Amplitude Equation for a Dynamic Strain Aging Model: Beyond Linear Stability Analysis of Serrated Flow in Metallic Alloy
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Amplitude Equation for a Dynamic Strain Aging Model: Beyond Linear Stability Analysis of Serrated Flow in Metallic Alloys Sergey N. Rashkeev1, Michael V. Glazov2, Frédéric Barlat2 and Daniel J. Lege2 1 Dept. of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235, USA 2 Alcoa Technical Center, 100 Technical Drive, Alcoa Center, PA 15069-0001, USA ABSTRACT A method for construction of “processing windows” to avoid negative strain rate sensitivity and associated serrated flow in some aluminum alloys is described. The method is based on the amplitude Ginzburg-Landau (GL) equations and analysis of bifurcation diagrams. The mathematical technique developed in the present work was applied to a specific aluminum alloy, Al-0.4%Mg-0.2%Si considered earlier in the literature [1-3], and yielded good results in terms of predicting the negative strain rate sensitivity regions in the “strain rate –temperature” parameter space. Using the GLanalysis it was demonstrated that even though the instability area is located in the region of intermediate strain rates, a qualitative difference exists between the areas of (relatively) fast and (relatively) slow strain rates. In the first case the dynamic behavior of the system is supercritical, in the second case it is subcritical. The second case is highly undesirable because it causes a sudden onset of stable stress serrations that are difficult to suppress, while in the first case the development of instability is gradual and, consequently, more easily controllable. 1. INTRODUCTION AND FORMULATION OF THE PROBLEM Strain hardening and strain rate sensitivity have been identified as important factors determining formability of aluminum alloys in the aluminum industry’s efforts to compete with steel for automotive applications. Strain localization, coarse slip bands, Luders fronts, Portevin-Le-Chatelier bands of different kinds, and surface roughening are among the detrimental phenomena which can result in either fracture or unacceptable visual appearance of formed parts [4,5]. Because of tremendous complexity of formability problems, so-called “limit diagram” approach has been developed in industry. Such diagrams (e.g., extrusion limit diagrams, forming limit diagrams) can be constructed experimentally, or calculated In all cases the ultimate goal is the same: to outline an “area” in the parameter space that can be safely accessed in the course of the corresponding processing operations. The goal of this work is to describe several analytical and computational tools that can be used to construct the “PLC-Effect Limit Diagrams”, and also to perform such a construction for a 6000 series aluminum alloy. The paper is organized as follows. In Section 2 a brief description of the McCormick model used in calculations is given. Data have been taken from the literature [1,2] for a 6063 aluminum alloy (Al-0.4%Mg-0.2%Si), which fit the McCormick model and provide possibilities for further detailed analysis. The construction of the PLC-free “processing windows” is performed in Section 3. PL
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