赫茲接觸理論於大壓痕深度之適用性研究

Applicability of Hertz Contact Theory in Large Indentation

吳承恩(Chen-En Wu)

指導教授 ： 莊嘉揚

國立臺灣大學/工學院/機械工程學研究所/碩士(2015年)

https://doi.org/10.6342/NTU.2015.00931

摘要

赫茲接觸理論（Hertz contact theory）是接觸力學中最基本的理論之一，並由Heinrich Hertz 於1882年發表。理論的內容為探討彈性體的接觸，並以數學模型描述接觸力、壓痕深度、接觸範圍及接觸面應力分布等。由於，赫茲理論必須滿足小接觸範圍及小壓痕深度的前提假設。然而根據實驗的結果卻發現，太小的壓痕深度會造成雜訊的影響過大，無法得到準確的結果量測。因此為了解決這個實驗上的問題，增加壓痕深度為一項解決的方法。然而，目前卻還沒有完整的研究指出增加壓痕深度，對於赫茲理論適用性的影響。因此本研究試圖討論增加壓痕深度對於赫茲理論的影響，並試圖量化適用的範圍。
本研究利用有限元素軟體建構二維軸對稱的模型，並計算彈性體接觸之結果。其中，接觸的情形包括剛體球與彈性半無限域、有限厚度的彈性平板以及不同半徑之彈性球等的接觸。試圖將赫茲理論延伸至大範圍的接觸。根據模擬的結果，發現當剛體球與彈性半無限域接觸時，壓痕深度與剛體球半徑的比例可以達到0.66，都還符合赫茲理論中負載與壓痕深度的關係。同時，本研究也針對此接觸情形進行聚二甲基矽氧烷（polydimethylsiloxane，PDMS）的奈米壓痕試驗，並且得到與模擬相同趨勢的結果。此外，根據模擬本研究指出，當彈性平板的厚度為剛體球半徑的12倍時，負載與壓痕深度的關係在壓痕深度比1以下，皆會完全符合赫茲理論。另外，根據模擬剛體球只要和曲率半徑為剛體球本身之半徑10倍以上的彈性球接觸，負載與壓痕深度的結果都會相同；且當彈性與剛體球半徑為10倍時，負載與壓痕深度的關係在壓痕深度比1以下，會完全符合赫茲理論。同時本研究也針對材料表面特性對於奈米壓痕試驗的影響作了許多的探討，如表面摩擦係數與親疏水性等。
根據以上結果，當進行奈米壓痕試驗量測材料的機械性質時，壓痕深度只要在本研究提出之範圍內，皆可以得到不受雜訊干擾的準確結果。

關鍵字

赫茲接觸理論 ； 奈米壓痕試驗 ； 有限元素法 ； 聚二甲基矽氧烷

並列摘要

Hertz contact theory, published by Heinrich Hertz in 1882, is one of the most fundamental theories in contact mechanics. The theory concerns about the contact of elastic bodies, and provides a mathematical model to describe the relationship between load, indentation depth, contact area and contact pressure. However, Hertz model should abide by assumptions of small indentation depth and contact region. Based on experiments, shallow depth indentation usually suffers from excessive noises, and it makes measurement inaccurate. Increasing the indentation depth is a way to avoid this problem. However, current studies did not point out the extension of large indentation in Hertz model. In this study, we concern influences of large indentation depth in Hertz model, and try to quantify the applied conditions.
This study utilizes finite element method (FEM) software to construct a two-dimensional axisymmetric model and compute results of the contact of elastic bodies, including contact of rigid sphere and elastic half-space, finite thickness elastic substrate and elastic spheres in different radius. It attempts to extend Hertz model in large indentation. Based on simulation, load-indentation relationship conforms to Hertz model as the ratio of indentation depth and radius of rigid sphere reaches 0.66 in the contact of rigid sphere and elastic half-space. Result of nanoindentation with polydimethylsiloxane (PDMS) has the same trend with simulation. Moreover, when the thickness of elastic substrate becomes 12 times of the radius of the rigid sphere, load-indentation relationship conforms to Hertz model until the ratio of indentation depth and sphere radius becomes one. In addition, contacts of rigid and elastic sphere with radius ratio of elastic and rigid sphere more than 10 have identical load-indentation relationship by simulation. When radius ratio of elastic and rigid sphere becomes 10, load-indentation relationship conforms to Hertz model. Besides, this study also focuses on different surface characteristic of samples in nanoindentation. In conclusion, as long as the indentation depth of nanoindentation lies in the applicable region mentioned above, you will obtain an accurate measurement without noise.

並列關鍵字

Hertz contact theory ； nanoindentation ； finite element method (FEM) ； polydimethylsiloxane (PDMS)

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