薄膜涂层内部受集中力的基本解及其应用
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摘要
随着现代材料制备技术的迅速发展和工程实际应用的需要,作为多种学科交叉而发展起来的薄膜涂层材料愈来愈受到人们的重视。通常薄膜涂层材料的破坏形式为涂层的剥落,能否正确评价涂层与基体材料的界面结合强度,是保证该类材料安全性、可靠性的关键。对薄膜涂层材料界面的力学行为的评价,首先要能精确地分析结合材料界面上的应力分布和应变分布。由于涂层往往极薄,这不是一个容易的问题。近几十年来诸多研究者从不同途径如理论分析、数值计算、实验测试等入手开展了很多研究。理论分析的结果,一方面不仅可以为数值计算等提供一个比较检验的依据,也可以作为格林函数,去进一步求解较为复杂的问题,另一方面可以作为数值计算方法如边界元法的基本解,改善数值计算的精度和效率。这种理论基本解或基于理论解的半解析解无疑具有重要的意义。
本论文在针对结合材料问题特别有效的镜像点法基础上作了改进,将其推广应用到研究薄膜涂层材料内部受力的问题,完善了薄膜涂层结构材料受集中力作用下的基本理论解体系,并讨论了理论解在解决实际问题时的一些拓展应用。具体研究内容和创新性工作总结如下:
首先针对薄膜涂层材料应用的日益广泛性,对其研究概况进行了总结回顾,介绍了目前常用的一些研究途径如理论研究、数值分析和实验测试等,第二章中系统回顾了在理论研究涂层材料非常有效的镜像点法原理及其优势特点。
研究了薄膜涂层材料结构形式中薄膜涂层材料内部受集中力作用的平面和空间问题。不同于以往的镜像点法,本文通过将集中力作用点关于界面和表面进行映射,引入两个系列的镜像点,求解了薄膜涂层结合材料内部受力的问题。设定各材料的分析函数为固定于力作用点和各镜像点的局部坐标系下的形式。在严格满足界面连续性条件、自由边界条件的基础上,利用Dirichlet单值性原理求解微分方程。对应于高阶镜像点的分析函数通过递推的方法从对应于低阶镜像点的分析函数求得。在无限体内受集中力作用的基本解基础上,求得了问题完备的显式理论解,以分析函数的无穷级数形式表示。由于不同的推导过程,本文分别分析了集中力作用的平面问题、集中力垂直界面作用的轴对称问题以及集中力平行界面作用的空间问题,完善了薄膜涂层材料问题的基本解体系。在求解集中力作用于界面上的问题时,首次求得了两个半无限体结合界面上受集中力作用的基本解。
对理论解进行了深入的讨论,首先通过数值算例对理论解的验证,证明了理论解的正确性,并且从推导过程易知理论解可以退化到一些特殊情况下的经典基本解。为了方便理论解的实际应用,归纳出理论解的规律,给出了多项式形式的递推公式。从多项式形式和数值算例验证中可以发现理论解虽以无穷级数形式给出,但具有非常快速的收敛性,针对一些常用结合材料往往只需要考虑前面几阶的镜像点就可以达到足够的精确度。借助于理论解易于编程求解的特性,讨论了不同材料组合形式对理论解结果收敛性的影响,发现随着结合材料力学属性之间差异的增大,达到要求精度所需考虑的镜像点数目也会相应增加。
在求得的理论解基础上研究了一些更接近实际的问题,如圆周载荷作用情况下材料内的应力场和位移场,以及结合压痕实验分析了Hertz压力分布下的薄膜涂层强度评价问题,显示出理论解除了本身的独立应用外,也可以作为一种格林函数或基本解用于分析更复杂的问题,为目前处于薄膜涂层研究工作中处于相对弱势的理论分析作出了一定的贡献。最后对未来一些关于薄膜涂层材料比较重要的研究方向进行了展望。
With the wide application of surface-treated material in many industries, the problems of coating materials have become unavoidable and therefore drawn considerable attentions. Effective analysis of stress or strain for such a thin-layer structure is very important. Many efforts have been made on obtaining the stress and displacement field quantitatively by different approaches such as theoretical analysis, numerical analysis, and experimental research etc. The theoretical solutions of some typical cases are very meaningful and expected since they can be used not only to solve more sophisticated problems by the Green’s function technology, but also can be used as the fundamental solution to improve the numerical accuracy and efficiency.
The problems of coating materials with concentrated forces applied in the interior of materials has been analyzed by using the improved image point method. This method has special advantages in dealing with dissimilar material problems. The extensive application of the theoretical solutions are also discussed. With our previous work, the fundamental solution system of thin-filmed materials are completed basically. The detailed research contents and the innotative achievements are summarized as follow.
In Chap 1 and 2, considering the wide application of coating materials, recent researches on such material structure are reviewed in perspectives of different analysis approaches. Therefore, the image point method which is very effective in dealing with coating materials problems is introduced in detail.
The planar and spatial problems of concentrated forces applied in the interior of semi-infinite coating materials are discussed. Through the reflections of the loading point about the interface and the free surface, two infinite series of image points are introduced, which is different from the previous application of the image point method. The theoretical solutions are given by the analytical functions defined under the corresponding local coordinates with their origins placed at the loading point and image points. Based on the boundary conditions and the interchange law, the analytical functions corresponding to the mirror point of higher order can be determined from that to the lower order ones by the recurrence relationship. The fundamental solution of a concentrated force loading at the interface of two bonded semi-infinite bodies is also obtained and published firstly.
The theoretical solutions are discussed. Through the comparisons between the analytical and numerical results, the theoretical solutions are verified. It is found that the theoretical solutions can degrade to some classical fundamental solutions of special problems. For the extensive application of the theoretical solutions, the solutions presented in the polynomial form are deduced and listed in the appendix. From the solution expressions, we can find the convergence of theoretical solution is very rapid, and only the first several image points are sufficient to ensure the accuracy in the most practical cases. The effect of different material properties combinations on the convergence of theoretical solution is also discussed. It is found that the bigger the mismatch between the Young’s modulus is, the more image points are needed to get satisfactory accuracy.
Based on the previous theoretical solutions, the problems of ring load and Hertz pressure in a coated half-space are studied. The results are discussed in connection with strength evaluation found experimentally.
本论文在针对结合材料问题特别有效的镜像点法基础上作了改进,将其推广应用到研究薄膜涂层材料内部受力的问题,完善了薄膜涂层结构材料受集中力作用下的基本理论解体系,并讨论了理论解在解决实际问题时的一些拓展应用。具体研究内容和创新性工作总结如下:
首先针对薄膜涂层材料应用的日益广泛性,对其研究概况进行了总结回顾,介绍了目前常用的一些研究途径如理论研究、数值分析和实验测试等,第二章中系统回顾了在理论研究涂层材料非常有效的镜像点法原理及其优势特点。
研究了薄膜涂层材料结构形式中薄膜涂层材料内部受集中力作用的平面和空间问题。不同于以往的镜像点法,本文通过将集中力作用点关于界面和表面进行映射,引入两个系列的镜像点,求解了薄膜涂层结合材料内部受力的问题。设定各材料的分析函数为固定于力作用点和各镜像点的局部坐标系下的形式。在严格满足界面连续性条件、自由边界条件的基础上,利用Dirichlet单值性原理求解微分方程。对应于高阶镜像点的分析函数通过递推的方法从对应于低阶镜像点的分析函数求得。在无限体内受集中力作用的基本解基础上,求得了问题完备的显式理论解,以分析函数的无穷级数形式表示。由于不同的推导过程,本文分别分析了集中力作用的平面问题、集中力垂直界面作用的轴对称问题以及集中力平行界面作用的空间问题,完善了薄膜涂层材料问题的基本解体系。在求解集中力作用于界面上的问题时,首次求得了两个半无限体结合界面上受集中力作用的基本解。
对理论解进行了深入的讨论,首先通过数值算例对理论解的验证,证明了理论解的正确性,并且从推导过程易知理论解可以退化到一些特殊情况下的经典基本解。为了方便理论解的实际应用,归纳出理论解的规律,给出了多项式形式的递推公式。从多项式形式和数值算例验证中可以发现理论解虽以无穷级数形式给出,但具有非常快速的收敛性,针对一些常用结合材料往往只需要考虑前面几阶的镜像点就可以达到足够的精确度。借助于理论解易于编程求解的特性,讨论了不同材料组合形式对理论解结果收敛性的影响,发现随着结合材料力学属性之间差异的增大,达到要求精度所需考虑的镜像点数目也会相应增加。
在求得的理论解基础上研究了一些更接近实际的问题,如圆周载荷作用情况下材料内的应力场和位移场,以及结合压痕实验分析了Hertz压力分布下的薄膜涂层强度评价问题,显示出理论解除了本身的独立应用外,也可以作为一种格林函数或基本解用于分析更复杂的问题,为目前处于薄膜涂层研究工作中处于相对弱势的理论分析作出了一定的贡献。最后对未来一些关于薄膜涂层材料比较重要的研究方向进行了展望。
With the wide application of surface-treated material in many industries, the problems of coating materials have become unavoidable and therefore drawn considerable attentions. Effective analysis of stress or strain for such a thin-layer structure is very important. Many efforts have been made on obtaining the stress and displacement field quantitatively by different approaches such as theoretical analysis, numerical analysis, and experimental research etc. The theoretical solutions of some typical cases are very meaningful and expected since they can be used not only to solve more sophisticated problems by the Green’s function technology, but also can be used as the fundamental solution to improve the numerical accuracy and efficiency.
The problems of coating materials with concentrated forces applied in the interior of materials has been analyzed by using the improved image point method. This method has special advantages in dealing with dissimilar material problems. The extensive application of the theoretical solutions are also discussed. With our previous work, the fundamental solution system of thin-filmed materials are completed basically. The detailed research contents and the innotative achievements are summarized as follow.
In Chap 1 and 2, considering the wide application of coating materials, recent researches on such material structure are reviewed in perspectives of different analysis approaches. Therefore, the image point method which is very effective in dealing with coating materials problems is introduced in detail.
The planar and spatial problems of concentrated forces applied in the interior of semi-infinite coating materials are discussed. Through the reflections of the loading point about the interface and the free surface, two infinite series of image points are introduced, which is different from the previous application of the image point method. The theoretical solutions are given by the analytical functions defined under the corresponding local coordinates with their origins placed at the loading point and image points. Based on the boundary conditions and the interchange law, the analytical functions corresponding to the mirror point of higher order can be determined from that to the lower order ones by the recurrence relationship. The fundamental solution of a concentrated force loading at the interface of two bonded semi-infinite bodies is also obtained and published firstly.
The theoretical solutions are discussed. Through the comparisons between the analytical and numerical results, the theoretical solutions are verified. It is found that the theoretical solutions can degrade to some classical fundamental solutions of special problems. For the extensive application of the theoretical solutions, the solutions presented in the polynomial form are deduced and listed in the appendix. From the solution expressions, we can find the convergence of theoretical solution is very rapid, and only the first several image points are sufficient to ensure the accuracy in the most practical cases. The effect of different material properties combinations on the convergence of theoretical solution is also discussed. It is found that the bigger the mismatch between the Young’s modulus is, the more image points are needed to get satisfactory accuracy.
Based on the previous theoretical solutions, the problems of ring load and Hertz pressure in a coated half-space are studied. The results are discussed in connection with strength evaluation found experimentally.
引文
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