Numerical studies of low cycle fatigue in forward extrusion dies
Introduction
The forward extrusion process is one of the basic cold-forging processes. In the most simple form, the die is designed as illustrated in Fig. 1. The die insert shown in Fig. 1 is pre-stressed by means of a strip-wound container. The process is widely used in industry for the manufacture of shafts and axles [1]. In many industrial applications, the forward extrusion process is combined with other forging processes, e.g. backward extrusion or spline forming [2].
On account of the large number of industrial applications involving forward extrusion dies, these have been studied to some extent throughout the years. A general overview of both practical and theoretical aspects is given by Lange [1]. In the present work, fatigue in the die material is of primary interest. This particular topic has been addressed by several authors.
Reviewing a number of experimental studies, Nielsen [3] states that the die insert typically fails due to transverse fatigue cracks originating near the fillet. In case of low radial pre-stressing, fatigue failure can instead be caused by longitudinal cracks and in other cases wear may play a dominant role. Fig. 2 shows an example of a single, transverse fatigue crack penetrating a forward extrusion die insert.
Hänsel [4], [5] and Knoerr et al. [6] developed methods for estimating the die life based on finite element simulations of the die insert and pre-stressing system. In both studies, only the first load cycle is described numerically. For a forward extrusion die insert, Knoerr et al. [6] simulated the material behaviour in both billet and die insert during the forming operation. Hereby, an enhanced description of the transient load on the die insert is obtained. Due to the excessive plastic deformation of the billet material, the elastic deformation is neglected. The die insert is described by a simple elastic-plastic material model which is not described in detail. The numerical analyses performed by Knoerr et al. [6] led to interesting results. However, some of the conclusions drawn by Knoerr et al. [6] are not supported by the work presented here. This discrepancy will be addressed later.
In the studies of Sonsöz and Tekkaya [7] and Ahn et al. [8], the die life is modelled using linear elastic fracture mechanics. Based on stress intensity factors, the growth of a single crack is simulated using the finite element method. Hereby, an estimate of the number of cycles to fracture is obtained, and a good correlation with experimental results was found.
The effect of different pre-stressing systems on the fatigue life of die inserts is discussed by Grønbæk and Nielsen [9], [10]. Various options in terms of pre-stressing of die inserts, including conventional stress rings and strip-wound containers, are discussed, and it is shown that increased pre-stress generally is attractive with respect to increasing die life. Strip-wound containers make it possible to accommodate higher interferences than conventional stress rings and, thereby, to increase the pre-stress. New concepts in the field of pre-stressing systems are also presented by Grønbæk and Nielsen [9], [10]. The use of new materials for the winding core (see Fig. 1) and the introduction of local or global axial pre-stressing makes it possible to postpone the initiation of fatigue cracks. Some of these concepts are discussed in detail in the present work.
Here, a material model combining non-linear kinematic hardening and isotropic hardening or softening is used to describe the elastic–plastic die material behaviour under cyclic loadings. The material model is described in Pedersen [11], where it also includes continuum damage mechanics for description of fatigue-damage development. An axi-symmetric finite element model is used to study the effects of different pre-stressing systems on the development of fatigue damage in critical regions of a forward extrusion die. The full material model including continuum damage mechanics was applied in Pedersen [12] to study fatigue damage development in a similar forward extrusion die. In the present work, continuum damage mechanics is not included in the analyses, but the finite element model is extended to include the pre-stressing system. Thus, a more realistic description of the boundary conditions for the die insert is obtained. Since the pre-stressing system is modelled, it is also possible to study some of the prestressing configurations described by Grønbæk and Nielsen [9], [10].
Section snippets
Constitutive equations
Throughout the loading history, only small regions of the die material are exposed to plastic flow. The bulk of the tool remains elastic and the strain components will remain small. Despite this, large effective plastic strains will generally be accumulated where plastic flow occurs due to the cyclic loading conditions. Hence, the analysis in the present work is based on a small strain formulation of the field equations. Furthermore, isothermal conditions are assumed as the temperature of the
Fatigue damage parameters
In the present work, a number of uncoupled damage measures have been used to describe material failure. The uncoupled damage measures are calculated from the stress–strain state in the material and do not affect the constitutive equations. The few criteria used in this study only represent a small part of the vast amount of empirical damage measures available in the literature, see [20], [21], [22]. Many of these criteria have been derived for specific problems and, here, only the most general
Computational model
In the present work, an axi-symmetric model of the die and the pre-stressing arrangement shown in Fig. 1 are used to study the influence of a range of parameters on the development of fatigue damage. The geometry of the model is shown in Fig. 3. It consists of four regions: Region A is the die insert, while the remaining regions make up the strip-wound container. The die insert is fitted into the winding core (Region B). The winding and the casing are regions C and D, respectively.
The numerical
Numerical results
In the following examples, the die insert material is described by σy/E=0.006, ν=0.3, R0/E=0.0006, b=1.1, R∞,s/E=0, αR=0.7, βR=160, γR=0.5, Cq=0.5, γ1=2000, γ2=180, γ3=50, X∞1/E=0.00125, X∞2/E=0.00375, X∞3/E=0.001, and m1=m2=m3=0. The material parameters correspond to the high-strength steel Calmax which was studied experimentally by Pedersen and Brøndsted [27].
The forward extrusion die studied in the present work has a geometry similar to the die insert studied by Hänsel [5], Knoerr et al. [6]
Conclusions
The numerical analyses presented here show that detailed modelling of the pre-stressing system, the die insert geometry or both can be used to lower the risk of fatigue failure in industrial applications. The results indicate that the stress–strain response in the die insert is highly dependent on the applied material model. For the material model and the set of material parameters used here, a stable response is almost reached after 50 load cycles. Fatigue life estimates should be based on
Acknowledgements
The author wishes to thank Professor Viggo Tvergaard at the Technical University of Denmark and Dr. Jens Grønbæk, STRECON® Technology, Danfoss A/S for valuable discussions during the course of this work. This work is supported by the MUP2 research programme Materials Processing, Properties and Modelling, financed by the Danish Agency for Development and Trade in Industry, the Danish Natural Science Research Council and the Danish Technical Research Council.
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Also at Department of Solid Mechanics, Technical University of Denmark.