On the utilization of radial extrusion to characterize fracture forming limits. Part II: testing and modelling

. This second part of the paper is focused on double-action radial extrusion testing and modelling to characterize material formability and failure in the bulk-to-sheet material flow transitions that are commonly found in metal forming. Results show that three-dimensional to plane-stress evolutions at the radially extruded flanges lead to different modes of fracture (by tension and by shear) that may or may not be preceded by necking. The use of double-action radial extrusion as a formability test also reveals adequate to characterize the failure limits of very ductile metallic materials, which cannot be


Introduction
Competition between cracking opening modes by tension (mode I) and by out-of-plane shearing (mode III) in conventional bulk formability tests give rise to an 'uncertainty region' in principal strain space, as it was recently observed by Sampaio et al. [1]. Although research on formability and crack opening modes has been taking the competition between crack opening modes into account, there are aspects related to material flow transitions in regions of bulk metal forming parts subjected to extensive plastic material flow that are not commonly taken into consideration. Accountability of these aspects means, in practical terms, to include the three-dimensional to plane stress material flow transitions (i.e., bulk-to-sheet evolutions) in the list of parameters that are responsible for the competition between the different crack opening modes.
Under these circumstances, the main objective of this second part of the paper is to investigate formability in three-dimensional to plane stress material flow transitions by means of a new formability test based on the working principle of double-action radial extrusion [2]. It must, however, be said that the relevance of the new proposed formability test goes beyond the abovementioned material flow transitions because it provides two significant advantages over the existing bulk formability tests.
Firstly, it allows carrying out tests with very slender cylindrical specimens, which would inevitably collapse by buckling during upset compression tests between flat parallel platens. This overcomes the difficulty of obtaining the experimental strains at fracture in very ductile materials, a topic that was also addressed by Bulzak et al. [3] when studying damage evolution in hot forming with rotary compression tests. However, the experimental evaluation of surface strains becomes very difficult for this test, due to the continuous movement of the specimen. Secondly, the fact that triggering and subsequent propagation of cracks in the double-action radial extrusion test will always lead to force drops in the force vs. time evolutions. This allows determining the instant of cracking and of the corresponding fracture strains by combination of the experimental force vs. time evolutions with the results obtained from digital image correlation (DIC). In fact, the force vs. time evolution of the double-action radial extrusion test is different from those found in conventional upset compression tests in which the combination of surface expansion with strain hardening commonly prevents the force from dropping at the onset of cracking.

Experimental testing conditions
The double-action radial extrusion formability tests were carried out in a multidirectional tool set that was installed in a hydraulic testing machine. Fig. 1 shows a detail of the tool set in which the movement of the extrusion punches (P) towards each other inside the dies (D) is obtained by means of two cam slide units consisting of a punch holder (H) and a die wedge actuator (A) with a working angle = 30°. Two guided pillars (G) ensure the alignment between the upper and lower tool halves. Further details on the tool set and on the conversion of the vertical force of the hydraulic testing machine into the horizontal force ℎ applied by the extrusion punches are given in the first part of this paper (Part I). The tests were carried out in a soft aluminum AA1050-O and a medium strength aluminummagnesium-silicon alloy AA6082 in the annealed (O) and solution heat-treated (T6) state. The specimens were machined from the supplied rods and a molybdenum disulphide (MoS2) based lubricant was utilized in the operating testing conditions provided in Table 1.

Results and Discussion
Strain-Loading Paths in Principal Strain Space. Fig. 2 shows the experimental evolutions of the major and minor in-plane strains with time determined by DIC and of the radial extrusion force ℎ with time after conversion of the vertical force measured by the load cell. The in-plane strain evolutions were taken from the outer flange surface for the three different materials.
As seen, the instant of fracture (refer to 'F' in Figs. 2a-c) is clearly recognized by a sudden drop in force and allows determining the major 1 and minor 2 in-plane strains at fracture from their evolutions with time ( 1 , 2 ) = ( ) obtained from DIC. Point 'I' is the location in the outer flange surfaces where the strain evolutions were obtained.   In case of the specimens made from aluminum AA1050 and AA6082 in the annealed (O) state, the instant of fracture is preceded by a change in the rate of force growth with time leading to the development of a neck on the outer flange surface (refer to 'N' in the force peaks of Figs. 2a, b). In contrast, the specimens made from aluminum AA6082-T6 only experience a sudden drop in force because cracking is not preceded by necking, as it was experimentally observed and will be shown later.
Application of the methodology that was described in the first part of this paper allows plotting the corresponding strain loading paths 1 = ( 2 ) in principal strain space. The result is shown in Fig. 2d and allows distinguishing between monotonic and non-monotonic evolutions up to fracture thanks to the absence of necking (AA6082-T6) and to the sharp bends of the strain loading paths (in point 'F') towards plane strain (AA1050-O) and pure shear (AA6072-O) deformation conditions that happen in the specimens where fracture is preceded by necking.
Finite element computed evolutions of the strain loading paths (refer to the dashed lines in Fig. 2d) show a good agreement with DIC. However, the numerical estimates for the soft aluminum AA1050 and the medium strength aluminum-magnesium-silicon alloy AA6082 in the annealed (O) state are only provided up to the onset of localized necking, which corresponds to approximately 80% of the total stroke. Beyond this point, the growth rate of the in-plane strains is very fast and the uncoupled finite element damage models utilized by the authors experience difficulties in handling the non-homogeneous plastic deformation and the elastic unloading caused by damage-induced softening. However, because the onset of localized necking and point 'F' occur almost instantly, finite elements will continue to be used in the following sections of the paper to analyze the deformation mechanics and the different modes of failure that were observed in the double-action radial extrusion formability test.
Deformation Mechanics. Figs. 3a-c shows the initial and intermediate rotationally symmetric meshes at 30% of the total stroke for specimens with an aspect ratio ℎ 0 ⁄ = 0.75 made from the different materials and supplied conditions that were utilized in the investigation. As seen in the predicted distribution of vertical stresses, the radially extruded flanges start showing signs of threedimensional to plane stress material flow transitions at the outer flange surfaces, which will become more pronounced as radial extrusion continues and the flange diameter increases.
The aspect ratio ℎ 0 ⁄ of the specimens plays a key role in the utilization of radial extrusion as a formability test capable of monitoring the three-dimensional to plane stress material flow transitions. In fact, larger aspect ratios than those shown in Figs. 3a-c, do not give rise to plane stress conditions (Fig. 3d) and may result in material flaws (Fig. 3e) or buckling, as reported in earlier investigations of the radial extrusion process [4]. In such situations not only the diameterto-thickness ratio of the radially extruded flanges deviates from typical sheet forming values as plane stress conditions no longer prevail on the outer flange surfaces. The asymmetric deformation of the three specimens with an aspect ratio ℎ 0 ⁄ = 0.75 that progressively develops during the final stage of the double-action radial extrusion test was simulated with the three-dimensional finite element model built upon the interaction between two and three-dimensional meshes that was earlier described. For this purpose, the authors artificially imposed small gauge imperfections of approximately 0.2 mm on the outer flange diameter located at the vicinity of the xz-plane. These imperfections account for less than 1% of the flange diameter and correspond to localized material weaknesses that resemble those utilized by Marciniak and Kuczynski [5] in their theory of plastic instability in sheet metal forming.
The result of the above-mentioned strategy combined with intermediate remeshing operations aimed at replacing elements with excessive distortion at the center of the specimens made from AA1050-O and AA6082-O is disclosed in Fig. 4. As seen, finite element computed geometries obtained shortly after the onset of cracking agree well with the experimental test specimens. Incipient crack propagation was numerically modelled by means of an element deletion (removal) technique based on the critical value of a damage function , where is a weight function and ̅ is the effective strain. Two different weight functions were used: (i) the McClintock [6] stress triaxiality ratio = � ⁄ , corresponding to failure by tension (mode I), where is the hydrostatic stress and � is the effective stress (Fig. 4a) and (ii) the Cockcroft-Latham [7] normalized principal stress ratio = 1 � ⁄ , corresponding to failure by out-of-plane shear (mode III), where 1 is the major principal stress (Figs. 4b, c).
The choice between the proper weight function to be used for each test case was based on the strain loading path evolutions disclosed in Fig. 2d, which clearly indicate failure by tension preceded by necking in AA1050-O, by out-of-plane shear preceded by necking in AA6082-O and failure without previous necking in AA6082-T6.  [6] and (b, c) Cockcroft-Latham [7] criteria and photographs of the specimens are enclosed for comparison purposes.
Further observation of the two different types of necks combined with fractography analysis by means of scanning electron microscopy (SEM) to be presented in the next section provides a better identification and understanding of the different types of failure and modes of deformation that are likely to occur in three-dimensional to plane stress material flow transitions.
Necking and Fractography. Combination of the finite element computed meshes with the images obtained from DIC and SEM (Fig. 5) allows concluding that fracture in the outer surface flange of the AA1050-O test specimens is by tension (mode I). Results also show evidence of plastic instability in the form of diffuse (Fig. 5a) and localized necking (Fig. 5c) in good agreement with the deformation mechanics of sheet forming. In fact, the localization of the neck along a direction inclined at an angle of approximately 53.2° (Fig. 5c) to the circumferential direction value is very close to the theoretical estimate of 54.7° (Fig. 5b) obtained for a sheet (plane stress) specimen made from an isotropic material, subjected to uniaxial tension [8].
The morphology of the cracks consists of circular dimple-based structures (Fig. 5d), and further corroborates the experimental observation of cracks opening by tension (mode I) and running radially along the flange. Application of the same methodology for the specimens made from AA6082-O (Fig. 6a) confirms that vertical cracks opened by shear stresses due to the occurrence of smooth parabolic dimple-based structures (Fig. 6e). Moreover, these cracks do not run radially along the flange (Fig. 4b), as it is commonly observed in the upset compression of cylindrical and tapered test specimens, due to significant distortion of outer flange surface placed within the neighborhood of the localized neck. All this is consistent with the strain path bend towards pure shear in principal strain space (Fig. 2d) before failing by through-thickness shearing (mode III).
The pure shear stress state acting at the outer flange surface (with = − > 0) gives rise to plane strain deformation along the radial direction ( = 0) due to material incompressibility. Thus, assuming these conditions to hold in-between the top and bottom flange surfaces due to combination of small thickness and plane stress material flow conditions, an analogy with uniaxial tension of a sheet test specimen can also be made. The analogy requires a similar approach to that taken for the specimen made from AA1050-O but now with diffuse necking taking place in the radial direction (Figs. 6b and 6c) and crack opening in the -plane after the occurrence of localized necking with angles of approximately 54° ~ 55° (refer to the top flange view in Fig. 6d). In what concerns the results obtained for the specimens made from AA6082-T6 (Fig. 7a) it is possible to conclude that vertical cracks open without previous necking (Fig. 7c). Crack propagation does not run radially, as it is schematically disclosed in Fig. 7b, due to the development of a mixed mode characterized by the simultaneous occurrence of circle and smooth parabolic dimple-based structures (Fig. 7d).

Conclusions
The work on the utilization of double-action radial extrusion as a new formability test to characterize failure in bulk-to-sheet material evolutions led to the following main conclusions: • The test can be successfully used to characterize material formability in the threedimensional to plane stress material flow transitions that are commonly found in bulk metal formed parts, • Material flow transitions give rise to uniaxial tension states of stress that eventually lead to crack opening with or without previous localized necking, • Cracks preceded by localized necking develop under plane strain or pure shear material flow conditions on the outer flange surface, • The morphology of the cracks reveals crack opening by tension (mode I), by shear (mode III) and by a mixed mode resulting from combination of modes I and III, • The new test is also adequate to characterize the formability limits of very ductile materials with large fracture strains, such as aluminum AA1050-O and AA6082-O, which cannot be easily determined by conventional upset compression tests.