# Identification of the Dynamic Properties of Al 5456 FSW Welds Using the Virtual Fields Method

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## Abstract

The present study focuses on the identification of the evolution of the dynamic elasto-plastic properties of Al 5456 FSW welds. An innovative method is proposed to make best use of the data collected with full-field measurements during dynamic experiments, and achieve identification of the mechanical properties of heterogeneous materials without requiring measurement of the load. Compressive specimens have been submitted to high strain-rate loading through a split Hopkinson pressure bar device while displacement fields were obtained using full-field measurement techniques. Two sets of experiments have been performed using two different methods: the grid method and digital image correlation. Afterwards, the identification of the elastic and plastic properties of the material was carried out using the Virtual Fields Method. Finally, identification of the evolution of the yield stress throughout the weld has been achieved for strain-rates of the order of 10^{3} s^{−1}.

## Keywords

Friction stir welding Virtual fields method Dynamic deformation Digital image correlation Grid method Split Hopkinson pressure bar## Introduction

Since its invention in 1991, the friction Stir Welding (FSW) process [1] has allowed the use of large aluminium structures for a wide range of applications, thanks to the high resistance of the welds thus produced. In various fields, such as automotive and aeronautics, these welds hold an important place. Therefore, the evolution of the mechanical properties at different strain-rates is of interest; with the knowledge that, depending on the process, the welded material can undergo important structural changes, ranging from different grain size to a total recrystallisation. However, the high strain-rate mechanical properties used in numerical simulations are still estimates. Indeed, different issues arise when dealing with dynamic experiments. It is not easy to obtain accurate measurements of the strain, the load and the acceleration at strain-rates of the order of 10^{3} s^{−1} or more.

Several tests have been used over the last century to carry out experiments at high strain-rates [2]. The split Hopkinson pressure bar (SHPB) was developed based on the work of Hopkinson [3] and Kolsky [4]. This system allows the realization of experiments at strain-rates up to 10,000 s^{−1}. Over the last decades, the SHPB and the tensile split Hopkinson bar [5] have become standards for the dynamic characterization of materials [6, 7, 8, 9, 10, 11, 12, 13, 14]. Starting with Hoge [15], the influence of the strain-rate on the mechanical properties of aluminium alloys, more specifically here, the tensile yield stress, has been investigated. For Al 6061 T6 and a strain-rate varying from 0.5 to 65 s^{−1}, Hoge measured an increase in yield stress of approximately 28 %. More recent work by Jenq et al. [8] showed the evolution of the stress-strain curve between compressive quasi-static and dynamic tests at strain-rates ranging from 1350 to 2520 s^{−1}. In that work, increases in yield stress of 25 % between the quasi-static test and the 1350 s^{−1} test and 60 % between the quasi-static and the 2520 s^{−1} test were measured. For Al 5083, Al 6061 and A356 alloys, it is also worth noting that Tucker [13] reported almost no evolution of the yield stress between tensile quasi-static and dynamic tests, also reaching similar conclusions for compression and shear. However, significant work hardening differences were recorded between tension and compression, with consistent increasing work hardening with strain-rate in compression.

To date, very few investigations have been conducted on the dynamic properties of welds. With SHPB experiments, it is possible to identify the average properties of a welded specimen [16, 17, 18]. However, there is no information about the local evolution of the dynamic properties within the weld. Due to the complex thermo-mechanical history of the welded material, the strain-rate dependence of the different areas of the weld could be quite different. Therefore, investigation of the evolution of the local properties of the material is of interest. Yokoyama et al. [19] proposed to carry out the identification of the dynamic local properties in a weld by cutting small specimens in the weld so as to consider each specimen as homogeneous. However, some issues remain due to the low spatial resolution and the assumption of the specimen homogeneity. This is also a very long and tedious process.

Developments in the field of digital ultra-high speed (UHS) cameras now allow the imaging of experiments at 10^{6} frames per second and above. The definition of ’ultra-high speed imaging’ is provided in [20]. Studies regarding the performance of high speed and ultra-high speed imaging systems have been reported in the past few years, e.g. [21, 22]. These technologies enable temporal resolutions on the order of a microsecond and below with good spatial resolution, making it possible to measure both full-field strains and accelerations with excellent temporal and spatial resolutions; this is essential for the current study. These cameras still have important drawbacks however: high noise level, low number of images and very high cost. Recently, the advent of in-situ storage cameras like the Shimadzu HPV-X or the Specialized Imaging Kirana has given new impetus to using ultra-high speed imaging for full-field deformation measurements. Image quality has improved considerably, as evidenced in [23].

Finally, in dynamic testing, the key issue relates to external load measurement. Indeed, inertial effects in standard load cells (‘ringing’) prevent accurate loads to be measured. The alternative is to resort to an SHPB set-up using the bars as a very bulky and inconvenient load cell. This procedure works well but within a very restrictive set of assumptions: specimen quasi-static equilibrium (no transient stress waves, requiring a short specimen) and uniaxial loading, in particular. The need for more complex stress states to identify and fully validate robust constitutive models requires investigators to move away from such stringent assumptions if at all possible. The current study is exploring this idea for welds, based on the Virtual Fields Method (VFM).

The VFM was first introduced in the late 1980s in order to solve inverse problems in materials constitutive parameter identification with the aid of full-field measurements. Since then, it has been successfully applied to the identification of constitutive parameters for homogeneous materials in elasticity [24, 25], elasto-plasticity [26, 27], and visco-plasticity [28]. The method has also been used for heterogeneous materials (welds) in quasi-static loading and elasto-plastic material response [29]. Recent developments by Moulart et al. [30] introduced the application of the VFM to the identification of the dynamic elastic properties of composite materials. The main idea in this case is to use the acceleration field as a load cell, avoiding the need to measure an external load. Since then, it has also been used to identify the damage process of concrete materials [31], and to analyze the deformation of a beam in dynamic three-point bending [32]. More recently, spectacular improvements in image quality has led to unprecedented quality of identification, as evidenced in [23] for the elastic response of a quasi-isotropic laminate at strain-rates above 2000 s^{−1}. However, until now, it is has never been attempted to identify an elasto-plastic model with this approach. Thus, the enclosed work breaks new ground by not only using acceleration fields instead of measured load data but also applying this approach in the more complex situation where heterogeneous plastic deformation is occurring in a weld.

The aim of this study is to explore new ways to use the VFM for the identification of the dynamic heterogeneous elasto-plastic properties of Al 5456 FSW welds. The nature of the paper is seminal in the way that it insists on the methodology and its potential. Many developments are still required to make this procedure a standard tool (including better UHS cameras, adapted test design etc.) but the authors feel that the current technique has great potential for future dynamic tests of materials. This study is part of a global long-term effort to design the next generation of high strain-rate tests based on rich full-field deformation information. The recent progress in UHS cameras reported above makes this contribution all the more timely, even if the results reported here are somewhat impaired by the fact that lower image quality cameras were used at the time that the experiments were performed.

## Specimens and Experiments

The identification of the dynamic properties of the weld was performed based on experimental results from SHPB tests. It is worth noting however that the set-up of the SHPB test is used here, but the SHPB data reduction procedures are not used. Moreover, the first images were taken when the transient stress wave was present in the specimens and the accelerations were at their maximum, preventing any use of the standard SHPB analysis anyway. Two series of tests have been carried out for this work: one at the University of Oxford on welded and base material specimens where the grid method (or ‘sampling moiré’) [33, 34, 35, 36, 37, 38, 39] was used, and a second one at the University of South Carolina on welded specimens only where digital image correlation [40] was used.

### Specimens

### SHPB Tests Using the Grid Method

These tests were performed on a SHPB set-up at the University of Oxford. Five tests were performed with the grid method, three on base material specimens and two on welded specimens.

#### Experiment

SHPB imaging parameters with SIM 16 camera

Camera | SIM 16 |

Sensor size | 1360 × 1024 pixel |

Field of view | 24.5 × 10 mm |

Interframe | 5 μs |

Shutter speed | 1 μs |

Total number of images | 16 |

Technique used | Grid method |

Period size | 150 μm |

Pixels per period | 9 |

Displacement | |

Smoothing method | Least square convolution |

Smoothing window | 31 × 31 measurement points |

Resolution | 0.048 pixels (0.8 μm) |

Strain | |

Differentiation method | Finite difference |

Resolution | 313 μstrain |

Acceleration | |

Differentiation method | Finite difference from smoothed displacements |

Resolution | 66,000 m s |

Both input and output bars were 500 mm long, 15 mm in diameter and made from steel. The impactor speed was up to 18 m s^{−1}. The strain-rate fields obtained by finite difference differentiation of the strain fields showed maximum local strain-rates of respectively 1300 and 1000 s^{−1} for welded and homogeneous specimens. It should be noted that the strain-rate maps are heterogeneous in space and variable in time. In particular, at the onset of plasticity, there is a sharp local increase of the strain-rate, as was also evidenced in [28]. In the standard SHPB approaches, this is ignored and only an average strain-rate is considered. Ideally, the heterogeneous strain-rate maps should be used to enrich the identification of the strain rate dependence, as was performed in [28]. This was not done here as the quality of the data does not currently allow for it, but this is a clear track to follow in the future to improve the procedure. The acquisition and lighting systems were triggered by a strain gauge bounded onto the incident bar. The images were taken with an interframe time of 5 μs, and a shutter speed of 1 μs. A total of 16 images were taken during each test. Indeed, the technology of the camera is based on the use of a beam splitter and 16 sensors. Therefore, each image was taken from a different sensor. As a result, there was a difference in light intensity between the different images, and it was not possible to accurately measure the displacement fields between images taken from different sensors. To address this issue, the displacement fields were computed between two images taken with the same sensor: one static reference image and the actual dynamic one. The displacement computation is based on the phase shift between the reference and deformed images [33]. In this study, a windowed discrete Fourier transform (WDFT) algorithm was used [34, 35, 42, 43]. It calculates the discrete Fourier transform of the intensity over a set of pixels over a triangular windowed kernel. However, the measured phase maps consist of values between \(-\pi\) and \(\pi\) . Therefore, it is not possible to measure a displacement associated to a phase shift that exceeds \(\pi\). In this case, it is necessary to unwrap the phase map in order to obtain the actual value of the displacement. Extensive work has been done in the past to address this problem [44, 45, 46]. The algorithm used in this study is presented in [47].

#### Smoothing, Acceleration and Strain Computation

In order to reduce the effect of measurement noise, displacement fields were smoothed using an iterative least square convolution method [48]. The smoothing was performed over a 31 × 31 pixels window using a second order polynomial function. Then, the strain fields were computed from the displacement fields by finite difference. Velocity and acceleration fields were calculated from the smoothed displacement field using a centred temporal finite difference scheme. This precluded reliable acceleration maps to be obtained for the two first and last images. Therefore, acceleration fields were only available for 12 steps of the experiment, when 16 steps were available for the strain and displacement fields. By recording two sets of images for the stationary specimen prior to testing, it is possible to compute the standard deviation of the resulting displacement, strain and acceleration maps. This provides an estimate of the ‘resolution’ as reported in Table 1 together with smoothing details.

### SHPB Tests Using Digital Image Correlation

This test was carried out on a welded specimen with digital image correlation on an SHPB set-up at the University of South Carolina.

#### Experiment

SHPB imaging parameters with DRS IMACON 200 camera

Camera | IMACON 200 |

Sensor size | 1340 × 1024 pixel |

Field of view | 24.5 × 10 mm |

Interframe | 4 μs |

Shutter speed | ~0.4 μs |

Total number of images | 16 |

Technique used | DIC |

Speckle pattern | Rub on transfer decal |

Subset | 55 |

Shift | 20 |

Displacement | |

Smoothing method | Least square convolution |

Smoothing window | 31 × 31 measurement points |

Resolution | 0.07 pixels (1.28 μm) |

Strain | |

Differentiation method | analytical |

Resolution | 484 μstrain |

Acceleration | |

Differentiation method | Finite difference from smoothed displacements |

Resolution | 45,000 m s |

Both input and output bars were 2388 mm long, 25.4 mm in diameter and made from steel. The 483 mm long steel impactor speed was 24 m s^{−1}. The strain-rate fields measured by finite difference of the strain fields showed a maximum local strain-rate of 1600 s^{−1}. The acquisition system was triggered by a piezo-electric sensor set on the incident bar. The images were taken with an interframe time of \(4\,\upmu s\), and a shutter speed of \(0.4\,\upmu s\). The DRS IMACON 200 uses the same type of technology as the SIM 16, therefore displacement fields were computed between a static reference image and the actual dynamic image from each sensor. Moreover, in order to reduce the influence of the difference of contrast between the different sensors, flat-field correction has been performed on the images [22]. All images were processed using the 2D-DIC software VIC-2D [49].

#### Noise Issues

#### Smoothing, Acceleration and Strain Computation

The strain fields were computed by analytical differentiation after least square quadratic fit over a 5 × 5 window of the displacement fields by the VIC-2D software [49]. Then the strain fields were smoothed using an iterative least square convolution method [48] over a 31 × 31 pixels window using a second order polynomial function. The calculation of the acceleration fields was performed with the method used for the computation of the measurement from grid method tests. The baseline information on the measurements can be found in Table 2.

### Results

Concerning the measurements with the grid method, the lower impact velocity could affect the identification of the mechanical properties of the material. Indeed, it results in lower values in the acceleration and strain fields and therefore, could hinder the identification process due to larger noise to signal ratio. One can notice that the average strain on the right hand side of the specimen barely reaches the estimated base material yield strain \(({\simeq}0.005)\). Therefore, it could affect the identified plastic parameters. This problem does not occur in the measurements realised with DIC due to the higher impact velocity and average strain over the specimen.

## Virtual Fields Method

*ρ*the density of the material,

*a*

_{ i }the acceleration vector,

*V*the volume where the equilibrium is written, \(u^{*}_{i}\) the virtual displacement field, \(\epsilon ^{*}_{ij}\) the virtual strain tensor deriving from \(u^{*}_{i}\), \(S_{V}\) the boundary surface of V, \(T_{i}\) the imposed traction vector over the boundary \(S_{V}\). In the case of dynamic experiments, load measurement is an issue. Therefore, in order to cancel out the contribution of the load in the principle of virtual work (PVW), a specific virtual field is used that must comply with the specification described in (4).

### Virtual Fields in Elasticity

*L*is the length of the identification area. By incorporating (7) and (8) into (6), the following system is obtained, assuming plane stress, linear isotropic elasticity and homogeneous elastic properties, as it has been shown in Sutton et al. [29] for quasi-static properties.

### Virtual Fields in Homogeneous Plasticity

Moreover, the stress-strain relationship being non-linear, a single virtual field is generally sufficient to perform identification when the number of parameters is low, which is the case here [27]. In order to calculate the value of \(\Phi (\sigma_{y},H)\), the stress field is computed at each step of the experiment using the method proposed by Sutton et al. [50]. This is an iterative method based on the radial return. The minimisation of the cost function is based on the Nelder-Mead simplex method [51].

### Virtual Fields in Plasticity for Heterogeneous Materials

## Results

### Elastic Parameters

Elastic parameters identified by the VFM

Reference | Grid: base material | Grid: welded specimen | |
---|---|---|---|

Young’s modulus (GPa) | |||

First test | 70 | 62 | 68 |

Second test | 70 | 32 | 72 |

Third test | 70 | 43 | |

Poisson’s ratio | |||

First test | 0.33 | 0.1 | 0.31 |

Second test | 0.33 | 0.1 | 0.37 |

Third test | 0.33 | 0.7 |

### Plastic Parameters

The reference value of Young’s modulus and Poisson’s ratio were used for the identification of the plastic parameters. The identification was carried out using the images from the plastic steps of the tests. Moreover, due to the fact that all areas of the weld do not yield at the same time and do not undergo the same amount of strain, the identification has been carried out using different numbers of images for each slice. Indeed, with the slices on the impact side of the specimen, the strain level is more important and yield occurs earlier. Therefore, 5–8 images were used to perform the identification, depending on the slice. Knowing that the identification makes use of a minimisation process, the starting values of the algorithm could have an impact on the identified parameters. The evolution of the cost function with the plastic parameters is represented in Fig. 12, for a slice in the centre of a base material specimen.

Plastic parameters identified by the VFM for base material specimens using the grid method

Reference | Base material 1 | Base material 2 | Base material 3 | |
---|---|---|---|---|

Yield stress (MPa) | 255 | 368 | 376 | 402 |

The same type of evaluation has been performed on a welded specimen (Fig. 14). As seen in Figs. 6 and 8, the strain levels are larger than for the base material so the identification can be carried out over the whole field of view. It can clearly be seen that an increased slice thickness (i.e. decreased number of slices) leads to a smoother spatial variation of the yield stress. However, when the number of slices goes over 12, it is not possible to identify the yield stress on the second half of the specimen where there is no convergence of the minimisation process anymore. The reason for this is that when the stress state is too uniform spatially, and accelerations are low, then both terms in Eq. 16 are close to zero and convergence cannot be reached (the \((2x_2-L)\) term has a zero mean over the slice where \(x_2\) varies between 0 and L). Increasing the slice width results in higher stress heterogeneity and convergence can be restored. This problem is not evidenced in the welded area where significant strain and stress heterogeneities are present because of the strain localization process. As a result of this compromise, the number of slices will be kept around 9 (and varies slightly for one specimen to the next as field of views and impact speeds are slightly different).

Finally, the current results have been collated with that from [53] to obtain an overview of the variations of yield stress profiles over a large range of strain-rates (Fig. 17). It is interesting to note that at the center of the weld, the identified values show a significant strain-rate sensitivity between \(83\,\upmu \,{\rm{s}}^{-1}\) and \(0.63\,{\rm{s}}^{-1}\), whereas for the base material, sensitivity is towards the larger strain-rates [53]. This is interesting but would need to be backed up with materials science studies to confirm if such an effect is expected. It is clear however that the very different micro-structures between the nugget, heat affected zones and base material could potentially lead to such differences. It must be emphasized that this kind of results would be extremely difficult to obtain by any current method and as such, the present methodology has great future potential to explore local strain-rate sensitivity in welds. This in turn can lead to the development of better visco-plastic constitutive models for such welds. Finally, more complex tests such as three or four point impact bending tests (as in [32]) could be used to identify elasto-visco-plastic models over a wider range of stress multi-axiality, which is currently another main limitation of the standard SHPB analysis.

## Conclusion

A new method for the identification of the dynamic properties of welds has been proposed in this study. It offers a significant contribution to the field of high strain-rate testing. In this work, the acceleration fields have been used as a load cell, in order to carry out the identification of the mechanical properties of the material. While previous work in this area [23, 30] was limited to the characterisation of the elastic properties, the identification of both elastic and plastic parameters has been carried out during this study. According to the authors knowledge, it is the first time that the identification of the dynamic yield stress of a material has been attempted without any external load measurement. Moreover, a local characterisation of the dynamic yield stress was performed on welded specimens. The repeatability of the process has been verified on two different set-ups and with two different full-field measurement techniques. The hardware, and more specifically, the high noise level of the cameras and the low number of available images currently remains the main weak point of the method. Moreover, it is essential that in the future, detailed uncertainty assessment of the identified data is performed so that error bars can be added to the yield stress evolutions in Fig. 17. This is a challenging task as the measurement and identification chain is long and complex with many parameters to set. This can only be addressed by using a realistic simulator as developed in [55] for the grid method and more recently [56] for Digital Image Correlation. This enables first to optimize the test and processing parameters (load configuration, subset and smoothing in DIC, virtual fields in the VFM) and then to provide uncertainty intervals for the identified parameters. This has recently been validated experimentally in [57]. The present case will be more computationally challenging but conceptually, the procedures in [55, 56, 57] can be used in exactly the same way as for elasticity. This will have to be investigated in the near future when tests with better images are available. It is a key issue for making this new procedure a standard technique for which users have confidence in the results.

It is believed that the demonstrated ability to extract local material properties without the requirement for external load measurement will open unprecedented opportunities to expand the range of experimental approaches that can be used in the field of high strain-rate testing. To develop the next generation of novel methodologies and make them available to researchers and engineers, significant additional research will be required, with the growth and continuous improvement of modern high speed imaging technology being the foundation for the effort.

## Notes

### Acknowledgments

The authors would like to thank the Champagne-Ardenne Regional Council for funding 50 % of the Ph.D. studentship of G. Le Louëdec. The authors would also like to acknowledge the support of the US Army Research Office through ARO Grants \(\sharp\) W911NF-06-1-0216 and Z-849901, and the NSF through the I/UCRC Center for Friction Stir Processing.

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