隨著加工技術逐漸的提昇，製造過程中許多微尺度問題漸漸浮現，例如：熱應力及材料相變化等問題。根據傳統傅立葉熱傳定律（Fourier’s law），熱於物體內部是以無限的速率傳遞。但於某些特殊情況下，熱傳遞速率可能為有限。基於上述因素，廣義熱彈性理論（Generalized thermoelectric theorem）便漸漸受到重視。其主要原因在於當材料受到高熱作用時，在極短時間下，該材料內部之應力會受溫度及位移影響。因此本文將引用Green-Lindsay的廣義熱彈性理論，來探討壓電材料之溫度、位移及應力之分佈情形。

　　本文將應用拉氏轉換法（Laplace transform method）移除統制方程式與邊界條件中的時間項，再利用控制體積法（Control volume method） 配合本文所提出之雙曲線型的形狀函數（Hyperbolic shape function） 來處理經過拉氏轉換後的方程式。提出此觀念主要目的在克服數值中可能有跳躍式的不連續現象（Jump discontinuity）。最後再利用逆拉氏轉換法（Numerical inversing of Laplace Transform）求得實際之位移、溫度及應力。為了欲驗證本文之數值分析的正確性，將以半無窮邊界條件作探討，求解出溫度、位移及應力，與解析解比較之，進而了解各參數對溫度、位移及應力之影響。

　　Owing to the improvements of the processing technology, there exist a lot of micro-scale problems during the fabrication, such as thermal stress and phase change of materials etc.. Under the assumption of the Fourier’s law, the speed of the thermal wave in a body is infinite. But this speed can be finite, such as the problem in a very short time. Under this circumstance, the generalized thermoelectric theorem can be introduced to analyze such problems. The main purpose is to investigate the effects of temperature and displacement on the thermal stress in a very short time. Therefore, Generalized thermoelectric theorem is used to investigate the temperature, the displacement, and the stress distribution in piezoelectricity material.

　　The governing differential equations and boundary conditions are transformed by using the Laplace transform method and then the control volume method is used to discrete the resulting differential equations. The hyperbolic shape functions. As introduced in order to overcome the numerical oscillations in the from Jump discontinuities Finally, the numerical inversion of the Laplace transform is used to obtained the temperature, the displacement and the stresses.

　　In order to evidence the accuracy of the numerical scheme, a comparison of the numerical results and the analytical solution is made for a semi-finite problem conditions is used to get the distributions of temperature, displacement, and stress, then compare with the analytical results.

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