Dynamic buckling analysis of functionally graded beam under thermal shock in Hamilton system
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摘要: 在Hamilton体系下，基于Euler梁理论研究了功能梯度材料梁受热冲击载荷作用时的动力屈曲问题；将非均匀功能梯度复合材料的物性参数假设为厚度坐标的幂函数形式，采用Laplace变换法和幂级数法解析求得热冲击下功能梯度梁内的动态温度场：首先将功能梯度梁的屈曲问题归结为辛空间中系统的零本征值问题，梁的屈曲载荷与屈曲模态分别对应于Hamilton体系下的辛本征值和本征解问题，由分叉条件求得屈曲模态和屈曲热轴力，根据屈曲热轴力求解临界屈曲升温载荷。给出了热冲击载荷作用下一类非均匀梯度材料梁屈曲特性的辛方法研究过程，讨论了材料的梯度特性、结构几何参数和热冲击载荷参数对临界温度的影响。Abstract: Based on the Euler beam theory, the dynamic buckling of the functionally graded beam subjected to thermal shock was investigated in the Hamilton system. The material properties of the functionally graded beam were assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The transient temperature fields were solved analytically using the Laplace transform and power series method. It was shown that the dynamic buckling problem can be reduced to a zero-eigenvalue problem in the symplectic space, the buckling loading and the buckling mode of the FGM beam correspond to the generalized eigenvalue and eigen solution. The buckling mode and the buckling thermal axial forces can be obtained through bifurcation condition, and the buckling temperature rise of the FGM beam can be obtained by inverse solution. In this research, the solution process for dynamic buckling of the FGM beam subjected to thermal shock using the symplectic method were given, and the effects of the material constitution, geometric parameters and the parameters of thermal shock load on the critical temperature were discussed.
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Key words:
- functionally graded materials /
- Euler beam /
- thermal shock /
- symplectic method /
- dynamic buckling
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图 5 FGM梁的1、2阶屈曲升温随k的变化
Figure 5. Variations of first and second order buckling temperature rise with k
图 6 不同a时FGM梁的临界屈曲升温(ΔT)l
Figure 6. Variations of critical temperature for specified values of a
表 1 陶瓷梁的静态热屈曲量纲一临界温度
Table 1. The static non-dimensional critical buckling temperature of ceramic beam
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表 2 功能梯度梁的各阶屈曲升温
Table 2. Buckling temperature rise of FGM beam
n θn ΔT/K SiC k=0.5 k=1 k=2 k=5 k=10 k=100 Ni 1 39.47 488.26 307.64 268.63 242.02 216.39 198.53 167.88 163.46 2 80.76 999.04 629.48 549.65 495.21 442.76 406.21 343.50 334.46 3 157.91 1953.43 1230.82 1074.73 968.29 865.73 794.27 671.66 653.97
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表 3 不同换热系数(hr)时FGM梁的临界屈曲升温
Table 3. Critical temperature rise of FGM beam for some specified values of hr
hr ΔT/K SiC k=0.5 k=1 k=2 k=5 k=10 k=100 Ni 10 487.31 306.98 268.09 241.57 216.01 198.19 167.6 163.18 30 487.79 307.31 268.36 241.8 216.2 198.36 167.74 163.32 50 488.26 307.64 268.63 242.02 216.39 198.53 167.88 163.46 70 488.73 307.97 268.9 242.25 216.58 198.69 168.02 163.59
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表 4 不同长细比(λ)下FGM梁的临界屈曲升温(ΔT)l
Table 4. Critical temperature rise of FGM beam for some specified values of λ
λ (ΔT)l/K SiC k=0.5 k=1 k=2 k=5 k=10 k=100 Ni 30 868.02 546.92 477.57 430.27 384.69 352.94 298.45 290.59 40 488.26 307.64 268.63 242.02 216.39 198.53 167.88 163.46 50 312.49 196.89 171.92 154.89 138.49 127.05 107.44 104.61 60 258.25 162.72 142.08 128.01 114.45 105.00 88.79 86.45
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表 5 热冲击载荷作用时间(Δt)不同时FGM梁的临界屈曲升温(ΔT)l
Table 5. Critical temperature rise of FGM beam for some specified values of Δt
Δt/s (ΔT)l/K SiC k=0.5 k=1 k=2 k=5 k=10 k=100 Ni 1 805.26 600.65 534.75 481.04 421.57 378.73 310.23 302.12 2 589.07 415.66 368.42 331.62 292.37 264.47 219.63 213.83 5 488.26 307.64 268.63 242.02 216.39 198.53 167.88 163.46 10 479.57 289.57 250.81 225.92 203.5 187.99 160.13 155.91 ∞ 479.39 288.26 249.28 224.58 202.55 187.3 159.69 155.48
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