In this paper, a three-dimensional transient heat conduction equation with an internal source as a thermal flux outflow, decreasing linearly with time from the surface of an elliptic plate is analyzed. Applying theory of integral transformations, the heat conduction model is explained, and its expression is obtained in the form of Mathieu functions. The large deflection of a plate when placed on the elastic foundation is formulated from the potential energy equation neglecting the second strain invariant. A new elegance of modified total strain energy is obtained in these formulations by incorporating the resulting moment and force within the energy term, thus reducing the computation step. The numerical calculations of distribution of the transient temperature, thermal deflection and the maximum normal bending stress are carried out on the outer elliptic boundary and illustrated graphically. Lastly, the corresponding results for circular plates have been presumed when the ellipse degenerates to a circle. The results reveal that the highest tensile stress exists on the major axis of the circular core relative to the elliptical core, which suggests the propagation of low heating due to inadequate heat penetration into the elliptical surface.

本文分析了一个以内部源为热通量流出的三维瞬态热传导方程，该方程从椭圆板表面随时间线性减小。应用积分变换理论，解释了热传导模型，并以Mathieu函数的形式得到了它的表达式。当放置在弹性基础上时，板的大挠度是由忽略第二应变不变量的势能方程制定的。通过在能量项中结合产生的力矩和力，在这些公式中获得了一种新的改进的总应变能，从而减少了计算步骤。瞬态温度分布的数值计算，热变形和最大法向弯曲应力在外椭圆边界上​​进行并以图形方式说明。最后，当椭圆退化为圆形时，已经假定了圆形板的相应结果。结果表明，相对于椭圆形核心，圆形核心的长轴上存在最高的拉伸应力，这表明由于热量对椭圆表面的渗透不足，导致低热量传播。