How accurate are Stoney's equation and recent modifications
How accurate are Stoney's equation and recent modifications
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DOI：
10.1063/1.1313776
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发表时间：
2000-11-01
影响因子：
3.2
通讯作者：
Klein, CA
中科院分区：
文献类型：
--
作者：
Klein, CA
斯托尼方程已经并且仍然被广泛用于评估作用在厚基底上沉积的涂层中的宏观应力。原则上，该公式仅适用于“薄膜近似”情况，即涂层比基底薄得多的情况。本文的主要目的是要证明，基于多层薄板中弹性相互作用的一般理论，应力的正确公式可以用斯托尼方程和一个等于\((1 + \gamma\delta(3))/(1 + \delta)\)的修正因子来表示，其中\(\gamma\)表示双轴模量之比，\(\delta\)是层厚度之比。据此表明：(a)对于厚度比\(\delta\)小于或等于\(0.1\)的情况，斯托尼方程不会产生严重误差；(b)阿特金森最近提出的修正方法（该方法不需要涂层模量的信息）对于厚度比直至\(\delta\)约等于或等于\(0.4\)的情况能得出好得多的结果；(c)布伦纳 - 森德罗夫类型的表达式可能会产生很大误导，应该避免使用。（C）2000年美国物理学会。[S0021 - 8979(00)05021 - 0]
Stoney's equation has been-and still is-extensively used to evaluate the macrostress acting in a coating deposited on a thick substrate. In principle, the formula applies only in the "thin-film approximation," that is, for coatings much thinner than the substrate. The main purpose of this communication is to demonstrate that, based on a general theory of elastic interactions in multilayer laminates, the correct formula for the stress can be expressed in terms of Stoney's equation and a correction factor equal to (1+gamma delta(3))/(1+delta), where gamma designates the ratio of the biaxial moduli and delta is the ratio of the layer thicknesses. In this light, it is shown that (a) Stoney's equation does not cause serious errors for thickness ratios delta less than or equal to 0.1; (b) Atkinson's recently proposed modification, which does not require information on the coating's modulus, yields much improved results for thickness ratios up to delta similar or equal to 0.4; and (c) Brenner-Senderoff-type expressions can be very misleading and should be avoided. (C) 2000 American Institute of Physics. [S0021-8979(00)05021-0].