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附錄1
絕熱高速切削有限元模型
摘要
二維正交切削過(guò)程的有限元模型正在發(fā)展。仿真是使用標(biāo)準(zhǔn)的有限元軟件結(jié)合一個(gè)特殊的電動(dòng)機(jī)，這種電動(dòng)機(jī)是能夠以網(wǎng)捕捉的形式用有規(guī)律的四邊形和三角形在剪切區(qū)域完整的捕捉事物。重現(xiàn)事物和確保數(shù)據(jù)收斂的這種技術(shù)已經(jīng)尋找到了。分割碎片有序排列和分割過(guò)程還在研究過(guò)程中。令人特別關(guān)注的是剪裂帶產(chǎn)生的問(wèn)題。彈性性能和切割速度的影響也正在討論。 Elsevier公司2002科技有限公司保留所有權(quán)利。
關(guān)鍵詞：加工； 有限元； 格； 芯片分割；絕熱剪切帶
1、言
鈦合金Ti6Al4V被廣泛應(yīng)用于航空航天及其他工業(yè)應(yīng)用。這些合金大部份是應(yīng)用于機(jī)械加工。 因此設(shè)計(jì)具有更好的可加工性鈦合金是一個(gè)值得研究的目的。
為了達(dá)到這一目的,找出嚴(yán)重影響材料可加工參數(shù)是非常必要的。這項(xiàng)工作可以利用有限元計(jì)算機(jī)模擬的參數(shù)研究方法來(lái)完成。一旦這最有前途的設(shè)計(jì)方法被肯定,實(shí)現(xiàn)合金改用的最后的一步材料設(shè)計(jì)過(guò)程就完成了。這種方式類似于標(biāo)準(zhǔn)的CAE生產(chǎn)周期，只有僅有少量幾個(gè)原型被建立。
創(chuàng)造一個(gè)金屬切削過(guò)程的可靠的計(jì)算機(jī)模型在這個(gè)過(guò)程中是一個(gè)關(guān)鍵步驟。在本文中，我們?cè)谝恍┘?xì)節(jié)描述了這種模型。它使用標(biāo)準(zhǔn)的有限元軟件來(lái)計(jì)算，從而保證可移植性和靈活性。隨著嚙合算法的要求迫切，特別預(yù)處理器已經(jīng)研制成功，這個(gè)特別的處理器是用C++編程，且可能應(yīng)用于不同平臺(tái)。
本文安排如下；第2部分在對(duì)模型要求的一個(gè)簡(jiǎn)短說(shuō)明之后，第3部分對(duì)有限元模型進(jìn)行詳細(xì)介紹。第4簡(jiǎn)述了一些模型的加工成果，重點(diǎn)放在細(xì)節(jié)的切屑形成過(guò)程。第5總結(jié)工作，并指出今后的研究目標(biāo)。
2、問(wèn)題
在金屬切削過(guò)程中，材料被切割工具從工件表面切除，碎屑形成。 這個(gè)問(wèn)題涉及塑性大變形，隨著刀具和工件、工具和碎屑之間的摩擦產(chǎn)生大量的熱量。在刀具前端工件材料的分離也已經(jīng)被模擬。
隨著材料參數(shù)的影響對(duì)材料設(shè)計(jì)的考慮比對(duì)加工過(guò)程本身更重要。這個(gè)切削過(guò)程的模擬就是指正交切削。這個(gè)過(guò)程是用二維模擬的,這大大減少了計(jì)算機(jī)所需要的計(jì)算時(shí)間。更進(jìn)一步簡(jiǎn)化是做非常嚴(yán)格的假設(shè)的工具。
在仿真中摩擦與熱流進(jìn)入工具已經(jīng)被忽略，但是可以很容易被包括在內(nèi)。 這種忽略的原因是，有必要盡可能簡(jiǎn)化切削過(guò)程，像下文將作解釋那樣，透視其背后的機(jī)制。 另外，毫無(wú)塞爾馬輻射從表面上的碎屑產(chǎn)生，在材料邊界也沒有的熱傳導(dǎo)。
綜上所述，高速切削是一個(gè)非線性問(wèn)題。它已經(jīng)被一個(gè)完全熱力耦合有限元模型模擬。 因此，編制了有限元法處理金屬切削加工時(shí)的劃傷問(wèn)題就成為一個(gè)艱巨的任務(wù)，利用商業(yè)有限元軟件是一個(gè)有吸引力的替代方案。 現(xiàn)代有限元軟件可以在原則處理這類強(qiáng)非線性問(wèn)題。在我們的研究中，我們決定用ABAQUS中/標(biāo)準(zhǔn)程式系統(tǒng)，這種系統(tǒng)允許定義復(fù)雜的接觸狀況，盡量避免定義材料屬性，在多方面保證程序可定制的，包括由用戶自定義子程序。我們假定以下大部分的方法可以應(yīng)用于同樣大的有限元包。 由于使用的標(biāo)準(zhǔn)化軟件，方程公式(有限元法，熱耦合，一體化計(jì)劃，等)可以在別的非常詳細(xì)的資料中找到[3]。
金屬切削過(guò)程中許多有限元模擬所用明確的方法(例如見[17] )都能被演示。這些公式方法都是有保證的。 (概述了切削過(guò)程中有有限元能夠被在[16]中好到)。盡管如此，決定用一個(gè)隱碼. 在模擬過(guò)程中匯總被檢查，但迭代過(guò)程不再保證銜接。利用ABAQUS /標(biāo)準(zhǔn)內(nèi)含編碼有一個(gè)好處，實(shí)行模擬過(guò)程中允許用戶在很大的范圍內(nèi)靈活的自定義子程序。 這種套路,可以用來(lái)執(zhí)行復(fù)雜的材料分離準(zhǔn)則。 此外，如果本地網(wǎng)有細(xì)化的需要，隱碼有較好的標(biāo)度行為。如果狹窄剪切帶形式, 命令執(zhí)行的單元尺寸為1鎊或不足1鎊是必要的(見第4.2 節(jié))優(yōu)勢(shì)，在CPU 使用時(shí)間有明確的算法，將大大降低。如果摩擦的影響較大，一個(gè)明確的方法可能是上好的，然而，并非如此。另一方面，明確方法往往需要改變一些物理參數(shù)，如密度或工具的速度，或用人工粘性。 我們認(rèn)為, 如果銜接能夠達(dá)到，沒有任何理由去考慮的一個(gè)隱模擬不亞于一個(gè)明確的。
也不同于許多其他的模擬,我們充分利用綜合階四邊形，它有優(yōu)于三角元素更好的收斂性能. 這個(gè)問(wèn)題的進(jìn)一步討論在第3.3節(jié)。
當(dāng)正交切削時(shí)，鈦合金形式分割碎屑 (見圖 9 )。金屬切削過(guò)程中任何詳細(xì)的模擬都必須能夠借此分割考慮。碎屑分割背后的機(jī)制仍然沒有完全弄懂[12，15，25，26]。顯然，所謂的絕熱剪切在分割過(guò)程起了突出的作用: 剪切帶材料熱軟化導(dǎo)致在這個(gè)區(qū)域產(chǎn)生變形。在軟化和變形之間的反饋引起狹長(zhǎng)區(qū)域附加巨大變形, 而周圍的材料只產(chǎn)生微小變形。然而，不知道絕熱剪切帶是否是由裂縫延伸到材料中引起的成的，這在[25]是作為假設(shè)。如果這是正確的，應(yīng)力集中在裂紋尖端誘引起剪切帶形變(見例如 [5] )。
在這里通過(guò)對(duì)該模型描述,我們假定碎屑分割是由純絕熱剪切,不是裂縫引起的。很顯然, 剪切帶材料點(diǎn)的有效塑性流動(dòng)曲線必須表明最大值。我們選用一個(gè)使溫流動(dòng)曲線出現(xiàn)最大值曲線流場(chǎng)，詳細(xì)解釋見4.1節(jié)。
如果分割碎屑形式,集中應(yīng)力導(dǎo)致了碎屑(近似)連續(xù)變形。必須采取措施，以避免有限元網(wǎng)格因扭曲太大而變形，尤其是在用四邊形元素仿真的的過(guò)程中。
綜上所述,模擬需滿足下列要求:
?盡可能定期使用四邊形，避免極端網(wǎng)格扭曲；
?剪切帶內(nèi)高密度網(wǎng)格；
?碎屑連續(xù)變形(分割)；
?隱式算法收斂；
?為得到可移植性和靈活性使用標(biāo)準(zhǔn)軟件。
在金屬切削模擬，為自動(dòng)形成網(wǎng)格算法的選用是固定的，如用拉格朗日方算法，元素扭曲變大，尤其是分割碎屑形式。頻繁重復(fù)分割以避免分子扭曲太大。在材料移出的剪切帶它也可以用來(lái)制造精確網(wǎng)格(見插圖6)。
然而,標(biāo)準(zhǔn)網(wǎng)格發(fā)電機(jī)是不能處理復(fù)雜的任務(wù)。因此，預(yù)處理程序已經(jīng)編輯了能夠分割曲率很大的被用四邊形剪切生成的區(qū)域程序。剪切帶的位置是用幾何判據(jù)和網(wǎng)格細(xì)化自動(dòng)決定的。預(yù)處理程序?qū)⒃谙乱还?jié)描述。隨后，對(duì)網(wǎng)格生成過(guò)程和建模的分割的詳細(xì)內(nèi)容作解釋。
3.有限元模型
3.1網(wǎng)發(fā)電的原理
過(guò)去的預(yù)處理程序(被稱為pre++ )都是用標(biāo)準(zhǔn)數(shù)據(jù)庫(kù)在C++中寫的，因此，可以移植到不同的平臺(tái)。 預(yù)處理程序被用來(lái)計(jì)算幾何參數(shù)數(shù)據(jù)，使模型參數(shù)輕松改變。它適用于二維三維空間中各種各樣的問(wèn)題。
生成四邊形最簡(jiǎn)單的辦法是劃分組件的物理區(qū)域，組件是被四條線和一個(gè)映射單位正方形限制的。單位正方形有規(guī)律的嚙合可以映射回用等角投影的該地區(qū)本身。細(xì)節(jié)的詳細(xì)敘述見〔23，24〕。
如果我們?cè)谡鎸?shí)空間內(nèi)用(x,y)，在平面內(nèi)用（ξ，η）定義坐標(biāo)，一般曲線坐標(biāo)系可以通過(guò)解拉普拉斯方程來(lái)定義
（1）
（2）
這里的表示，等等。這個(gè)方程系統(tǒng)的物理解釋:當(dāng)兩個(gè)對(duì)立邊攜帶不同的電壓，坐標(biāo)協(xié)調(diào)電場(chǎng)區(qū)域的等勢(shì)線。
把坐標(biāo)（，）作為獨(dú)立變數(shù)，這當(dāng)然是很容易求解的方程。在這種情況下方程已被顛倒過(guò)來(lái)，求解
（3）
（4）
這是一個(gè)半橢圓形的線性方程組求解，可以解決使用標(biāo)準(zhǔn)方法。嚙合算法通常是用來(lái)制造網(wǎng)在一個(gè)物理地區(qū)，是經(jīng)過(guò)了一個(gè)有限元計(jì)算的結(jié)果，因?yàn)樗挥脕?lái)自動(dòng)生成網(wǎng)格的過(guò)程。 因此，界限被計(jì)算步驟定義，因此已經(jīng)離散。求解方程，定期矩形網(wǎng)使用的網(wǎng)格大小的選擇應(yīng)小于最小距離，使等量的舊與新網(wǎng)相同。
由于不規(guī)則形狀的區(qū)域解點(diǎn)數(shù)目已是一個(gè)相當(dāng)大的數(shù)據(jù)，謹(jǐn)慎選擇算法是有優(yōu)勢(shì)的。我們已經(jīng)制定了一個(gè)多重算法，詳細(xì)介紹見勃蘭特[7]。 這種算法的優(yōu)點(diǎn)是快速，穩(wěn)定，而且也給出一個(gè)截?cái)嗾`差的估計(jì)。這種計(jì)算方法可以起到數(shù)值誤差是可比的截?cái)嗾`差。 由于方程是非線性的，只能用近似格式( FAS )的方法來(lái)進(jìn)行。多重技術(shù)依賴的事實(shí)標(biāo)準(zhǔn)松弛方法(如高斯-賽德爾)非常有效地減少振蕩解決部分誤差，而暢順，大部份波長(zhǎng)不影響不大。因此，我們經(jīng)過(guò)幾個(gè)步驟放寬任何涉及方程的誤差可以代表以及對(duì)粗網(wǎng)少點(diǎn)。放松對(duì)這個(gè)粗格再次降低小波長(zhǎng)組成，其中， 現(xiàn)在有一個(gè)較大的絕對(duì)波長(zhǎng)為電網(wǎng)是粗糙。因此，遞歸計(jì)劃是用在錯(cuò)誤的，是有效降低對(duì)所有尺度。 這種算法是一個(gè)標(biāo)準(zhǔn)的工具,用于解決橢圓型方程使讀者可參考文獻(xiàn)進(jìn)一步的詳細(xì)內(nèi)容[20]。它只需約一分鐘，一個(gè)標(biāo)準(zhǔn)的工作站，即使格數(shù)點(diǎn)約為250 000只要界限的區(qū)域不是太強(qiáng)烈彎曲。插圖1( a )顯示坐標(biāo)系在一個(gè)簡(jiǎn)單的區(qū)域被用描述算法創(chuàng)建。
附錄2
A finite element model of high speed metal cutting with adiabatic shearing
Abstract
A finite element model of a two-dimensional orthogonal cutting process is developed. The simulation uses standard finite element software together with a special mesh generator that is able to mesh the chip completely with regular quadrilateral elements and a strong mesh refinement in the shear zone for continuous and segmented chips. The techniques of remeshing and to ensure convergence of the implicit calculation is described. Results for the formation of segmented chips are presented and the segmentation process is studied. Of special interest is the occurrence of split shear bands. The influence of the elastic properties and of the cutting speed is also discussed.
Keywords: Machining; Finite elements; Remeshing; Chip segmentation;
Adiabatic shear bands
1. Introduction
Titanium alloys like Ti6Al4V are widely used in aerospace and other industrial applications. A large fraction of the production costs for components made of these alloys is due to machining. The design of titanium alloys with better machinability is therefore a worthwhile research aim.
To achieve this, it is necessary to identify the important material parameters that critically influence the machinability of the material. This can be done by parameter studies using finite element computer simulations. Once the most promising design avenues are determined, the actual alloy modification can be done, which is thus only the final step of the material design process. This approach is similar to the standard CAE production cycle, where only a few prototypes are built.
* Corresponding author.
E-mail addresses: martin.baeker@tu-bs.de (M. Baker), j.roesler@tu-bs.de (J. Rosler), c.siemers@tu-bs.de (C. Siemers). 1 Work supported by Deutsche Forschungsgemeinschaft.
Creating a reliable computer model of the metal cutting process is the first and crucial step in this process. In this paper, we describe such a model in some detail. It uses standard finite element software for the calculations, thus ensuring portability and flexibility. As the requirements on themeshing algorithm are quite strong, a special preprocessor has been developed, which is programmed in Ctt and is thus also portable to different platforms.
The paper is organized as follows: after a short description of the requirements on the model in Section 2, the details of the finite element model are given in Section 3. Some results produced with the model are shown in Section 4, focussing on the details of the chip formation process. Section 5 summarizes the work and points out future research aims.
2. The problem
In the metal cutting process material is removed from the surface of the workpiece by a cutting tool and a chip is formed. The problem involves large plastic deformations which generate a considerable amount of heat, as does the friction between tool and workpiece and also between tool and chip. The separation of workpiece material in front of the tool also has to be modeled.
As the influence of the material parameters is more important for material design considerations than are the details of the process itself, the cutting process simulated here is that of orthogonal cutting. The process is simulated as two-dimensional, which strongly reduces the computer time needed for the calculation. A further simplification is done by assuming the tool to be perfectly rigid.
Friction and heat flow into the tool have been neglected so far in the simulations, but can easily be included. The reason for this omission is that it is necessary to simplify the cutting process as much as possible to gain insights into the underlying mechanisms as will be explained below. Also, there is no thermal radiation from the free surface of the chip and no heat transfer at the boundary of the material is allowed.
Rapid machining is a strongly non-linear problem due to the effects described above and it has to be simulated using a fully coupled thermomechanical finite element model. It is therefore a formidable task to develop a finite element code to deal with the metal cutting problem from scratch, so that the use of commercial FE software is an attractive alternative. Modern finite element software can in principle handle such strongly non-linear problems. For our studies we have decided to use the ABAQUS/Standard program system, which allows the definition of complex contact conditions, leaves many possibilities to define material behaviour, and can be customized in many regards by including user-defined subroutines. We suppose that most of the methods described below would work with any similarly powerful FE package. Due to the use of standardized software, the formulation of the equations (finite element formulation,thermomechanical coupling, integration scheme, etc.) can be found in great detail elsewhere [3].
Many finite element simulations of the metal cutting process are performed using the explicit method (see for example [17]), which is guaranteed to converge. (An overview over finite element simulations of the cutting process can be found in [16].) Nevertheless, we have decided to use an implicit code. Here convergence is checked during the simulation, but the iterative solution process is no longer guaranteed to converge. One advantage of using the implicit code ABAQUS/Standard is that this allows a great range of flexible user-defined subroutines to be introduced in the simulation. Such routines can be used to implement complicated material separation criteria. In addition to that, the implicit code has a better scaling behavior if local mesh refinement is needed. If narrow shear bands form, element sizes of the order of 1 lm or less are necessary (see Section 4.2) and the advantage in CPU time of using an explicit algorithm will strongly diminish. An explicit method is probably superior if frictional effects are large, which is, however, not the case here. On the other hand, explicit methods often need to change some physical parameters like density or tool velocity, or have to use artificial viscosity. In our opinion, there is no reason to consider an implicit simulation inferior to an explicit one, if convergence can be achieved.
Also differently from many other simulations, we use fully integrated first-order quadrilateral elements, which have better convergence properties than triangular elements. This is discussed further in Section 3.3.
Titanium alloys form segmented chips when cut orthogonally (see Fig. 9). Any detailed simulation of the metal cutting process must be able to take this segmentation into account. The mechanisms behind chip segmentation are still not completely understood [12,15, 25,26]. It is clear that so-called adiabatic shearing plays a prominent role in the segmentation process: Thermal softening of the material in the shear zone leads to an increased deformation in this zone, which produces heat and leads to further softening. This positive feedback between softening and deformation causes a narrow band of extremely strong deformation, while the surrounding material is only slightly deformed. It is, however, not known whether the adiabatic shear bands are caused by cracks growing into the material, as assumed in [25]. If this is true, the stress concentration at the crack tip can then induce the formation of the shear band (see e.g. [5]).
For the model described here, we assume that chip segmentation is caused by pure adiabatic shearing, without cracks occurring. It is quite clear that the effective plastic flow curve of a material point in the shear band must show a maximum for this mechanism to hold. We have used a flow curve field where even the isothermal flow curves show a maximum. This is further detailed in Section 4.1.
If segmented chips form, the shear concentration leads to a (nearly) discontinuous deformation of the chip. Measures have to be taken to ensure that the finite element mesh is not too much distorted due to this deformation, especially in a simulation using quadrilateral elements.
To summarize, the simulation has to meet the following requirements:
. use of quadrilateral elements, as regular as possible, avoiding extremely distorted meshes;
. high mesh density in the shear zone;
. discontinuous deformation (segmentation) of the chip;
. convergence of the implicit algorithm;
. use of standard software for portability and flexibility.
The use of an algorithm for automatic remeshing is mandatory in a simulation of metal cutting, as element distortions become large in a Lagrangian approach, 2 especially if segmented chips form. A frequent remeshing ensures that the elements never become too distorted. It can also be used to create a refined mesh in the shear zone that moves with the material (see Fig. 6).
However, standard mesh generators are not able to handle the complex tasks involved in this problem without difficulties. Thus a preprocessor has been programmed that can mesh the strongly curved regions created by the cutting process using quadrilaterals. The position of the shear zone is automatically determined using a geometric criterion and the mesh is refined there. The preprocessor is described in the following section. Afterwards, details of the mesh creating process and of the modeling of the segmentation are explained.
3. The finite element model
3.1. Principles of mesh generation
The used preprocessor (called Pre++) is written in Ctt using standard class libraries and is thus portable to different platforms. The preprocessor can be used to calculate parametrized geometry data, so that model parameters can easily be changed. It is applicable to a wide range of problems in two and (with some restrictions) in three dimensions.
The easiest way of generating quadrilateral elements is to divide the physical region to be meshed into parts that are bounded by four lines and can be mapped onto the unit square. A regular meshing of the unit square can then be mapped back onto the region itself using a conformal map, as described in some detail in [23,24].
If we define the coordinates in real space with (x; y) and those on the square with (n; g), a general curvilinear coordinate system can be defined by solving the Laplace equation
（1）
（2）
Here nxx denotes the partial derivative o2n=ox2, etc. This system of equations has a physical interpretation: the coordinates correspond to the equipotential lines of an electric field on the region when two opposing sides are held on a different voltage.
It is of course much easier to solve the equation using the coordinates （n; g） as independent variables. In this case the equation has to be inverted, resulting in
（3）
（4）
This is a quasi-linear elliptic system of equations, which can be solved using standard methods. The meshing algorithm is usually used to create a mesh on a physical region that is the result of a finite element calculation, as it is used to automatize the remeshing process. Therefore, the bounding lines are defined by the node positions of the previous calculation step and are thus already discretized. To solve the equations, a regular rectangular mesh is used where the grid size is chosen to be smaller than the smallest distance between nodes on the bounding surfaces, so that the contour of the old and the new mesh closely agree.
As the number of solution points has to be quite large for irregularly shaped regions, it is advantageous to choose the solution algorithm with some care. We have decided on a fullmultigrid algorithm as introduced by Brandt [7]. This algorithm has the advantage that it is fast, robust, and that it also gives an estimate of the truncation error involved in the discretization, so that the calculations can be performed until the numerical error is comparable to the truncation error. As the equations are non-linear, a full approximation scheme (FAS) method has to be used. The multigrid technique relies on the fact that standard relaxation methods (like Gauss–Seidel) are very efficient in reducing the oscillating part of the solution error, whereas the smoother, large-wavelength part is not affected very much. Therefore after a few relaxation steps any equation involving the error can be represented as well on a coarser grid with less points. Relaxation on this coarser grid again reduces the small-wavelength components, which, however, now have a larger absolute wavelength as the grid is coarser. Therefore, a recursive scheme is used where the error is efficiently reduced on all length scales involved. This algorithm is a standard tool for the solution of elliptic equations, so that the reader is referred to the literature for further details [20]. It needs only about a minute on a standard workstation even when the number of lattice points is about 250 000 as long as the boundaries of the region are not too strongly curved. Fig. 1(a) shows the coordinate lines created with the described algorithm on a simple region.