層狀結構受移動負荷之暫態熱彈性問題研究

Abstract

The purpose of this work is to establish an analytical model to analyzc the transient behavior of the layered structures due to a moving source such as moving frictional load or moving line heat sources. The layered structures under consideration include infinitely long elastic layered media, finite length layered slab, composite hollow cylinder and the ring stiffened hollow cylinder. In this work, both classical parabolic heat conduction equation, which is derived by Fourier's law of conduction heat transfer, and hyperbolic heat conduction equation, which is derived by non-Fourier's equation of conduction heat transfer, are utilized for the analysis of temperature distribution of the layered structures. The thermoelastic field is based on the uncoupled thermoelasticity which neglects the coupling effects of temperature and strain rate. The basic equilibrium equation in stress field and the thermoelastic displaeement funetion combined with the Galerkin function (or Love function) are employed to analyze thermal stress distributions. The method of Laplace transform is applied first to handle the timedependent transient effect, then the Fourier transform or the eigenfunction expansion method is utilized to slove the associated boundary value problem in the Laplace transformed region. Finally, he numerical method, known as Fourier series technique, is adopted to obtain the inversion of the Laplace transform. Hybrid Laplace transform/finite element method (LTFEM) is also established in this work to calculate the transient state of temperature and thermal stress distributions. This method consists in formulating and solving the problem in the Laplace transformed region by the finite element method. Reliability and accuracy of the LTFEM are proven by the same problem that is solved by the theoretical model. Numerical results are presented for the effects of dimensionless material parameters on temperature and thermal stress distributions of the layered structures. Based on the numerical results, the substantial decrease of thermal stress distributions can be achieved by decreasing thermal conductivity ratio, increasing Biot number, increasing shear modulus ratio, and increasing the speed of moving load. 本文的目的在探討層狀結構受到移動負荷（例如移動摩擦負合或線狀移動熱源）時之 暫態溫度和熱應力分佈情形．文中考慮的層狀結構包括：無限長之彈性雙層結構、有 限長之雙層板、有限長之雙層空心圓柱及有加強環之空心圓柱．本文在求解溫度分佈 時所使用之熱傳導方程式有二種：一為由餞利葉熱傳導定律所導出的拋物線形熱傳導 方程式，另一為由非傅利葉熱傳導方程式所導出的雙曲線形熱傳導方程式．而求解熱 應力分佈時所利用的方程式則是基於不考慮溫度和應變率耦合效應之非耦合性熱彈性 定理，因此求解熱應力分佈時所使用的方程式亦有兩種：一為應力場中之平衡方程式 ，另一為結合熱彈性位移函數及葛勒金函數（或勒伏函數）的方程式． 為解析此種起始值及邊界值問題，本文首先利用拉普拉斯轉換對時間相關項作轉換， 然後利用傅利葉轉換或特徵函數展開法求解在轉換空間中有關溫度及熱應大分佈的解 析解，最後利用傅利葉級數法求出相關於溫度與熱應力解析解的拉普拉斯逆轉換，即 可得到相關熱彈性問題之暫態變化情形．本文同時利用複合拉普拉斯轉換及有限單元 法的數值方法求解相似問題，由結困相互比較發現由理論求解及使用數值方法所求得 的結果非常吻合． 關於求得之溫度及熱應力分佈之數值結果以相對於無因次參數之圖形匚示，從相關的 結果中可得到如下之結論：若要降低材料中之熱應力分佈可利用降低兩層材料之熱傳 導係數比、增高材料表面之熱對流係數值、增加兩層材料之剪力模式比、加快負荷速 度等方法達成．