Extension of the sensitivity-based virtual fields to large deformation anisotropic plasticity

Let us consider a body \(\mathscr{B}\), where the position of particles in the reference configuration is given by X and in the deformed one by x. The motion of each material point can be described by a function x = ϕ(X,t), which maps the position of every particle in the reference configuration to the current deformed configuration. The displacement field is defined as the difference between the current and the reference positions: The deformation gradient is defined as: where I is the second order identity tensor. Using polar decomposition, the deformation gradient can be written as the product of two second order tensors: where V is the left stretch tensor and R is the rotation tensor. The left stretch tensor can be conveniently calculated as: where the root operator refers to the root of a matrix. A consequence of such mathematical description is that for every point, a local coordinate system rotates during deformation, as outlined in Fig. 1. This is an important feature to consider when the body includes a texture, as its orientation will follow any local rotations.

A convenient measure of strain, called Hencky strain, can be constructed from the left stretch tensor: This strain measure can be used to formulate constitutive laws within the finite deformation framework. For further details on continuum mechanics the reader is referred to [12].

Quasi-static equilibrium can be expressed in so-called ‘weak form’ in which it is enforced as a weighted average over the entire domain, expressed here in the current configuration \(\mathscr{B}_{t}\), in absence of volume forces: where \(\partial \mathscr{B}_{t}\) is the boundary of \(\mathscr{B}_{t}\), n is the outwards vector of \(\partial \mathscr{B}_{t}\) and σ is the Cauchy stress tensor.

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.

The PVW can be alternatively formulated in the reference configuration. In that case, another stress tensor is defined, called the first Piola-Kirchhoff stress tensor P: Notably, the first Piola-Kirchhoff stress tensor is asymmetric due to the asymmetry of the deformation gradient.

As a result, in the reference configuration, the PVW can be expressed as: where \(\mathscr{B}_{0}\) is the considered body in the reference configuration and \(\partial \mathscr{B}_{0}\) its boundary. This form is much more suitable for practical implementation in case of the proposed method, as the virtual fields U^∗, defined in the reference configuration \(\mathscr{B}_{0}\), do not need updated virtual boundary conditions, as will become apparent later in the article. This approach has been used by most of the VFM community [36, 38, 39, 41]. Noticeably, it was demonstrated that the current configuration formulation could be successfully applied to the case of hyperelasticity as well [2, 23, 32].

The VFM uses the PVW to identify material parameters from kinematic data and loading. Generally, kinematic fields are measured by means of full-field techniques such as DIC over the entire domain. This data is then used to reconstruct the stress field using a set of material parameters, denoted here as χ. The calculated stresses must be in equilibrium with the measured loading, which is enforced through either Eq. 6 or 8. As the correct values of the constitutive parameters are unknown at the start of the process, the stress field is first estimated with a guessed set of parameters and the equilibrium can be checked by means of Eq. 8:

The material parameters are found through an iterative minimisation of the cost function, i.e. the correct material parameters produce a stress field that minimises the gap in the PVW. Since the full-field measurements provide spatially dense data, the integral of the stresses can be approximated by a discrete sum. Additionally, multiple load levels (time steps) have to be included to involve all the parameters of the constitutive model. Multiple independent virtual fields can be used to involve data in the cost function in various ways, which generally leads to better conditioning of the cost function, resulting in more robust minimisation and more accurate identification. Finally, a general form of the cost function can be expressed as: where W_int is the virtual work of internal forces and W_ext is the virtual work of external forces, S^j is the surface area of j-th point and h is the thickness of specimen.

Since in most of cases the measurements are only performed at the surface of specimen, some assumption about the through-thickness distribution of the mechanical fields must be made. Often, plane stress is assumed, provided the thickness of specimen is small in comparison to the other two dimensions and the loading is in plane. The measurements provide in-plane kinematic quantities and the out-of-plane stresses are considered negligible (σ₁₃ = σ₂₃ = σ₃₃ = 0). However, when the PVW is formulated in the reference configuration (8), the 2D Cauchy stress must be pulled-back from \(\mathscr{B}_{t}\) to \(\mathscr{B}_{0}\), as shown in Eq. 7. In reality, the gradient of deformation is fully 3D and so some additional assumptions must be made. Assuming that for a thin specimen the out-of-plane shearing is negligible (F₁₃ = F₂₃ = F₃₁ = F₃₂ = 0), the Jacobian (det(F)) can be now expressed as: The in-plane values are measured, however, the out-of-plane term is still unknown. It can be approximated by calculating the out-of-plane strain during stress reconstruction using the elasto-plastic constitutive law: Here, the Hooke’s law was assumed for the elastic part and the isochoric flow for the plastic part of the strain increments. This strain can be then used to calculate the missing component: It should be noted that if the isochoric flow assumption becomes questionable, back-to-back camera systems can be used to determine an average value of ε₃₃, as shown in [14]. Finally, with known Jacobian it is possible to pull-back the Cauchy stress to the reference configuration.

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 main idea behind the sensitivity-based virtual fields is to find areas during the test, both in space and time, where the information about each parameter is contained. A separate virtual field is constructed automatically for each constitutive parameter which allows the cost function to represent each parameter with maximised sensitivity. In order to locate areas where the information is encoded for the i-th parameter, an incremental stress sensitivity is calculated. By perturbing the value of the i-th parameter during the stress reconstruction for a given set of current parameters, a change in stress field is noted, highlighting areas where the parameter is active in influencing the stress. This information is encoded in a map called stress sensitivity: where δχ_i is a perturbation of the i-th parameter, typically − 0.2χ_i ≤ δχ_i ≤− 0.1χ_i. The negative sign is taken to expand the VFs over points currently not active, as opposed to penalizing points that just became active (e.g. in the case of yield stresses). Furthermore, incremental stress sensitivity maps are found as: These sensitivity maps, either incremental (15) or total (14) can be used as virtual strains to provide relevant weight to the stresses in the PVW equation. This incremental approach was proposed in order to effectively decouple the influence of yielding-related parameters from hardening parameters which act at different time scales.

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.

An effective way to solve the matching problem is to construct a virtual mesh, which consists of virtual nodes connected through virtual elements. The virtual mesh defines the virtual displacements at the nodes and interpolate them within an element using classical FE shape functions. The nodes are connected with 4-node quadrilateral elements and the interpolation is done using standard bi-linear shape functions (N). The virtual fields can be calculated using the values of virtual displacements at each of the node and the spatial derivatives of the shape functions. For every point of measurement, the virtual strain fields can be found as a function of virtual displacements of the neighbouring nodes. This function is linear with respect to the displacements and can be expressed using the strain-displacement matrix B_el, which relates virtual displacements at the nodes to the virtual strains at the considered data point: where U^∗ is a vector containing the virtual displacements of the element nodes and \(\frac {\partial \mathbf {U}^{*}}{\partial \mathbf {X}}\) is a vector containing values of virtual strains¹ at the data point. It is chosen here to express the virtual fields in the reference configuration, as a results B_el does not change with deformation. It can be expressed as:

For every point, matrix B_el can be generated and then assembled into a global strain-displacement matrix (B_glob) which relates virtual displacements from all virtual nodes to the virtual strains at all data points. The global strain-displacement matrix has then to be modified to account for virtual boundary conditions. Often, it is necessary to constrain the virtual displacement in the direction of applied loading to be constant across the loading edge, so that only the resultant load appears in the PVW equation and not its (unknown) distribution. This simplifies the traction contribution to the PVW (6, 8) to become: where F^load is the total force applied. Imposing these constraints into B_glob, a modified global strain-displacement matrix is found, \(\bar {\mathbf {B}}_{glob}\). In order to solve the least-square problem, a pseudo-inverse of this matrix is found which can be used to generate the corresponding virtual displacements: These virtual displacements now produce virtual strain fields that ‘look like’ the incremental stress sensitivity maps and obey the necessary virtual boundary conditions. The virtual strain fields are finally found with the following formula:

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.

The test proposed by Rossi et al. involved tensile loading applied to a flat coupon with two circular notches, which introduce heterogeneous deformation [39]. It was simulated in Abaqus (v. 6.13) to generate synthetic data which were then used to test the identification algorithm. The geometry of the specimen is presented in Fig. 2. The mesh density was chosen according to a convergence study. The thickness of the plane stress elements was chosen as 0.74 mm, similar to that used in [38] and typical of thin anisotropic metal sheets for the automotive industry. The bottom edge was fixed, while a constant vertical displacement of 6.75 mm was applied across the top edge, which was additionally fixed in the lateral direction to simulate the effect of the grip of a test machine. The initial material orientation was defined by specifying the angle (𝜃) between the rolling direction and the horizontal axis of the model, indicating the principal material axes (Ξ,H) needed to describe the anisotropic properties (see Fig. 2). Overall, total vertical strains of about 40% were obtained with the model.

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.

The elastic response was modelled with Hooke’s law extended to finite deformation: where Δσ is the rate of change of the Cauchy stress, D is the elastic operator for plane stress and \({\Delta }{\boldsymbol {\varepsilon }}^{e}_{L}\) is an increment of the elastic part of the Hencky strain. Here, we assume that both Young’s modulus E, and Poisson’s ratio ν needed to construct D are known. They can be identified from full-field data using the noise-optimised virtual fields in elasticity [4], considering only the initial elastic steps.

In order to ensure the objectivity of the model, a corotational frame was adopted. The reference frame at each material point rotates with the material. Fig. 1 presents the corotational and material frames in both reference and current configurations. Initially, the corotational frame (i,j) is co-linear with the global frame. The angle between the material coordinate system (Ξ,H) and the global reference frame is known. This angle can be used to construct a tensor (R_mat) rotating the local coordinate system to the material one. Due to deformation, the corotational frame rotates with R and in the current configuration is denoted (1,2). The angle between the current material coordinate system (ξ,η) and the corotational frame is fixed and related to R_mat. The strain increment in the current material frame (ξ,η) can be expressed in terms of the strain increment in the global frame (i,j): More complex strategies could be employed to account for the evolution of material texture such as that presented in [31]. They would have to be implemented in the constitutive routine used to reconstruct stress (Fig. 4). Consequently, an inverse rotation could be performed in order to represent the reconstructed stress tensor in the global coordinate system, in which the PVW is expressed.

The strain increments in the global coordinate system are simply calculated as the difference between two consecutive total strains:

For each of the two plastic models investigated, Hill48 and Yld2000-2D, an associated flow rule was assumed, as well as an additive decomposition of strain increments: where \({\Delta }{\boldsymbol {\varepsilon }}_{L}^{e}\) is the elastic strain increment and \({\Delta }{\boldsymbol {\varepsilon }}_{L}^{p}\) is the plastic strain increment.

A yield criterion, in general, can be written as: where σ_eq is an equivalent stress formulated differently for each plasticity model and σ_y is a yield stress evolving according to a hardening law. A shared feature of both models is that they are formulated in terms of Cauchy stress expressed in the current material frame (ξ,η in Fig. 1).

Hill48 is a popular model for anisotropic plasticity [19]. When plane stress is assumed the equivalent stress can be expressed as: This model depends on 4 independent parameters defining the anisotropy. A suitable way to express the governing parameters is to relate them to mechanical quantities measured in an experiment. In this paper the plastic potentials (R_ij) are used to define the criterion: where \(R_{ij} = \frac {\sigma _{ij}^{y}}{\sigma _{0}}\). Note that \(\sigma _{11}^{y}\) and \(\sigma _{22}^{y}\) are the yield stresses identified in planar uniaxial tests conducted at 0^∘ and 90^∘ respectively and \(\sigma _{33}^{y}\) is the through-thickness yield stress. Finally, \(\sigma _{12}^{y}\) is the yield stress identified under pure shear. Furthermore, it was assumed that the yield stress in the hardening law is equal to \({\sigma }^{y}_{11}\), i.e. \(\sigma _{0} = {\sigma }^{y}_{11}\), as this reduces the number of variables to be identified by one, and does not affect the formulation of model.

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].

Yld2000-2D [8] was developed strictly for plane stress conditions for which the equivalent stress can be calculated as: where a is an exponent based on the metal micro-structure (a = 8 for FCC and a = 6 for BCC) and \({X}_{1}^{\prime }\), \({X}_{2}^{\prime }\) and \({X}_{1}^{\prime \prime }\), \({X}_{2}^{\prime \prime }\) are principal values of two stress tensors X^′, X^″ which are defined as linear combinations of the Cauchy stress: Matrices L^′ and L^″ are given by:

The model involves 8 independent parameters, α₁–α₈, and can accurately represent the behaviour both in simple tension as well as biaxial loading.

Two different hardening laws were adopted here: linear (32) and the power law hardening (33): Linear hardening is defined with two parameters: initial yield stress σ₀ and hardening modulus H. The non-linear power law includes 3 parameters: initial yield stress σ₀, hardening modulus H, and an exponent n. The equivalent plastic strain \(\bar {\varepsilon }^{p}\) is integrated over the history of deformation by means of summing the equivalent plastic strain increments obtained at each increment:

The reference parameters used to generate the models are presented in Tables 1 and 2.

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.

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.

The displacements were then corrupted with a Gaussian white noise, with standard deviation of 0.3 μ m, representative of a real experiment. For each data point, displacements were smoothed using spatial and temporal filters, as typically done for experimental data. Temporal smoothing was performed using the Savitzky-Golay method with a polynomial order m_temp and a window size of w_temp. This was complemented with a spatial smoothing using a Gaussian filter of standard deviation σ_spat, with the window size adjusted to be the maximum odd number smaller than 3 × σ_spat × 2. While temporal smoothing was performed on all 400 time steps obtained from Abaqus, only some of the temporal data points were passed to the identification procedure. The main reason for that was to decrease the computational effort, but also this increased the size of the strain increments mitigating the impact of strain noise on the error in the stress reconstruction. The time steps used for the identification are graphically presented on the global force-displacement curves in Fig. 3.

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 kinematic data was then used as an input to an in-house Matlab code implementation of the VFM. The stresses were calculated using the deformation data and an initial guess for the material parameters. The SBVFs were calculated during the stress reconstruction process which allowed the values of the cost function to be calculated with Eq. 10. The material parameters were refined iteratively by means of the Matlab function fmincon minimising (10) with a sequential quadratic programming algorithm (SQP). Starting points were selected randomly from the interval between 50% and 200% of the reference values using a random number generator. Two starting points were used for each data set in order to ensure that the identified parameters corresponded the the global minimum. The identification algorithm is summarized in Fig. 4 in the form of a flowchart. Finally, the identified parameters were compared against the reference values to quantify the accuracy of procedure.

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).

This works aims at extending the SBVFs to large deformation anisotropic plasticity. As mentioned earlier, the specimen design was previously proposed by Rossi et al. [39] who employed UDVFs to identify Hill48 parameters using a single test, as well as Yld2000-2D parameters using a combination of three tests. In this work, their user-defined virtual fields were used as a benchmark to test the suitability of the SBVFs. They showed that suitable VFs for the test consisted of a combination of three virtual fields, defined with the following virtual displacements: The virtual displacements are defined in the coordinate system presented in Fig. 5. The virtual fields are constructed in a way to include all stress components in the cost function.

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.

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.

Because of the rather large number of parameters and their individual lack of physical meaning, a direct comparison between reference and identified parameters on a one to one basis is not always relevant to draw meaningful conclusions. The apparent uniaxial yield stress \(\hat {\sigma }\) as a function of material orientation was found to provide a convenient and physical-based way of evaluating identification errors. Assuming a yield criterion of the form of Eq. 25 and a uniaxial state of stress at angle 𝜃 with respect to the material coordinate system, the stress state in the material coordinate system can be expressed as: This stress can be introduced into the expression of σ_eq and the criterion can now be algebraically transformed into: Using Eq. 39, it is possible to reconstruct the variation of the yield stress with angle 𝜃 and equivalent plastic strain \(\bar {\varepsilon }^{p}\). Such map can be calculated for both the reference and identified parameters, so that they can be meaningfully compared. The error can be quantified by looking at the map of relative error: The map can be used to investigate whether the error is associated with the variation of yield stress with material orientation and/or hardening. Alternatively, the mean of the error map, \(\bar {r}\), gives a single numerical value for the accuracy of the identification, later referred to as the global error: This approach enables the reference and identified sets of parameters to be compared in a more meaningful way as opposed to parameter by parameter and furthermore, it also allows for meaningful comparisons across different material models.

As a first validation of the methodology, the six raw FE data sets generated for the Hill48 model were used. Since the data did not contain any noise, no smoothing was used. The following parameters were selected for generation of the sensitivity-based virtual fields: δχ_i = − 0.15χ_i, virtual mesh of 14 × 14 elements, and the scaling parameters based on the mean of top 30^th percentile of the internal virtual work terms. As shown before, the choice of the parameters has a limited impact on the identification and mostly affects the random portion of the error [28]. The results of the identifications are presented in Table 3.

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 σ₁₁ or σ₂₂ 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.

Scaling of UDVFs is of crucial importance. Without introducing the scaling discussed in Section “User-defined virtual fields”, the UDVFs lead to large identification errors. In particular, the yield stress is correctly identified only near the material orientation angle, while the remaining orientations are characterised incorrectly, as shown in Fig. 6a and b. The reason for this is most likely due to the first set of UDVFs (35) being overrepresented in the cost function. As a result, the vertical component of the stress field represented in the global frame is the major source of information, leading to insensitivity of the cost function to the other two stress components. The identification can be improved by introducing the scaling which restores the balance between the three virtual fields. In that case the anisotropy is recognized much better, as shown in the difference between Fig. 6b and c.

Having validated the methodology, the effect of noise on the identified parameters was studied. Since in practice, spatial and temporal smoothing are always implemented, we also implemented this here to simulate the experimental process more realistically. Without any smoothing, the difference between UDVFs and SBVFs was even larger, but this was thought to be a somewhat unfair comparison. First, a sensitivity study was conducted to determine the optimal smoothing parameters. The study was performed using fixed settings, i.e. sensitivity-based virtual fields, linear hardening and the same level of noise in each run. In total, 30 copies of noise were processed. The spatial smoothing parameters were chosen under fixed temporal smoothing, likewise the temporal parameters were chosen under fixed spatial smoothing. As presented in Table 4, the smoothing has a limited effect on the magnitude of both systematic and random errors (global error value and its spread respectively), which suggests that the choice of parameters is not critical for the identification here. Generally, the stronger the smoothing, the smaller the random portion of error, at a cost of increased systematic error. However, it was found previously that significant noise can cause spurious elastic unloadings, which strongly affect both errors [26]. As a result, some smoothing settings can improve both errors at the same time. In order to balance systematic and random errors, spatial smoothing with a window of 9 was selected, combined with 11 points temporal smoothing using 3^rd order polynomial.

The selected smoothing parameters were used to study the effect of the virtual fields on the systematic and random errors. The identification was run on 30 copies of noise, as this gives enough statistical representation to establish the random part of the error. The results are presented in Tables 5 and 6 for the linear and the power law respectively.

In the case of the Yld2000-2D model, the main goal was to explore the comparative robustness of the UD and SB VFs when the model is richer in parameters to be identified. Only exact data from the 45^∘ test were used, as it is already a good example of the UDVFs underperforming. The identified parameters are presented in Table 7. It is worth noting, that in the case of Yld2000-2D, the anisotropic parameters do not have any obvious physical meaning, thus comparing them on a parameter to parameter basis is not so relevant to draw conclusions as discussed in Section “Error quantification”. The distribution of initial and final yield stresses are presented in Fig. 10 and the identified yield loci are shown in Fig. 11. The superiority of the SBVFs is spectacular. The distribution of anisotropy is very well identified and the main source of error comes from the hardening modulus, which is consistent with the trend observed for Hill48. In the case of the UDVFs, the identification was highly inaccurate. The parameters found were significantly different from the reference, and four out of eight reached identification constraints (set to 50% and 200% of the reference). Most likely, this is due to the generic nature of the fields. Yld2000-2D is a more complex model in which each parameter has a small effect on the overall yield surface, thus extracting data correctly challenges the virtual fields much more, compared to the case of Hill48, for which the used UDVFs were specifically developed. This can be observed by looking into Hessians, computed with Eq. 42 and presented graphically in Fig. 12.

In both cases, there is a strong sensitivity to the shearing components (α₇,α₈) and a moderate one for α₄. When the UDVFs were used, there was very little sensitivity to the remaining parameters, except for the cross-correlation between α₇ and three other parameters: α₃,α₅ and α₈. In contrast, for the SBVFs, many more parameters were active. There was a cross-correlations between α₇ 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.

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.

The identification was run on a standard PC with an Intel Core i5 processor (3.20GHz) and 24 GB of RAM memory. The stress reconstruction routines were coded in Matlab, however they were automatically translated to C language and called as mex file using Matlab coder tool. This procedure improved the efficiency of the stress reconstruction routine, which is the most computationally demanding process. The approximate running times of identification for Hill48 model can be found in Fig. 13. In case of Yld2000-2D the time needed to obtain the results were 122 and 278 hours respectively for the UDVFs and SBVFs.

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].