*2.4. Strain Measurement Using Circle Grid Analysis*

The in-plane strains of the incrementally expanded tubes were determined by CGA using the automatic measurement system ARGUS ® v.6.2 by GOMTM equipped with a camera having a resolution of 1624 × 1236 pixels. For this purpose, the outer tube surfaces were electrochemically etched with a grid of circles with 0.75 mm of diameter and a distance between centers of 1.5 mm.

Measurement and classification of the deformed circles into different colors by ARGUS at the end of incremental tube expansion (Figure 5a) allowed determining the in-plane distribution of strains along the longitudinal direction from the undeformed lower tube region to the upper end of the plastically expanded tube surface (Figure 5b). The results for a typical longitudinal cross section marked with a black line in Figure 5a are given by the corresponding black line in principal strain space (Figure 5b).

**Figure 5.** (**a**) Experimental determination of the in-plane strains by CGA using the automatic measuring system ARGUS ® , and (**b**) representation of these strains in principal strain space.

#### *2.5. Numerical Modelling*

Numerical modelling of the conventional and incremental tube expansion processes was carried out with the commercial finite element computer program DEFORM™-3D. DEFORM™-3D was chosen due its capability to obtain a good agreement between numerical and experimental strains in incremental sheet forming processes [20,21].

The tube material was assumed as isotropic, elastic and plastic, and its initial geometry was discretized by means of solid tetrahedral elements. Tube material properties were taken from a previous work of the authors [9]. The tools were modelled as rigid (non-deformable) bodies and discretized by means of spatial triangular elements.

A penalty contact algorithm was utilized to model the interaction between the tools and the tube material.

Discretization of the tube and tool in case of conventional tube expansion with a rigid tapered conical punch took advantage of the rotational symmetry conditions of the process to create a simple three-dimensional model built upon an angular sector of 18 ◦ (1/20 of the full three-dimensional model). A total of 11,530 tetrahedral elements were utilized with an average side length of 1 mm and a reduced side length of 0.25 mm in the upper tube regions where mesh refinement was needed. Figure 6 shows the initial and final deformed meshes with the predicted contour of effective strain at the end of the process.

Typical CPU time to complete the numerical modelling of conventional tube expansion was approximately 3 min in a personal computer equipped with an Intel I7-4749 CPU (3.6 GHz) processor.

**Figure 6.** Finite element modeling of conventional tube expansion with a rigid tapered conical punch having a semi-angle of 15◦ . (**a**) Initial mesh with a detail of mesh refinement and (**b**) predicted distribution of effective strain at the end of the process.

Discretization of the tube material and of the single point hemispherical tool in case of incremental tube expansion required a full three-dimensional finite element model. The initial mesh consisted of 50,000 tetrahedral elements distributed along a finer mesh region at the upper tube end, which initially comes into contact with the tool, and a coarser mesh region for the remaining regions of the tube (Figure 7a). The final mesh at the end of the process (Figure 7b) consisted of approximately 120,000 tetrahedral elements due to several remeshings (based on critical element distortion) that were automatically performed to keep the numerical simulation from stopping because of excessive element distortion.

**Figure 7.** Finite element modeling of incremental tube expansion. (**a**) Initial mesh and (**b**) final mesh after eight forming stages.

The trajectory of the single point hemispherical tool was set identical to that utilized in the experiments and the total CPU time to complete the eight forming stages of the multi-stage incremental tube expansion process was approximately equal to 600 h. Cyclic strain loading paths of incremental tube forming cannot be obtained from experimental strain analysis; therefore, finite element predicted strain loading paths become the closest as one can be from the physical phenomenon, despite the long computational time.
