The work, carried out by Slavomír Parma, Tereza Vaňková, René Marek, Heidi P. Feigenbaum, Dušan Gabriel, Jan Štefan, Matouš Uhlík, Jan Džugan, Jiří Plešek and Yannis F. Dafalias, assembles a plasticity model from established constitutive building blocks and adds new features. The yield behavior is described by the classical quadratic Hill yield function, extended with internal variables for hardening, which captures anisotropy in the yield strength of 3D-printed metals. On top of that, the authors implement an advanced multicomponent kinematic hardening rule known as MAFTr, developed by Dafalias and Feigenbaum, together with a multicomponent isotropic hardening rule after Chaboche. The kinematic hardening rule was originally designed for cyclic plastic loading, and previous work has shown it can fit cyclic loading experiments for additively manufactured metals.

The MAFTr rule has a specific structure. The backstress, an internal variable that tracks the translation of the yield surface during plastic flow, is built from four independent components. The first three follow Armstrong–Frederick-type evolution equations, while the fourth contains a threshold parameter that delays dynamic recovery: as long as the norm of that component stays within the threshold, it obeys Prager’s linear hardening rule, and once it exceeds the threshold it behaves as an Armstrong–Frederick-type rule. The “r modification” in the rule’s name refers to a constitutive feature shown to improve predictions of multiaxial ratcheting, the progressive accumulation of plastic strain under cyclic loading.

The genuinely new element in this implementation is an orthotropy factor, denoted g, which the authors introduce to make the kinematic hardening rate direction-dependent. The factor is computed from the direction of plastic loading and the Hill tensor, and it acts as a multiplier on the rate parameters of the backstress evolution. When the material is isotropic, the factor equals one and the original hardening rules are recovered exactly. The motivation is experimental: prior evidence shows that ultimate strength, initial hardening rate, ratcheting rates and yield stress vary with material direction, a pattern also seen in forged steels even after tempering. The authors deliberately did not extend the factor to the isotropic hardening rule, expecting cyclic plastic loading predictions to be governed primarily by kinematic hardening.

Turning a constitutive theory into working finite element code is where much of the paper’s effort lies. The model is formulated for small strains with an additive decomposition of total strain into elastic and plastic parts, and an associative flow rule in a normalized form. The authors reduce the governing equations to an initial value problem for the stress, the hardening variables and a loading index, and they integrate this system with an explicit forward Euler scheme combined with a trial stress algorithm that decides, at each step, whether the material response is purely elastic or elasto-plastic. Because the model is rate independent, the total strain rate cannot act as a parameter that changes the solution; the result after integration must be the same regardless of loading speed.

Explicit integration of plasticity models has a well-known weakness: the yield condition is not guaranteed to hold at the end of each step, so the stress state can drift away from the yield surface, corrupting the interpretation of that surface as a limit bounding the elastic domain and threatening numerical stability. To counter this, the implementation applies a stress correction after every forward Euler step. The authors derive closed-form formulas for two return projection algorithms suited to Hill’s orthotropic yield function, and they employ the computationally less expensive central projection, which exploits the fact that Hill’s criterion is deviatoric and therefore insensitive to the spherical part of stress. They note that for extreme orthotropy the central projection may induce errors, in which case closest point projection would be preferable.

The paper also addresses integration accuracy and stability in depth. A formula for the local truncation error is derived from the Jacobian of the system of differential equations, for which the authors provide analytical derivative formulas, and the global truncation error of the forward Euler method is shown to be of first order with respect to step size. A convergence test using a single element with a prescribed strain trajectory confirmed this prediction: the global relative error components for the stress and backstress converged at rates of 1.0059 and 1.0070, essentially linear with respect to step size, when the exact solution was approximated by an extremely fine integration step of 10 to the power minus 9 seconds. For stability, the authors derive a linearized error equation and a global error contribution operator whose spectral radius must not exceed one. A numerical stability example showed a remarkably strong and consistent relation between that spectral radius and actual stability: with integration steps up to 22 microseconds the spectral radius stayed at or below one and no stability loss was observed, while at 23 microseconds the spectral radius exceeded one and stability was lost.

Verification was carried out in the commercial finite element code Abaqus/Standard, version 2019, through the user subroutine UMAT written in FORTRAN 77. Because analytical solutions for the full model are not known, the authors verified the implementation by reducing it to models already built into Abaqus. When the model was simplified to a von Mises yield condition with isotropic and kinematic hardening and compared against Abaqus’s built-in equivalent, the maximum error was 6 times 10 to the power minus 7. When it was simplified to Hill’s model without hardening and compared against Abaqus’s built-in Hill model, the maximum error was 10 to the power minus 3. Two parameter sets were then used for full-model verification: material A, isotropic in both elasticity and plasticity, and material B, orthotropic in both, with elastic parameters differing from isotropic values by up to 15 percent and plastic parameters by up to 30 percent, a setting chosen to be realistic and of practical use.

The verification examples demonstrate how strongly orthotropy shapes the response. In uniaxial cyclic loading of a single 20-node hexahedral element, the stress–strain and backstress–strain curves for the two materials showed similar trends but clear quantitative differences, attributable primarily to the rotation of the material properties; repeating the orthotropic analysis without rotation diminished the differences considerably. In biaxial cyclic loading, the influence of orthotropy was more pronounced: the maximum difference in one normal strain component reached approximately 30 percent with an average of 17 percent, while a shear strain component showed a maximum difference over 45 percent and an average of 18 percent. A separate comparison showed that the new orthotropy factor g predominantly reduces strain at the same stress, with an average difference of about 5 percent in the normal strain and about 11 percent in the shear strain, and its influence grows with plastic loading. The authors also note that the results suggest even a deviation from isotropy of about 10 percent causes significant quantitative deviation in results, though the results remain qualitatively similar, and that for a problem with symmetric geometry and loading, symmetry breaks down when the material properties are rotated.

To demonstrate applicability beyond single elements, the authors simulated a plate with a hole, meshed with 1280 reduced-integration hexahedral elements and loaded in tension and compression to an extreme value of plus or minus 150 megapascals, with the material orientation rotated from the geometric coordinate system. The simulation ran for almost 3.5 hours on a standard desktop computer with an Intel Core i7-12700 processor and 64 gigabytes of RAM, and the implementation worked well on this large mesh. The observed asymmetry in the results was not particularly pronounced, which the authors attribute to the relatively small orthotropy of the elastic parameters used. Importantly, the authors are explicit about the limits of the work: no simulation of real experimental data is presented. The model’s constitutive features have been validated separately—the MAFTr rule against multiaxial ratcheting experiments for isotropic materials, and Hill’s yield criterion in numerous prior works—but full validation of the combined orthotropic model requires experimental data combining multiaxial ratcheting with yield strength anisotropy, or ratcheting tests on specimens printed or cut in different directions, and such data are rare in the literature. Developing calibration procedures that combine cyclic loading data from specimens loaded in several directions is identified as a key next step toward a fully predictive model for 3D-printed metals.

Subject of Research: Finite element implementation of an orthotropic elasto-plasticity model with advanced kinematic hardening for additively manufactured metals

Article Title: Implementation of an orthotropic plasticity model with advanced kinematic hardening rule for additively manufactured materials

Article References: Parma, S., Vaňková, T., Marek, R., Feigenbaum, H. P., Gabriel, D., Štefan, J., Uhlík, M., Džugan, J., Plešek, J., & Dafalias, Y. F. (2026). Implementation of an orthotropic plasticity model with advanced kinematic hardening rule for additively manufactured materials. Results in Engineering, 32, Article 113187. https://doi.org/10.1016/j.rineng.2026.113187

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DOI: 10.1016/j.rineng.2026.113187

Keywords: additive manufacturing, orthotropic plasticity, Hill yield criterion, kinematic hardening, ratcheting, finite element analysis, Abaqus UMAT, forward Euler integration, numerical stability, 3D-printed metals, cyclic loading, return projection