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为了研究二维功能梯度板的平面瞬态温度场及影响因素,假设热导率和容积比热容在板的高和宽方向为同一类型指数函数分布,基于该板的瞬态热传导基本方程,采用分离变量法,导出在初始和边界不同恒温时该板平面瞬态温度场的级数解析解.通过两种方法的对比,验证了论文研究结果的正确性.结果表明:方板内的等温线,在梯度参数α=β时,对称于板的45°对角线分布,在α=β=0时,对称于板的中心分布;在α≠β时,板的等温线不具对称性;随着冷却时间的增加,非均匀板的等温线向左下(或右上)角移动,右上(或左下)角等温线间距变宽,左下(或右上)角等温线间距变窄,且矩形板的冷却过程比方板更快.因此,可选择适合的梯度参数和几何形状来满足设计、应用和热应力分析的需要,该解析解可作为检验其他近似方法的参考标准.
In order to study the planar transient temperature field and the influencing factors of two-dimensional functional gradient plate, assuming that the thermal conductivity and volumetric specific heat capacity are the exponential distribution of the same type in the height and width directions of the plate, based on the basic equation of transient heat conduction of the plate, Variable method is used to derive the series analytic solution of the transient temperature field in the plane of the plate at different initial and boundary temperatures.The comparison between the two methods verifies the correctness of the research results.The results show that the isotherms, When the gradient parameter α = β, the 45 ° diagonal distribution symmetrical to the plate is symmetric to the central distribution of the plate when α = β = 0; the isothermal plate is not symmetrical when α ≠ β; When the cooling time increases, the isothermal line of the non-uniform plate moves to the lower left corner (or upper right corner), the spacing between the upper right (or lower left) isothermal lines becomes wider, the spacing between the lower left (or upper right) isothermal lines narrows, and the cooling process So it is possible to choose suitable gradient parameters and geometries to meet the needs of design, application and thermal stress analysis, which can serve as a reference for testing other approximations.