Mechanical response and texture evolution of AZ31 alloy at large strains for different strain rates and temperatures

In order to study the behavior of material under finite deformation at various strain rates, the responses of AZ31 Mg sheet are measured under uniaxial (tension and compression) and multiaxial (simple shear) loadings along rolling direction (RD), 45° to rolling direction (DD), 90° to rolling direction (TD), and normal to the sheet (ND) to large strains. The material exhibits positive strain rate sensitivity (SRS) at room and elevated temperatures; the SRS is more pronounced at high temperatures and lower strain rates. The r-value of the material under tensile loading at room temperatures is higher in TD at lower strain rate. Texture measurements on several failed specimens are reported under tension and simple shear after finite plastic deformation of about 20% equivalent strain. The as-received material exhibits a strong fiber with equal fractions of grains having the c-axis slightly tilted away from the sheet normal towards both +RD and −RD. Pole figures obtained after tensile loading along the rolling direction (RD) show that the texture of the material strengthens even at low strains, with c-axis perpendicular to the sheet plane and prism planes lining up in a majority of grains. However, the tensile loading axis along TD does not lead to similar texture strengthening; the c-axis distribution appears to be virtually unchanged from the virgin state. The pole figures obtained after in-plane compression along RD brings the c-axes of the grains parallel to the loading direction. The pole figures after simple shear loading show that the c-axis rotates to lie on the sheet plane consistent with a compression axis 45° away on the sheet plane.     

Forming simulation of aluminum sheets using an anisotropic yield function coupled with crystal plasticity theory

This paper describes the application of a coupled crystal plasticity based microstructural model with an anisotropic yield criterion to compute a 3D yield surface of a textured aluminum sheet (continuous cast AA5754 aluminum sheet). Both the in-plane and out-of-plane deformation characteristics of the sheet material have been generated from the measured initial texture and the uniaxial tensile curve along the rolling direction of the sheet by employing a rate-dependent crystal plasticity model. It is shown that the stress–strain curves and R-value distribution in all orientations of the sheet surface can be modeled accurately by crystal plasticity if a “finite element per grain” unit cell model is used that accounts for non-uniform deformation as well as grain interactions. In particular, the polycrystal calculation using the Bassani and Wu (1991) single crystal hardening law and experimental electron backscatter data as input has been shown to be accurate enough to substitute experimental data by crystal plasticity data for calibration of macroscopic yield functions. The macroscopic anisotropic yield criterion CPB06ex2 (Plunkett et al., 2008) has been calibrated using the results of the polycrystal calculations and the experimental data from mechanical tests. The coupled model is validated by comparing its predictions with the anisotropy in the experimental yield stress ratio and strain ratios at 15% tensile deformation. The biaxial section of the 3D yield surface calculated directly by crystal plasticity model and that predicted by the phenomenological model calibrated with experimental and crystal plasticity data are also compared. The good agreement shows the strength of the approach. Although in this paper, the Plunkett et al. (2008) yield function is used, the proposed methodology is general and can be applied to any yield function. The results presented here represent a robust demonstration of implementing microscale crystal plasticity simulation with measured texture data and hardening laws in macroscale yield criterion simulations in an accurate manner.     

Texture development and plastic anisotropy of B.C.C. strain hardening sheet metals

A Taylor-like polycrystal model is adopted here to investigate the plastic behavior of body centered cubic (b.c.c.) sheet metals under plane-strain compression and the subsequent in-plane biaxial stretching conditions. The <111> pencil glide system is chosen for the slip mechanism for b.c.c. sheet metals. The {110} <111> and {112} <111> slip systems are also considered. Plane-strain compression is used to simulate the cold rolling processes of a low-carbon steel sheet. Based on the polycrystal model, pole figures for the sheet metal after plane-strain compression are obtained and compared with the corresponding experimental results. Also, the simulated plane-strain stress—strain relations are compared with the corresponding experimental results. For the sheet metal subjected to the subsequent in-plane biaxial stretching and shear, plastic potential surfaces are determined at a given small amount of plastic work. With the assumption of the equivalence of the plastic potential and the yield function with normality flow, the yield surfaces based on the simulations for the sheet metal are compared with those based on several phenomenological planar anisotropic yield criteria. The effects of the slip system and the magnitude of plastic work on the shape and size of the yield surfaces are shown. The plastic anisotropy of the sheet metal is investigated in terms of the uniaxial yield stresses in different planar orientations and the corresponding values of the anisotropy parameter R, defined as the ratio of the width plastic strain rate to the through-thickness plastic strain rate under in-plane uniaxial tensile loading. The uniaxial yield stresses and the values of R at different planar orientations from the polycrystal model can be fitted well by a yield function recently proposed by Barlat et al. (1997b).     

On the evolution of lattice deformation in austenitic stainless steels—The role of work hardening at finite strains

In this work, a three dimensional crystal plasticity-based finite element model is presented to examine the micromechanical behaviour of austenitic stainless steels. The model accounts for realistic polycrystal micromorphology, the kinematics of crystallographic slip, lattice rotation, slip interaction (latent hardening) and geometric distortion at finite deformation. We utilise the model to predict the microscopic lattice strain evolution of austenitic stainless steels during uniaxial tension at ambient temperature with validation through in situ neutron diffraction measurements. Overall, the predicted lattice strains are in very good agreement with those measured in both longitudinal and transverse directions (parallel and perpendicular to the tensile loading axis, respectively). The information provided by the model suggests that the observed nonlinear response in the transverse {200} grain family is associated with a competitive bimodal evolution of strain during inelastic deformation. The results associated with latent hardening effects at the microscale also indicate that in situ neutron diffraction measurements in conjunction with macroscopic uniaxial tensile data may be used to calibrate crystal plasticity models for the prediction of the inelastic material deformation response.     

Anisotropy of martensitic transformations in modeling of shape memory alloy polycrystals

Based on the knowledge of the anisotropy associated with the martensitic transformations obtained from tension/compression experiments with oriented CuAlNi single crystals, a simple constant stress averaging approach is employed to model the SMA polycrystal deformation behaviors. Only elastic and inelastic strains due to the martensitic transformation, variant reorientations in the martensite phase and martensite to martensite transformations in thermomechanical loads are considered. The model starts from theoretical calculation of the stress-temperature transformation conditions and their orientation dependence from basic crystallographic and material attributes of the martensitic transformations. Results of the simulations of the NiTi, NiAl, and Cu-based SMA polycrystals in stress–strain tests are shown. It follows that SMA polycrystals, even with randomly oriented grains, typically exhibit tension/compression asymmetry of the shape of the pseudoelastic σ−ε curves in transformation strain, transformation stress, hysteresis widths, character of the pseudoelastic flow and in the slope of temperature dependence of the transformation stresses. It is concluded that some macroscopic features of the SMA polycrystal behaviors originate directly from the crystallography of the undergoing MT's. The model shows clearly the crystallographic origin of these phenomena by providing a link from the crystallographic and material attributes of martensitic transformations towards the macroscopic σ−ε−T behaviors of SMA polycrystals.     

Rate-dependent quasi-flow corner theory for elastic/visco-plastic materials

A rate-dependent quasi-flow plastic constitutive model with punch-speed sensitivity is proposed for the large-deformation sheet metal forming process, which is based on the quasi-flow corner theory and U–L formulation for the virtual work-rate equation. Three kinds of constitutive theories with strain rate dependence, classical flow theory, deformation theory with rate form obeying non-orthogonality rule, and the present quasi-flow corner theory, are introduced into the U–L finite element formulation to simulate the deformation localization processes of plane strain tension in order to investigate effects of strain rate sensitivity on the localizing deformation characters. Furthermore, three kinds of typical forming processes sheet metals, one being an uniaxial stretching and another being a square cup drawing with circular blank, and third being a deep drawing of an oil pan, actual industrial forming part, are also numerically simulated by the present model and compared with experimental results. Good agreement between numerical simulation and experimental ones exhibits the validity of the quasi-flow corner theory.     

Microstructural effects on yield surface evolution in cubic metals using the viscoplastic ?-model

Polycrystalline yield surfaces of metals are a good way to characterize the anisotropy of plastic deformation. The evolution of these surfaces is impossible to accurately reproduce without taking into account the evolution of the material microstructure such as texture development. In this paper, a numerical computation of yield surfaces using the viscoplastic ?-model is proposed. Results concerning face-centered cubic metals subjected to a plane strain compression test are presented. The influence of several mechanical parameters (strain hardening, strain rate sensitivity coefficient and accumulated deformation) on subsequent yield surfaces evolution is studied. The analysis of the change in the shape and size of the yield surfaces shows that the results depend strongly on the parameter ? which controls the strength of the interactions in the polycrystal. In addition, the predictions are compared to the widely used viscoplastic self-consistent model as well as to experimental yield loci taken from the literature for various aluminum alloy sheets. A fairly good qualitative agreement between our ?-model results and the experimental ones is found. The probable links between the parameter ? and the microstructural features such as the stacking fault energy and the grain size of the polycrystal are also briefly discussed.     

Multiaxial yield behavior of 1100 aluminum following various magnitudes of prestrain

The multiaxial yield and flow behavior of metals has been of interest for many years. Recently, the experimental work of Phillips & Lee [1979], Shiratori et al. [1979] and Ohashi [1982] has been quite notable in this field. These authors have concentrated their efforts in measuring yield loci after small to moderate prestrains (≤0.06). In this paper we discuss small strain yield loci we have measured after prestrains between 0.03 and 0.05 in torsion. These experiments on 1100 aluminum are in general agreement with the literature. They show a translation, distortion and expansion of the yield loci. A rounded nose forms in the direction of prestrain with the yield locus flattering opposite the prestrain. We observed that the distortions change to match the strain direction after very small reversals in prestrain.The subsequent yield locus has also been measured after a large torsional prestrain of γ=0.5. Using a 5 × 10^?6 offset criterion for yielding, the shape, distortion and translation of the yield locus was very similar to that found after the smaller prestrains. In addition a large-strain yield locus, using a back extrapolation technique, was determined for the same sample. This yield locus exhibited close to von Mises isotropic expansion. The observed deviations, while slight are extremely important. They match those predicted by a polycrystal slip model. Thus, the small-strain yield locus, after a large prestrain, appears to be determined largely from dislocation considerations only, where as the large-strain yield locus is determined by the developing texture. Finally, aluminum sheet was deformed by rolling to larger prestrains ?^{von Mises} = 0.5, 1.0, 1.5, 2.0 and 2.5 and subsequently tested in plane strain compression. Two types of compression experiments were done, one such that there was no deformation mode change from rolling, the other rotating the direction of zero strain by 90° producing a stress path change. The large strain yield and flow behavior of these experiments was again predicted using the relaxed constraint polycrystal model of Kocks & Canova [1981]. For these very large prestrains the experiments and texture theory differ. Micrstructural observations have shown the presence of micro-shear bands which resulted from the rolling prestrain. We speculate that these features are responsible for the deviation from crystal plasticity theory.We believe that this work points to several operative mechanisms of deformation. Small-strain yielding (5 × 10^?6) appears to be controlled purely by dislocation mechanisms and interactions even after relatively large prestrains. Large-strain yielding, on the other hand, is controlled by texture after moderate prestrains (at least to γ = 0.5). After large prestrains, obtained by rolling, the experiments deviate from texture based predictions. This is possibly the result of microstructural deformation mechanisms, for example micro-shear bands, playing a role in the deformation process.     

Plastic flow for non-monotonic loading conditions of an aluminum alloy sheet sample

Non-linear deformation paths obtained using uniaxial tension followed by simple shear tests were performed for a 1050-O aluminum alloy sheet sample in different specimen orientations with respect to the material symmetry axes. In order to eliminate the time influence, the time interval between the first and second loading steps was kept constant for all the tests. Monotonic uniaxial tension tests interrupted during loading were used to assess the recovery that takes place during this time. In order to eliminate the influence of the initial plastic anisotropy and to compare the results as if the material hardening was isotropic, the flow stress was represented as a function of the plastic work. The behavior of the material after reloading was analyzed in terms of dislocation microstructure and crystallographic texture evolutions. For more quantitative assessment, the full constraints [Int. J. Plasticity 13 (1997) 75] and visco-plastic self-consistent [Acta Metall. Mater. 41 (1993) 2611] polycrystal models were used to simulate the material behavior in the non-linear deformation paths. Based on experimental and simulation results, the relative contributions of the crystallographic texture and dislocation microstructure evolution to the anisotropic hardening behavior of the material were discussed.     

Crack separation energy-rates for the DBCS model under biaxial modes of loading

A Modified version of the Dugdale-Bilby-Cottrell-Swinden (DBCS) model simulating the effect of plasticity at the tip of a crack in an infinite region was used by kfouri and rice (1978) to calculate the crack separation energy-rate G^Δ corresponding to a finite crack growth step Δa during plane strain mode I crack extension. The loading consisted of a remote uniaxial tension σp applied normally to the plane of the crack. Using Rice's path-independent integral J to characterize the applied load in the crack tip region, and assuming the length R of the crack tip plastic zone to be small compared with the length of the crack, an analytical expression was derived relating the ratios (J/G^Δ) and (2a/R) for small values of (2a/R). The analytical solution was incomplete in itself in that the value assumed in the plastic region of the DBCS model for the normal stress Y acting on the extending crack surfaces was the product of the yield stress in uniaxial tension σ_Y and an unknown parameter C, the value of which depends on the effect of the local hydrostatic stresses in the case of plane strain conditions. The analytical solution was compared with a numerical solution obtained from a plane strain elastic-plastic finite element analysis on a centre-cracked plate (CCP) of material obeying the von Mises yield criterion. The value used for the yield stress was 310 MN/m² and moderate isotropic linear hardening was applied with a tangent modulus of 4830 MN/m². A uniaxial tension σp was applied on the two appropriate sides of the plate. The comparisons showed that the analytical and finite element solutions were mutually consistent and they enabled the value of C to be established at 1.91. In the present paper similar comparisons are made between the analytical solution and the finite element solutions for the CCP of the same material under different biaxial modes of loading. By assuming further that the form of the analytical solution does not depend on the details of the geometry and of the loading at remote boundaries, a comparison has also been made with the results of a finite element analysis on a compact tension specimen (CTS) made of the same material as the CCP. The different values of C obtained in each case are consistent with investigations by other authors on the effect of load biaxiality on crack tip plasticity.     

Numerical modeling of formability of extruded magnesium alloy tubes

In this paper, a constitutive framework based on a rate-dependent crystal plasticity theory is employed to simulate the large strain deformation phenomena in hexagonal closed-packed (HCP) metals such as magnesium. The new framework is incorporated into in-house codes. Simulations are performed using the new crystal plasticity model in which crystallographic slip and deformation twinning are the principal deformation mechanisms. Simulations of various stress states (uniaxial tension, uniaxial compression and the so-called ring hoop tension test) for the magnesium alloy AM30 are performed and the results are compared with experimental observations of specimens deformed at 200 °C. Numerical simulations of forming limit diagrams (FLDs) are also performed using the Marciniak–Kuczynski (M–K) approach. With this formulation, the effects of crystallographic slip and deformation twinning on the FLD can be assessed.     

Study of a drawing-quality sheet steel. I: Stress/strain behaviors and Lankford coefficients by experiments and micromechanical simulations

The sheet-metal industry uses Lankford coefficients and the forming-limit curve, FLC, as standards for characterizing a sheet’s ability to be stretched and deep drawn. Investigators have recently made significant advances in computer codes that predict these measures of formability. However, complete experimental data sets that provide input properties and verification data for the simulations rarely exist for a single material. The current investigation focused on obtaining such data for a single drawing-quality steel sheet. Measurements intended for the calibration and initial verification of the simulation code include uniaxial-tension tests, through-thickness and plane-strain compression experiments, and quantitative texture – orientation distribution function – evaluations, while a comparison between measured and simulated Lankford coefficients, Part I, and an FLC, Part II, provide a rigorous verification of the computer simulations. In order to initially verify the simulations, we performed through-thickness and plane-strain compression measurements. A key experimental result was that the flow curve in free, through-thickness compression – an experiment that corresponds to biaxial stretching – lies 18% above the uniaxial tensile data. The plane-strain compression curve is another 11% above the free-compression stress/strain data. We measured the Lankford coefficients, as a function of angle to the rolling direction, for the same steel sheet, finding the maximum values in and at 90° to the rolling direction, 1.59 and 1.89 respectively. A minimum Lankford coefficient of 1.19 was measured at 45° to the rolling direction. For calibrating a rate-dependent visco-plastic self-consistent polycrystal model we needed only to measure the material’s initial texture and to fit power-law and saturation-hardening laws to our tensile data. This kept the set of adjustable parameters to a minimum. Without other adjustments to the model, we predicted the correct stress levels in the free and channel-die compression experiments as well as values of Lankford coefficients. These successes indicate that the polycrystal model should be capable of simulating the entire FLC, Part II.     

Incorporation of twinning into a crystal plasticity finite element model: Evolution of lattice strains and texture in Zircaloy-2

A crystal plasticity finite element code is developed to model lattice strains and texture evolution of HCP crystals. The code is implemented to model elastic and plastic deformation considering slip and twinning based plastic deformation. The model accounts for twinning reorientation and growth. Twinning, as well as slip, is considered to follow a rate dependent formulation. The results of the simulations are compared to previously published in situ neutron diffraction data. Experimental results of the evolution of the texture and lattice strains under uniaxial tension/compression loading along the rolling, transverse, and normal direction of a piece of rolled Zircaloy-2 are compared with model predictions. The rate dependent formulation introduced is capable of correctly capturing the influence of slip and twinning deformation on lattice strains as well as texture evolution.     

A finite strain elastic-viscoplastic self-consistent model for polycrystalline materials

A large strain elastic-viscoplastic self-consistent (EVPSC) model for polycrystalline materials is developed. At single crystal level, both the rate sensitive slip and twinning are included as the plastic deformation mechanisms, while elastic anisotropy is accounted for in the elastic moduli. The transition from single crystal plasticity to polycrystal plasticity is based on a completely self-consistent approach. It is shown that the differences in the predicted stress-strain curves and texture evolutions based on the EVPSC and the viscoplastic self-consistent (VPSC) model proposed by Lebensohn and Tomé (1993) are negligible at large strains for monotonic loadings. For the deformations involving unloading and strain path changes, the EVPSC predicts a smooth elasto-plastic transition, while the VPSC model gives a discontinuous response due to lack of elastic deformation. It is also demonstrated that the EVPSC model can capture some important experimental features which cannot be simulated by using the VPSC model.     

Numerical failure analysis of a stretch-bending test on dual-phase steel sheets using a phenomenological fracture model

Advanced High Strength Steels (AHSS) are increasingly used in automotive industry due to their superior strength and substantial weight advantage. However, their compromised ductility gives rise to numerous manufacturing issues. One of them is the so-called ‘shear fracture’ often observed on tight radii during stamping processes. Since traditional approaches, such as the Forming Limit Diagram (FLD), are unable to predict this type of fractures, great efforts have been made to develop failure criteria that could predict shear fractures. In this paper, a recently developed Modified Mohr–Coulomb (MMC) ductile fracture criterion (Bai and Wierzbicki, 2010) is adopted to analyze the failure behavior of a Dual Phase (DP) steel sheet during stretch-bending operations. The plasticity and ductile fracture of the present sheet are fully characterized by a Hill’48 orthotropic model and a MMC fracture model, respectively. Finite element models with three different element types (3D, shell and plane strain) were built for a Stretch Forming Simulator (SFS) test (Shih and Shi, 2008), numerical simulations with four different R/t values (die radius normalized by sheet thickness) were performed. It has been shown that the 3D and shell element simulations can predict failure location/mode, the upper die load–displacement responses as well as wall stress and wrap angle at the onset of fracture for all R/t values with good accuracy. Furthermore, a series of parametric studies were conducted on the 3D element model, and the effect of tension level (clamping distance), tooling friction, mesh size and fracture locus on failure modes and load–displacement responses were investigated.     

Effects of texture on shear band formation in plane strain tension/compression and bending

In this study, effects of typical texture components observed in rolled aluminum alloy sheets on shear band formation in plane strain tension/compression and bending are systematically studied. The material response is described by a generalized Taylor-type polycrystal model, in which each grain is characterized in terms of an elastic–viscoplastic continuum slip constitutive relation. First, a simple model analysis in which the shear band is assumed to occur in a weaker thin slice of material is performed. From this simple model analysis, two important quantities regarding shear band formation are obtained: i.e. the critical strain at the onset of shear banding and the corresponding orientation of shear band. Second, the shear band development in plane strain tension/compression is analyzed by the finite element method. Predictability of the finite element analysis is compared to that of the simple model analysis. Third, shear band developments in plane strain pure bending of a sheet specimen with the typical textures are studied. Regions near the surfaces in a bent sheet specimen are approximately subjected to plane strain tension or compression. From this viewpoint, the bendability of a sheet specimen may be evaluated, using the knowledge regarding shear band formation in plane strain tension/compression. To confirm this and to encompass overall deformation of a bent sheet specimen, including shear bands, finite element analyses of plane strain pure bending are carried out, and the predicted shear band formation in bent specimens is compared to that in the tension/compression problem. Finally, the present results are compared to previous related studies, and the efficiency of the present method for materials design in future is discussed.     

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.     

Multiaxial constitutive model accounting for the strength-differential in Inconel 718

The nickel-base alloy Inconel 718 exhibits a strength-differential, that is, a different plastic flow behavior in uniaxial tension and uniaxial compression. A phenomenological viscoplastic model founded on thermodynamics has been extended for material behavior that deviates from classical metal plasticity by including all three stress invariants in the threshold function. The model can predict plastic flow in isotropic materials with or without a flow stress asymmetry as well as with or without pressure dependence. Viscoplastic material parameters have been fit to pure shear, uniaxial tension, and uniaxial compression experimental results at 650°°C. Threshold function material parameters have been fit to the strength-differential. Four classes of threshold functions have been considered and nonproportional loading of hollow tubes, such as shear strain followed by axial strain, has been used to select the most applicable class of threshold function for the multiaxial model as applied to Inconel 718 at 650 °C. These nonproportional load paths containing corners provide a rigorous test of a plasticity model, whether it is time-dependent or not. A J₂J₃ class model, where J₂ and J₃ are the second and third effective deviatoric stress invariants, was found to agree the best with the experimental results.