**1. Introduction**

The plastic strain ratio r is a parameter that indicates the ability of a sheet metal to resist thinning or thickening when subjected to either tensile or compressive forces in the plane of the sheet [1,2]. It has been used as an important parameter to evaluate the formability of automotive sheet metal. It is typically advantageous for the material to experience a minimal reduction in the area when subjected to a force; this means that it is a good drawing material with a high R-value. Therefore, an accurate measurement of R-value is of great significance to the study of the tensile and compressive properties of materials. Stickels [3] predicted the plastic-strain ratio of low-carbon steel sheets using the Young's Modulus, while Ghosh [4] calculated the R-value of Al-Mg-Si alloy sheets based on crystal plasticity models. However, since the crystal structure of polycrystalline materials is often difficult to model accurately, the accuracy of the current theoretical estimates is not high [5,6]. Some scholars [7–9] employed laser-ultrasound resonance spectroscopy to measure the texture, thickness, and plastic strain ratio on-line. Although the laser-ultrasonic method has the advantages of being fast and non-destructive, the detection accuracy is greatly affected by the vibration of the tested strip and changes in the environment temperature. In some traditional R-value measurement methods [10,11], the thickness strain is calculated using length strain and width strain. The stated measurement methods are based on the assumption of constant volume, which states that the volume of the material will not change during the deformation process. However, some materials do not always follow

**Citation:** Fang, S.; Zheng, X.; Zheng, G.; Zhang, B.; Guo, B.; Yang, L. A New and Direct R-Value Measurement Method of Sheet Metal Based on Multi-Camera DIC System. *Metals* **2021**, *11*, 1401. https://doi.org/ 10.3390/met11091401

Academic Editors: Gabriel Centeno and Ricardo Jose Alves de Sousa

Received: 10 July 2021 Accepted: 2 September 2021 Published: 4 September 2021

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the constant volume assumption, especially inside the necking band. This makes the exact determination of the R-value challenging [12]. Because of this, development of an experimental method for direct measurement of R-value becomes meaningful.

Digital Image Correlation (DIC), a non-contact measurement method, has achieved great success in the field of optical measurement and is still developing rapidly [13–17]. A first attempt to experimentally measure R-value using DIC was reported by Xie in 2017 [18]. Instead of observing the front surface of the sheet metal, this method observed the sheet metal at a 45-degree angle. As a result, both the front surface and the depth side of the sheet metal can be observed at the same time by two DIC cameras, and the three strains *ε*1, *ε*<sup>2</sup> and *ε*<sup>3</sup> can be measured simultaneously. A shortcoming of this method is that the thickness strain can only be measured on the edge. Additionally, the edge needs to be cut, which will introduce a residual strain and ultimately affect the measurement results.

Thickness strain measurement at any point on the sheet metal surface using DIC was first reported by our lab at Oakland University together with Dantec-Dynamics GmbH, Germany [19]. This method was based on using multiple multi-camera DIC systems to measure strains in three directions simultaneously. Two sets of stereo vision DIC measurement systems, each consisting of two GigE cameras, were used in this method. One stereo vision DIC system was placed on the front side of the test sample, while the other system was placed on the back side of the sample. The use of the double-sided calibration strategy unified the world coordinate system of the front and back DIC measurement systems to one coordinate system, thus enabling the measurement of the thickness strain. This paper describes the use of this technology to measure the R-value directly without any assumptions. A tensile test of DP980 was conducted to directly measure the R-value. The theory of multi-camera DIC and the experimental setup will briefly be explained. Section 2 of this paper explains the DIC related theories, the two-sided calibration strategy, and the algorithms to smooth the thickness strain, while Section 3 presents the experimental setup and experimental process. The two R-value calculation methods will be presented, as well as the experimental results of DP980 and a comparison between the R-value determined through experimental results determined using the constant volume assumption. The potential and limitation of the method, as well as conclusions, will be discussed and presented in Section 4.
