A comparative study of kinematic hardening rules at finite deformations

Kinematic hardening models describe a specific kind of plastic anisotropy which evolves with the deformation process. It is well known that the extension of constitutive relations from small to finite deformations is not unique. This applies also to well-established kinematic hardening rules like that of Armstrong-Frederick or Chaboche. However, the second law of thermodynamics offers some possibilities for generalizing constitutive equations so that this ambiguity may, in some extent, be moderated. The present paper is concerned with three possible extensions, from small to finite deformations, of the Armstrong-Frederick rule, which are derived as sufficient conditions for the validity of the second law. All three models rely upon the multiplicative decomposition of the deformation gradient tensor into elastic and plastic parts and make use of a yield function expressed in terms of the so-called Mandel stress tensor. In conformity with this approach, the back-stress tensor is defined to be of Mandel stress type as well. In order to compare the properties of the three models, predicted responses for processes with homogeneous and inhomogeneous deformations are discussed. To this end, the models are implemented in a finite element code (ABAQUS).     

A polycrystal plasticity model based on the mechanical threshold

A temperature and rate-dependent viscoplastic polycrystalmodel is presented.It uses a single crystal constitutive response that is based on the isotropic Mechanical Threshold Stress continuum model. This combination gives us theability to relate the constitutive model parameters between the polycrystaland continuum models. The individual crystal response is used to obtain themacroscopic response through the extended Taylor hypothesis. A Newton-Raphsonalgorithm is used to solve the set of fully implicit nonlinear equations for each crystal. The analysis also uses a novel state variable integration method which renders the analysis time step size independent for constant strain rate simulations. Material parameter estimates are obtained through an identification study, where the error between experimental and computed stress response is minimized. The BFGS method, which is used to solve theidentification problem, requires first-order gradients. These gradients arecomputed efficiently via the direct method of design sensitivity analysis.Texture augmentation is performed in a second identification study by changing crystal weights (volume fractions).     

Texture predictions using a polycrystal plasticity model incorporating neighbor interactions

A viscoplastic model is presented for distributing the deformation applied to a polycrystal in a non-uniform fashion among the constituent crystals. Interactions with surrounding crystals are incorporated in the calculation of the deformation rate of each crystal through an appropriately defined local neighborhood. A compliance tensor is computed for each crystal based on a viscoplastic constitutive relation for deformation by crystallographic slip. The compliance of the crystal relative to that of its neighborhood provides a means for partitioning the macroscopic deformation rate among the crystals. The deviation of the crystal deformation rate from the macroscopic value is suitably scaled to obtain the crystal spin. Polycrystal simulations of crystallographic texture development using this model are compared to the results of similar calculations using the Taylor model, to finite element simulations of a model polycrystal, and to experimental data. The model incorporating neighbor interactions is shown to result in improved texture predictions, in terms of both the intensity levels and the locations of certain texture components.     

Texture and strain localization prediction using a N-site polycrystal model

Most polycrystal models of plastic deformation rely on the assumption that strain and stress are uniform within the domain of each grain. Comparison between measured and predicted textures suggests that this assumption is realistic for most single-phase aggregates and crystal symmetries. In this paper, we implement a self-consistent N-site model that allows one to account for strain localization and local misorientation near grain boundaries. We apply this model to face centered cubic (fcc) and hexagonal close packed (hcp) aggregates, and analyze the similarities and differences with a one-site model that assumes uniform stress and strain-rate within a grain. We find that the assumption of uniformity is justified in first order. We discuss the implications of the N-site model for the simulation of systems with hard inclusions, orientation correlations, and recrystallization mechanisms.     

A generalized self-consistent polycrystal model for the yield strength of nanocrystalline materials

Inspired by recent molecular dynamic simulations of nanocrystalline solids, a generalized self-consistent polycrystal model is proposed to study the transition of yield strength of polycrystalline metals as the grain size decreases from the traditional coarse grain to the nanometer scale. These atomic simulations revealed that a significant portion of atoms resides in the grain boundaries and the plastic flow of the grain-boundary region is responsible for the unique characteristics displayed by such materials. The proposed model takes each oriented grain and its immediate grain boundary to form a pair, which in turn is embedded in the infinite effective medium with a property representing the orientational average of all these pairs. We make use of the linear comparison composite to determine the nonlinear behavior of the nanocrystalline polycrystal through the concept of secant moduli. To this end an auxiliary problem of Christensen and Lo (J. Mech. Phys. Solids 27 (1979) 315) superimposed on the eigenstrain field of Luo and Weng (Mech. Mater. 6 (1987) 347) is first considered, and then the nonlinear elastoplastic polycrystal problem is addressed. The plastic flow of each grain is calculated from its crystallographic slips, but the plastic behavior of the grain-boundary phase is modeled as that of an amorphous material. The calculated yield stress for Cu is found to follow the classic Hall-Petch relation initially, but as the gain size decreases it begins to depart from it. The yield strength eventually attains a maximum at a critical grain size and then the Hall-Petch slope turns negative in the nano-range. It is also found that, when the Hall-Petch relation is observed, the plastic behavior of the polycrystal is governed by crystallographic slips in the grains, but when the slope is negative it is governed by the grain boundaries. During the transition both grains and grain boundaries contribute competitively.     

Stress integration method for a nonlinear kinematic/isotropic hardening model and its characterization based on polycrystal plasticity

Sheet metal forming processes generally involve non-proportional strain paths including springback, leading to the Bauschinger effect, transient hardening, and permanent softening behavior, that can be possibly modeled by kinematic hardening laws. In this work, a stress integration procedure based on the backward-Euler method was newly derived for a nonlinear combined isotropic/kinematic hardening model based on the two-yield’s surfaces approach. The backward-Euler method can be combined with general non-quadratic anisotropic yield functions and thus it can predict accurately the behavior of aluminum alloy sheets for sheet metal forming processes. In order to characterize the material coefficients, including the Bauschinger ratio for the kinematic hardening model, one element tension–compression simulations were newly tried based on a polycrystal plasticity approach, which compensates extensive tension and compression experiments. The developed model was applied for a springback prediction of the NUMISHEET’93 2D draw bend benchmark example.     

Predictions of forming limit diagrams using a rate-dependent polycrystal self-consistent plasticity model

In the sheet-metal forming industry, forming-limit strains have been a useful tool for quantifying metals formability. However, the experimental measurement of these strains is a difficult, time consuming and expensive process. It would be useful if strains calculated with a theoretical model could replace many of the experimental measurements. In this research, we analyze forming-limit strains of metals using a rate-dependent plasticity, polycrystal, self-consistent (VPSC) model in conjunction with the Marciniak–Kuczynski (M–K) approach. Previous researchers have studied forming limit diagrams (FLDs) based on the full-constraints Taylor model. This is the first time, to the authors’ knowledge, that the self-consistent approach has been introduced to simulate the polycrystal FLD behavior. Numerous microstructural factors characterizing the material have a strong influence on the FLD, so our model includes the effects of slip hardening, strain-rate sensitivity, anisotropy and initial texture. Finally, the calculation of the FLD with a more realistic scale transition successfully predicts some of the experimental tendencies that the Taylor model cannot reproduce for aluminum alloys AA6116-T4 and AA5182-O.     

A hardening model for anisotropic materials

Results from proportionate-loading experiments under plane-stress states illustrate the existence of a field of uniform-hardening potentials for the yield and creep deformation behavior of isotropic, slightly anisotropic, aelotropic and orthotropic polycrystalline materials in the initially strain-free condition. For two different plane-stress states, it is shown that a linear functional relationship holds between the plastic-strain increment ratio and the stress ratio in these materials and that, consequently, the field is adequately modeled by a uniform-hardening anisotropic function that is quadratic in the components of deviatoric stress. Anisotropic plane-stress yield functions are formulated for any stage in the deformation process by combining the uniform-hardening function with the kinematic-hardening rule. The resulting functions, which correspond to rigid translations of initial yield loci according to Ziegler's rule, provide good agreement with experimental observations on a marked Bauschinger effect and an absence of cross hardening.     

A computational study of a model of single-crystal strain-gradient viscoplasticity with an interactive hardening relation

The behavior of a model of single-crystal strain-gradient viscoplasticity is investigated. The model is an extension of a rate-independent version, and includes a new hardening relation that has recently been proposed in the small-deformation context (Gurtin and Reddy, 2014), and which accounts for slip-system interactions due to self and latent hardening. Energetic and dissipative effects, each with its corresponding length scale, are included. Numerical results are presented for a single crystal with single and multiple slip systems, as well as an ensemble of grains. These results provide a clear illustration of the effects of accounting for slip-system interactions.     

Application of a discrete polycrystal model to the analysis of cyclic straining in copper

A discrete polycrystal model, designed to simulate a metal aggregate macro-element, is applied to the study of cyclic straining in copper. The numerical method of solution (an adaptation of the “finite element method”) incorporates a convergent discrete Green's function within the constrained minimum principle which governs the (crystallographic) plastic shear increments at each load step. Isothermal elastic moduli of copper crystals and Taylor's hardening rule with constant hardening modulus are used in the calculations. Numerical results are obtained for macroscopic elastic properties, cyclic stress-strain curves (which indicate the contribution of aggregate heterogeneity to macroscopic hardening), macroscopic plastic work, and residual (latent) strain energy through four loading cycles between fixed macrostrain limits. Other estimates for elastic properties also are included, and all results are compared, both qualitatively and quantitatively, with published experiments. The predictions of the model are in general satisfactory.     

A comparative study of central and upwind difference schemes using the primitive variables

The use of the velocity-pressure formulation of the Navier-Stokes equations for the numerical solution of fluid flow problems is favoured for free-surface problems, more involved flow configurations, and three-dimensional flows. Many engineering problems involve such features in addition to strong inertial effects. The computational instabilities arising from central-difference schemes for the convective terms of the governing equations impose serious limitations on the range of Reynolds numbers that can be investigated by the numerical method. Solutions for higher Reynolds numbers Re > 1000 could be reached using upwind-difference schemes. A comparative study of both schemes using a method based on the primitive variables is presented. The comparison is made for the model problem of the driven flow in a square cavity. Using a central scheme stable solutions of the pressure and velocity fields were obtained for Reynolds numbers up to 5000. The streamfunction and vorticity fields were calculated from the resulting velocity field and compared with previous solutions. It is concluded that total upwind differencing results in a considerable change in the flow pattern due to the false diffusion. For practical calculations, by a proper choice of a small amount of partial upwind differencing the vorticity diffusion near the walls and the global features of the solutions are not sigificantly altered.     

Numerical implementation of non-local polycrystal plasticity using fast Fourier transforms

We present the numerical implementation of a non-local polycrystal plasticity theory using the FFT-based formulation of Suquet and co-workers. Gurtin (2002) non-local formulation, with geometry changes neglected, has been incorporated in the EVP-FFT algorithm of Lebensohn et al. (2012). Numerical procedures for the accurate estimation of higher order derivatives of micromechanical fields, required for feedback into single crystal constitutive relations, are identified and applied. A simple case of a periodic laminate made of two fcc crystals with different plastic properties is first used to assess the soundness and numerical stability of the proposed algorithm and to study the influence of different model parameters on the predictions of the non-local model. Different behaviors at grain boundaries are explored, and the one consistent with the micro-clamped condition gives the most pronounced size effect. The formulation is applied next to 3-D fcc polycrystals, illustrating the possibilities offered by the proposed numerical scheme to analyze the mechanical response of polycrystalline aggregates in three dimensions accounting for size dependence arising from plastic strain gradients with reasonable computing times.     

A strain space nonlinear kinematic hardening/softening plasticity model

A strain space plasticity theory based on the nonlinear kinematic hardening and softening rule is developed in order to accommodate work-hardening, work-softening, and elastic-perfectly plastic materials with one set of constitutive equations, and to facilitate strain controlled calculations. A generalized hardening/softening parameter is proposed, and the potential of linking the parameter to micro-mechanical material changes is discussed. The theory is used to investigate work-softening materials numerically and highlights a need for additional experimental results in this area.     

A study of directionality of latent hardening behavior in aluminium single crystals

By defining the yield stress in a latent hardening test as the first deviation from the elastic straight line, the yielding and hardening behavior on a latent system in the positive and negative slip direction was studied in aluminium single crystals. It is shown that the yield stresses on both the positive and negative latent systems are about equal to or a little lower than the maximum resolved shear stress in the primary test, but much higher than that of the active system. The influence of relative orientation and prestrain on latent hardening and initial work-hardening in the secondary test was also investigated, and it was found that there is a considerable effect on initial work-hardening, but none on latent hardening. With reasonable approximation, a hardening rule for single crystal could be proposed from the experimental results, that is, except for the yield stress on the system negative to the active system that is very low, hardening on the other systems is nearly the same as self-hardening.     

A model to predict the coupled axial torsion properties of ACSR electrical conductors

The modeling of ACSR (aluminum-conductor steel-reinforced) electrical conductors for dynamic analysis requires some knowledge of the mechanical properties of the conductor. It was found both experimentally and theoretically, using a simple strength of materials approach, that the axial-torsional behavior of ACSR conductors is highly coupled; i.e., axial motion causes torsional motion and vice versa. Although wind-induced oscillation of ACSR power lines has been observed for years, the importance of axial-torsional coupling has not been generally recognized, nor studied. A simplified mathematical model correlated well with experimental measurements for this type of coupled mechanical behavior. It is hoped that being able to control the amount of coupling through cable design may lead to better control of wind-induced oscillations.     

On the derivation and comparative analysis of large rotation shell theories

Summary For the geometrically nonlinear first approximation theory of elastic shells three energy-consistent large rotation shell variants are constructed. The governing shell equations are derived as Euler-Lagrange equations of an associated variational principle of stationary total potential energy. The numerical applicability is considered for a highly nonlinear shell problem. To incorporate the presented theories into the frame of shell models published in the literature a comparative analysis is carried out for a large number of shell equations.

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Zur Herleitung und vergleichenden Untersuchung von Schalentheorien für große Rotationen

Übersicht Im Rahmen der geometrisch-nichtlinearen ersten Schalenapproximation werden drei energiekonsistente Schalentheorien für große Rotationen hergeleitet. Die Schalengleichungen werden als Euler-Lagrange Gleichungen eines zugehörigen Variationsprinzips vom stationären Wert der potentiellen Gesamtenergie hergeleitet. Die numerische Anwendbarkeit wird anhand eines stark-nichtlinearen Schalenbeispiels nachgewiesen. Um die in dieser Arbeit hergeleiteten Theorien entsprechend einordnen zu können, wird für eine größere Zahl von in der Literatur angegebenen Schalengleichungen eine vergleichende Untersuchung mit numerischer Auswertung durchgeführt.

     

On the calibration of the Chaboche hardening model and a modified hardening rule for uniaxial ratcheting prediction

A systematic mathematical approach is developed in the context of uniaxial cyclic ratcheting for the parameter determination of the decomposed Chaboche hardening rule. This is achieved by deriving the relation between the evolution of the backstress and the plastic strain accumulation. Unlike current calibration techniques where a trial–error approach is employed to fit the simulation results to experimental data, the proposed method determines the parameters directly from uniaxial ratcheting experiments. Numerical results indicate that Chaboche’s hardening model is much more efficient than what has been demonstrated before. Finally, as an improvement to the decomposed model, a modification is made to one of the backstress components. This improved component enables the model to predict uniaxial ratcheting with more accuracy.