K.W. Neale

Evaluation of anisotropic yield functions for aluminum sheets

Abstract The anisotropic behaviour of some rolled aluminum alloys is investigated using the phenomenological approach via some of the recently proposed 3-D yield functions. The directional variations of the yield stress and the so-called R-values in the plane of the sheet are predicted using the 3-D yield functions and the results are compared to previously reported experimental measurements. The hardening properties of the alloys examined are extracted from the experimental uniaxial curves along the rolling direction (RD). The forming limit curves are produced using the Marciniak and Kuczynski (Marciniak, Z., Kuczynski, K., 1967 Limit strains in the process of stret-forming sheet metals. Int. J. Mech. Sci. 9, 609–620) approach and the predictions of different yield functions are compared to the experimental curves. In general, although the refinements incorporated to the yield function proposals in chronological order produced better agreement with the experimental observations (at the expense of some complexity in the formulation), the overall performance of each criteria examined could nevertheless vary with the type of the application, as can be anticipated from the phenomenological nature of the approach.

Limit strain predictions for strain-rate sensitive anisotropic sheets

Abstract The combined effects of material strain-rate sensitivity and anisotropy on necking or “limit” strain predictions are examined for thin sheets with transversely isotropic properties. Various rate dependent constitutive laws based on flow theory and deformation theory of plasticity are considered. The strong effect of material strain-rate sensitivity in increasing the amount of straining prior to localized necking is first emphasized. We then discuss the joint influence of rate dependence and anisotropy on the theoretical limit strains and forming limit curves. Both strain-rate sensitivity and the local shape of the anisotropic yield surface are shown to significantly affect the predicted limit strains. A necking-band bifurcation analysis is also carried out to reveal in an explicit manner the remarkable sensitivity of overall forming limit diagram shapes to the parameters in the anisotropic yield function.

On crystal plasticity FLD analysis

This paper is concerned with the computation of forming limit diagrams (FLDs) using a rate-sensitive polycrystal plasticity model together with the Marciniak–Kuczynski 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 needs only to be applied to two polycrystalline aggregates, one inside and one outside the band. Each grain is modelled as an FCC crystal with 12 distinct slip systems. The response of an aggregate comprised of many grains is based on an elastic–viscoplastic Taylor‐type polycrystal model developed by Asaro and Needleman (1985). The effects of initial imperfection intensity and orientation, initial distribution of grain orientations, crystal elasticity, strain rate sensitivity, single slip hardening and latent hardening on the FLD are discussed in detail. The predicted FLD is compared with experimental data for an aluminium alloy sheet.

SIMULATION OF THE BEHAVIOUR OF FCC POLYCRYSTALS DURING REVERSED TORSION

Abstract Taylor-type polycrystal plasticity models with various single slip hardening laws are evaluated by studying the large strain behaviour of FCC polycrystals during reversed torsion. The hardening laws considered include the model of Asaro and Needleman (“Texture Development and Strain Hardening in Rate Dependent Polycrystals,” Acta Metall. (1985), 34 , 1553) as well as a power-law and an exponential version of that, and a more recent model by Bassani and Wu (“Latent Hardening in Single Crystals II. Analytical Characterization and Predictions,” Proc. R. Soc. Lond. (1991), A435 , 21). The material parameters for the various hardening laws are fitted to experimental compression data for copper and then used to predict reversed large strain torsion of tubes. Differences under “free-end” (axially unconstrained) or “fixed-end” (axially constrained) conditions between predictions and experimental observations are discussed in detail. In addition to the torque response, the Swift effects upon twist reversal are studied.

Predictions of forming limit diagrams using a rate-sensitive crystal plasticity model

Abstract A rate-sensitive crystal plasticity model is applied directly in conjunction with the Marciniak-Kuczynski approach to predict forming limit diagrams (FLDs) for annealed FCC sheet metals having various initial textures. Sheets with ideal orientations are first analysed in order to investigate analytically the influence of microstructure and microscopic properties on the FLD. Two ideal orientations of rolling textures, (101) [010] (Goss) and (001) [OTO] (cube), are considered. In the limit of zero material strain-rate senstivity (m → 0), a simple analytic expression for the localized necking strains is obtained. It indicates that localized necking occurs when the total accumulated shear over all slip systems in the band of localized deformation reaches a critical value. The FLDs of annealed sheets having arbitrary initial textures are then determined numerically. Three particular initial textures are considered. Among these, the predicted FLD for the R-type initial texture shows the best agreement with the experimental ones. Finally, the effects of various factors on the forming limit diagrams are discussed such as: slip hardening, strain-rate sensitivity, initial imperfections, initial texture and yield surface shape.

On the stability of the ideal orientations of rolling textures for F.C.C. polycrystals

Abstract The stability of the ideal orientations of rolling textures for f.c.c. metals (cube, Goss, brass, copper, Taylor and S) and their influence on texture formation are investigated. The characteristics of the three-dimensional lattice rotation fields at and in the vicinity of these orientations, as well as the development of preferred orientations during deformation, are simulated numerically using a rate-sensitive crystal plasticity model. Three types of boundary conditions are considered: plane strain compression, “lath” compression, and “pancake” compression. The investigation shows that, with the help of the three-dimensional rotation fields, the rotation characteristics of an orientation can be described by its rate of change in Euler space, the corresponding gradients and the relative rate of change of its ODF intensity. The formation of rolling textures depends on the characteristics of the α (Goss-brass) and β fibres. Boundary conditions affect the composition of β and the flow velocity of orientations towards and along α and β, and consequently result in different texture components after large deformation.

Large strain shear and torsion of rate-sensitive FCC polycrystals

Abstract The behaviour of FCC polycrystals subjected to large strain uniform simple shear is first investigated. A rate-sensitive polycrystal model is used to compute to compute the resulting axial and shear stresses on the polycrystal, and to simulate the texture evolution. Both “full constraint” and “relaxed constraint” versions of this model are considered. A rigorous analysis is then carried out to extend the simple shear results to the case of axially-constrained solid circular bars in torsion, where it is assumed that the solid bar has the same polycrystalline properties as the element analysed for simple shear. It is shown that the predicted trends for solid bar torsion can differ significantly from those for uniform simple shear. The simulated textures obtained with the full constraint polycrystal model are seen to be in better agreement with experimental textures for copper than those predicted by the relaxed constraint model.

A mesoscopic approach for predicting sheet metal formability

A mesoscopic approach for constructing a forming limit diagram (FLD) is developed. The approach is based on the concept of a unit cell. The unit cell is macroscopically infinitely small and thus represents a material point in the sheet, and is microscopically finitely large and thus contains a sufficiently large number of grains. The responses of the unit cell under biaxial tension are calculated using the finite element method. Each element of a mesh/unit cell represents an orientation and the constitutive response at an integration point is described by the single crystal plasticity theory. It is demonstrated that the limit strains are the natural outcomes of the mesoscopic approach, and the artificial initial imperfection necessitated by the macroscopic M–K approach is not relevant in the mesoscopic approach. The effects of strain-rate sensitivity, single slip hardening and latent hardening, texture evolution, crystal elasticity and spatial orientation distribution on necking are discussed. Numerical results based on the mesoscopic approach are compared with experimental data.

Finite element analysis of localization in FCC polycrystalline sheets under plane stress tension

Abstract Localization phenomena in thin sheets subjected to plane stress tension are investigated. The sheet is modelled as a polycrystalline aggregate, and a finite element analysis based on rate-dependent crystal plasticity is developed to simulate large strain behaviour. Accordingly, each material point in the specimen is considered to be a polycrystalline aggregate consisting of a large number of FCC grains. The Taylor model of crystal plasticity theory is assumed. This analysis accounts for initial textures as well as texture evolution during large plastic deformations. The numerical analysis incorporates certain parallel computing features. Simulations have been carried out for an aluminum sheet alloy, and the effects of various parameters on the formation and prediction of localized deformation (in the form of necking and/or in-plane shear bands) are examined.

A modified model for simulating latent hardening during the plastic deformation of rate-dependent FCC polycrystals

A strain hardening model for the plastic deformation of rate-dependent FCC crystals is proposed based on experimental observations previously reported for single crystals. This model, which is an extension of that employed by Peirce et al. [1983], includes both the self-hardening and latent hardening of the slip systems. The differential hardening of the latent systems is assumed to arise from the interaction between glide dislocations and forests. With this hardening model and a rate-sensitive crystal plasticity theory, the deformation behavior of FCC polycrystals can be predicted from the deformation response of the constituent single crystals. As examples, the uniaxial tensile behaviour of pure aluminum and copper polycrystals is simulated using the extended model, and the results are compared with published experimental data. The effects of latent hardening on polycrystal deformation, especially on flow stress and the formation of tensile textures, are discussed.