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The elastoplastic deformation behaviors of thin-walled tubes made of pure aluminum and steel were measured under various tension–torsion combined loadings. The ratio between the displacement and rotation of a grip was held constant for the linear-loading experiment. In the nonlinear-loading experiments, a specimen was subjected to uniaxial tension followed by the simultaneous application of tension and torsion. It was determined that the associated flow rule predicts the plastic flow behaviors observed in the experiments with sufficient accuracy, provided that the specimen was subjected to linear loadings. Meanwhile, under nonlinear loadings, the plastic flow behaviors were markedly dissimilar to those in the linear loadings, and the direction of plastic flow rotated toward the direction of the stress/strain rate. By analyzing the experimental results, a linear relationship between the plastic flow direction and strain rate direction was determined. Therefore, the experimental data demonstrated that the stress state and the strain rate direction are essential parameters characterizing a plastic flow rule. Eventually, a pseudo-corner model capable of reproducing experimentally observed plastic deformation behavior was proposed.

Elastoplastic responses of rate-independent polycrystals are simulated under linear and various nonlinear strain paths, and the influence of the loading condition on the plastic flow behavior is investigated. It is observed that the shear components of elastoplastic tangent moduli start decreasing when the plastic deformation occurs, and the direction of the plastic flow depends on the direction of the stress/strain rates. A phenomenological plastic flow rule capable of reproducing these behaviors is developed. The flow rule relates the direction of plastic strain rate to the direction of strain rate, and yields the reduction of shear modulus after plastic yielding. The proposed constitutive model is then integrated into a finite element program, and the plane strain tension and pure bending are analyzed. The advantage of the proposed flow rule for simulating the occurrence of a shear band is demonstrated.

The formability of AZ31 sheets was experimentally measured for uniaxial tension, plane strain stretching, and equi-biaxial stretching modes. The limit strain of the specimen was reasonable under uniaxial tension but was drastically reduced in the biaxial stretching modes. Full-field strain analysis and the observation of the fracture surface revealed that the specimen fractured without the formation of a visible local necking. To predict the fracture behavior using a crystal plasticity-based finite element method, a mean field model was developed to solve the mechanical interplay between the parent and twins. The destabilization of the specimen was simulated by introducing fracture criteria as functions of the accumulated slip inside the contraction twin region. Simulations revealed that strong strain heterogeneity forms in the case of the biaxial stretching modes, and this heterogeneity is apparently reduced under uniaxial tension as a result of the activation of prismatic slip, which produces a uniform strain distribution. Contraction twins nucleate at smaller strain levels in the biaxial stretching modes, and premature failure is predicted in the biaxial stretching modes as a result of the accumulation of slip in the twinned regions. Parametric studies show that even when the volume fraction of twins is small, they have a significant impact on the ductility of AZ31 sheets.

Plastic flow rule is a fundamental equation that describes the plastic behavior of polycrystalline aggregate. To clarify plastic response at the point where the loading mode is suddenly changed, uniaxial tension followed by combined tension–torsion was applied to an A3003-O tube. The loading mode was altered at different plastic strains. It was observed that the direction of the plastic strain rate rotated immediately when the loading mode was changed. The greater the deviation in the subsequent loading mode from the uniaxial tension, the higher is the extent of rotation of the direction of plastic strain rate. The extent of instantaneous rotation of the plastic flow direction was the same regardless of the magnitude of the plastic strain at the alteration of the loading mode. Thus, the experimental results showed that the direction of the plastic flow depended on the direction of the stress/strain rate tensor; however, it was not affected by the magnitude of the plastic strain. Then, the effects of the hardening behavior of a slip system on the direction of the plastic flow were examined through a crystal plasticity simulation that considers the latent and kinematic hardenings. The crystal plasticity simulations reproduced the experimentally observed rotation of the plastic flow direction. Moreover, the simulations showed that the latent and kinematic hardenings did not affect the direction of the macroscopic plastic flow, even though they altered the activities of the slip systems. Therefore, it became clear that the plastic flow behavior of the polycrystal depends on the stress state and the direction of the stress/strain rate, and it is not affected by either the magnitude of the plastic strain or the hardening behavior of a slip system.

Strain localization of sheet metals subjected to plane-strain stretching was simulated by finite-element analysis based on a crystal plasticity model. The ratio of specimen thickness to grain size, denoted by Ng, is varied from 1 to 70, and its influence on the evolution of surface roughness and the occurrence of sheet necking is investigated. Roughening of the free surface of the specimen is induced by the grain-scale strain heterogeneity associated with local grain misorientation. The magnitude of surface roughness depends mainly on the grain size and is less sensitive to Ng. As Ng decreases while maintaining the thickness, the magnitude of surface roughness becomes large with respect to the thickness. As a result, large geometrical imperfection is generated. Therefore, the formability of sheet metal is reduced as Ng decreases. Such an effect is found to be particularly considerable when Ng is less than 30. It is also found that the localization mode shifts from a sharp shear banding mode to a localized thinning mode as Ng increases.

A thin-walled tubular specimen of A3003-O is subjected to uniaxial, biaxial, and triaxial stress paths using an axial load-internal pressure-torsion type test machine. For linear multiaxial stress paths, the ratios of axial, circumferential, and shear stresses are kept constant, and the stress–strain relations for various stress paths are measured. The work-hardening behavior of the specimen is evaluated based on the plastic work per unit volume, and contours of equal plastic work are constructed. The shape of the contour changes progressively with increasing plastic strain. Therefore, the amount of work hardening of the specimen depends on the plastic work and the applied stress path. In order to clarify the source of such work-hardening behavior, numerical simulations are performed using the crystal plasticity model. Two hardening models are adopted. In one model, the slip resistance is given as a function of accumulated slip, and, in the other model, the slip resistance is given as a function of dislocation density. The evolution of macroscopic flow stress depends only on the plastic work for the accumulated-slip-based model, and this model cannot predict the experimental trend. On the other hand, the dislocation-density-based model reproduces the stress-path dependent work-hardening behavior observed in the experiments, although quantitative agreement is not fully achieved. In the simulation, the evolution rate of the dislocation density varies depending on the stress path, which is identified as the source of the stress-path-dependent work-hardening behavior.

A novel framework to predict the onset of plastic flow localization is presented. The proposed framework combines a classical strain localization analysis with a homogenization-based finite element method, and has high applicability to various types of material with a characteristic microstructure that may have significant heterogeneity as long as its representative volume element can be represented by a finite element discretization. According to the proposed method, a plastic flow localization analysis can be performed taking only one or two material points in macroscopic analysis. This means that localization analysis of materials involving very complex microstructures, which is hard to be satisfactorily treated in a fully micro-macro-coupled finite element analysis with the homogenization approach, can be carried out with a reasonable computational cost. As a practical application of the proposed general framework, a plane stress version, that is, a Marciniak–Kuczynski-type (M–K) approach, is considered, then the forming limit strains of FCC polycrystalline sheets are evaluated. Crystal plasticity theory is adopted as a constitutive model for each crystal grain, and the homogenization-based finite element method is used to evaluate the average material response to be used in M–K-type sheet necking analysis. A numerical convergence analysis is conducted to determine the appropriate size of the representative volume element in the homogenization, and the effect of the geometrical configuration of crystal grains is studied. Then, the forming limit strains of a textured material are evaluated. The computational results are compared with those obtained using the conventional Taylor-type polycrystalline model.

Plastic deformation characteristics and limit strains are simulated for textured face-centered cubic polycrystalline sheets using a generalized Taylor-type crystal plasticity model. The r- values are predicted to be 1.04, 7.74, and 0.17 for the pseudo-random, {1 1 1}〈 uvw 〉, and {0 0 1}〈 uvw 〉 textures, respectively. The {1 1 1}〈 uvw 〉 texture gives limit strains as large as the random texture, whereas the {0 0 1}〈 uvw 〉 texture yields limit strains apparently higher than the other two, even though its r -value is extremely low. Thus, the r -value cannot act as an indicator of the stretchability of sheet metals. For the {0 0 1}〈 uvw 〉 texture, a superior strain-hardening ability under the plane-strain stretching mode is found to be responsible for the increase in the limit strains under plane strain and equi-biaxial stretching modes. It is concluded that the enhancement of the strain hardening ability for the plane-strain stretching mode is one of the key factors of high stretchability sheets.

We investigated the effects of crystal orientation on the bendability of aluminum alloy sheets by experiments using single crystal specimens and finite element analysis using a crystal plasticity model. In the experimental investigation, single crystal specimens having cube and Goss orientations were made from a coarse-grained Al–Mg–Si alloy sheet. The cube single crystal specimens have shown an excellent bendability regardless of the bending direction. Meanwhile, the bendability of the Goss single crystal specimens strongly depended on the bending direction. The finite element analysis results are remarkably consistent with the experimental results.

Individual grains in a polycrystal rotate during plastic deformation. This leads to a change in the crystallographic texture, and results in an increase or decrease of the macroscopic flow stress of the material. Such a change of strength as a result of grain rotations is called geometrical or texture hardening/softening. In the present study, for textured aluminum alloy sheets, the geometrical hardening/softening effect in the in-plane plane-strain stretching mode is numerically investigated using a generalized Taylor-type polycrystalline model. It is found that the cube texture ( { 1 0 0 } 〈 0 0 1 〉 ) exhibits significant geometrical hardening when the major stretching direction is inclined at 45° relative to the orthotropic axes, and that a cube texture rotated about the normal direction (ND) shows a notable degree of geometrical hardening for any in-plane orientation of the sheet. Using the Marciniak-Kuczynski-type approach, forming limits for these textured sheets are analyzed. It is found that geometrical hardening definitely enhances the formability. It is, therefore, strongly suggested that texture control guided by the present results may be highly effective in producing aluminum alloy sheets with higher formability.

Forming limit stresses of sheet metals subjected to linear and combined stress paths are analyzed using the M-K model in conjunction with two anisotropic work-hardening models: a work-hardening model which is capable of describing Bauschinger and cross-hardening effects, and a work-hardening model which cannot predict the cross-hardening effect. It is found that the forming limit stress is path-independent when the stress–strain curves for the linear and combined stress paths agree well with each other. On the other hand, the forming limit stress for the combined stress path depends on the strain path when the prestrain changes the subsequent stress–strain relation. We conclude that the stress-based forming limit criterion is efficient only for a material with a work-hardening behavior that is not affected by strain path change. The influence of the work-hardening behavior on the forming limit stress is discussed in detail.

The effects of typical texture components observed in rolled aluminum alloy sheets (i.e., copper, brass, S, cube and Goss texture components) on forming limit strains are numerically studied, utilizing the Marciniak–Kuczynski-type approach and a generalized Taylor-type polycrystal model. It is shown that only the cube texture component yields forming limits much higher than those for a random texture in a biaxial stretch range, while the other four texture components tend to yield lower forming limits. Furthermore, it is found that when the orthotropic axes of the cube texture material are inclined at 45° relative to the major stretch direction, abnormally high forming limits are obtained for near plane strain forming paths.

The strain path dependence of forming limit strains and stresses of a steel tube subjected to combined axial load and internal pressure are investigated for linear and combined stress paths using a tension–internal pressure testing machine. The combined stress paths consist of two (first and second) linear stress paths and include unloading between the first and second loadings. The strain hardening behavior of the steel tube for many linear and combined stress paths is observed in terms of equivalent stress–equivalent plastic strain ( σ ¯ – e ¯ ) curves. The forming limit stresses for the linear and some combined stress paths fall on a single curve in stress space irrespective of the stress paths when the corresponding σ ¯ – e ¯ curves observed for the given stress paths are on a single curve. On the other hand, the forming limit stresses for some combined stress paths are lower than those for the linear stress paths when the specimen exhibits a low strain hardening rate immediately after the change in stress paths. It is, therefore, concluded that the forming limit stress of the steel tube is not fully path-independent, and that the path dependence of forming limit stress is strongly affected by the strain hardening behavior of the material for given loading paths.

The effect of changing strain paths on the forming limit stresses of sheet metals is investigated using the Marciniak–Kuczynski 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.

Forming Limit Curves (FLCs) of 5000 series aluminum alloys with different Mg contents, 2.5 and 5.5 mass%, are experimentally determined in detail using the Marciniak test. The forming limit strains of the Al–Mg 5.5% alloy are larger than those of the Al–Mg 2.5% alloy from uniaxial tension to near plane-strain tension regions, whereas the order of formability is reversed in the biaxial stretching region. To evaluate the work-hardening behavior of the test materials under biaxial tension and to determine appropriate anisotropic yield functions, biaxial tensile tests of the test materials are carried out using cruciform specimens. Moreover, forming limit strains of the test materials under proportional loading are calculated using the Marciniak-Kuczynski approach based on a phenomenological plasticity model with non-normality effects [Kuroda and Tvergaard, 2001, J. Mech. Phys. Solids, 49, pp. 1239–1263]. The validity of the phenomenological plasticity analysis is discussed by comparing the observed FLCs with calculated ones.

The anisotropic plastic deformation behavior of extruded 5000 series aluminum alloy tubes, A5154-H112, of 76 mm outer diameter and 3.9 mm wall thickness is investigated, using a servo-controlled tension-internal pressure testing machine. This machine is capable of applying arbitrary stress or strain paths to a tubular specimen using an electrical, closed-loop control system. Detailed measurements were made of the initial yield locus, contours of plastic work for different levels of work-hardening, and the directions of the incremental plastic strain vectors for both linear and combined stress paths. It is found that the measured work contours constructed in the principal stress space are similar in shape, and that the directions of the incremental plastic strain vectors remain almost constant at constant stress ratios. The work-hardening behavior predicted using Hosford's or the Yld2000-2d yield functions under the assumption of isotropic hardening agrees closely with the observations for both linear and combined stress paths. The material is thus found to work-harden almost isotropically. Both yield functions are effective phenomenological plasticity models for predicting the anisotropic plastic deformation behavior of the material.

The quartz crystal microbalance (QCM) technique is capable of detecting small mass changes in the region of nanograms per square centimetre from resonant frequency changes of the quartz crystal. In this study, the QCM technique, combined with electrochemical measurements, was applied to the minute corrosion of iron thin films in deaerated neutral solutions. An iron thin film with a thickness of 200 nm was electroplated on the gold electrode of a quartz crystal. The mass changes of the iron thin film during natural immersion or galvanostatic polarization in deaerated pH 6.48 borate solution, pH 6.42 borate solution with 10−2 M chloride ions, pH 6.0 borate solution with 0.5 M chloride ions and pH 6.48 phosphate solution were measured as a function of time or potential to evaluate the iron dissolution rate or iron dissolution current. The corrosion rate of the iron thin film on natural immersion increased in the order pH 6.48 phosphate > pH 6.0 borate with 0.5 M chloride ions > pH 6.42 borate with 10−2 M chloride ions > pH 6.48 borate solution. The net current flowing through the external circuit during galvanostatic polarization near the corrosion potential was successfully separated into the iron dissolution current and hydrogen evolution current. Tafel plots of the iron dissolution current and hydrogen evolution current were made to evaluate the corrosion mechanism of the iron thin film. The Tafel slopes of iron dissolution and hydrogen evolution thus obtained depend on the electrolyte solutions, from which conclusions can be drawn on the corrosion mechanism.

This lecture emphasizes the usefulness of material modeling to improve the predictive accuracy in the forming simulations of aluminum alloy sheets, and consists of part I and II. In part I, we present a numerical simulation of sheet necking on the basis of an elastic-viscoplastic crystal plasticity model. In this model, a sheet possessing an initial imperfection in the form of a reduced thickness band is postulated, and the growth of the band caused by the strain localization is analyzed. Our attention is focused on the impact of the r-value for the stretchability of aluminum alloy sheet. In part II, we demonstrate the effect of phenomenological material modeling on the predictive accuracy of finite element analysis (FEA). It is concluded that the biaxial tensile testing method using a cruciform specimen is an effective material testing method for accurately detecting and modeling the deformation behavior of sheet metals under biaxial tension.

Plastic deformation characteristics and limit strains are simulated for textured face‐centered cubic polycrystalline sheets using a generalized Taylor‐type crystal plasticity model. The r‐values are predicted to be 1.04, 7.74 and 0.17 for the random, {111}〈uvw〉 and {001}〈uvw〉 textures, respectively. The {111}〈uvw〉 texture gives limit strains as large as the random texture, whereas the {001}〈uvw〉 texture yields limit strains evidently higher than the other two even though its r‐value is extremely low. Thus, the r‐value cannot act as an indicator to the stretchability of sheet metals. For the {001}〈uvw〉 texture, a superior strain‐hardening ability under plane‐strain stretching mode is found to be responsible for the increase in the limit strains under plane‐strain and equi‐biaxial stretching modes. We conclude that the enhancement of the strain‐hardening ability for plane strain mode is one of the key factors for high stretchability sheets.

Work-hardening behavior of single-phase steel and dual-phase steel which is made of hard martensite surrounded by soft ferrite is analyzed by using an elastoplastic crystal plasticity model in conjunction with the incremental self-consistent model. Two-stage loading paths consisting of uniaxial tension, unloading and subsequent uniaxial tension/compression for various directions are applied. Bauschinger effect and transitional re-yielding behavior, which depends on the direction of the second loading path, are predicted and analyzed with respect to the distribution of the residual resolved shear stresses within the material. These features, which are caused by the inhomogeneity of the residual stress field, are especially pronounced in the case of the dual-phase steel because of the strong mechanical contrast between ferrite and martensite phases.

Forming limit strains of 270 MPa grade cold rolled steel sheet have been precisely measured under linear and combined strain paths using a newly developed, double‐action punch‐stretching testing apparatus. In the combined strain paths, the test material is subjected to equibiaxial tension followed by plane‐strain tension. It is found that the forming limit strains are path‐dependent and that those measured for linear strain paths are higher than those for the combined strain paths. Furthermore, in order to check the path‐independency of the forming limit stresses [Arrieux, R., Bedrin, C. and Bovin, M.: Proc. 12th IDDRG Congress, (1982), 61–71], forming limit stresses of the test material have been calculated for the linear and combined strain paths using the numerical method proposed by Stoughton [Stoughton, T. B.: Int. J. Mech. Sci., 42 (2000), 1–27]. The forming limit stresses calculated from the combined strain paths have almost coincided with those determined for linear strain paths. The FLSC concept is therefore valuable for predicting the failure of metal sheets in plane stress states, particularly in multistage forming.

The effects of changing strain paths on forming limit stresses of sheet metals are investigated using the Marciniak‐Kuczynski model. Forming limits are analyzed for proportional loading and two types of combined loadings: combined loading which includes unloading between the first and second loadings and that which includes an abrupt strain path change without unloading between the first and second loadings. The forming limit stress curves in stress space calculated for the combined loading with unloading are in good agreement with those calculated for the proportional loading, while the forming limit curves in strain space are strongly dependent on the strain paths. The forming limit stresses calculated for combined loading with an abrupt strain path change, however, do not coincide with those calculated for proportional loading. The strain path dependence of the forming limit stresses is discussed in detail.