擠壓模具概述外文資料翻譯

1、<p><b>　　外文資料翻譯</b></p><p>　　Overview of adaptable die design for extrusions</p><p>　　W.A. Gordon.C.J. Van Tyne.Y.H. Moon</p><p><b>　　ABSTRACT</b></p

2、><p>　　The term “adaptable die design” is used for the methodology in which the tooling shape is determined or modified to produce some optimal property in either product or process. The adaptable die design me

3、thod, used in conjunction with an upper bound model, allows the rapid evaluation of a large number of die shapes and the discovery of the one that produces the desired outcome. In order for the adaptable die design metho

4、d to be successful, it is necessary to have a realistic velocity field for th</p><p>　　© 2006 Elsevier B.V. All rights reserved.</p><p>　　Keywords: Extrusion; Die design; Upper bound approa

5、ch; Minimum distortion criterion</p><p>　　1. Introduction</p><p>　　New metal alloys and composites are being developed to meet demanding applications. Many of these new materials as well as trad

6、itional materials have limited workability. Extrusion is a metalworking process that can be used to deform these difficult materials into the shapes needed for specific applications. For a successful extrusion process, m

7、etalworking engineers and designers need to know how the extrusion die shape can affect the final product. The present work focuses on the design of appr</p><p>　　rion needs to be established for the optimiz

8、ation of the die shape. The criterion must be useable within an upper bound model. The full details of the method are presented elsewhere [1–6]. In the present paper, following a review of previous models for extrusion,

9、the flexible velocity field for the deformation region in a direct extrusion will be briefly presented. This velocity field is able to characterize the flow through a die of almost any configuration. The adaptable equati

10、on, which descri</p><p>　　2. Background</p><p>　　2.1. Axisymmetric extrusion</p><p>　　Numerous studies have analyzed the axisymmetric extrusion of a cylindrical product from a cylin

11、drical billet. Avitzur[7–10] proposed upper bound models for axisymmetric extrusion through conical dies. Zimerman and Avitzur [11] modeled extrusion using the upper bound method, but with generalized shear boundaries. F

12、inite element methods were used by Chen et al. [12] and Liu and Chung [13] to model axisymmetric extrusion through conical dies. Chen and Ling [14] and Nagpal [15] analyzed other die sh</p><p>　　2.2. Distort

13、ion and die shape analysis</p><p>　　Numerous analytical and experimental axisymmetric extrusion investigations have examined the die shape and resulting distortion. Avitzur [9] showed that distortion increas

14、es with increasing reduction and die angle for axisymmetric extrusion through conical dies. Zimerman and Avitzur [11] and Pan et al. [21] proposed further upper bound models, including ones with flexibility in the veloci

15、ty field to allow the distorted grid to change with friction. They found that increasing friction causes more</p><p>　　2.3. Three-dimensional non-axisymmetric extrusion analysis</p><p>　　Both th

16、e upper bound and finite element techniques have been used to analyze three-dimensional non-axisymmetric extrusions. Nagpal [27] proposed one of the earliest upper bound analyses for non-axisymmetric extrusion. Upper bou

17、nd and finite element models were developed Basily and Sansome[28] , Boer et al.[29] , and Boer and Webster [30] . Kiuchi [31] studied non-axisymmetric extrusions through straight converging dies. Gunasekera and Hoshino

18、[32–34] used an upper bound model to study the extrus</p><p>　　3. The adaptable die design method</p><p>　　The adaptable die design method has been developed and is described in detail in a seri

19、es of papers [1–5]. The method has been extended to non-axisymmetric three-dimensional extrusion of a round bar to a rectangular shape [6]. The major criterion used in developing the method was to minimize the distortion

20、 in the product. The present paper provides a brief overview of the method and results from these previous studies. </p><p>　　Fig. 1. Schematic diagram of axisymmetric extrusion using spherical coordinate sy

21、stem through a die of arbitrary shape</p><p>　　3.1. Velocity field</p><p>　　An upper bound analysis of a metal forming problem requires a kin matically admissible velocity field. Fig. 1 shows th

22、e process parameters in a schematic diagram with a spherical coordinate system (r, θ , φ ) and the three velocity zones that are used in the upper bound analysis of axisymmetric extrusion through a die with an arbitrary

23、die shape. The material is assumed to be a perfectly plastic material with flow strength,.he friction, which exists between the deformation zone in he work piece </p><p>　　The material starts as a cylinder o

24、f radius Ro and is extruded into a cylindrical product of radius . Rigid body flow occurs in zones I and III, with velocities of and , respectively. Zone II is the deformation region, where the velocity is fairly comple

25、x. Two spherical surfaces of velocity discontinuity Γ 1 and Γ 2 separate the three velocity zones. The surface Γ 1 is located a distance from the origin and the surface Γ2 is located a distance from the origin.</p>

26、;<p>　　The coordinate system is centered at the convergence point of the die. The convergence point is defined by the intersection of the axis of symmetry with a line at angle α that goes through the point where t

27、he die begins its deviation from a cylindrical</p><p>　　shape and the exit point of the die. Fig. 1 shows the position of the coordinate system origin. The die surface, which is labeled ψ(r) in Fig. 1, is gi

28、ven in the spherical coordinate system. ψ(r) is the angular position of the die surface as a function of the radial distance from the origin. The die length for the deformation region is given by the parameter L.</p&g

29、t;<p>　　The best velocity field to describe the flow in the deformation region is the sine-1 velocity field [1,2] . This velocity field uses a base radial velocity, , which is modified by an additional term compri

30、sed to two functions with each function containing pseudo-independent parameters to determine the radial velocity component in zone II:</p><p><b>　　(1)</b></p><p>　　The ε function pe

31、rmits flexibility of flow in the radial, r, direction, and the γ function permits flexibility of flow in the angular, θ , direction. The value of is determined by assuming proportional distances in a cylindrical sense f

32、rom the centreline:</p><p><b>　　(2)</b></p><p>　　This velocity field was found to be the best representation of the flow in the deformation region of an extrusion process for an arbi

33、trarily shaped die.</p><p>　　The ε function is represented as a series of Legendre polynomials that are orthogonal over deformation zone. The representation of ε is:</p><p><b>　　(3)</b&

34、gt;</p><p><b>　　Where </b></p><p>　　being the coefficients of the Lengendre polynomials Pi(x) and being the order of the representation. There is a restriction that:</p><p

35、>　　， (4)</p><p>　　The remaining higher order coefficients (A2 to A) are the pseudo-independent parameters, with values determined by minimization of the total power. Legendre

36、polynomials are used so that higher order terms can be added to the function without causing significant changes in the coefficients of the lower order polynomials. This feature of the Legendre polynomials occurs because

37、 they are orthogonal over a finite distance.</p><p>　　The γ function that satisfies the boundary conditions and allows the best description of the flow is:</p><p><b>　　(5)</b></p&

38、gt;<p><b>　　where</b></p><p>　　and the high order coefficients B1 to B are pseudo-independent parameters with values determined by minimization of the total power. The order of the represe

39、ntation is . It has been shown [3] that = 6 and = 2 are usually sufficient to provide reasonable flexibility for the flow field in the deformation region.</p><p>　　3.2. Die shape</p><p>　　The di

40、e shape is described by the functionψ (r). The adaptable die shape is described by a set of Legendre polynomials:</p><p><b>　　(6)</b></p><p><b>　　where</b></p><

41、;p>　　and being the coefficients of the Legendre polynomials Pi(x). The order of the Legendre polynomial representation is . The boundary conditions at the entrance and exit of the deformation region require that:</

42、p><p>　　At r = , ψ= α</p><p>　　At r = , ψ= α (7)</p><p>　　If a streamlined die is used then this function must meet two additional boundary co

43、nditions:</p><p>　　At r = , </p><p>　　At r = , (8)</p><p>　　3.3. Distortion criteria</p><p>　　The criterion that was found to minimize th

44、e distortion in the extrusion product involves minimizing the volumetric effective strain rate deviation [4,5] . The volumetric effective strain rate deviation in the deformation zone is:</p><p><b>　　W

45、here</b></p><p><b>　　with:</b></p><p><b>　　(10)</b></p><p>　　and are the components of the strain rate field.</p><p>　　3.4. Determining t

46、he adaptable die shape</p><p>　　The search for the optimal coefficients for the Legendre polynomials representing the die shape is not constrained. A nested optimization routine is used with the velocity fie

47、ld (inside loop) being minimized with respect to the externally supplied power for the process, and the die shape (outside loop) being adapted to minimize the distortion criterion. The final shape is called an adaptable

48、die shape, since the shape has adapted to meet the specified criterion.</p><p>　　Fig. 2. Streamlined adaptable die shape with no adaptation in the rotational directiowith red = 0.60, L/Ro= 1.0,mf= 0.1,Rr/Ro=

49、 0.1 and μ = 1.5. The area reduction ratio is red, Rr is the cornered radius of the rectangular product, andμ is the height to width ratio of the rectangular product. (For interpretation of the references to colour in th

50、is figure legend, the reader is referred to the web version of the article.)</p><p>　　3.5. Extension to three-dimensional non-axisymmetric shapes</p><p>　　In extending the adaptable die design m

51、ethod from axisymmetric flow to non-axisymmetric three-dimensional flow in the deformation region requires several special considerations [6]. First, the velocity field needs to be modified to allow for rotational moveme

52、nt in the deformation region. Second, the bearing region on the exit side of the die needs to be analyzed properly. Third, the functions used to describe the die shape need to have some flexibility in the rotational dire

53、ction (i.e. ψ(r, φ ).</p><p>　　4. Die shape to minimize distortion</p><p>　　To illustrate the adaptable die design method a specific three-dimensional example is presented. An extrusion upper bo

54、und model was used to determine adaptable die surface shapes, which minimize distortion through minimizing the volumetric effective strain rate deviation in the deformation zone for the extrusion of a cylindrical billet

55、into a round cornered rectangular product. Two schemes were used. In the first method there was no flexibility allowed in the rotational direction, ψ (r), whereas</p><p>　　In Fig. 4, the extrusion die surfac

56、e shape on the two rectangular symmetry planes of the product is presented. The adaptable die shape with rotational flexibility is different from the die shape obtained without adaptation in the rotational direction espe

57、cially along the φ = π /2 symmetry plane. The adaptable die shape geometry along the φ = π /2 symmetry plane increases the speed of the material in the deformation zone near the exit. Fig. 5 shows the resulting distorted

58、 grid in the extrudate on</p><p>　　Fig. 3. Streamlined adaptable die shape with adaptation of the die shape in the rotational direction with red = 0.60, L/Ro= 1.0,mf= 0.1,Rr/Ro= 0.1 and μ = 1.5.</p>&

59、lt;p>　　Fig. 4. Streamlined die shape with no adaptation as the rotational direction</p><p>　　compared to streamlined die shape with adaptation as a function of the rotational direction—plotted along recta

60、ngular symmetry planes.</p><p>　　Fig. 5. Extrudate distorted grid along rectangular symmetry planes for extrusion through a streamlined die shape with no adaptation as the rotational direction compared to a

61、streamlined die shape with adaptation as a function of the rota-tional direction—plotted along rectangular symmetry planes. Ar is the width of he extrudate and Br is the height of the extrudate.</p><p>　　5.

62、Summary</p><p>　　This paper presented an overview of the “adaptable die design” methodology. The full details of the method are given elsewhere [1–6]. In order to use the adaptable die design method in conju

63、nction with an upper bound analysis, it is necessary to have a velocity field in the deformation region with sufficient flexibility so that the model can be closer to the real flow. The specific criterion of producing a

64、product with minimal distortion involves minimizing the volumetric strain rate deviation. </p><p><b>　　擠壓模具設(shè)計(jì)概述</b></p><p><b>　　摘 要</b></p><p>　　術(shù)語“適應(yīng)”是用于模具設(shè)

65、計(jì)的方法，在結(jié)果或過程中確定加工形狀或改良來產(chǎn)生最佳的性能。模具設(shè)計(jì)方法的適應(yīng)性，使用一個(gè)上界，允許快速評(píng)價(jià)一大批二維模型和發(fā)現(xiàn)那個(gè)想要的結(jié)果。為了成功得到合適的擠壓模具的設(shè)計(jì)方法，擁有一個(gè)現(xiàn)實(shí)的速度場，通過擠壓模具變形過程中的任何形狀和速度場一定要讓彈性材料的運(yùn)動(dòng)，并達(dá)到要求的物料流量的描述，這是非常必要的。各種各樣的標(biāo)準(zhǔn)，可用于適應(yīng)性擠壓模具的設(shè)計(jì)方法，例如產(chǎn)生輕微變形的模具。一個(gè)雙優(yōu)化過程是用來確定測(cè)量值在速度場的靈活變量，而且決

66、定最符合給定的標(biāo)準(zhǔn)的模型。該方法已擴(kuò)展到非完全軸對(duì)稱模具產(chǎn)品的形狀的設(shè)計(jì)中。</p><p><b>　　1、介紹</b></p><p>　　新合金和復(fù)合材料正在開發(fā)中，以滿足應(yīng)用的要求。許多這些新材料以及傳統(tǒng)材料擁有有限的施工性能。擠壓加工過程是使這些難加工的材料加工成有具體應(yīng)用的形狀。一個(gè)成功的擠壓工藝、金工工程師和設(shè)計(jì)師需要知道什么樣的擠壓模具形狀的影響最終的

67、產(chǎn)品。目前的工作重點(diǎn)是設(shè)計(jì)合適的擠壓模型。給出了一個(gè)方法來確定有特殊標(biāo)準(zhǔn)的模型：生產(chǎn)產(chǎn)品或優(yōu)化形狀的指定的屬性,例如最低失真,或在生產(chǎn)過程中通過優(yōu)化產(chǎn)品，如最低擠壓力產(chǎn)生的形狀。術(shù)語“適應(yīng)”是用于模具設(shè)計(jì)的方法,確定了模具形或修改產(chǎn)生一定的最優(yōu)屬性的產(chǎn)品或過程。這種適應(yīng)性強(qiáng)的擠壓模具的設(shè)計(jì)方法,是使用一個(gè)上界的快速評(píng)估模型,使得大量模型并且發(fā)現(xiàn)了一個(gè)能優(yōu)化的理想結(jié)果。有幾種情況必須符合適應(yīng)力強(qiáng)的模具設(shè)計(jì)方法才是可行的。首先,一個(gè)普遍但現(xiàn)

68、實(shí)的速度流場中需要使用到數(shù)學(xué)模型通過對(duì)任意形狀的擠壓模具，描述上界材料的流動(dòng)。第二,一個(gè)堅(jiān)固的標(biāo)準(zhǔn)則需要建立優(yōu)化的模具形狀。這個(gè)標(biāo)準(zhǔn)必須利用在一個(gè)上界的模型中。</p><p>　　其他方法的詳細(xì)資料。摘要本文對(duì)先前的模型、靈活的速度場在一個(gè)直接的擠壓變形區(qū),將會(huì)簡略介紹。這個(gè)速度場能描述流體通過一個(gè)任何配置的模具。描述模具形狀的合適的方程，同樣提到。這個(gè)模具形狀方程的常量是關(guān)于標(biāo)準(zhǔn)的優(yōu)化。這個(gè)標(biāo)準(zhǔn)可以用來減少變

69、形。最后，合適的模具的形狀可以以最少程度的失真擠壓生產(chǎn)產(chǎn)品。本論文的目的是提供一個(gè)簡單的概述合適的模具的設(shè)計(jì)方法。</p><p><b>　　2、背景</b></p><p><b>　　2.1軸對(duì)稱擠壓</b></p><p>　　無數(shù)的研究分析了一個(gè)圓柱形坯料從到一個(gè)圓柱形產(chǎn)品的軸對(duì)稱擠壓。Avitzur提議上界的軸

70、對(duì)稱模型是通過錐形的模具。Zimerman和 Avitzur通過上界最大值辦法模仿了擠壓，但廣義地修改了邊界。Chen和Liu還有Chung等人已經(jīng)使用了有限元方法通過錐形模具模仿軸對(duì)稱模型擠壓。Chen和 Ling 還有Nagpal分析了其他的模具形狀。他們?yōu)橥ㄟ^任意形狀的軸對(duì)稱擠壓模具發(fā)展速度場。Richmond首次提出這一流線形模具的概念作為一個(gè)模具輪廓形狀的模具最優(yōu)化的扭曲。Yang和Han為流線形模具發(fā)展了最大界模型。Srin

71、ivasan等提出了控制模具的應(yīng)變速率作為流線形，這改善了材料有限工作性的擠壓過程。Lu和Lo用一種改進(jìn)的應(yīng)變速率控制提出了一種模具形狀。</p><p>　　2.2變形及模具形狀的分析</p><p>　　眾多的分析和實(shí)驗(yàn)性的軸對(duì)稱擠壓的調(diào)查已經(jīng)探討了模具形狀和造成的失真。Avitzur展示了失真的增加通過越來越大的下降和通過錐形模具的軸對(duì)稱擠壓的模角。Zimerman和Avitzur

72、還有 Pan提出了進(jìn)一步上界的模型包括那些在速度場允許歪曲的網(wǎng)格通過摩擦改變的靈活性.他們發(fā)現(xiàn)增加的摩擦引起在擠壓產(chǎn)品中更多的失真，Chen等人確證了模角跟摩擦持續(xù)下降的提高。</p><p>　　其他的研究工作主要集中在非錐形模具的形狀。Nagpal提煉了上界方法的研究或軸對(duì)稱模具的形狀。Chen 和 Ling使用上界的方法來研究流經(jīng)余弦、橢圓形和雙曲線的模具在試圖在尋找一個(gè)模型使力量和多余壓力最小化。Rich

73、mond 和 Devenpeck不是假設(shè)一個(gè)特殊類型的模型，而是設(shè)計(jì)一種基于擠塑產(chǎn)品一些特點(diǎn)的模型。用滑移線分析與理想的假設(shè)和無摩擦的情況，Richmond提出了一種流線形的C形模具，并且在模具入口和出口平衡過渡。流線形模具以很多軸對(duì)稱模具設(shè)計(jì)的轉(zhuǎn)向力為基礎(chǔ)。Yang 和 Han 還有 Ghulman等人用流線形模具發(fā)展了上界模具。</p><p>　　某些材料,如金屬基復(fù)合材料,可以在很小的應(yīng)變率范圍內(nèi)被成功擠

74、壓,導(dǎo)致發(fā)展的主要控制應(yīng)變速率消逝。這個(gè)控制應(yīng)變速率的變形區(qū)來自于一項(xiàng)研究表明纖維斷裂在擠壓的觸須增強(qiáng)復(fù)合材料在峰值應(yīng)變率最小化的研究(25)。Srinivasan 等人是最初開發(fā)者，(19)流線型外形模具試圖在一個(gè)大的區(qū)域的變形區(qū)產(chǎn)生恒應(yīng)變速率。lu和lo(20)用精制板的方法來解釋摩擦材料特征在變形區(qū)的變動(dòng)。kim 和其他人[26]采用有限元方法,設(shè)計(jì)了軸對(duì)稱控制應(yīng)變速率的模具。他們用貝塞爾曲線描述的模具形狀并且在變形區(qū)最小化了用體

75、積應(yīng)變速率的偏差。</p><p>　　2.3 三維非完全軸對(duì)稱擠壓分析</p><p>　　上界和有限元技術(shù)都用于分析三維非完全軸對(duì)稱擠壓。Nagpal[27]提出的最早的上界分析引入非完全軸對(duì)稱擠壓方法之一。Sansome，Basily（28）和Boer等人（29）發(fā)明了上界和有限元模型。Boer和Webster[30]，Kiuchi[31]通過直聚集模具研究了非完全軸對(duì)稱擠壓模具。G

76、unasekera和Hoshino[32-34]使用一個(gè)上界模型研究了多邊形的形狀通過雙螺桿擠出機(jī)模具以及通過流線型的模具。Wu 和 Hsu(35)提出了一個(gè)靈活的速度場,通過擠壓多邊形形狀來直集聚模具。Han和其他人（36) 為了研究擠壓模具通過流線型產(chǎn)生clover-shaped部分，從先前的軸對(duì)稱模型[37]的上界創(chuàng)建一個(gè)速度場。Yang 和其他人(37)應(yīng)用一般上界模型來研究的橢圓和矩形部分雙螺桿擠出機(jī)。Han 和Yang [3

77、8]模擬了齒輪的擠壓。Yang 和其他人 (39)也使用有限元分析證實(shí)實(shí)驗(yàn)與分析的上界優(yōu)越的性能。非完全軸對(duì)稱三維被運(yùn)用上限元素的技術(shù)[40]和空間元素固定區(qū)域[[41,42]而得以進(jìn)一步研究。</p><p>　　流線型模具已經(jīng)是大部分三維擠壓所使用的模具。模具的形狀在入口和出口之間要靠經(jīng)驗(yàn)的選擇并且要講究嚴(yán)密的工程原理。Nagpal和其他人(43)認(rèn)為一個(gè)具有最初位置的點(diǎn)的最終位置是通過確定局部片段減少的面積

78、和整體減少的面積一樣來決定的。一旦一個(gè)物質(zhì)點(diǎn)的位置得以確定，一個(gè)三次多項(xiàng)式則可適用于模具進(jìn)出口的點(diǎn)上。Gunasekera[44]和其他人講該方法僅限于允許二次幾何體。Ponalagusamy和其他人[45]提出了使用貝塞爾曲線來進(jìn)行流線型的擠壓模具的設(shè)計(jì)。Kang 和Yang[46)使用了有限元模型來預(yù)測(cè)為最優(yōu)軸承的長度“L”形擠壓。對(duì)三維擠壓模具的研究設(shè)計(jì)是很有限的。控制應(yīng)變率的概念僅僅被應(yīng)用于機(jī)械擠壓而不能應(yīng)用于軸對(duì)稱三維擠壓。&

79、lt;/p><p>　　3.適應(yīng)性模具設(shè)計(jì)方法</p><p>　　適應(yīng)性模具設(shè)計(jì)方法已經(jīng)在-5頁進(jìn)行詳述并作以發(fā)展。</p><p>　　Fig. 1. 通過鑄模任意形狀用球面坐標(biāo)系的軸對(duì)稱擠壓系統(tǒng)的示意圖</p><p>　　這個(gè)方法一直延伸到一個(gè)圓棒的非完全軸對(duì)稱三維擠壓到一個(gè)長方形[6]。用于開發(fā)方法的一個(gè)主要標(biāo)準(zhǔn)就是是減少產(chǎn)品的變形。這

80、篇文章簡要介紹了方法和從先前的研究成果。</p><p><b>　　3.1.速度場</b></p><p>　　金屬成形問題的一個(gè)上界值分析需要給定一個(gè)動(dòng)速度場。圖1顯示圖表中工藝參數(shù)與球形坐標(biāo)系統(tǒng)(r,θ、φ)和三個(gè)速度的區(qū)域通過一個(gè)任意形狀的模具適用于上界的軸對(duì)稱擠壓分析。所使用的材料被假設(shè)是一個(gè)完美的塑料材料強(qiáng)度σo。存在于工件和模具之間變形區(qū)的摩擦具有十分大

81、的摩擦力,mfσo /√τ3,在那里持續(xù)不斷的磨擦系數(shù)、物流、即可從0到1值。</p><p>　　物質(zhì)隨著氣缸的半徑羅和擠壓成圓柱產(chǎn)品的半徑射頻()。剛體流出現(xiàn)在一區(qū)域和三區(qū)域，速度分別為何。二區(qū)域是變形區(qū),那里的速度相當(dāng)復(fù)雜。兩個(gè)球形表面Γ1和Γ2將這三個(gè)區(qū)域分開。Γ1分開,這三位于距離r0的起源，Γ2位于射頻(），離起點(diǎn)遠(yuǎn)。</p><p>　　坐標(biāo)系統(tǒng)集中在模具聚合點(diǎn)。聚合點(diǎn)被定義

82、為交匯對(duì)稱軸的角α與線,經(jīng)過在模具開始偏離圓柱形狀及模具的出口點(diǎn)的死亡。圖1所示的位置是坐標(biāo)系統(tǒng)的起源。模具表面在圖1中標(biāo)有ψ(r),認(rèn)為在球面坐標(biāo)系統(tǒng)。ψ(r)是模具表面一個(gè)角位置起到了遠(yuǎn)離源點(diǎn)的作用。變形的模具區(qū)域的長度由參數(shù)L標(biāo)記。</p><p>　　用來描述變形區(qū)的變動(dòng)最好的流場是sine-1速度場[1,2]。這個(gè)速度場的使用向速度，該速度由一個(gè)附加的用于組成每組功能都包含獨(dú)立的pseudo參數(shù)的術(shù)語組

83、成，該參數(shù)決定了區(qū)域2的向速度成分:</p><p>　　Ur = + εγ (1)</p><p>　　函數(shù)ε允許徑向流,r,方向,γ函數(shù)允許靈活性的流動(dòng)角θ方向。vr的值是由假定在一個(gè)圓柱距離中決定的:</p><p><b>　　(2)</b></p

84、><p>　　這個(gè)速度場被發(fā)現(xiàn)有最好的呈現(xiàn)流動(dòng)變形區(qū)內(nèi)形成任意形狀的模具的擠壓工藝。</p><p>　　這個(gè)ε功能是被描繪成一系列勒這是正交多項(xiàng)式對(duì)變形區(qū)。這個(gè)</p><p><b>　　ε表示的是：</b></p><p><b>　　(3)</b></p><p><

85、;b>　　在這個(gè)公式里</b></p><p><b>　　和 ρ = r/</b></p><p>　　Ai 成為多項(xiàng)式系數(shù)Pi (x)，. 這里有一個(gè)限制:</p><p>　　剩下的高階系數(shù)(A2到Ana)獨(dú)立的pseudo參數(shù)的確定與價(jià)值最小的總功率。勒讓德多項(xiàng)式在有限的距離正交。</p><p&g

86、t;　　函數(shù) γ 滿足了有限條件，最佳描述如下</p><p><b>　　在這個(gè)公式里</b></p><p>　　和高解析數(shù)B1 t到是獨(dú)立的pseudo系數(shù)，所有值由整體力量的大小決定。展示的順序是. 這表明[3]</p><p>　　= 6 和= 2 足以為在變形區(qū)的流動(dòng)領(lǐng)域提供合理的靈活性。</p><p>&

87、lt;b>　　3.2. 模具形狀</b></p><p>　　模具形狀是由ψ(r)功能所描述的。適應(yīng)性模具形狀由一系列的Legendre 多項(xiàng)式描述：</p><p><b>　　在這個(gè)公式里</b></p><p><b>　　和</b></p><p>　　、是Legendre

88、 多項(xiàng)式的高階系數(shù)Pi (x).</p><p>　　Legendre 多項(xiàng)式的展示順序是. 變形區(qū)的入口和出口之間的有限條件要求：</p><p>　　at r =, ψ= α</p><p>　　at r =, ψ= α (7)</p><p>　　如果一個(gè)流線型

89、模具是被使用過的，那么該功能則滿足兩條附加有限條件：</p><p><b>　　3.3. 變形標(biāo)準(zhǔn)</b></p><p>　　在擠壓產(chǎn)品中減少變形的的標(biāo)準(zhǔn)包括減小體積應(yīng)變速率的偏差e</p><p>　　[4,5]. 體積效果的應(yīng)變速率在變形區(qū)如下：</p><p>　　3.4. 決定適應(yīng)性模具的形狀</p&g

90、t;<p>　　尋找最優(yōu)多項(xiàng)式系數(shù)為勒表明模具的形狀不拘泥。一套使用的速度場(內(nèi)浸)優(yōu)化路徑被最小化的外部提供電力，對(duì)于這個(gè)過程,模具(外環(huán))可以用來最小化變形的程度。最后的形狀叫做可接受的模具的形狀,因?yàn)樾螤钜呀?jīng)適應(yīng)了滿足指定標(biāo)準(zhǔn)。</p><p>　　圖2. 流線型外形與非適應(yīng)性模具在轉(zhuǎn)動(dòng)方向用紅=0.60、L/駛上=1,物流= 0.1 Rr=0.1μ/駛上/和1.5。這個(gè)地區(qū)減薄率是紅色的,R

91、r是長方形的產(chǎn)品的徑長，μ是高寬比的矩形的產(chǎn)品。(對(duì)于解釋在對(duì)色彩的參考圖中,讀者可參考網(wǎng)絡(luò)版本。)</p><p>　　3.5. 對(duì)三維非完全軸對(duì)稱延伸形狀</p><p>　　在沖壓模具的設(shè)計(jì)方法,從非軸對(duì)稱適應(yīng)性的非完全軸對(duì)稱三維流流中變形區(qū)需要幾個(gè)特殊考量[6]。首先,速度場需要修改,以允許在變形區(qū)內(nèi)旋轉(zhuǎn)移動(dòng)。第二,軸承地區(qū)出口需要適當(dāng)分析模具。第三,用于描述模具形狀的功能需要在旋

92、轉(zhuǎn)方向（例如ψ(r,φ))上有靈活性。這種靈活性是的模具形狀與旋轉(zhuǎn)協(xié)調(diào)一致。</p><p>　　4.模具形狀的失真程度最小化</p><p>　　為了解釋模具的設(shè)計(jì)方法，我們舉一個(gè)具體的三維例子。在擠壓上界模型被用來確定可變性模具表面的形狀,通過減小體積來把失真程度降低到最小，在應(yīng)變速率偏差的變形區(qū)把一個(gè)圓柱形擠壓成圓鋼坯一隅矩形產(chǎn)品。兩種方案。第一種方法不允許在旋轉(zhuǎn)方向(r)內(nèi)靈活反應(yīng)

93、。第二種方法是模具形狀能夠適應(yīng)的旋轉(zhuǎn)方向,(r,φ)。兩種方法的實(shí)現(xiàn)條件都是在模具的進(jìn)口和出口地區(qū)。這個(gè)節(jié)段(φ= 0 / 2π）?？谛螤畹慕o出了圖形。2號(hào)和3號(hào)。圖2是模具形狀沒有旋轉(zhuǎn)靈活在圖3模具可以去適應(yīng)它的形狀并且在轉(zhuǎn)動(dòng)方向內(nèi)來減少失真。以上的例子中的占地減小到60%,產(chǎn)品為一個(gè)矩形，寬高比,μ、1.5。圖4中,擠出模具表面形狀對(duì)兩個(gè)長方形對(duì)稱平面的產(chǎn)品。模具的適應(yīng)性和旋轉(zhuǎn)的靈活性是不同形狀的模具未經(jīng)改編的外形獲得旋轉(zhuǎn)方向特別是

94、沿φ=π/ 2對(duì)稱平面。模具的適應(yīng)性和幾何形狀沿φ=π/ 2對(duì)稱平面增加了速度的材料變形區(qū)附近的出口。圖5。最終產(chǎn)生扭曲的網(wǎng)格對(duì)稱的飛機(jī)。模具的適應(yīng)性更小的差異表明形狀相比,在變形形態(tài)內(nèi)的模具旋轉(zhuǎn)靈活。</p><p><b>　　5.總結(jié)</b></p><p>　　本文對(duì)“適應(yīng)力強(qiáng)的模具設(shè)計(jì)”的方法論。具體給出了應(yīng)用該方法的地方(1-6)。為了使用強(qiáng)模具的設(shè)計(jì)方法