical shell decreases significantly, and the upper critical load is almost unchanged. -
表 1 功能梯度材料扁球壳组分材料性能
特性参数 铝(Al) 粘土(Al2O3) 氮化硅(Si3N4) 氧化锆(ZrO2) 弹性模量E/GPa 70 380 300 151 热膨胀系数α/K−1 23×10−6 7.4×10−6 3.1×10−6 10×10−6 导热系数K/[W/(m·K)] 204 10.4 15 2.09 泊松比v 0.3 0.3 0.3 0.3 
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表 2 各向同性扁球壳的无量纲外侧压力Q对应的无量纲中心挠度w(0)
挠度$ w(0) $ $ {{K = {a^2}\sqrt {12(1 - {v^2})} } \mathord{\left/ {\vphantom {{K = {a^2}\sqrt {12(1 - {v^2})} } {Rh}}} \right. } {Rh}} $ 16 20 文献[25] 本文 文献[25] 本文 0.100 64.985 83 65.135 36 101.431 08 101.693 18 0.300 166.073 11 166.426 41 263.520 99 264.143 79 0.600 254.774 03 255.260 21 409.333 98 410.151 89 0.964 — 289.156 45 458.060 30 458.871 95 1.028 289.191 16 289.723 92 — 457.801 18 2.000 243.607 78 244.214 86 366.196 05 367.096 63 注：表中的挠度取向内侧为正。
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