Extension of the sensitivity-based virtual fields to large deformation anisotropic plasticity
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Abstract
The virtual fields method is an approach to inversely identify material parameters using full-field deformation data. In this work, we extend the sensitivity-based virtual fields to large deformation anisotropic plasticity. The method is firstly generalized to the finite deformation framework and then tested on numerical data obtained from a finite element model of a deep-notched specimen subjected to a tensile loading. We demonstrated the feasibility of the method for two anisotropic plasticity models: Hill48 and Yld2000-2D, and showed that all the parameters could be characterise from such a test. The sensitivity-based virtual fields performed better than the currently accepted standard approach of user-defined ones in terms of accuracy and robustness. The main advantage of the sensitivity-based virtual fields comes from the automation of virtual fields generation. The process can be applied to any geometry and any constitutive law.
Keywords
The virtual fields method Anisotropic plasticity Sensitivity-based virtual fields Material testing Full-field measurementsIntroduction
Accurate modelling of metal forming processes is of key interest to industries such as automotive. One of the main issues in simulating processes such as deep drawing of metal sheets is ensuring that the chosen constitutive model represents the material accurately. Many of the metallic materials used in this industry exhibit anisotropic properties due to texture induced during cold rolling which highly affect deformation of it during forming processes (e.g. earrings formation during deep drawing [20]).
One of the most popular anisotropic plasticity yield criteria is Hill48 [19], which in case of plane stress conditions requires four parameters, generally identified from three uniaxial tests performed in three directions: rolling (RD), transverse (TD) and 45^{∘} to RD. In many cases, experimental results suggest that Hill48 is not capable of predicting biaxial yield behaviour accurately, and thus has limited applicability for forming predictions. Numerous models were proposed to capture the biaxial behaviour of sheet metals more accurately, such as: Yld89 [9], Stoughton’s model of 2002 and further refinement of 2009 [43, 44], BBC2000 [6], BBC2005 [5], Yld2000-2D [8] and Yld2004-18 [7]. Often, the usefulness of these complex models is limited by the significant effort required to accurately identify their parameters experimentally. In particular, many of these models require performing an additional biaxial test, such as bulge or equibiaxial tension on cruciform specimens, increasing the cost of the procedure. Therefore, there is a drive to improve testing techniques and a promising way to achieve this goal is to collect experimental data using more advanced methods, such as full-field measurements e.g. digital image correlation (DIC).
New tests can be designed in order to collect more data within a single run, compared to the standard methods. The use of full-field measurements makes it possible to choose complex geometries for the test specimens, introducing heterogeneous strain fields, thereby enabling the yield envelope to be probed at thousands of different stress states at once. One of the main challenges in such approach is to extract the material parameters from the collected data. Two of the most used inverse techniques capable of doing this are finite element model updating (FEMU) and the virtual fields method (VFM). It was demonstrated that these approaches can reduce the number of tests needed to fully characterize anisotropic models, in particular Hill48, or Yld2000-2D [11, 13, 18, 21, 35, 38, 39].
The virtual fields method is a very efficient technique for extraction of material parameters from full-field measurements. One of the main advantages of the VFM over the FEMU is that it is significantly faster in terms of the computational time. In fact, some authors reported that for their particular application the VFM was 125 times faster than FEMU [47]. This is especially important as the complexity of material models and the number of data points available from the measurements grow. Another advantage of the VFM is that it acts directly on collected data and no numerical simulations are required. As a result, the method can be integrated directly into a DIC platform making it more accessible to practising engineers. The method has already been applied to a range of materials and constitutive laws such as arteries [2], rubbers [17, 45, 46], composites [15], and metals [22, 24, 25, 33, 39].
One of the main challenges in the VFM is the choice of virtual fields. These are test functions that act upon the reconstructed stress fields to check for stress equilibrium. Their choice strongly affects the accuracy of the identification. Until recently, no structured method was available to generate high quality virtual fields for non-linear problems. Currently, the standard approach is to rely on user-defined virtual fields (UDVFs), using standard expansion bases such as polynomials or harmonic functions. The effectiveness of these user-defined virtual fields strongly depends on their choice, and requires the user to understand the method in depth to be able to select these fields in an informed way. Recently, a new approach for generating high quality virtual fields has been developed, leading to the so-called sensitivity-based virtual fields (SBVFs) [28]. They outperformed UDVFs in case of isotropic plasticity, and are generic enough to be applied to any constitutive model.
In this work, we have extended the SBVFs to the case of large deformation anisotropic plasticity, and demonstrated their feasibility to calibrate Hill48 and Yld2000-2D yield functions from a deep notched specimen subject to tensile loading.
Theory
Brief recall of the finite deformation framework
The virtual fields method
Equation 6, called the principle of virtual work (PVW), is satisfied for any continuous function u^{∗} (called virtual displacements) that is piecewise-differentiable. Both stress and test function (virtual displacements) are expressed in the current configuration in the case of Eq. 6.
Sensitivity-based virtual fields
The test functions (virtual fields) in Eq. 10 are arbitrary and must be selected before the identification is conducted. The selection of virtual fields has a significant impact on the accuracy of the identification. These functions influence the amount of error introduced to the cost function by selecting which data points, and with what weight, are introduced to the cost function. The main difference between virtual fields arises in the way they propagate experimental noise.
In linear elasticity, an automated procedure has been published in 2004, relying on the minimization of the impact of noise on the identified parameters, i.e., finding the virtual fields leading to the maximum likelihood solution for a given basis of functions to expand the virtual fields [3]. This is now routinely used by the VFM community and also implemented on the commercial DIC/VFM platform MatchID [29]. An attempt at extending this to non-linear laws, namely, isotropic elasto-plasticity, was published in 2010 [34]. The idea there was to use a piecewise linear definition of the virtual fields based on the tangent stiffness matrix. The method did improve results but was found to lack flexibility as it required an expression for the tangent matrix. Also, for non-linear models where strains are generally larger, sensitivity to noise is not necessarily the most relevant criterion to select virtual field.
Recently, a new type of virtual field for the non-linear laws was proposed [28]. They are automatically generated during the identification procedure with very limited user input. These fields, called sensitivity-based virtual fields, are based on the reconstructed stress field and so, easily and automatically adapt to any geometry and material model. They were shown to outperform the user-defined virtual fields for isotropic small-strain plasticity and seemed very promising for more complex problems. They were also shown to outperform the tangent-matrix fields from [34], though only marginally for this particular case.
The incremental sensitivity maps cannot however be directly used in the PVW as the corresponding virtual displacements, needed in the PVW, are unknown. However, it is possible to construct virtual displacements such that the resulting virtual strain fields ‘look like’ the incremental sensitivity maps. This can be achieved by performing a least-square match, under some constraints, between virtual strain fields and incremental stress sensitivity.
Note that the construction of the sensitivity-based virtual fields must be performed at every time step. However, as mentioned before, if the reference configuration is chosen for the PVW, the matrix B_{glob} is assembled only once for the entire identification.
Computing stress sensitivities significantly increases the identification time, as it virtually doubles the number of necessary stress reconstructions. To improve the computational efficiency, a selective updating scheme can be employed. Recall that any continuous virtual displacement fields constitute a valid choice, including the sensitivity-based virtual fields based on incorrect (e.g. initial) parameters. Effectively, these can be used to put the minimisation algorithm in the neighbourhood of the solution without updating them, but carrying across the iterations. As the algorithm converges, the virtual fields can be updated with parameters much closer to the correct values, saving many stress reconstructions and significantly improving computational efficiency.
Finally, in order to balance the contributions from each virtual field, a scaling is introduced. The virtual displacements are scaled by a factor dependent on the current internal virtual work contributions (W_{int}). For each iteration, W_{int} is calculated (10), and then sorted according to the absolute values over all time steps. The scaling parameter is calculated as a mean out of the top x^{th} percentile of the sorted values. This ensures that virtual fields contributions associated with each parameter are of similar orders of magnitude.
Numerical simulations
The standard way to test anisotropic materials is to conduct tensile tests on dog-bone specimens cut at different angles to the rolling direction of the sheet (typically 0^{∘}/45^{∘}/90^{∘}). If the material model under inspection includes a parameter related to biaxial yield stress, an additional test is required, either a bulge test or equibiaxial tension on a cruciform specimen. The major limitation of this approach is that a single test provides only one data point on the yield locus and many tests are needed to match the yield surface.
An alternative is to run a test with enough stress heterogeneity to identify all necessary parameters at once. Some of the heterogeneity in the tensile test can be obtained by means of material orientation, geometric features, and loading. Rossi et al. [39] proposed a test on a deep-notched specimen under tensile loading capable of identifying the Hill48 model using a single specimen. The test is replicated here and combined with the sensitivity-based virtual fields to test their applicability to large strain anisotropic plasticity. This does not mean that this test is optimal in any way, but it can serve as a clear comparison on how VFs selection impacts the identification. Many different geometries have been proposed in the literature to produce heterogeneous states of stress and strain, it is beyond the scope of the present paper to investigate this. However, future work will look at specimen optimization, in the same spirit as for composites testing in elasticity [16].
FE model
Constitutive models
In the VFM, the stress field is reconstructed explicitly from the kinematic measurements through an assumed constitutive law. In this work, two different large strain plasticity models were considered: Hill48 and Yld2000-2D.
Although the model is popular in literature, it suffers from poor performance when used in context of sheet materials subject to biaxial loading [30, 39, 44]. This is mostly due to the quadratic nature of the law which does not represent real materials accurately. It was found that non-quadratic models such as Yld2000-2D or Stoughton2009 predict the behaviour of sheet metals such as steel or aluminium more accurately [44].
The model involves 8 independent parameters, α_{1}–α_{8}, and can accurately represent the behaviour both in simple tension as well as biaxial loading.
Reference parameters defining the plastic anisotropy of the material
Hill48 | Yld2000-2D | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
R _{11} | R _{22} | R _{33} | R _{12} | α _{1} | α _{2} | α _{3} | α _{4} | α _{5} | α _{6} | α _{7} | α _{8} |
1.000 | 1.054 | 1.276 | 0.738 | 1.11 | 1.35 | 1.21 | 1.11 | 1.07 | 0.96 | 1.21 | 1.15 |
Reference parameters for the hardening laws based on properties of BH340 steel alloy
Linear | Power law | |||
---|---|---|---|---|
σ _{0} | H | σ _{0} | K | n |
256 | 855 | 203 | 439 | 0.3195 |
For each of the constitutive models a routine was produced integrating the constitutive equations using an implicit scheme with a radial-return algorithm, returning the state of stress and out-of-plane strain at each time step, with both being rotated back to the global coordinate system.
For Hill48, 6 data sets were generated in total, considering both hardening models at the three different material orientations (30^{∘},45^{∘},60^{∘}). Due to its computational intensity, only one model was considered for Yld2000-2D, simulating linear hardening with a material orientation of 45^{∘}.
While in this work the constitutive modelling was kept simple, the proposed methodology is valid for any chosen material model. In practice, the investigator supplies a constitutive law to be identified (see Fig. 4). The model can be arbitrarily complex, as long as it is capable of reconstructing the stress field based on the measured kinematic data (e.g. deformation gradient) and internal state variables that can be carried over between different load levels. By choosing more complex models that account for e.g. multiplicative decomposition of deformation gradient [27], hyperelasticity [42], anisotropic hardening [10] or even complete anisotropic elastoplasticity [40] more complex material description could be reached, but this has not been considered in this work.
Data processing
Following the FE simulations, raw displacements and positions of data points were extracted from Abaqus and exported into Matlab (2016a). The cloud of points was then interpolated onto a rectangular grid of 409 × 349 data points, corresponding to the data density typical of a DIC measurement. The region of interest was then trimmed to a grid of 362 × 202 points spanning the region of interest (ROI) (Fig. 2), which produced 57,392 data points. In the case of Yld2000-2D model, to reduce the computational time, a coarser data grid was used: 214 × 119 producing 20,024 points in total.
For each model, a different number of time steps was taken. In the case of Hill48 with linear hardening, the calculations were relatively fast, thus having many temporal points was not a problem. Ultimately, 83 frames were considered. In the case of the power law hardening, during deformation a plastic instability occurred and the strains began to localize in a very small band, causing a geometrical softening. This data was discarded, as shown in the figure. Overall, only 60 frames were used, with a maximum plastic strain of about 30%. Finally, for Yld2000-2D only 27 time steps were used, as the constitutive law itself is much more computationally demanding, and having more points would just lead to inconvenient computational times, unnecessary at this first validation stage.
After smoothing and down-sampling were performed, the deformation gradient F was calculated using central finite difference, which was then used to calculate V with Eq. 4, R with Eq. 3 and ε_{L} with Eq. 5. The set of the three quantities (F,R,ε_{L}) for all data points constitutes the kinematic data used to identify the material parameters.
The SBVFs were updated when the first-order optimality [1] fell below certain threshold (1 × 10^{− 4}), indicating convergence of the procedure. The virtual fields were recomputed and stored in memory, and the threshold was scaled down by a factor of 2.3 to allow further refinement. Sometimes, after selective updating was performed, the value of cost function would increase, creating an apparent local minimum terminating the minimisation. To prevent this, the value of cost function was offset below previous iteration when the update was performed. This scheme leads to an increasing rate of updates as the solution converges, providing valid values of virtual fields at the optimum. In total, about 10–15 virtual fields were computed throughout the identification consisting of more than 100 iterations, with the number controlled indirectly by the two parameters (first-order optimality threshold and threshold refinement parameter).
User-defined virtual fields
It is worth noticing, that only the first virtual field (35) includes the virtual work of external forces. This impacts the balance of contributions from each virtual field to the cost function. Traditionally, the residual is scaled by the virtual work of external forces to provide a dimensionless value [33]. This was not possible here, as two of the virtual fields do not include the contribution of the external forces. To overcome this problem, scaling by the maximum value of the internal virtual work was introduced, which normalises the peaks of the internal virtual work to 1. To balance all three fields even further, the residual coming from the first virtual field was scaled by a factor 500, which was found to produce the best results. This workaround scaling shows that the manually defined virtual fields lack generality, and must be individually tailored for each application, which is a significant disadvantage.
Error quantification
As the number of material parameters in the constitutive models increases, quantifying the accuracy of identification becomes a challenging task. Often, these models are defined with parameters that lack physical meaning, and on its own have limited impact on the model outcome, which is driven by the compound action of all of the parameters, as for Yld2000-2D. It is therefore important to establish tools to compare different sets of identified parameters in order to quantify the accuracy of the identification meaningfully.
Results
Validation of the method
Identified parameters and mean error as defined in Eq. 41 for the Hill48 model using exact data for both linear and power law hardening
Orientation | Hardening | VFs | \(\sigma ^{0}_{11}\) | \(\sigma ^{0}_{22}\) | \(\sigma ^{0}_{33}\) | \(\sigma ^{0}_{12}\) | H | n | Mean error [%] |
---|---|---|---|---|---|---|---|---|---|
45^{∘} | Linear | UD | 1.007 | 1.003 | 1.001 | 0.990 | 1.037 | [-] | 0.6 |
45^{∘} | SB | 1.002 | 1.000 | 1.001 | 0.997 | 0.996 | [-] | 0.2 | |
30^{∘} | UD | 1.020 | 1.029 | 0.997 | 0.985 | 1.025 | [-] | 1.2 | |
30^{∘} | SB | 1.003 | 1.000 | 1.001 | 0.997 | 0.994 | [-] | 0.3 | |
60^{∘} | UD | 1.012 | 1.053 | 1.019 | 0.999 | 1.021 | [-] | 1.6 | |
60^{∘} | SB | 1.000 | 1.009 | 1.003 | 1.002 | 0.995 | [-] | 0.3 | |
30^{∘} + 60^{∘} | UD | 1.030 | 1.030 | 1.002 | 0.984 | 1.049 | [-] | 1.5 | |
30^{∘} + 60^{∘} | SB | 1.001 | 1.001 | 1.000 | 0.996 | 0.993 | [-] | 0.4 | |
45^{∘} | Power | UD | 1.000 | 1.018 | 0.989 | 0.955 | 1.020 | 0.973 | 1.7 |
45^{∘} | SB | 1.020 | 1.015 | 1.014 | 1.005 | 1.039 | 1.036 | 1.1 | |
30^{∘} + 60^{∘} | UD | 1.104 | 1.102 | 1.086 | 1.057 | 1.136 | 1.206 | 3.3 | |
30^{∘} + 60^{∘} | SB | 1.003 | 1.003 | 1.003 | 0.999 | 1.005 | 1.008 | 0.1 |
Clearly, for linear hardening, all tests were successful at identifying the material parameters. Although marginal here, it is worth noting that the mean error for sensitivity-based VFs is already consistently smaller compared to the user-defined ones. The single tests performed at either 30^{∘} or 60^{∘} were only performed usig exact data and the linear hardening. As they are unlikely to be practical when experimental and modelling errors are introduced, they were disregarded for the more complex cases. For the power law hardening, the identified values were also very close to the reference. With the exception of the combined 30^{∘} + 60^{∘} tests using the user-defined VFs, the mean error was consistently below 2%, showing that the methodology adopted here has a potential of identifying Hill48 parameters using a single test. The combined test was adopted in order to obtain more data over uniaxial states of stress, in comparison to the 45^{∘} test. The latter, probes the yield envelope under combined shear and normal loading, with very little data points containing information about pure σ_{11} or σ_{22} behaviour. By including both 30^{∘} and 60^{∘} tests, more information is available about these regions, as well about shearing behaviour enabling identification all of the parameters.
Effect of noise in the data
Identified parameters and mean errors as defined by Eq. 41 for Hill48 model with linear hardening using different smoothing combinations
σ_{spat}/ window | w_{temp}/ m_{temp} | \(\sigma ^{0}_{11}\) | \(\sigma ^{0}_{22}\) | \(\sigma ^{0}_{33}\) | \(\sigma ^{0}_{12}\) | H | Mean error [%] |
---|---|---|---|---|---|---|---|
1.0/5 | 11/3 | 0.974 ± 0.27 | 0.961 ± 0.26 | 0.998 ± 0.07 | 1.108 ± 0.15 | 0.858 ± 0.43 | 4.08 ± 0.09 |
1.3/7 | 11/3 | 0.967 ± 0.26 | 0.965 ± 0.242 | 0.988 ± 0.06 | 1.031 ± 0.15 | 0.941 ± 0.43 | 1.99 ± 0.11 |
1.5/9 | 11/3 | 0.966 ± 0.29 | 0.968 ± 0.25 | 0.984 ± 0.06 | 1.008 ± 0.16 | 0.963 ± 0.46 | 1.45 ± 0.13 |
1.85/11 | 11/3 | 0.963 ± 0.31 | 0.967 ± 0.25 | 0.980 ± 0.07 | 0.991 ± 0.17 | 0.975 ± 0.51 | 1.70 ± 0.15 |
1.5/9 | 11/3 | 0.966 ± 0.29 | 0.968 ± 0.25 | 0.984 ± 0.06 | 1.008 ± 0.16 | 0.963 ± 0.46 | 1.45 ± 0.13 |
1.5/9 | 15/3 | 0.977 ± 0.24 | 0.971 ± 0.21 | 0.985 ± 0.06 | 1.026 ± 0.13 | 0.975 ± 0.36 | 1.50 ± 0.08 |
1.5/9 | 21/3 | 1.035 ± 0.33 | 1.013 ± 0.35 | 0.999 ± 0.10 | 1.011 ± 0.54 | 1.055 ± 0.66 | 2.33 ± 0.08 |
Identified parameters for Hill48 model with linear hardening
Orientation | VFs | \(\sigma ^{0}_{11}\) | \(\sigma ^{0}_{22}\) | \(\sigma ^{0}_{33}\) | \(\sigma ^{0}_{12}\) | H | Mean error [%] |
---|---|---|---|---|---|---|---|
45^{∘} | UD | 0.954 ± 0.15 | 1.021 ± 0.20 | 0.994 ± 0.03 | 1.034 ± 0.11 | 0.903 ± 0.35 | 2.13 ± 0.04 |
45^{∘} | SBs | 0.9656 ± 0.29 | 0.968 ± 0.25 | 0.984 ± 0.06 | 1.008 ± 0.16 | 0.963 ± 0.46 | 1.45 ± 0.13 |
30^{∘} + 60^{∘} | UD | 1.041 ± 0.33 | 1.042 ± 0.03 | 1.010 ± 0.01 | 1.009 ± 0.03 | 1.048 ± 0.05 | 2.56 ± 0.01 |
30^{∘} + 60^{∘} | SB | 0.993 ± 0.08 | 0.992 ± 0.08 | 0.994 ± 0.02 | 1.036 ± 0.09 | 0.962 ± 0.16 | 1.29 ± 0.04 |
Identified parameters for Hill48 model and the power law hardening
Orientation | VFs | \(\sigma ^{0}_{11}\) | \(\sigma ^{0}_{22}\) | \(\sigma ^{0}_{33}\) | \(\sigma ^{0}_{12}\) | H | n | Mean error [%] |
---|---|---|---|---|---|---|---|---|
45^{∘} | UD | 0.842 ± 1.11 | 0.891 ± 1.55 | 0.925 ± 0.68 | 0.995 ± 0.47 | 0.849 ± 1.12 | 0.900 ± 1.18 | 4.67 ± 0.34 |
45^{∘} | SB | 0.929 ± 0.43 | 0.929 ± 0.44 | 0.962 ± 0.38 | 1.031 ± 0.33 | 1.045 ± 0.55 | 1.028 ± 0.75 | 4.35 ± 0.10 |
30^{∘} + 60^{∘} | UD | 1.020 ± 0.26 | 1.021 ± 0.26 | 1.032 ± 0.28 | 1.053 ± 0.30 | 1.042 ± 0.24 | 1.074 ± 0.59 | 2.60 ± 0.08 |
30^{∘} + 60^{∘} | SB | 0.967 ± 0.18 | 0.968 ± 0.18 | 0.996 ± 0.17 | 1.045 ± 0.17 | 1.026 ± 0.27 | 1.043 ± 0.40 | 3.15 ± 0.06 |
Hill48 identification
Linear hardening
Power law hardening
The results are markedly different for the power law hardening. Firstly, the effect of noise is significantly larger compared to the linear hardening case, as shown in Fig. 7 (note the difference in scaled between Fig. 7a and b). The mean errors for the 45^{∘} tests are between 4 and 5%, compared to about 2% for the simpler hardening law (Fig. 7). While level of error is still satisfactory, it must be noted that it represents a lower bound as the simulation of experimental uncertainties remains very basic. Unlike the linear hardening case, the random error was the smallest when the SBVFs were used and the smallest systematic error was obtained with the UDVFs. Significant improvement was found when the single test was replaced by the combined tests, both in terms of systematic and random errors. This indicates that the additional test contributes significant data to the cost function. Since the smoothing parameters were chosen for the linear hardening model, it is possible that a different combination of parameters would provide smaller errors, and differentiate between the two virtual fields types in a different way.
Yld2000-2D identification
Identified values of parameters for the Yld2000-2D from a 45^{∘} test using exact data
VFs | α _{1} | α _{2} | α _{3} | α _{4} | α _{5} | α _{6} | α _{7} | α _{8} | σ _{0} | H | Global error [%] |
---|---|---|---|---|---|---|---|---|---|---|---|
Reference | 1.11 | 1.35 | 1.21 | 1.11 | 1.07 | 0.96 | 1.21 | 1.15 | 256.00 | 855.00 | [-] |
UDVFs | 2.22 | 0.68 | 1.68 | 1.49 | 0.54 | 0.48 | 1.04 | 0.77 | 241.39 | 783.14 | 14.6 |
SBVFs | 1.17 | 1.41 | 1.24 | 1.15 | 1.11 | 1.04 | 1.26 | 1.20 | 264.74 | 934.26 | 1.6 |
In both cases, there is a strong sensitivity to the shearing components (α_{7},α_{8}) and a moderate one for α_{4}. When the UDVFs were used, there was very little sensitivity to the remaining parameters, except for the cross-correlation between α_{7} and three other parameters: α_{3},α_{5} and α_{8}. In contrast, for the SBVFs, many more parameters were active. There was a cross-correlations between α_{7} and all other parameters, as well as small to moderate sensitivities for all α parameters, showing that most of the material parameters were active during the test and the SBVFs were capable of balancing their contribution.
Computational efficiency
One of the important practical aspects of the type of methodology presented here concerns computational efficiency. Information on computing times are often left out in publications on this topic and therefore, it is difficult to compare computational efficiency with competing techniques motivating our decision to report this information.
In the case of Hill48 model, the identification took approximately six hours for linear hardening and between eight and ten hours in the case of power law hardening, due to the additional unknown parameter. The SBVFs are not significantly slower compared to the UDVFs, and the difference is mostly due to the additional stress reconstructions needed to calculate the incremental stress sensitivities. The number of these reconstructions can be controlled by the number of times the SBVFs are updated, and so the difference could be brought down even more. It should also be emphasized that these computing times would be significantly reduced using a compiled programming language instead of Matlab.
In the case of the Yld2000-2D model the identification times were much longer, mostly due to very slow stress reconstruction procedure, which on average took ten minutes for the chosen data density. Additionally, as 10 unknown parameters were sought, the calculations of the gradients in the minimisation problem call for many stress reconstructions, further increasing the disparity between Hill48 and Yld2000-2D. As a comparison, the FEM model used to generate the data, took 4 hours to complete.
Although the running times were very high, there are many ways in which they could be improved. First, and as stated before, efficient implementation in a fast language would lead to a significant improvement. Second, replacing the implicit stress reconstruction algorithm with a direct method such as the one used in [39] would make the stress reconstruction faster, especially in the case of Yld2000-2D for which computing derivatives for the implicit scheme is a very computationally intensive process. The direct method is valid only for large deformation data (with plastic flow well established), however it could be coupled with the implicit algorithm for elasto-plastic transition. It was found that when the SQP algorithm was replaced with the Levenberg-Marquardt algorithm, the computations were up to 10 times faster, indicating that choosing a proper tool for minimising the cost function is crucial. For instance, the identification of the Yld2000-2D model using the SBVFs was reduced from 278 hours with SQP to merely 26 with Levenberg-Marquardt. The major advantage of the Levenberg-Marquardt algorithm is that it requires significantly fewer iteration to find a minimum, compared to the SQP algorithm. In the case of Hill48 identifications the former converged in approximately 8 iterations, compared to 130 of the latter. However, the efficiency of the Levenberg-Marquardt algorithm heavily relies on the initial guess which must be close to the solution to obtain fast convergence. This is not required by SQP which can find the solution from any starting point.
Choosing an optimal data density remains an open problem. In this work, about 60,000 spatial data points were used for the identification problem. However, this could potentially be reduced. For DIC measurements, the number of data points could be effectively controlled by means of the stepsize. Additionally, the number of load steps taken into consideration could be varied. It is worth noting that if temporal smoothing is performed on the entire collected data, even the time steps not used explicitly in the identification affect the outcome, as the information is passed through the temporal filter. The limiting factor for the temporal resolution is related to the stress reconstruction algorithm, i.e. in the case of implicit algorithms the larger the strain increments, the larger the reconstruction error. More studies are needed on the optimal number of data points for accurate reconstruction of material parameters. To define optimal spatial and temporal sampling leading to acceptable systematic and random errors, the impact of DIC parameters, and smoothing would need to be assessed with a synthetic image deformation procedure as in [37].
Conclusions and future work
In this work we extended the sensitivity-based virtual fields, originally proposed in [28] to large deformation anisotropic plasticity, which is the main novelty of this contribution. We tested the performance of the fields using a deep-notched tensile test, already used in that context by Rossi et al. [39], and found that the proposed virtual fields can be successfully applied to the problem.
It was found that for the Hill48 model both UDVFs and SBVFs were capable of correctly identifying the parameters from a single test. It must be noted that the UDVFs were developed using a trial-and-error approach by the lead author of [39]. In that context, the systematic procedure of SBVFs seems to be especially appealing as it removes the need of an informed input by the user to arrive at correct parameters. This is especially important with even more complex models such as Yld2000-2D where deriving appropriate UDVFs is a challenging task.
The main advantage of the new fields comes from the automatic generation procedure, with limited input from the investigator, involved only in setting virtual mesh density and scaling parameters. It was found before that these parameters have very limited effect on the overall identification and can be chosen a priori without a significant impact on the parameter values [28]. As a result, high quality virtual fields are generated for any material model, regardless of the test configuration. This opens up possible implementation in a user-friendly VFM software for non-linear model identification, like the MatchID DIC/VFM platform [29]. We demonstrated the effectiveness of the method on both Hill48 and Yld2000-2D, with the latter especially successful in comparison to the standard approach.
The identifiability of Hill48 model from a single test has been already established before [13, 39]. Interestingly, the results suggest that a single test performed at 45^{∘} may contain enough information to characterise the Yld2000-2D criterion as well. In fact, if confirmed experimentally, it would give an exciting alternative to the standard test protocol involving three uniaxial tests and one biaxial, significantly reducing the experimental effort to characterise a material. This would be possible due to the ability of the SBVFs to identify in space and time when each parameter is active and focus exclusively on those regions when identifying the parameters.
The method is currently being validated experimentally on an automotive DC04 steel alloy. The results from both UDVFs and SBVFs will be compared to the parameters identified with the standard multi-test protocol to confirm whether the Yld2000 criterion can indeed be identified from a single heterogeneous test, which would be a significant step forward to reduce identification costs and time scales.
Although a relatively simple material model was employed in this work, the presented method is general and can be used with any constitutive model. There are no limitations on the complexity of material model used within the VFM framework, given that it reconstructs stress field from measured kinematic fields (deformation gradient) and some internal state variables (that can be resolved by considering history of deformation). It must be stressed that usually the more complex the model, the more material parameters must be identified experimentally. The method proposed here allows for complete identification of material parameters given the experiment contains sufficient information. To the authors’ best knowledge there is no systematic way of assessing the level of information that a test contains given a constitutive model. Investigating this in future would be certainly of importance for designing better experiments.
The new route to virtual field selection demonstrated here has potential in many applications, in particular for non-linear models with large numbers of parameters. An obvious extension would be hyper-visco-elastic models. The VFM has been applied to such materials in the past, see [17] for instance, but always with UDVFs, limiting the complexity of the considered models. Another area of interest concerns transient dynamic tests to identify the high strain rate elasto-plastic response of materials. Finally, now that a systematic route has been clearly identified to generated virtual fields automatically for non-linear laws, the problem of test optimization can be addressed. This has been studied for linear elasticity in the past, thanks to the availability of the noise-optimized virtual fields from [3], and can now be addressed for non-linear models using a procedure similar to that in [16].
Footnotes
- 1.
‘virtual strain’ is used here inaptly, to relate to the application of the PVW in small deformation framework. In fact, here, we should say ‘virtual displacement derivatives’ or ‘virtual displacement gradients’ but ‘virtual strain’ is more compact and convenient.
Notes
Acknowledgements
Dr. Frances Davis and Prof. Fabrice Pierron acknowledge support from Engineering and Physical Sciences Research Council (EPSRC) through grant EP/L026910/1. Prof. Fabrice Pierron also expresses gratitude to the Wolfson Foundation for support through a Royal Society Wolfson Research Merit Award. Mr Aleksander Marek acknowledges funding from EPSRC through a Doctoral Training Grant studentship. The authors also acknowledge the use of the IRIDIS High Performance Computing Facility, and associated support services at the University of Southampton, in the completion of this work. Dr. Davis also acknowledges support from the Leverhulme Early Career Fellowship.
Compliance with Ethical Standards
Conflict of interests
Prof. Pierron is a co-founder of the MatchID company
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