Viscoplastic Self-consistent Formulation Research Articles

Viscoplastic self-consistent formulation as generalized material model for solid mechanics applications

The viscoplastic self-consistent (VPSC) formulation is extended into a generalized material model/code (VPSC-GMM) that can be easily called by different explicit and implicit boundary-value problem solvers, including high-performance parallel computing implementations. To that end, necessary numerical, algorithmic and programming improvements to the baseline VPSC formulation were developed, and are thoroughly presented in this work. The novel VPSC-GMM is coupled with a Lagrangian finite element (FE) hydrodynamics code, and validated by comparison with stand-alone VPSC predictions for one-element simulations of uniaxial compression and tension and simple shear. Finally, VPSC-GMM coupled with the hydrodynamics FE code is applied to simulate Taylor impact of a polycrystalline tantalum Taylor cylinder. Good agreement between the predicted deformed cylinder shape and the experimental measurement is observed. In terms of parallel computing performance, linear scaling of the computational time with the number of threads is observed.

Modeling of the thermo-mechanical response and texture evolution of WE43 Mg alloy in the dynamic recrystallization regime using a viscoplastic self-consistent formulation

This paper presents a microstructure sensitive model for predicting mechanical response and texture evolution of metals in the dynamic recrystallization regime. A recently proposed viscoplastic self-consistent (VPSC) formulation for the prediction of recrystallization driven by strain energy and intragranular misorientation is extended to hexagonal close-packed (hcp) metals. The model is applied to the dynamic recrystallization of magnesium alloy WE43 at different temperatures and strain rates. Model predictions in terms of stress-strain response and texture evolution are compared to the experimental measurements and acceptable agreement is achieved. According to the model predictions, superplastic behavior of nuclei was found to be the dominant softening mechanism at high temperatures and low strain rates. High concentration of precipitates at the grain boundaries and presence of alloying elements are the likely causes of low boundary mobility, resulting in nucleation dominated dynamic recrystallization. Relatively strong basal compression textures indicate dominant activity of basal slip, which can be achieved only through large difference in slip resistance between soft basal and hard prismatic and pyramidal modes.

Modelling recrystallization textures driven by intragranular fluctuations implemented in the viscoplastic self-consistent formulation

This paper presents a recrystallization model driven by intragranular orientation gradients and strain energy fields calculated by means of the viscoplastic self-consistent (VPSC) formulation. The VPSC model is extended for calculation of the coupling between intragranular stress fluctuations with corresponding second moments of lattice spin and misorientation fields in the grains. Access to these quantities allows modelling of transition bands and nucleation kinetics. In the proposed recrystallization model, grain growth is assumed to be proportional to the difference between the stored energy of each grain and that of the effective medium. Recrystallization textures for several cubic metals are simulated, showing good agreement with corresponding experiments. The model reveals the importance of considering appropriate, microstructurally-based and orientation-dependent recrystallization nucleation mechanisms. The recrystallization texture of heavily rolled copper with a strong cube texture component is found to be a consequence of nucleation at transition bands, which is also the cause of the recrystallization textures in compressed iron and drawn copper wire. In contrast, the recrystallization texture of rolled interstitial-free steel is found to be caused by grain boundary nucleation occurring in grains with the highest strain energy.

A new visco-plastic self-consistent formulation implicit in dislocation-based hardening within implicit finite elements: Application to high strain rate and impact deformation of tantalum

Modeling deformation processes of materials under high strain rate and impact conditions in which the strain rates vary spatiotemporally over several orders of magnitude is challenging, especially in terms of constitutive description. Visco-plastic power-law flow rule, commonly used within crystal plasticity constitutive models, usually introduces superfluous strain rate effects, entering numerically via the slip activation criterion, which result in artificially high flow stress predictions under high strain rate deformation conditions. This paper presents a novel, implicit finite element implementation of a visco-plastic self-consistent polycrystal model that is implicit in dislocation-based hardening with removed strain rate sensitivity introduced by the power-law flow rule. In the model, the strain rate dependence of the flow stress is defined solely by the rate sensitivity of Peierls stress and the evolution of dislocation density within the hardening law, enabling the prediction of high strain rate and impact deformation. As a result, the predicted flow stress is accurate in its magnitude. The physically based strain rate and temperature sensitive model also features, in addition to Schmid, the non-Schmid activation contribution for slip, which is necessary for understanding and modeling the plastic deformation of body-centered cubic metals. The model is calibrated to simulate the strain rate- and temperature-sensitive monotonic deformation of tantalum and is subsequently applied to a Taylor impact test of the same material. Since the impact simulation required remeshing and interpolation of state variables, the spherical linear interpolation algorithm in the space of quaternions is implemented to facilitate the interpolation of texture. Predictions of the simulation were found to compare favorably with experimental measurements of post-test geometrical changes and texture evolution. The implementation and insights from these predictions are presented and discussed in the paper.

Modeling of intragranular misorientation and grain fragmentation in polycrystalline materials using the viscoplastic self-consistent formulation

The recently established methodology to use known algorithmic expressions of the second moments of the stress field in the grains of a polycrystalline aggregate for calculating average fluctuations of lattice rotation rates and the associated average intragranular misorientation distributions using the mean-field viscoplastic self-consistent (VPSC) formulation is extended to solve the coupled problem of considering the effect of intragranular misorientations on stress and rotation rate fluctuations. In turn, these coupled expressions are used to formulate and implement a grain fragmentation (GF) model in VPSC. Case studies, including tension and plane-strain compression of face-centered cubic polycrystals are used to illustrate the capabilities of the new model. GF-VPSC predictions of intragranular misorientation distributions and texture evolution are compared with experiments and full-field numerical simulations, showing good agreement. In particular, the inclusion of misorientation spreads reduced the intensity of the deformed texture and thus improved the texture predictions. Moreover, considering that intragranular misorientations act as driving forces for recrystallization, the new GF-VPSC formulation is shown to enable modeling of microstructure evolution during deformation and recrystallization, in a computationally efficient manner.

Simulation of Texture Evolution during Uniaxial Deformation of Commercially Pure Titanium

The evolution of texture in commercially pure (CP) titanium during uniaxial tension and compression through VPSC (Visco-plastic self-consistent) simulation is reported in the present study. CP-titanium was subjected to both uniaxial tension and compression upto 35% deformation. During uniaxial tension, tensile twin of type and compressive twin of type were observed in the samples. However, only tensile twin of type and compressive twin of type was observed in the samples during uniaxial compression. Volume fractions of the twins were increased linearly as a function of percentage deformation during uniaxial tension. Whereas, during uniaxial compression the twinning volume fraction was increased up to 20% deformation and then decreased rapidly on further increasing the percentage deformation. During uniaxial tension, the general t-type textures were observed in the samples irrespective of the percentage deformation. The initial non-basal texture was oriented to split basal texture during uniaxial compression of the sample. VPSC formulation was used for simulating the texture development in the material. Different hardening parameters were estimated through correlating the simulated stress-strain curve with the experimental stress-strain data. It was observed that, prismatic slip operated as the primary deformation mode during uniaxial tension whereas basal slip acquired the leading role during deformation through uniaxial compression. It was also revealed that active deformation modes were fully depending on percentage deformation, loading direction, and orientation of grains.

Predicting intragranular misorientation distributions in polycrystalline metals using the viscoplastic self-consistent formulation

In a recent paper, we reported the methodology to calculate intragranular fluctuations in the instantaneous lattice rotation rates in polycrystalline materials within the mean-field viscoplastic self-consistent (VPSC) model. This paper is concerned with the time integration and subsequent use of these fluctuations to predict orientation-dependent misorientation distributions developing inside each grain representing the polycrystalline aggregate. To this end, we propose and assess two approaches to update the intragranular misorientation distribution within the VPSC framework. To illustrate both approaches, we calculate intragranular misorientations in face-centered cubic polycrystals deformed in tension and plane-strain compression. These predictions are tested by comparison with corresponding experiments for polycrystalline copper and aluminum, respectively, and with full-field calculations. It is observed that at sufficiently high strains some grains develop large misorientations that may lead to grain fragmentation and/or act as driving forces for recrystallization. The proposed VPSC-based prediction of intragranular misorientations enables modeling of grain fragmentation, as well as a more accurate modeling of texture using a computationally efficient mean-field approach, as opposed to computationally more expensive full-field approaches.

Twinning effects in a rod-textured AM30 Magnesium alloy

We experimentally and numerically investigated the effect of twinning on plasticity using an extruded rod-textured magnesium alloy. The rod-texture is a 〈 1 0 1 ¯ 0 〉 -axis fiber texture that presents a fundamentally different anisotropy correlated to twinning with respect to the widely discussed c-axis fiber texture generated by clock rolling. We quantified a profuse { 1 0 1 ¯ 2 } 〈 1 0 1 ¯ 1 〉 extension twinning along the extrusion direction (ED) that consumed the entire parent before the inflection point in the stress–strain behavior. However, under compression along the extrusion radial direction (ERD), the twinning model in the viscoplastic self-consistent formulation still predicts substantial extension twinning. However, in this case the stress–strain curve did not inflect, and Regime II hardening was absent. We demonstrate via EBSD analyses that the absence of Regime II hardening along the ERD was due to a non-Schmid effect by multivariant “stopped” twinning. The intersecting variants of stopped twins incurred twin–twin interactions that limited the twin growth. Profuse { 1 0 1 ¯ 1 } 〈 1 0 1 ¯ 2 〉 double twinning occurs both under ED and ERD but peculiarly triggered earlier under ERD than under ED, so the Voce model under VPSC could not capture their effect. The complex networks of stopped twins in the ERD clearly negate a possible Hall–Petch effect on Regime II by twin segmentation, since otherwise Regime II would be more marked in the ERD. Rather, the stopped twins suggest preferential latent hardening within the twinned regions by parent dislocation transmutation upon their incorporation in the twins. In fact, since twin–twin interactions mitigate the growth rates of sweeping extension twin boundaries, dislocation transmutation could be limited to the extent that Regime II hardening will be eliminated.

Multiscale modeling of plasticity based on embedding the viscoplastic self-consistent formulation in implicit finite elements

This paper is concerned with the multiscale simulation of plastic deformation of metallic specimens using physically-based models that take into account their polycrystalline microstructure and the directionality of deformation mechanisms acting at single-crystal level. A polycrystal model based on self-consistent homogenization of single-crystal viscoplastic behavior is used to provide a texture-sensitive constitutive response of each material point, within a boundary problem solved with finite elements (FE) at the macroscale. The resulting constitutive behavior is that of an elasto-viscoplastic material, implemented in the implicit FE code ABAQUS. The widely-used viscoplastic selfconsistent (VPSC) formulation for polycrystal deformation has been implemented inside a user-defined material (UMAT) subroutine, providing the relationship between stress and plastic strain-rate response. Each integration point of the FE model is considered as a polycrystal with a given initial texture that evolves with deformation. The viscoplastic compliance tensor computed internally in the polycrystal model is in turn used for the minimization of a suitable-designed residual, as well as in the construction of the elasto-viscoplastic tangent stiffness matrix required by the implicit FE scheme. Uniaxial tension and simple shear of an FCC polycrystal have been used to benchmark the accuracy of the proposed implicit scheme and the correct treatment of rotations for prediction of texture evolution. In addition, two applications are presented to illustrate the potential of the multiscale strategy: a simulation of rolling of an FCC plate, in which the model predicts the development of different textures through the thickness of the plate; and the deformation under 4-point bending of textured HCP bars, in which the model captures the dimensional changes associated with different orientations of the dominant texture component with respect to the bending plane.