Design of Streamline Dies for Drawing Driven by Fracture SERGEI ALEXANDROV 1 , YUSOF MUSTAFA 2 and ELENA LYAMINA 1 1 A.Yu. Ishlinskii Institute for Problems in Mechanics, Russian Academy of Sciences, 101-1 Prospect Vernadskogo, 119526 Moscow RUSSIA sergei_alexandrov@spartak.ru http://www.ipmnet.ru 2 Faculty of Mechanical Engineering Universiti Teknologi Malaysia 81310 Skudai, Johor Darul Ta’zim MALAYSIA mustafa51059@gmail.com http://www.utm.my Abstract: - This paper presents an efficient analytical method for design of streamline dies driven by fracture. The method is based on Bernoulli’s theorem relating pressure and velocity along any streamline extended to ideal flows in plasticity. The Cockroft-Latham criterion is adopted to predict the initiation of ductile fracture. In order to apply the method developed, it is not necessary to know the solution to the boundary value problem of plasticity. The final result is a simple relation between geometric parameters of the process and the constitutive parameter involved in the fracture criterion. Since the latter is supposed to be known for a given material, the relation determines a safe domain for drawing without fracture. Key-Words: - drawing, streamline die, fracture, ideal flow, Bernoulli’s theorem, perfectly plastic material 1 Introduction Ideal flows in plasticity are widely used as the basis for inverse methods for the preliminary design of metalworking processes (see, for example, [1-5]). A comprehensive review on the ideal flow theory and its applications has been provided in [6]. Fracture is one of the most important modes of failure in metalworking processes. However, no fracture criterion is involved in the ideal flow theory. In the present paper, design solutions of stationary processes based on the ideal flow theory are supplemented with design driven by fracture. Empirical ductile fracture criteria are often used to predict fracture in metal forming. In particular, such criteria are included in modern commercial finite element packages. Reviews of empirical ductile fracture criteria are provided, for example, in [7 – 9]. In the present paper, the fracture criterion proposed in [10] is adopted. Note that a modified version of this criterion has been introduced in [11]. However, the original and modified criteria coincide for rigid perfectly/plastic materials. Therefore, both criteria are referred to as the Cockroft and Latham criterion in the present paper. This criterion has been used and/or evaluated for several metals in [9, 12 - 19]. In particular, it has been indicated in [12, 13, 15] that the Cockroft and Latham criterion is the best amongst the various existing criteria under the conditions investigated in these papers. Various approximate methods have been adopted to predict the initiation of fracture in drawing. In particular, the upper bound theorem has been used in [20, 21] and the solution for plastic flow through an infinite conical channel given in [22] has been adopted in [23] for drawing through conical and wedge-shape dies. Numerical solutions have been given in [17, 24]. The present approach is based on the extended Bernoulli theorem proven in [25] assuming that the shape of the die has been found using the ideal flow theory. A remarkable property of this approach is that there is no need to know the solution to the plasticity problem to apply the Cockroft and Latham criterion. The final expression is extremely simple and can be directly used for preliminary design of drawing driven by fracture. 2 Ideal Flows and Fracture Critrion Ideal flows constitute a wide class of solutions in the theory of rigid perfectly/plastic solids. A comprehensive review on ideal flows has been provided in [6]. They are of particular interest in Tresca’s solids, i.e. solids satisfying Tresca’a yield criterion and its associated flow rule. Let 1 σ , 2 σ and 3 σ be the principal stress. It is possible to assume with no loss of generality that 1 2 σ σ > and 1 3 σ σ > . In what follows, it is sufficient to consider Recent Advances in Applied and Theoretical Mechanics ISBN: 978-1-61804-304-7 73