**1. Introduction**

The route for characterizing the sheet formability limits started in the late 1960s when Keeler [1] and Goodwin [2] developed the circle grid analysis (CGA) technique for determining the in-plane strains on the surface of sheet metal formed parts. The use of principal strain space to plot these strains and to identify their critical values at the onset of failure by Embury and Duncan [3] in the early 1980s paved the way to what they called "formability maps", which are nowadays known as the forming limit diagrams (FLDs) [4].

A typical FLD for a sheet metal forming material is built on three different types of failure limit curves [5]: (i) the forming limit curve (FLC) corresponding to failure by necking, (ii) the fracture forming limit lines corresponding to failure by cracking and (iii) the wrinkling limit curve (WLC) delimiting the onset of wrinkling in the lower lefthand of the second quadrant. In sheet metal forming, there are two fracture forming lines corresponding to crack opening by tension (mode I of fracture mechanics, hereafter referred to as FFL) and crack opening by in-plane shear (mode II of fracture mechanics, hereafter referred to as SFFL) [6]. The experimental determination of the FFLs and SFFLs was comprehensively explained by the authors in previous publications [4,5], who also described the different methods and procedures to obtain the FLCs.

The route for establishing the formability limits of tube materials starts with the determination of the onset of necking (FLC) by means of tube of hydroforming [7,8]. No

**Citation:** Suntaxi, C.; Centeno, G.; Silva, M.B.; Vallellano, C.; Martins, P.A.F. Tube Expansion by Single Point Incremental Forming: An Experimental and Numerical Investigation. *Metals* **2021**, *11*, 1481. https://doi.org/10.3390/met11091481

Academic Editor: Zhengyi Jiang

Received: 18 August 2021 Accepted: 14 September 2021 Published: 17 September 2021

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methodologies for characterizing the crack opening modes and determining the fracture forming lines were proposed until 2016, when Centeno et al. [9] utilized CGA to plot the FLC and the FFL corresponding to tube cracking by tension.

Subsequent research work combining numerical methods, digital image correlation (DIC) and making use of a broader range of tube forming processes comprising expansion [10], inversion [11] and bulging [12] allowed obtaining the FLC and the FFL of tube materials for a wider range of strain paths running from uniaxial tension up to equal biaxial stretching (e.g., from strain ratios *β* = *dε*2/*dε*<sup>1</sup> ranging between −1/2 to 1). These efforts were recently complemented by the work of Magrinho et al. [13], who proposed an experimental procedure to determine the SFFL of tube materials (i.e., the fracture forming limit line corresponding to tube cracking by in-plane shear).

In view of the aforementioned work, recent developments in incremental tube expansion, reduction, wall grooving and hole flanging using a single point hemispherical tool by Wen et al. [14] and Movahedinia et al. [15] raise the question of whether their deformation mechanics and formability limits remain the same as those of conventional tube forming processes. The answer to this question was firstly addressed by Cristino et al. [16], who presented an analytical model based on membrane analysis for tube expansion by single point incremental forming (hereafter referred to as incremental tube expansion). The model reveals the main differences between conventional and incremental tube expansion in terms of stress/strain states and damage accumulation to explain the greater formability of incremental tube expansion compared to that of conventional tube expansion with a rigid tapered conical punch.

The analytical model of Cristino et al. [16] is based on a rigid, perfectly plastic tube material and assumes near-proportional (equal biaxial stretching) experimental strain loading paths in principal strain space to facilitate algebraic treatment.

Under these circumstances, it is important to revisit the accumulation of damage by means of a numerical simulation model capable of accounting for material strain hardening and for the loading/unloading cycles of incremental tube expansion. In this paper, we provide a novel perspective on the formability and failure of incremental tube forming processes subjected to non-proportional loading. We analyze different methodologies to account for stress triaxiality and accumulation of damage, and discuss if the FFLs of tube materials determined by means of conventional tube forming processes subjected to near proportional loading paths are still valid for incremental tube expansion characterized by non-proportional loading paths that oscillate cyclically from shearing to biaxial stretching, as the single point hemispherical tool approaches, contacts and moves away from a specific location of the plastically deformed tube surface.

Experimental and numerical simulation results plotted in the space of effective strain vs. stress triaxiality [17] give support to the discussion, which is of paramount importance to infer about the FFLs of tubes being material properties, in contrast to their FLCs, which are dependent on the applied loading paths.

#### **2. Methods and Procedures**

The investigation was carried out in AA6063-T6 extruded aluminum tubes with an outer radius *r*<sup>0</sup> = 20 mm and a wall thickness *t*<sup>0</sup> = 2 mm. The first part of this section summarizes the methods and procedures that were utilized to determine the material flow curve and the formability limits by necking (FLC) and by fracture under tension (FFL) using conventional tube forming processes. The data provided in the figures were retrieved from previous publications of the authors [9–12].

In the second part of this section, we present the experimental testing conditions of incremental tube expansion, describe the methodology that was used to determine the strain paths using circle grid analysis (CGA), provide an analytical framework to transform the formability limits from principal strain space into the effective strain vs. stress-triaxiality space and summarize the numerical modelling conditions utilized in finite element analysis.

#### *2.1. Flow Curve*

The flow curve of the AA6063-T6 tubes is shown in Figure 1 and was obtained by merging the stress–strain evolutions that were previously obtained by the authors using tensile and stack compression tests [9]. Tensile tests were carried out in specimens machined out from the tube longitudinal direction and provided the material stress response for values of effective strain below 0.23 (refer to the vertical dashed line). Stack compression tests were performed in cylindrical specimens that were assembled by pilling up disks that were also machined out from the supplied tubes and allowed characterizing the strain hardening behavior of the tube material for the remaining values of effective strain.

**Figure 1.** Flow curve of the aluminum AA6063T6 tubes (adapted from [9]).
