Three-Dimensional Virtual Fields Method Accurately Identifies Sheet Metal Hardening Parameters

A computational study led by researchers Miguel G. Oliveira, Sandrine Thuillier, and António Andrade-Campos has demonstrated that a fully three-dimensional formulation of the virtual fields method can accurately identify post-necking strain hardening parameters in sheet metal tension tests where conventional two-dimensional approaches fail. Published in Advanced Modeling and Simulation in Engineering Sciences, the research evaluates the calibration of plastic material behavior beyond the point of maximum uniform elongation up to rupture using synthetic data generated from detailed finite element simulations. The elastoplastic behavior of sheet metals is conventionally characterized through standard uniaxial tensile tests to extract the true stress-strain curve required for numerical simulations in manufacturing processes. However, standard extensometers underestimate resulting strains after the onset of necking because deformation concentrates non-uniformly within a localized zone, while a complex triaxial stress state develops. Although inverse identification methodologies based on surface measurements have been applied widely, the researchers established that the standard two-dimensional plane stress assumption compromises kinematic validity after necking, producing non-negligible errors that correlate directly with specimen thickness.

Standard sheet metal characterization methods typically struggle once diffuse necking initiates after the point of maximum uniform elongation, leading to localized thinning and width contraction that cannot be captured accurately through classical extensometry. Historical post-necking corrections, such as the analytical formulation developed by Bridgman for circular cross-section specimens, require extensive experimental effort and cannot be adapted to thin sheets with rectangular cross-sections. Subsequent approaches proposed by researchers like Zhang and Mirone introduced area-reduction relationships or material-independent formulations to reduce experimental effort, but challenges remain in capturing non-uniform full-field states. While finite element model updating methodologies allow the use of full-field surface measurements, building numerical models that accurately predict the onset and exact spatial location of necking without severe element distortion requires substantial computational time. In contrast, energy-based methods and the linear stress-strain curve identification method bypass repeated numerical simulations by evaluating work balances directly from measured surface strain fields, yet they remain bound by two-dimensional kinematic and stress assumptions that disregard through-thickness stress components.

The virtual fields method provides an inverse identification framework grounded in the principle of virtual work, balancing internal virtual work against external virtual work across continuous and kinematically admissible virtual displacement fields. Expressed in the undeformed Lagrangian configuration, the principle of virtual work eliminates the need to update virtual fields during large deformations, as the initial volume of each material point remains constant throughout the test. In two-dimensional formulations of the virtual fields method, out-of-plane stress components are assumed to be zero under plane stress conditions, while out-of-plane normal strain is approximated using elastic Hooke’s law and the isochoric plastic flow rule. However, when diffuse and localized necking occur, a severe triaxial stress state develops in the specimen interior, violating the plane stress assumption. Because experimental measurement techniques cannot currently capture full volumetric displacements inside opaque sheet metals, the authors investigated whether extending the virtual fields method to a fully three-dimensional framework resolves post-necking discrepancies across varying sample thicknesses.

To evaluate the intrinsic mechanics of the two formulations in a controlled environment, the investigators constructed three-dimensional finite element simulations of a tensile specimen featuring a smoothly varying cross-section using Abaqus Standard. This specimen geometry localizes deformation within a central volume of interest while generating a heterogeneous strain field over an extended area, providing rich mechanical data without requiring artificial geometry perturbations. The virtual material was modeled using elastic isotropic properties governed by Hooke’s law and a von Mises yield criterion combined with a Swift isotropic hardening law, calibrated against experimental data from a DP780 dual-phase steel sheet with a nominal thickness of 1.5 millimeters. Across all simulated tests, a constant vertical displacement of 5 millimeters was imposed on the top edge, while the bottom edge remained fully fixed. The numerical model was discretized with 197,955 eight-node brick elements with reduced integration, with three element layers defined along the thickness direction to achieve a balance between through-thickness strain resolution and computational expense.

To rigorously investigate the influence of geometry on the inverse identification, the researchers simulated eight sample thicknesses ranging from 1.5 to 10 millimeters, specifically evaluating values of 1.5, 2.0, 2.5, 3.0, 4.0, 6.0, 8.0, and 10.0 millimeters. Because the imposed longitudinal displacement remained constant at 5 millimeters across all models, necking occurred at different stages of the tests depending on specimen geometry. For the 1.5-millimeter sample, severe localized necking developed toward the end of loading, reaching a maximum equivalent plastic strain comparable to the rupture strain of 0.65 recorded experimentally for DP780 steel. In contrast, thicker specimens exhibited delayed necking, with the 8-millimeter and 10-millimeter samples failing to reach localized necking within the 5-millimeter displacement range.

The numerical implementation of the virtual fields method was programmed in Python using array-based operations to streamline matrix calculations and approach the execution speeds of compiled Fortran code. Stress reconstruction from the logarithmic strain tensor was executed using the Unified Material Modeling Driver for Plasticity, which integrated the constitutive equations through an implicit backward-Euler scheme with an elastic predictor and plastic corrector. Parameter optimization was driven by the gradient-free Nelder-Mead simplex algorithm available in the SciPy library, selected for its ability to execute custom callback functions and manage parameter updates at each iteration. For the three-dimensional virtual fields method, displacement fields were extracted directly from the interior and boundary nodes of the volume of interest. For the two-dimensional formulation, in-plane displacements were extracted exclusively from surface nodes, and connectivity was restructured into a surface mesh that discarded all interior volumetric nodes, reducing the total number of evaluated material points by two-thirds.

To formulate the virtual work balance, the researchers defined three kinematically admissible virtual displacement fields. The first virtual field featured an in-plane displacement linear along the tensile loading axis and independent of transverse coordinates, producing a constant virtual strain gradient across the volume of interest and yielding non-zero external virtual work proportional to the applied tensile load. The second virtual field incorporated non-constant spatial distributions that vanished along the loaded boundary, isolating internal virtual work and focusing exclusively on in-plane stress components while maintaining a uniform distribution across the thickness. The third virtual field represented a novel three-dimensional extension that varied along the thickness coordinate, assigning non-zero weights to all nine components of the virtual displacement gradient tensor and coupling in-plane and out-of-plane stresses. By evaluating different combinations of these virtual fields, the researchers examined how specific spatial weighting affected the stability and parameter sensitivity of the least-squares objective function.

In the first phase of results, the authors analyzed the accuracy of the predicted load by computing the internal virtual work using reference material parameters and comparing it against the reference load recorded in the numerical simulations. Across all eight sample thicknesses, the three-dimensional virtual fields method achieved exceptional fidelity, keeping the maximum relative error below 0.03 percent, with error values tending to decrease slightly as thickness increased. Conversely, the two-dimensional formulation produced non-negligible discrepancies that grew substantially with increasing specimen thickness. In the 1.5-millimeter sample, the load prediction error remained near zero during uniform deformation but rose sharply to approximately 1.0 percent once localized necking initiated at 4 millimeters of displacement. For the 10-millimeter specimen, the relative error of the two-dimensional virtual fields method reached 6.0 percent at the final load increment, demonstrating that out-of-plane stress development causes progressive degradation of two-dimensional models.

Further investigation of the stress states revealed that the failure of the two-dimensional method stems directly from the rapid growth of out-of-plane normal stress components within the necked region. Comparing a material point located at the center of the specimen front surface against a corresponding point on the mid-thickness plane demonstrated that the transverse out-of-plane stress on the mid-plane was double the magnitude observed at the surface. Because the two-dimensional formulation assumes plane stress across the entire cross-section and relies solely on surface kinematic measurements, it cannot account for through-thickness stress gradients or triaxial stress states, leading directly to deviations between internal virtual work and external reaction forces.

When applied to inverse parameter identification of the Swift hardening law, starting from the known reference values, the three-dimensional virtual fields method converged precisely to the reference parameters across all tested thicknesses. In contrast, the two-dimensional virtual fields method diverged significantly from the true parameters for the 3-millimeter and 10-millimeter samples, yielding an initial yield stress parameter for the 10-millimeter specimen that was physically unrealistic. When starting from initial guesses 1.5 times greater than the reference values, the three-dimensional method successfully identified the strain hardening exponent exactly and recovered strength coefficients and yield stresses within 1.0 percent of reference values across all virtual field combinations. The authors emphasized that while these simulated tests confirm the necessity of a fully three-dimensional framework when calibrating materials subjected to severe necking, experimental implementation will require emerging volumetric measurement techniques to capture internal displacement fields directly.

Subject of Research: Inverse identification of sheet metal hardening behavior using two-dimensional and three-dimensional virtual fields methods

Article Title: Inverse identification of the hardening behavior with a fully 3D virtual fields method

Article References: Oliveira, M. G., Thuillier, S., & Andrade-Campos, A. (2025). Inverse identification of the hardening behavior with a fully 3D virtual fields method. Advanced Modeling and Simulation in Engineering Sciences, 12(1), Article 10. https://doi.org/10.1186/s40323-025-00293-7

Image Credits: AI Generated

DOI: 10.1186/s40323-025-00293-7

Keywords: virtual fields method, sheet metal forming, strain hardening, inverse identification, Swift hardening law, plasticity, diffuse necking, localized necking, finite element analysis, constitutive modeling, triaxial stress state, full-field measurement, DP780 dual-phase steel, mechanical characterization, plane stress, Nelder-Mead optimization, large deformation