定常温度热弹性梁中精化理论和分解定理的等价性

Equivalence of Refined Theory and Decomposed Theorem of Thermoelastic Beams under Steady Temperature

根据分解定理,把自由表面热弹性梁内的应力状态分解为三部分:内应力状态、热应力状态和Papkovich-Fadle应力状态(简称P-F应力状态)。随后给出了定常温度热弹性Biot通解的一种新的简化形式。不作预先假设,从热弹性理论出发,利用热弹性通解和Lur'e算子方法构造了梁的精化理论,得出了自由表面热弹性梁的三个精确方程:四阶方程、温度方程和超越方程。最后分别证明了分解定理的三个应力状态与精化理论的三个方程一一等价。

The decomposition theorem of thermoelastic beams without transverse surface loadings is proposed, and that the general state of stress of beams can be decomposed into three parts, ie., the interior state, thermal state and Papkovich-Fadle state (shortened form the P-F state) is concluded. Based on thermoelasticity theory, the refined theory of thermoelastic beams is derived by using thermoelastic solution and Lure method without ad hoc assumptions, and the exact equations for the beam without transverse surface loadings consist of three governing differential equations: the fourth-order equation, temperature equation and transcendental equation. It is then proved that the refined beam theory and the decomposed beam theorem are equivalent-the fourth-order equation, temperature equation and transcendental equation are equivalent to the interior state, thermal state and P-F state respectively.