Mathematical definition of the 3D strain field of the ring in the radial-axial ring rolling process Luca Quagliato n , Guido A. Berti Department of Management and Engineering, University of Padua, Italy article info Article history: Received 29 March 2016 Received in revised form 4 July 2016 Accepted 7 July 2016 Available online 15 July 2016 Keywords: Metal forming Ring rolling Strain analysis Analytic functions FE analysis abstract The paper focuses on the radial-axial ring rolling process and details a new mathematical approach for the determination of the evolution of the ring geometry during the deformation process, taking into account separately the sequence of incremental deformations occurring when the ring passes through the mandrel-main roll gap and through the axial rolls gap. Based on the determined geometry of the ring, the three strain components of the strain tensor are estimated and the equivalent plastic strain is computed. The proposed approach, taking into account a third strain in each deformation gap, allows an estimation of the equivalent plastic strain, which is a required parameter for the analytical estimation of the flow stress of the material, needed to compute the forming force. Since a direct validation of the strain components is not possible in the industrial RARR process, authors’ models for the determination of geometry and strain, together with preliminary authors’ models for the estimation of strain rate and temperature drop along the process, have been applied to a literature case for the estimation of the radial forming force in order to obtain a validation of the proposed models. Prediction of radial forming force utilizes a literature model based on slip line theory adapted to the ring rolling process. To extend the validation of the approach and to explore the quality of its predictions to other process configurations, different geometry of the ring have been considered and compared with FEM predictions. These com- parisons resulted in good agreement between analytical and FEM results as concerns ring geometry evolution, strain tensor prediction and effective strain estimation. & 2016 Elsevier Ltd. All rights reserved. 1. Introduction Radial-axial ring rolling (RARR) is widely used in the produc- tion of seamless rings for the automotive and aerospace industries, where a ring work-piece is drawn into the mandrel-main roll gap and the axial rolls gap, causing expansion of the diameter as well as reduction in the thickness and height [1,2]. In the literature, many efforts have been spent in the in- vestigation of the radial-axial ring rolling process considering different points of view, generally aiming to improve the knowl- edge of the interactions among the ring, the tools, the process set up and the production environment. As concerns the process de- sign, Hua et al. [3] defined useful rules for the estimation of the ring stiffness, which is an important factor to avoid collapses or unexpected deformations during the forming process. Zhou et al. [4], utilizing ABAQUS/Explicit solver, studied the influence of the tools dimensions, providing rules to optimize their choice. Zhou et al. [5] also analyzed the influence of the axial rolls motion laws on the radial-axial ring rolling process and provided guidelines for their set-up. Furthermore, Qian et al. [6] developed a method to optimize the dimensions of the ring blank and investigated the influence of this choice on the process. In addition to that, many efforts have been also made for the estimation of the process forces in RARR, either adopting analytical approaches such as SLAB (Parvizi et al. [7]) and upper bound method (Parvizi et al. [8]), or FE methodology (Guo et al. [9]). In the first case, the main approximation is given by the adoption of a rigid plastic model for the behavior of the material; this choice seems to be far from the reality of forming processes, where the flow stress generally depends on temperature, strain, and strain rate. The FE approach overcomes this limit, taking into account the dependency of flow stress on temperature, strain, and strain rate, which are calculated for each element of the discretized domain. However, FE simulation is not so straightforward and is highly time-consuming. In the design stage of a ring rolling process, the planner should make a preliminary decision based on the feasi- bility of the process, which can be expressed by the stability of the forming process, the cycle time, and the required forming forces. However, due to the intrinsic nature of the RARR process, where thermal and mechanical effects are combined with the motions of Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ijmecsci International Journal of Mechanical Sciences http://dx.doi.org/10.1016/j.ijmecsci.2016.07.009 0020-7403/& 2016 Elsevier Ltd. All rights reserved. n Corresponding author. E-mail address: quagliato@gest.unipd.it (L. Quagliato). International Journal of Mechanical Sciences 115-116 (2016) 746–759