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畢業(yè)論文（設計）
外文翻譯
題 目： 三維接觸問題的有限元分析
2015 年 3 月 9
二維有限元接觸解決方案的驗證
2.1赫茲接觸
對于不同材質(zhì)的經(jīng)典三維接觸解決方案的驗證，正確的地方是先從一個簡單的二維模型開始。然后通過這個實驗獲取準確的結(jié)果，再由二維模型轉(zhuǎn)移到三維維問題。目前關于兩個氣缸之間的赫茲接觸分析,它們之間的接觸壓力是由赫茲發(fā)現(xiàn)表面的彈性變形與牛頓光學干涉條紋之間的差距玻璃鏡片產(chǎn)生的。他假設一般接觸面積是橢圓的干涉條紋。由于先進的方法可用于彈性 半空間的邊值問題,然后他簡化和假定每個相互接觸是一個彈性半無限。這使得它可以計算在接觸變形。為了滿足上述條件的要求是:接觸面必須是小的尺寸相比,每個物體和接觸面積必須小于相對的表面的曲率半徑。第一個要求是必要的,為了不影響高強度的邊界接觸地區(qū)。第二個要求是:以確保外表面接觸是大致平面。另一個要求是應變接觸區(qū)必須小。最后但并非最不重要的假設是假定為無摩擦接觸的表面,所以沒有切向牽引力在接觸區(qū)域中的赫茲問題。
赫茲接觸假設的摘要：
1.每個個體在接觸中被認為是彈性半空間應力計算。
2.接觸被假定為無摩擦。
3.這兩個機構(gòu)之間的接觸是相容的接觸。
4.接觸區(qū)域附近的應變非常小。
5.相比接觸體的尺寸的接觸面積比較小。
二維平面應變分析
在一些工程問題，例如在內(nèi)部壓力的管道，一壩進行水負荷或圓柱滾子強行壓縮，如圖2.1，有顯著應變僅在一個平面上;應變在一個方向上比在另外兩個方向所述應變小得多。在這種情況下，更小的應變會被忽略，工程問題就解決了作為二維平面應變問題。
圖2.1：二維平面應變問題的說明
應變垂直于XY平面εz作用示于圖2.2和剪切應變εXZ和εYZ被假定為零。假設為平面應變是沿Z方向具有作用于僅在X或Y的負載均勻的橫截面方向的，不以Z方向變化。對于本圖進行正常負載假設為2D平面應變問題，并且解決了如圖2.1厚度有限元模型。
圖2.2:應變在x - y平面
兩個氣缸之間的聯(lián)系
本數(shù)值本案涉及兩個氣缸中接觸，其軸線平行于彼此。力共同沿著接觸的方向上施加。氣缸因接觸面變形'2A'的負載。
圖2.3：在赫茲接觸變形的機構(gòu)
如果具有曲率半徑R1和R2的兩個氣缸等于作為曲率半徑R的氣缸.
兩個氣缸的材料特性是楊氏模量為E1和E2，泊松比為υ1分別υ2
來自Johnson理論結(jié)果[20]對于上述分析是半接觸長度和最大接觸壓力。半接觸長度'A'由下式給出：
用于在接觸區(qū)域的壓力分布的如下：
接觸壓力是最大的觸點的中心，并且由給定的方程在接觸的邊緣的壓力等零。
2.1.1有限元關于無摩擦的解決方案
圖2.4：有限元二維接觸模型
使用商業(yè)有限元軟件ANSYS[21]進行了上述數(shù)值案例的分析。這個問題的建模涉及每個氣缸二維分析?th和運用對稱邊界條件建模。 PLANE42元素被用來嚙合二維維度圓柱體。PLANE42是二維四邊形元件具有4個節(jié)點，每個節(jié)點具有兩個自由度的，如在一個平面上的兩個方向的平移。關于嚙合，該接觸區(qū)域附近的嚙合精細，具有最小元素大約0.26毫米和網(wǎng)孔是粗如移離具有最大元件尺寸0.92毫米接觸區(qū)域。正常負載沿著共同接觸點施加。兩個圓柱體和接觸模型的載荷數(shù)的材料性能示于表2.1
表2.1：無摩擦有限元二維接觸模型數(shù)據(jù)
材料特性
尺寸數(shù)據(jù)
氣缸1（B1）
1.彈性模量 ( E1 )=30000 X 106 Pa
2.泊松比 (ν 1 )=0.25
半徑
( R1 )=10
mm
氣缸2（B2）
1.彈性模量( E 2 )=29120 X 106 Pa
2.泊松比(ν 2 )=0.3
半徑
( R2 )=13
mm
負載數(shù)據(jù)： 負載（P）=4300N
用于嚙合的接觸區(qū)域，從[21]是CONTA172和TANGE169，并且在[21]接觸的向?qū)нx項在這里用來創(chuàng)建這些接觸上的接觸區(qū)域的元件。 CONTA172是用3個節(jié)點和2度的二維元件在一個平面上，自由在每個節(jié)點和用于模型表面到表面的接觸。TARGE169是用于表示與所述接觸相關目標表面的二維元件元素。在半接觸長度和最大接觸被壓問題解決之后，接觸長度是使用節(jié)點的解決方案，從一般處理程序菜單中選擇。接觸區(qū)是認可的色差，接觸的狀態(tài)從列表中每個節(jié)點可以讀取結(jié)果，選擇一般處理程序。接觸長度測量接觸邊界的節(jié)點的距離。有限之間的百分比差異元件解決的結(jié)果和分析解決方案的結(jié)果是1％,它證明了有限元素的解決方案的結(jié)果與赫茲分析結(jié)果一致。比較了有限元解結(jié)果和赫茲解析解結(jié)果所示表2.2。
表2.2：對于分析結(jié)果的有限元分析與赫茲接觸模型的比較
分析結(jié)果
解析結(jié)果
差異的％
半接觸長度（mm）
1.1872
1.1197
1.04
峰值接觸長度（mm）
1711.2
1698.1
0.7
2.2斯彭斯解決方案
在上一節(jié)中有一些假設的赫茲接觸分析，如摩擦接觸，小應變的接觸等。本節(jié)討論的非赫茲接觸問題的分析。在非赫茲接觸問題，我們已經(jīng)通過假設放寬了引入摩擦接觸之間的摩擦的情況下，赫茲摩擦接觸氣缸之間的效果將存在兩個氣缸，具有用于正常的不同的材料性質(zhì)接觸問題。由于機構(gòu)之間接觸不同與切線位移，滑移將發(fā)生與總接觸區(qū)域，成為組合中央?yún)^(qū)包圍滑移區(qū)。
機構(gòu)滑移的數(shù)量取決于摩擦系數(shù)之間的聯(lián)系等,如果摩擦的數(shù)量較多,滑移會減少,如果摩擦的數(shù)量少,那么滑移將更多。還可以增加摩擦系數(shù)的數(shù)量,使得滑移是零。存在一個切線牽引力，由于摩擦滑動區(qū)這些切線牽引力在相反的方向，兩個接觸面的接觸區(qū)域和由下式給出：
（2.5）
因為正常負荷增加，接觸尺寸也增加。在任意兩點在兩個接觸面接觸區(qū)進行不同的切向位移。一旦他們進入?yún)^(qū)之間沒有相對切向位移點。應力和應變的大小會增加接觸長度的比例。
斯賓塞[10]溶液指出，對于相同的材料特性，即，楊氏模量，泊松比和摩擦系數(shù)滑移區(qū)域的量是相同的。所有的物體所具有的形狀輪廓Z = AX n和等于平放穿孔。他還得出的結(jié)論是在單調(diào)載荷下，堅持區(qū)域大小的比率和接觸區(qū)域大小只取決于材料性能。例如摩擦μ和系數(shù)泊松比ν。他還介紹了不同數(shù)值的不同泊松比ν范圍從0到0.5的求解，并觀察到更準確的結(jié)果ν接近0.5區(qū)段尺寸c給出由下式：
（2.6）
是第一類的完整橢圓積分和，解決了使用命令ELLIPKE從數(shù)學規(guī)劃工具Matlab[22]。第一類橢圓積分可配制成
其中β是等于：
參數(shù)β的數(shù)值用來衡量不同的彈性常數(shù)不同的材料。 β的值等于零時兩種材料在接觸是類似的材料，為零時兩個材料具有泊松比等于0.5，即對于不可壓縮的材料，對于β極端值±0.5時，一個主體和其它具有零泊松比的材料是不可壓縮的。
2.2.1有限元接觸摩擦的解決方案
圖2.5：有限元二維模型接觸摩擦
非赫茲接觸問題有限元的解決方案也進行了同樣的方式，赫茲接觸除了ANSYS改變一些選項。系統(tǒng)的建模使用?氣缸進行分析。 PLANE42二維4聯(lián)接的QUAD元件被用來接觸兩個網(wǎng)絡體，元件CONTA172和TARGE169被用于在接觸區(qū)域靠近嚙合。附近的接觸網(wǎng)狀區(qū)優(yōu)良具有最小單元尺寸4.5e-2毫米。粗網(wǎng)格距離具有最大元件尺寸0.95毫米接觸。兩者之間的摩擦接觸是在ANSYS材料模型選項指定。斯賓塞[9]提出了獨特的接觸問題的解決方案和獨立的加載路徑。后來斯賓塞[10]為冪律硬度的計算提出了解決方案，材料性能，摩擦系數(shù)和載荷，獨立的加載路徑。因此，我們沒有考慮負載介入這個摩擦接觸問題的有限元解。
表2.3:有限元二維接觸與摩擦模型數(shù)據(jù)
材料特性
維度數(shù)據(jù)
柱面1（B1）
1. 彈性模量 ( E1 )=30000 X 106 Pa
2. 泊松比(ν 1 )=0.25
半徑 ( R1 )=10 mm
柱面2（B2）
1. 彈性模量( E 2 )=29120 X 106 P
2. 泊松比(ν 2 )=0.3
半徑 ( R2 )=13 mm
負載數(shù)據(jù)： 負載 (P)=3200 N
摩擦： 摩擦系數(shù)(μ)=0.01-0.05
解決問題之后,結(jié)果是使用節(jié)點從通過處理程序菜單中選擇的區(qū)域大小和接觸大小獲得的。接觸的區(qū)域,區(qū)域被認可的顏色的區(qū)別和聯(lián)系每個節(jié)點的狀態(tài)也可以從中讀出，從結(jié)果列表中選擇后處理。接觸長度和根區(qū)大小測量之間的距離分別為接觸和根區(qū)邊界。接觸模型解決了不同摩擦系數(shù)。表中給出的數(shù)值結(jié)果李[2]和坎波斯[23]表明,摩擦系數(shù)的增加會導致增加區(qū)域大小和導致減少接觸的大小，觀察到這些結(jié)果，摩擦系數(shù)是導致了粘區(qū)的大小增加。有限元件的比較結(jié)果與分析結(jié)果繪制在圖2.6。該有限元結(jié)果與所述彭斯解析協(xié)議結(jié)過相同，根據(jù)有限元計算結(jié)果和斯賓塞之間的最大百分比差異分析的結(jié)果，觀察到的摩擦值系數(shù)低。與這個幅度百分比差19.23％，更多的錯誤（19.23％）相比于在該元件的尺寸，是由于更小的區(qū)域大小和區(qū)域位置。這種錯誤可能是由附近的接觸網(wǎng)格細化減少。在這項工作中網(wǎng)格細化由軟件版本的限制。有限元計算結(jié)果會更準確的在更高的系數(shù)值。
圖2.6.FEA和斯彭斯的圖像比較結(jié)果
本文摘譯自美國辛辛那提大學3D接觸問題的有限元分析理學碩士
2. Validation of 2D Finite Element Contact Solution
2.1 Hertz Contact
For validation of the classic 3D contact solution with different materials, the right place to start with a simple 2D model. Then use this experience gained in obtaining accurate results for 2D model to move to the 3D problem.
The present section deals with the analysis of the Hertz contact between two cylinders. Hertz found the elastic deformation of the surfaces from Newton’s optical interference fringes in the gap between the glass lenses due to the contact pressure between them. He made the hypothesis that in general contact area is elliptical from the observations of the interference fringes. Due to well developed methods available for the boundary value problem for the elastic half-space, he then simplified and assumed that each body in contact as an elastic half-space. This makes it possible to calculate the deformation at the contact.
In order to satisfy the above conditions the necessary requirements are: contact area must be small compared to the dimension of each body, and contact area must be small compared to the relative radii of the curvature of the surfaces. The first requirement is necessary in order to not to influence the boundaries from the highly stressed contact region. The second requirement is necessary to make sure that the surface outside the contact region is approximately plane. Another requirement is that strains in the contact region must be small. The last but not least assumption is the contacting surfaces are assumed to be frictionless, so there are no tangential tractions at
the contact region in the Hertz problem.
SUMMARY OF THE ASSUMPTIONS FOR HERTZ CONTACT
1. Each body in contact is considered as elastic half-space for stress calculations.
2. The contact is assumed to be frictionless.
3. The contact between the two bodies is non-conforming contact.
4. The strains near the contact region are assumed to be small.
5. The contact area is small compared to the dimensions of the contacting bodies.
2D Plane Strain Analysis
In some engineering problems, such as a pipe under internal pressure, a dam subjected to water loading, or cylindrical roller compressed by force as in Figure 2.1, have significant strain only in one plane; strain in one direction is much less than the strains in other two directions. In such cases smaller strain is ignored and the engineering problem is solved as a 2D plane strain problem.
Strain normal to the X-Y plane εas shown in Figure 2.2 and shear strains ε and ε are assumed to be zero. The assumptions for plane strain are, long bodies along the Z-direction having uniform cross section subjected to loads that act only in X and/or Y directions and do not vary in Z-direction. For the present case cylinders in contact subjected to normal load is assumed as 2D plane strain problem and solved the unit thickness finite element model as shown in Figure 2.1.
Contact between Two Cylinders
The present numerical case involves the two cylinders in contact, whose axes are parallel to each other. Force is applied in the direction of common normal of contact along the axis of the cylinder. Cylinders deformed at the contact surfaces of size ‘2a’ due to the Load.
If the two cylinders having the radii of curvature as R 1 and R 2 then relative radius of curvature is equal to R.
(2.1)
Material properties of the two cylinders are, Young’s modulus as E 1 and E 2 , Poisson’s ratio as υ and υ respectively.
(2.2)
Theoretical results from Johnson [20] for the above analysis are semi contact length and the peak contact pressure. Semi contact length ‘a(chǎn)’ is given by:
(2.3)
The expression for pressure distribution in the contact area is given by:
(2.4)
Contact pressure is maximum at the center of contact and is given by the equation. At the edge of contact the contact pressure equals to zero.
2.1.1 Finite Element Contact Solution without Friction
The analysis of the above numerical case was performed using the commercial finite element software ANSYS [21]. Modeling of this problem involves modeling of ?th of each cylinder for the present 2D analysis and applying symmetry boundary conditions. PLANE42 elements were used to mesh the two quarter cylindrical bodies. PLANE42 is 2D quad element with 4 nodes and each node has 2 degrees of freedom as translations in two directions in a plane. Regarding meshing, the mesh near the contact region is fine, having minimum element size 0.26 mm and the mesh is coarse as moved away from the contact region having maximum element size 0.92 mm. Normal load is applied along the common normal at the point of contact. Material properties of the two cylindrical bodies and load data of the contact model are shown in Table 2.1.
Table 2.1: Finite element 2D Contact model data without friction
Material Properties
Dimensional Data
Cylinder 1 (B1)
1. Modulus of Elasticity(E )=30000X 10Pa
2. Poisson’s ratio (ν )=0.25
Radius(R )=10mm
Cylinder 2 (B2)
1. Modulus of Elasticity(E )=2912010 Pa
2. Poisson’s ratio (ν )=0.3
Radius()=13 mm
Load Data: Load (P)=3200 N
Elements used to mesh the contact region, from [21] are CONTA172 and TANGE169, and the contact wizard option in [21] is used here to create these contact elements on the contact region. CONTA172 is a 2D element with 3 nodes and 2 degrees of freedom in a plane at each node and is used to model surface to surface contacts.TARGE169 is a 2D element used to represent target surfaces associated with the contact elements.
After solving the problem, the results semi contact length and the peak contact pressure were postprocessed. Contact length was postprocessed using the nodal solution option from the general postprocessor menu. The contact region was recognized by the color difference and also contact status of the each node can be read from the list results option in the general postprocessor. Contact length was measured as distance between the nodes that are in contact boundary. The percentage difference between the finite element solution results and the analytical solution results is 1% and it proves that finite element solution results are in agreement with Hertz analytical results. Comparison for finite element solution results and Hertz analytical solution results are shown in the able 2.2.
Table 2.2: Comparison of finite element analysis results with analytical results for Hertz contact model
ANSYS RESULTS
ANALYTICAL
RESULTS
% DIFFERENCE
SEMI CONTACT
LENGTH (mm)
1.1872
1.1997
1.04
PEAK CONTACT
PRESSURE (XPa)
1711.2
1698.1
0.7
2.2 Spence Solution
In the previous section there are some assumptions for the Hertz contact analysis such as frictionless contact, small strains at the contact, etc. The present section deals with the analysis of non-Hertzian contact problems. In the non-Hertzian contact problem, we have relaxed the frictionless contact assumption by introducing the friction between the contacts for the Hertz case. The effect of friction between the contacting cylinders will exist only if the two cylinders have the different material properties for the normal contact problem. Due to different tangential displacements between the contacting bodies, slip will take place and the total contact region becomes the combination of central stick region surrounded the slip region.
The amount of slip depends upon the coefficient of friction between the contacting bodies such that if the amount of friction is more, the slip will be less and if the amount of friction is less, then the slip will be more. It is also possible to increase the coefficient of friction to an amount such that slip will be zero. There exists a tangential traction due to the friction in the slip zone. These tangential tractions act in opposite directions on the two contacting surfaces at the contact region and is given by:
(2.5)
As the normal load increases, the contact size also increases. Any two points in the contact region on two contact surfaces undergo different tangential displacements. Once they come into the stick zone there is no relative tangential displacement between those points. The magnitude of stress and strain grows in proportion to the contact length.
The Spence [10] solution states that for the same material properties, i.e., Young’s modulus, Poisson’s ratio and coefficient of friction the amount of slip region is same for all the bodies those have the shape profile and is equal to the flat punch. He also concluded that under monotonic loading, the ratio of stick zone size to contact zone size depends only on the material properties, such as coefficient of friction μ and Poisson’s ratio ν . He was also presented the numerical solution with varying values of Poisson’s ratio ν ranging from 0 to 0.5 and observed more accurate results as ν approaches to 0.5The stick zone size c is given by the following equation:
(2.6)
is the complete elliptic integral of first kind and.Here kwas solved using the command ELLIPKE from mathematical programming tool Matlab [22]. The elliptic integral of first kind can be formulated as:
(2.7)
where β is equal to the:
(2.8)
The parameter β is the measure of the differences in elastic constants of the two different materials. The value of β is equal to zero when both the materials in contact are similar materials and also zero when both the materials have the Poisson’s ratio equal to 0.5, i.e., for incompressible materials. The extreme values for β are ±0.5 when one body is incompressible and other has zero Poisson’s ratio.
2.2.1 Finite Element Contact Solution with Friction
The finite element solution of non-Hertzian contact problem is also carried out in the same way as Hertzian contact except with changing some options in ANSYS. The system was modeled using ? of the cylinders for analysis. PLANE42 2D four noded QUAD elements were used to mesh the two contacting bodies. Elements CONTA172 and TARGE169 were used to mesh near the contact region. The mesh near the contact region is fine having the minimum element size 4.5e-2 mm and coarse mesh away from the contact having the maximum element size 0.95 mm. Friction between the two contacting bodies was specified in the material models option in ANSYS. Spence [9] presented unique solution for the contact problem with fiction independent of load path. Later Spence [10] presented solution for power law indenter that depends on geometry, material properties, friction coefficient and loading, independent of load path. We therefore have not considered load stepping in the finite element solution of this frictional contact problem.
Table 2.3: Finite element 2D Contact model data with friction
Material Properties
Dimensional Data
Cylinder 1 (B1)
1. Modulus of Elasticity()=30000XPa
2. Poisson’s ratio (ν )=0.25
Radius ( )=10 mm
Cylinder 2 (B2)
1. Modulus of Elasticity()=29120 X Pa
2. Poisson’s ratio (ν )=0.3
Radius ( )=10 mm
Load Data:
Friction:
Load (P)=3200 N
Coefficient of friction (μ)=0.01-0.05
After solving the problem, the results were postprocessed using the nodal solution option from the general postprocessor menu for the stick zone size and contact size. Contact region and stick region were recognized by the color difference and contact status of the each node can also be read from the list results option in general postprocessor. Contact length and stick zone size were measured as the distance between the nodes that are in contact and stick zone boundaries respectively. The contact model was solved for different coefficient of friction values. The numerical results presented in Li [2] and Campos [23] shows that, increase in coefficient of friction causes increase in stick zone size and causes decrease in contact size. It was also observed in these results,that coefficient of friction causes increase in the stick zone size. Comparison of the finite element results with the analytical results was plotted in the Figure 2.6. It was seen from the results that the finite element results are in agreement with the Spence analytical results. The maximum percentage difference between finite element results and Spence analytical results was observed at low coefficient of friction value. The magnitude of this percentage difference is equal to 19.23%. The reason for more error (19.23%) compared to other cases is due to smaller stick zone size as compared to the element size in that location. This error can be reduced by mesh refinement near the contact. In this work mesh refinement was limited by the software version. Finite element results are more accurate at higher coefficient values.
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