1. Introduction
A forming limit diagram (FLD) is a criterion for judging whether the obtained sheet metal component is safe or necked. The accuracy of its representation directly affects decisions to approve the forming process. Localized instability analysis, proposed by Hill [
1] in 1952, laid the theoretical foundation for the prediction of the tensile instability of sheet metals. In 1965, Keeler [
2] proposed the concept of forming limit diagrams in an SAE report, which was further supplemented by Goodwin [
3] in 1968. When localized instability (or the bifurcation phenomenon) occurs, a groove appears on the surface, and the crack expands rapidly, causing the failure of the sheet [
4,
5]. The bifurcation phenomenon of the sheet metal means a localized deformation in a narrow band, while the deformation remains homogeneous elsewhere [
6,
7]. Therefore, the localized instability stage (also known as localized necking) is generally considered as the maximum allowable deformation of the sheet, and the strain at the beginning of the localized instability stage is used as the limit strain for judging whether the sheet has failed [
8].
The most commonly used test method for establishing a forming limit diagram at present is the Nakazima hemispherical bulging test [
9] or the modified flat punch bulging test by Marciniak et al. [
10]. The acquisition of a reliable limit strain is one of the most concerning issues in the study of forming limits. A new development in strain measuring, referred as digital image correlation (DIC), allows the strain evolution to be recorded continuously and accurately for the entire experimental procedure. This certainly provides the possibility of obtaining more accurate necking limits in the strain space. However, the core challenge is to propose a method to determine the onset of localized necking.
DIC technology tracks the speckle movement on the top surface of the specimen and further calculates the strain field according to time. In general, two essential points should be determined through the experimental procedure: (i) the instant of localized necking to indicate the timing and (ii) the location of localized necking to indicate the limit strain.
One popular method is the so-called space-dependent method, where the major strain of the points on the cross section perpendicular to the crack growth direction after fracture is used to fit an inverse parabola, and the limit strain is determined by the vertex of the inverse parabola [
9]. Generally, the space-dependent method only considers a certain deformation state of the specimen and does not pay attention to the deformation history. The emergence and development of digital image correlation technology have led to the development of analysis and measurement methods based on strain history.
Huang et al. [
11] obtained the relationship between the second derivative of strain and time. They believed that the increase in the second derivative near the fracture indicates the beginning of localized necking, and the limit strain is determined correspondingly. Merklein et al. [
12] performed regression analysis on the major strain rate in the center of the necking region. The extreme point of specific parameters obtained from the regression analysis was considered to be the start of localized necking. Hotz et al. [
13] used the first derivative of the thickness strain versus the punch position to determine the onset of localized necking and put forward constructive comments on the calculation of the derivative. Martínez-Donaire et al. [
14] suggested that the major strain rate at the boundary point of the instability zone first increases and then decreases with the deformation of the sheet. The extreme point of the major strain rate corresponds to the onset of localized necking. Min et al. [
15] compared the forming limit diagrams of several different sheet metals. It was found that the circular mesh strain analysis technique always overestimates the limit major strain of the material, but the method of the ISO standard [
9] underestimates the limit strain. Therefore, they proposed that if the limit strain is to be more accurately obtained, it is necessary to consider the spatial and time factors in order to capture the subtle changes in the sample when localized necking occurs, due to the fact that localized necking is an unstable physical process with both spatial and temporal characteristics.
From the perspective of physical understanding, parameters related to the thickness strain (ε
3) may be more worthy of attention. The physical interpretation of localized necking is the sudden thinning of the sheet in the thickness direction (or bifurcation phenomenon); thus, relevant parameters of ε
3 directly reflect the deformation of the polycrystalline sheet metal along the thickness direction. Wang et al. [
16] selected two points on the cross-section of their sample perpendicular to the crack growth direction. One point was selected at the center of the crack, and the other was away from the center. The difference between the heights of the two points was recorded. The first derivative of the height difference (ΔZ) could then be calculated, and it was considered that the sudden increase in the first derivative represented the onset of localized necking. Di et al. [
17] used the local surface curvature of the sample to determine the onset of localized necking. Min et al. [
18] proposed a two-dimensional curvature criterion based on the ordinate value of the cross-section of the crack growth direction.
In many deterministic approaches to localized necking, the first derivation is necessary to capture the timing and the strain limit. From a practical point of view, if the DIC signal is not prepared properly, the derivation will be too noisy to use. Thus, such an approach requires extensive operator experience and could be tedious and labor-intensive. This paper aims to develop an approach to determine the onset of localized necking in polycrystalline sheet metals, while eliminating the derivative calculation. The physical interpretation of localized necking under uniaxial tension is also utilized to shed light on this necking deterministic approach. The experimental results and theoretical prediction are compared to help us achieve an in-depth understanding of the correlation between the bifurcation in strain evolution and localized necking.
To achieve this purpose, we (i) first introduce the tested materials and the experimental procedures used in this study; (ii) investigate the bifurcation phenomenon in the strain evolution under uniaxial tension tests; (iii) provide the detailed detection procedure based on the bifurcation phenomenon to capture the limit strains; and (iv) compare and analyze the final forming limits to illustrate the effectiveness of the proposed method.