本文主要以有限元素ABAQUS的模擬，分析薄膜-基板系統間存在mismatch strain讓整個系統產生的變形，並且找出曲率的數值。由於薄膜內的應力不易量測，一般求得應力的方法都是以量測基板曲率來反算薄膜內的應力，但是由於實驗的對象都是微小的結構，尺度的關係使得實做上相當容易有人為的誤差，在本文裡我們利用有限元素的方式，使用ABAQUS來幫助模擬，先試著建立較為基礎的平板模型，利用平板理論來驗證模型的可行性，再將基礎的模型延伸成為薄膜-基板系統，在Stoney formula的理論假設範圍內比較模型分析數值與理論解，探討其差異並且改變模型的型態，讓可以模擬的範圍更加的廣泛。

　In this work we employ the finite element software, ABAQUS, to simulate the stress evolution in film-substrate system subject to intrinsic mismatch strains. The mismatch strain could be origined from surface stress, different thermal expansion coefficients, and others. In pratical situations it is difficult to measure the stress in situ. The renowned Stoney formula provides a simple analytical connection between the internal stress and the curvature of the substrate, and thus could serve as a convenient tool to estimate stress through the measured curvature. However, the derivation of the Stoney formula was based on classical plate assumptions. In addition, the energy terms associated with the film deformation are all neglected. Naturally, this formula, though simple, could only be valid under certain conditions. Here we employ the finite element analysis to examine the range of validity of Stoney formula.

Cammarata, R.C. (1994), Surface and interface stress effects in thin films. Prog. Surf. Sci. 46, pp.1-38.

Cammarata, R.C., Sieradzki, K. and Spaepen, F. (2000), Simple model for interface stresses with application to misfit dislocation generation in epitaxial thin films. J. Appl. Phys., 87, pp.1227-1234.

Cammarata, R.C., Trimble, T.M. and Srolovitz, D.J. (2000), Surface stress model for instrinsic stresses in thin films. J. Mater. Res.15, pp.2468-2474.

Freund, L.B., Suresh, S. (2003), Thin Film Materials: Stress, Defect Formation and Surface Evolution. Cambridge Univ. Press, Cambridge.

Herbert, R (1988), Elastic Plates：Theory and Application. New York :Wiley.

Kirchhoff, G., (1850), Über das Gleichgewicht unt die Bewegung einer elastischen Scheibe. Journal für die Reine und Angewandte Mathematik (Crelle), Vol.40, pp51-58

Maan, H.J. (2004), Design of Plate & Shell Structures. ASME press.

Nix, W.D. and Clement, B.M. (1999), Crystallite coalescence: A mechanism for intrinsic tensile stresses in thin films. J. Mater. Research 14, pp.3467-3473.

Nix, W.D., and Doerner, M.F. (1988), Stresses and deformation processes in thin films on substrates. Critical Reviews in Solid State and Materials Sciences. 14, pp.225-268.

Nix, W.D. (1989), Mechanical properties of thin films. Metall. Trans. 20A, pp.2217.

Spaepen, F. (2000), Interfaces and stresses in thin films. Acta Materialia 48, pp.31–42.

Stoney, G.G. (1909), The tension of metallic films deposited by electrolysis. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 82, No.533, pp.172-175.

Timoshenko, S. (1940), Theory of Plates and Shells. New York ;McGraw-Hill Book Company, Inc., London

Ugural, A.C. (1999), Stresses in Plates and Shells, Second Edition, McGraw-Hill, New York.

洪維恩, 數學運算大師Mathematica 4, 碁峯資訊, 2001.

張智星, Matlab程式設計與應用, 清蔚科技, 2000.

愛發股份有限公司, ABAQUS實務入門引導, 全華科技, 2005.