Forming limit comparisons for FCC and BCC sheets

In this paper, numerical simulations of forming limit diagrams (FLDs) are performed based on a rate-sensitive polycrystal plasticity model together with the Marciniak–Kuczynski (M–K) approach. Sheet necking is initiated from an initial imperfection in terms of a narrow band. The deformations inside and outside the band are assumed to be homogeneous, and conditions of compatibility and equilibrium are enforced across the band interfaces. Thus, the polycrystal model need only be applied to two polycrystal aggregates, one inside and one outside the band. Both FCC and BCC crystals are considered with 12 distinct slip systems for an FCC crystal and 24 distinct slip systems for a BCC crystal. The response of an aggregate comprised of many grains is based on an elastic–viscoplastic Taylor-type polycrystal model. With this formulation, the effects of initial imperfection intensity and orientation, crystal elasticity, strain-rate sensitivity, single slip hardening, and latent hardening on the FLD can be assessed. Identical initial textures are considered for both FCC and BCC polycrystals and the predicted FLDs are compared with each other.     

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.     

Evaluation of self-consistent polycrystal plasticity models for magnesium alloy AZ31B sheet

Various self-consistent polycrystal plasticity models for hexagonal close packed (HCP) polycrystals are evaluated by studying the deformation behavior of magnesium alloy AZ31B sheet under different uniaxial strain paths. In all employed polycrystal plasticity models both slip and twinning contribute to plastic deformation. The material parameters for the various models are fitted to experimental uniaxial tension and compression along the rolling direction (RD) and then used to predict uniaxial tension and compression along the traverse direction (TD) and uniaxial compression in the normal direction (ND). An assessment of the predictive capability of the polycrystal plasticity models is made based on comparisons of the predicted and experimental stress responses and R values. It is found that, among the models examined, the self-consistent models with grain interaction stiffness halfway between those of the limiting Secant (stiff) and Tangent (compliant) approximations give the best results. Among the available options, the Affine self-consistent scheme results in the best overall performance. Furthermore, it is demonstrated that the R values under uniaxial tension and compression within the sheet plane show a strong dependence on imposed strain. This suggests that developing anisotropic yield functions using measured R values must account for the strain dependence.     

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.     

Nanocontact between BCC tungsten and FCC nickel using the quasicontinuum method

The nanocontact between BCC tungsten and FCC nickel is investigated by using the multiscale simulation method (quasicontinuum method). The new proposed interatomic potential – Extended Finnis-Sinclair potential is adopted and compared with the conventional Ackland Finnis-Sinclair potential in the nanoindentation model. A series of multiscale simulations of contact formation, indentation, subsequent pulling, and separation between a tungsten indenter and a nickel substrate are performed by means of the quasicontinuum method. Different plastic mechanisms are observed for different crystal directions and indenter shapes. Besides the adherence between the tungsten indenter and the nickel substrate, the geometrical shape, which is formed by the slip planes inside the substrate, plays a vital role in the material transfer between the indenter and the substrate. The formation and evolution of necking and cracking in the nanocontact model are also discussed.     

Embedded polycrystal plasticity and adaptive sampling

A simulation capability for multi-scale embedded polycrystal plasticity is demonstrated with over two orders of magnitude wall-clock speedup compared to direct embedding. In the coarse-scale material model, the visco-plastic part of the material response is based on parameters determined from polycrystal level fine-scale calculations. Polycrystal plasticity parameters are approximated from fine-scale calculations using adaptive sampling to substantially reduce the total number of expensive fine-scale calculations which must be performed. The adaptive sampling method uses Kriging models for local interpolation of the fine-scale plasticity parameters and a metric-tree database for storage and retrieval of the fine-scale response models. Efficacy of the method is demonstrated through a variety of example problems involving both quasi-static and dynamic loading scenarios.     

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.     

A plasticity model for interface friction: application to sheet metal forming

Successful numerical simulations of forming operations require robust and accurate tool-workpiece interface friction models. In this paper we extend the rate-independent, isotropic, isothermal interface friction model proposed by Anand (Anand, L., 1993. A constitutive model for interface friction. Computational Mechanics 12, 197–213) to a rate-dependent formulation. Material parameters in the friction model are determined for lubricated interfaces between Al6111-T4 sheet and D2 tool steel. The lubricants used are MP404 and boric acid; the MP404 lubricant is currently used in industry, whereas boric acid has recently been proposed as a solid-film lubricant for sheet forming by Erdemir (Erdemir, A., 1991. Tribological properties of boric acid and boric acid-forming surfaces. Part i: crystal chemistry and mechanisms of self-lubrication of boric acid. Lubrication Engineering 47, 168–173). The interface friction model is implemented in the finite element code ABAQUS/Explicit (ABAQUS Reference Manual., 1999. Providence, RI), and the finite element program is used to simulate two sheet forming operations: axisymmetric cup-drawing and square pan-drawing. The predictions from the finite element simulation are shown to be in very good agreement with experimental results.     

A finite element formulation based on non-associated plasticity for sheet metal forming

In the present paper, a finite element formulation based on non-associated plasticity is developed. In the constitutive formulation, isotropic hardening is assumed and an evolution equation for the hardening parameter consistent with the principle of plastic work equivalence is introduced. The yield function and plastic potential function are considered as two different functions with functional form as the yield function of Hill [Hill, R., 1948. Theory of yielding and plastic flow of anisotropic metals. Proc. Roy. Soc. A 193, 281–297] or Karafillis–Boyce associated model [Karafillis, A.P. Boyce, M., 1993. A general anisotropic yield criterion using bounds and a transformation weighting tensor. J. Mech. Phys. Solids 41, 1859–1886]. Algorithmic formulations of constitutive models that utilize associated or non-associated flow rule coupled with Hill or Karafillis–Boyce stress functions are derived by application of implicit return mapping procedure. Capabilities in predicting planar anisotropy of the Hill and Karafillis–Boyce stress functions are investigated considering material data of Al2008-T4 and Al2090-T3 sheet samples. The accuracy of the derived stress integration procedures is investigated by calculating iso-error maps.     

Path-dependence of the forming limit stresses in a sheet metal

The effect of changing strain paths on the forming limit stresses of sheet metals is investigated using the Marciniak–Kuczyński model and a phenomenological plasticity model with non-normality effects [Kuroda, M., Tvergaard, V., 2001. A phenomenological plasticity model with non-normality effects representing observations in crystal plasticity. J. Mech. Phys. Solids 49, 1239–1263]. Forming limits are simulated for linear stress paths and two types of combined loading: a combined loading consisting of two linear stress paths in which unloading is included between the first and second loadings (combined loading A), and combined loading in which the strain path is abruptly changed without unloading (combined loading B). The forming limit stresses calculated for combined loading A agree well with those calculated for the linear stress paths, while the forming limit curves in strain space depend strongly on the strain paths. The forming limit stresses calculated for the combined loading B do not, however, coincide with those calculated for the linear stress paths. The strain-path dependence of the forming limit stress is discussed in detail by observing the strain localization process.     

Investigating the limits of polycrystal plasticity modeling

A material model which describes the rate-dependent crystallographic slip of FCC metals has been implemented into a quasistatic, large deformation, nonlinear finite element code developed at Sandia National Laboratories. The resultant microstructure based elastic–plastic deformation model has successfully performed simulations of realistic looking 3-D polycrystalline microstructures generated using a Potts-model approach. These simulations have been as large as 50,000 elements composed of 200 randomly oriented grains. This type of model tracks grain orientation and predicts the evolution of sub-grains on an element by element basis during deformation of a polycrystal. Simulations using this model generate a large body of informative results, but they have shortcomings. This paper attempts to examine detailed results provided by large scale highly resolved polycrystal plasticity modeling through a series of analyses. The analyses are designed to isolate issues such as rate of texture evolution, the effect of mesh refinement and comparison with experimental data. Specific model limitations can be identified with lack of a characteristic length scale and oversimplified grain boundaries within the modeling framework.     

Anisotropic plasticity for sheet metals using the concept of combined isotropic-kinematic hardening

Hill's 1948 anisotropic theory of plasticity (Hill, R., 1948. A theory of yielding and plastic flow of anisotropic metals. Proc. Roy. Soc. London A193, 281–297) is extended to include the concept of combined isotropic-kinematic hardening, and the objective of this paper is to validate the model so that it may be useful for analyses of sheet metal forming. Isotropic hardening and kinematic hardening may be experimentally observed in sheet metals, if yielding is defined by the proportional limit or by a small proof strain. In this paper, a single exponential term is used to describe isotropic hardening and Prager's linear kinematic hardening rule is applied for simplicity. It is shown that this model can satisfactorily describe both the yield stress and the plastic strain ratio, the R-ratio, observed in tension test of specimens cut at various angles measured from the rolling direction of the sheet. Kinematic hardening leads to a gradual change in the direction of the plastic strain increment, as the axial strain increases in the tension test; while in the traditional approach for sheet metal, this direction does not change due to the use of isotropic hardening.     

Analysis for polycrystal nonproportional cyclic plasticity with a dissipated energy based rule for cross-hardening

Based on a nonclassical hardening law and the Hill’s self-consistent scheme, a new approach is proposed for the analysis of polycrystal nonproportional cyclic plasticity. A novel parameter related to the plastic dissipation on each slip system is proposed and embedded in the Bassani’s definition of cross-hardening. The tangential elastoplastic tensor relating the increments of stress and strain in a single crystal is derived and the corresponding numerical algorithm for polycrystal plasticity is developed. The elastoplastic response of 316 stainless steel subjected to typical biaxial nonproportional strain cycling is analyzed, and the main features are well replicated. The validity of the proposed approach is demonstrated by the satisfactory agreement between the computed results and experimental observation.     

Three-dimensional deformation process simulation with explicit use of polycrystal plasticity models

The combination of massive parallel processing and polycrystal plasticity theory offers the potential for applying detailed microstructural models to macroscopic deformation processes. In this work the finite element method is used to solve for the three-dimensional deformation of a plastic workpiece. The elemental constitutive response is derived from the microstructural response of a polycrystal aggregate situated in the element. Crystal orientations and their respective weighted contributions to the aggregate response are selected to approximate the orientation distribution derived from experimental pole figure measurements. The interaction of the material symmetry adopted in analysis of pole figures and the boundary conditions posed in the plasticity boundary value problem are examined. Through the introduction of distinct aggregates with decreasing crystal to aggregate ratio, an inhomogenous material response is developed where: (1) the orientation distribution becomes well approximated only by a collection of spatially distinct aggregates, and (2) these aggregates experience deformation paths of increasing variation. It is shown that the use of spatially distinct aggregates in a material experiencing local kinematic inhomogeneities throughout its deformation history leads to texture predictions that compare favorably with experimental measurements.     

A hybrid finite element formulation for polycrystal plasticity with consideration of macrostructural and microstructural linking

A hybrid finite element formulation for the plastic deformation of FCC metals with anisotropy is outlined. Polycrystal plasticity theory is used to develop the constitutive response. The hybrid approach facilitates introduction of the microscale stress in the macroscopic statement of equilibrium. Convergence of the hybrid formulation is contrasted with that of a velocity-pressure formulation. It is demonstrated that the hybrid formulation is well suited for studies where significant spatial variations in constitutive response result from having only one, or a very few, crystal orientations represented in each finite element. A simulation of channel die compression is made with one crystal per finite element. The resulting texture evolution is compared with other texture evolution models and experimental data for cold rolled aluminum. It is demonstrated that the brass texture component, observed in the experimental data, is developed through shear deformations arising from grain-to-grain interactions.     

FE-analysis of sheet metal forming processes using continuous contact treatment

Discrete meshes cause stepwise propagation of the contact nodes of a sheet despite the fact that the contact region in the actual forming process is altered very smoothly. This can cause problems of convergence and accuracy in contact-sensitive processes, such as a bending process. In this study, a scheme for a continuous contact treatment is proposed in order to consider the more realistic behavior of the contact phenomena during the forming process. For verification of the proposed method, the contact pressures and forming load are evaluated during the compression forming of a tube. The analysis of a hemispherical dome formed without a blank holder is also presented in order to investigate the effects of the proposed algorithm. The results show that the precise deformation mode is predicted by the utilization of the proposed method.     

An examination of the Lin model for polycrystal plasticity by means of two-dimensional finite element analysis

A two-dimensional inhomogeneous problem is calculated by the finite element method, where a nonhardening infinite medium containing inclusions is subjected to a simple tension. The results of plastic strains are substituted into the Lin and KBW models to get the stress fields, which are compared with the results of FEM. The Lin model agrees well with the FEM, but the KBW model does not. Even though the variation of plastic strain exceeds the order of elastic strain, the stresses still remain within the same order as the yielding stress. This fact may suggest that the coefficient matrix relating to the variation of plastic strain to that of stress in the Lin model is near-singular.     

HILL屈服准则与晶体塑性模型对FCC单晶材料塑性各向异性描述能力的比较

采用宏观HILL模型和晶体塑性模型对面心立方单晶(FCC)材料的非均匀交形进行了数值模拟，意在比较两种不同尺度的模型对塑性各向异性的描述能力的差异。为了使两种模型具有可比性，对于FCC单晶材科，本文提出一种用晶体塑性模型来确定HILL模型中各向异性参数的标定方法。数值分析表明，两类模型对单晶体塑性各向异性的描述能力存在着差异。对FCC单晶材料，HILL模型对各向异性的预测能力没有晶体塑性模型细致，晶体塑性模型更能追踪塑性各向异性的变化。但两种模型对应力应变响应预测的趋势是一致的。对两种模型描述的差异，做了详细的分析。     

On the homogenization of metal matrix composites using strain gradient plasticity

The homogenized response of metal matrix composites（MMC） is studied using strain gradient plasticity.The material model employed is a rate independent formulation of energetic strain gradient plasticity at the micro scale and conventional rate independent plasticity at the macro scale. Free energy inside the micro structure is included due to the elastic strains and plastic strain gradients. A unit cell containing a circular elastic fiber is analyzed under macroscopic simple shear in addition to transverse and longitudinal loading. The analyses are carried out under generalized plane strain condition. Micro-macro homogenization is performed observing the Hill-Mandel energy condition,and overall loading is considered such that the homogenized higher order terms vanish. The results highlight the intrinsic size-effects as well as the effect of fiber volume fraction on the overall response curves, plastic strain distributions and homogenized yield surfaces under different loading conditions. It is concluded that composites with smaller reinforcement size have larger initial yield surfaces and furthermore,they exhibit more kinematic hardening.     

Rate-independent crystalline and polycrystalline plasticity,application to FCC materials

This paper deals with the simulation of the mechanical response and texture evolution of cubic crystals and polycrystals for a rate-independent elastic–plastic constitutive law. No viscous effects are considered. An algorithm is introduced to treat the difficult case of multi-surface plasticity. This algorithm allows the computation of the mechanical response of a single crystal. The corresponding yield surface is made of the intersection of several hyper-planes in the stress space. The problem of the multiplicity of the slip systems is solved thanks to a pseudo-inversion method. Self and latent hardening are taken into account. In order to compute the response of a polycrystal, a Taylor homogenization scheme is used. The stress–strain response of single crystals and polycrystals is computed for various loading cases. The texture evolution predicted for compression, plane strain compression and simple shear are compared with the results given by a visco-plastic polycrystalline model.