Plastik şekil verme işlemlerinde deformasyon analizi için önerilen bir yöntem ve levha haddeleme problemine uygulanması

Abstract

Geçmişi çok eskilere dayanan plastik şekil verme işlemleri son elli yılda diğer şekil verme işlemlerine kıyasla oldukça popüler olmuştur. Modern dünyanın ihtiyaçları ve işlemler arası rekabet artarken, işlem süresince malzemenin şekil değiştirme durumunun bilinmesi ilgili alanda temel bilgi olarak kabul edilmektedir. Bu çalışmada, bir plastik şekil verme işlemi olan haddeleme sırasında malzemenin şekil değişimini incelemek için çok güçlü bir yöntem olan ve plastisitede özel olarak Üst Sınır Teoremi adı ile bilinen ilkeye dayanan Üst Sınır Yönteminin daha etkin bir biçimde kullanılmasını temin için bu yöntemin uygulanmasında gerekli olan kinematikçe uyumlu uzaysal hız alanlarının kurulabilmesi için sistematik bir yöntem tanıtılmış ve levha haddeleme problemine uygulamaları verilmiştir. Giriş bölümünde, plastik şekil verme işlemlerinde şekil değiştirme analizinin önemi vurgulanmış ve işlem sırasında malzemenin şekil değiştirmesi hakkında bilgi sahibi olmanın gereklerinden bahsedilmiş, bu alanda çalışmalar yapmış olan bilim adamlarının; gerek plastisite teorisinin sınır değer problemlerinin lineer olmamasından gerekse de plastikleşen bölgenin sınırlarının bilinmemesinin yanısıra haddelemede ortaya çıkan problemlerin geometrilerinin karmaşıklığından dolayı sınır koşullarındaki karmaşıklıklardan kaynaklanan güçlüklerin üstesinden gelmek üzere, bu alanda ortaya çıkan problemleri çözmeye yönelik olarak ortaya koydukları yaklaşık yöntemler hakkında bilgi verilmiştir. İkinci bölümde, konu ile bütünlük sağlamak için plastisite teorisinin kısa bir özeti yapılmıştır. Üçüncü bölümde, plastik şekil verme işlemlerinde şekil değiştirme analizi için önerilen yöntemin uygulamasının yapıldığı levha haddeleme probleminin tanımı yapılmıştır. Dördüncü bölümde, kahcı(steady-state) düzlem şekil değiştirme probleminde kinematikçe uyumlu uzaysal hız alanlarının sistematik bir biçimde kurulabilmesini temin etmek için önerilen yöntemin tanıtımı detaylı bir biçimde açıklanmıştır. Beşinci bölümde, önerilen yöntem kullanılarak, ele alınan problemin geometrisine uygun olarak seçilmiş parabolik,dairesel ve eliptik akım çizgileri aileleri ve deformasyon bölgesinin seçilen giriş veya çıkış yüzeylerine bağlı olarak elde edilen kinematikçe uyumlu beş tip hız alanı ve deformasyon bölgesi modellemesi elde edilmiştir. Altıncı bölümde, beşinci bölümde elde edilen hız alanlarının Üst S ınır Yönteminde kullanılması suretiyle üst sınır fonksiyonelleri elde edilmiştir. Yedinci bölümde, altıncı bölümde elde edimiş olan üst sınır fonksiyonellerinin ekstremum koşulları elde edilmiş ve analitik olarak çözümlerinin mümkün olmadığı bu ifadelerin nümerik olarak çözümlerinde kullanılan yöntemden kısaca bahsedilmiş ve bu çalışmanın sonunda elde edilen sonuçlar grafik ve tablo biçiminde verilmiştir. Sonuçlar ve öneriler kısmında, yapılan bu çalışmanın bilimsel yönü tartışılmış ve bu alana ne gibi katkıları olabileceği sunulmuştur.

Most of the problems arising in the mathematical theory of plasticity cannot be solved analytically, except for a few idealized cases which are of minor importance from applica tions point of view. This situation is felt strongly as far as the plastic deformation proc esses are concerned where the precise knowledge on the deformation modes of material is quite important. Nowadays, information about the deformation of the material during a plastic forming process is used in every area associated with the relevant process, from plant planning to the quality of the final product. The difficulty which comes from the non-linearity inherent in the plasticity theory in ob taining the analytical solutions to the boundary value problems arising in the plastic forming processes forced the scientists to develop some approximate methods, especially appropriate for those problems. Efforts spent by researchers in this direction resulted in several methods. Among them the slab method, the slip-line method, the flow potential method, the finite element method, the finite difference method, visioplasticity method, upper bound method and the boundary element method are the ones used widely. We strongly believe that the Upper Bound Method has a special place among these methods, because Upper Bound Theorem is a general statement for the boundary value problems in plasticity and is used both alone and to support the other methods, especially the finite element method, as well. The difficulty in employing the upper bound theorem in the analysis of the deformation of material during a plastic forming process is to construct a kinematically admissible velocity field appropriate for the problem under consideration. The subject of this study is to propose a rigorous, systematic method for constructing kinematically admissible velocity fields for plane strain problems, especially for strip rolling. First of all, as is suggested by the title, what it is dealt with in this paper is not to analyze the deformation in symmetric rolling, but to present a method to analyze these kinds of problems. The main work presented in this study is to introduce a rigorous, systematic method to construct kinematically admissible spatial velocity fields which are required to employ the Upper Bound Method. The main assumption is to take the flow lines (Stream lines, in Eringen's terminology) as a one parameter family of curves. It is shown that the spatial velocity field which satisfies the incompressibility condition (eqn (3)) can be written as eqn (11) and eqn (12), in which f(x,y) and F( ?) are unknowns. Various plastic metal forming processes such as rolling, extrusion, drawing, forging etc., are called free boundary problems. The boundaries of the deformation region as well as the configuration of the free surface is not uniquely predictable. IX In the frame of the presented method, the function of flow lines family and one of the en try or exit surfaces can be chosen arbitrarily. In this study, it is shown that kinematically admissible spatial velocity field and the other boundary of the deformation region can be obtained systematically in relation with the chosen quantities. The selection of the flow lines family and of the boundaries affects the results seriously. Therefore, by proposing more realistic deformation models in accordance with the experimental results, better kinematically admissible spatial velocity fields can be constructed, by using the method outlined above. Although the presentation of the method is restricted to plane strain problems, it can be generalized to 3D problems. KINEMATICALLY ADMISSIBLE VELOCITY FIELDS It is assumed that the trajectories followed by the material point (the flow lines) in the plastic deformation zone for the symmetric plane strain rolling problem can be represented as a one-parameter family of curves in Cartesian coordinates: ğ=f(x,y), OZğZl. (1) y Figure v(x,y) j f(*,y)H -*¦ x 0. A typical flow line in Cartesian coordinates Since the velocity vector at a point in the deformation zone is tangential to the flow line passing through that point dy df '/dx v dx df/dy u (2) where u and v are the components of the velocity vector in the x and y direction, respec tively. If v is eliminated between eqn (2) and the incompressibility condition ssl = 0 or âc oy (3) then df du ^df du J a2/ ufldx d2f\_ Q dy ax dx cy \dy2 dfl$y dxdy) (4) or by the elimination of u between eqn (2) and eqn (3) dy dx dx dy \ dx2 df/dx dxdy) (5) is obtained. Equations (4) and (5) are quasi-linear partial differential equations of first order. Their general solutions is S(Ç,Ç) = 0 or Ç = F(Ç), (6) where S and F are arbitrary differentiable functions, and £, and Ç are the solutions of the following system of ordinary differential equations: dx dy du % & \dy2 $1% dxdy ) (7) or dx dx dv Q2f dflty + #2/^ dx2 dfjât dxdy. (8) It can be easily verified that the solutions of eqn (4) and eqn (5) can be expressed as f(*,y)=ç > (9) (10) Substituting eqn (9) and eqn (10) into eqn (3) or eqn (2) it is seen that G(ğ) is equal to -F(Ç). Therefore, eqn (9) and eqn (10) can be written as follows: (11) (12) where F(Ç) is an arbitrary differentiable function which will be determined by the initial boundary conditions of the problem under consideration, the natural initial boundary conditions being the velocity field on the entry or on the exit surface of the deformation zone. The conditions will be obtained from the continuity of the velocity components in the direction normal to the boundaries of the deformation zone. As is shown in fig (2), the deforming material consists of three zones. In zone I and m the material moves rigidly parallel to the symmetry axis with velocities v0 and \f re spectively. In zone II the material flows obeying a kinematically admissible velocity field XI along the flow lines. There are three boundaries (rItr2,r3 ), the entry surface, the exit surface and the contact surface, respectively. y Figure 2. A typical strip rolling through a pair of cylindrical rigid rolls In each individual zone the velocity field and its derivatives should be continuous. The expressions describing the velocity vector in two neighboring zones are not identical, but because of volume constancy, the normal component of velocity across boundaries be tween two zones should be continuous (fig. (3)). Parallel to the surface, velocity disconti nuity may exist. zone I Figure 3. A typical discontinuity surface in strip rolling The velocity components which are normal to the boundaries r} and r3 must be equal on both side of these surfaces. This condition can be expressed as v. n7 = v0. n, on.T,, v* n2=vfn2 on r2, (13) (14) where v* is the velocity vector in zone n with the components u and v and n; and n2 are the vectors normal to the boundaries r} mdr2, respectively. Since the boundary r3 is a flow line by itself, the continuity condition which is stated for the other boundaries is sat isfied without any restriction. To determine the unknown function F(Ç) which appears in eqn (11) and eqn (12), only one of the conditions (eqn (13) or eqn (14) ) can be employed. Therefore, either rt or r2 can be chosen arbitrarily. After F(Ç) is obtained, the velocity field is determined. The other condition will be used to determine the related boundary of the deformation zone. Let the entry surface be chosen. It is assumed that the parametric representation of it is x = Xj(s), y = yj(s). So, the normal vector nt of the entry surface can be expressed as x cfyj dxj. (15) Since the velocity vectors x*and\0 can be written as v*=ai+vj or y*=-^-F(x)i+-^-F(x)\ and v0=- 0> ox then the continuity condition ( eqn (13) ) can be written as follows: v0i, x=x,(s) ds ax y=y,(s) dx, dy. ds (16) If eqn (1) is considered, then the derivative of £ with respect to s on the chosen surface leads to: dğ_= (19) Equating F(E,)\ of eqn (18) and eqn (19) to each other the following expression is ob tained: dy2 -=v" dyj or Vfdy2=v0dy, vf dç~v° dÇ From eqn (20) the ordinate of the exit surface can be found as (20) y2=j-y,(Ç)- (21) Eqn (21) expresses that any flow line denoted by ğ which passes through the entry sur face at yt ordinate without cutting any member of the family of flow lines exits from the deformation zone at y2 ordinate of the exit surface. This expression is nothing but the global incompressibility condition. xi From eqn (21), expressing £ as a function of y2, the following relation can be written: Ç~S6>2). (22) which means that if the exit surface were known such as x = x2(y2) and it were put into eqn(l), eqn (22) would have been obtained. So, the following relation exist. ş=f(x3(y2),y3)=4(y2) ¦ (23) From eqn (23) x = x2 (y2) can be solved as the equation of the exit surface. DISCUSSION AND CONCLUSION In this study, a rigorous, systematic method to construct kinematically admissible velocity fields to analyze deformation in plastic deformation processes is introduced and some ex amples are given to demonstrate applications of the method. The method presented here essentially reduces the problem of constructing kinematically admissible velocity fields to choose the flow lines and one of the boundaries of the deformation zone. Once the flow lines in the plastic deformation zone and one of the boundaries of the deformation zone are chosen the method presented here describes how to find the velocity field and the other boundary of the deformation zone. Early applications of the upper bound theorem were limited because of the uncertainity in constructing kinematically admissible velocity fields appropriate for the problem under considerations. The main purpose of this study is to remove this limitation on the upper bound analyses of plastic forming processes. With the method presented here at hand, it is possible to construct a family of kinematically admis sible velocity fields flexible enough (depending on some parameters) so that the family can include (or can be as close as desired to) the real solution. The method can be extented to the three-dimensional problems. What is needed to do is to replace the condition (2) which states that the velocity is tangent to the flow lines, with its three-dimensional equivalent and to consider the three-dimensional version of the incom- pressibility condition (3). Although the mathematics involved in this sheme is much more voluminous compared to the two-dimensional case, there is no any essential difficulty in arriving at the results similar to (11) and (12). The method can be also extended to con struct kinematically admissible velocity fields for transient problems. One of the ways in constructing time-dependent kinematically admissible velocity fields is to consider the pa rameters which appear in the model, being dependent on time in a certain way. One of the major motives in proposing the method presented in this study is the velocity discontinuity in the early models which were used to analyze deformation in rolling, ex trusion, wire drawing, etc.. The kinematically admissible velocity fields with discontinu ity are not physically realistic. It is not difficult to claim that none of the kinematically admissible velocity fields with discontinuity can be a model for real deformation pattern. On the other hand, kinematically admissible velocity fields with discontinuity restrict the application of the upper bound theorem to the rigid-ideally plastic materials. Since it is possible to construct some kinematically admissible velocity fields without discontinuity by employing the method presented in this study, now, the upper bound theorem can be extended to analyze the deformation in rolling (in other processes, too) of the elastic- plastic, strain-hardining materials. In this case, an appropriate variational principle should be employed.