Toshihiko Kuwabara

Measurement and analysis of differential work hardening in cold-rolled steel sheet under biaxial tension

Abstract Biaxial tensile tests of cold-rolled steel sheet were carried out using newly designed cruciform specimens. The specimens were deformed under linear loading paths in a servo-controlled biaxial tensile testing machine. The maximum equivalent strain attained was 0.04. Plastic orthotropy remained coaxial with the principal stresses throughout every experiment. However, the successive contours of plastic work in biaxial stress space changed their shapes progressively, exemplifying a phenomenon which has been termed differential work hardening by Hill and Hutchinson (Trans. ASME J. Appl. Mech. 59 (1992) 1) and by Hill et al. (Int. J. Solids Struct. 31 (1994) 2999). The geometry of the entire family of the work contours was compared with the yield loci calculated from several existing yield criteria. Hill’s quadratic yield criterion overestimated the measured work contours; in particular, in the neighborhood of balanced biaxial tension, the discrepancy was large, while the other yield criteria described the behavior of the work contours well. The only yield criterion that could describe the general trends of the work contours as well as the in-plane r -value distribution with good accuracy was Gotoh’s biquadratic yield criterion. Moreover, it was observed that the components of an increment of logarithmic plastic strain are proportional to the components of the associated normal to the current work contour in stress space. Accordingly, it appears that the work contours act instantaneously as plastic potentials, at least under linear loading paths.

Use of abrupt strain path change for determining subsequent yield surface : Experimental study with metal sheets

A basic idea for a method for determining the subsequent yield surface in the vicinity of a current loading point by using an abrupt strain path change has been proposed recently by Kuroda and Tvergaard (Acta mater., 1999, 47, 3879). The proposed method is applied to real experimental studies. In a biaxial tensile testing apparatus, a cruciform specimen is used, with the strains measured by a biaxial-strain gauge. Then, with the hydraulic pressure of two sets of opposing hydraulic cylinders servo-controlled independently, the testing apparatus can be used to prescribe an abrupt change of the strain path. Both a cold-rolled steel sheet and an aluminum alloy sheet are investigated. The differences between the yield surface shapes found by the strain path change procedure and the shapes found by probing the yield points from the elastic region are shown and discussed for different cases.

Measurement and analysis of yield locus and work hardening characteristics of steel sheets wtih different r-values

Abstract Biaxial tensile experiments of six kinds of steel sheet with different r-values are carried out using a servo-controlled biaxial tensile testing machine. Successive contours of plastic work in the biaxial stress space and the plastic strain rate vectors are precisely measured for linear loading paths. The measured data are compared with the theoretical predictions based on the Taylor–Bishop–Hill model with the full constraints and relaxed constraints assumptions using the experimentally determined crystallographic textures. Comparisons with the Hill quadratic and Hosford yield criteria are also made. It is found that (1) the TBH model with the full constraints assumption is superior to that with the relaxed constraints assumption in predicting the plastic deformation characteristics of steel sheets; (2) The Hosford yield criterion is an effective phenomenological model for predicting the plastic deformation characteristics of steel sheets.

Non-quadratic yield criterion for orthotropic sheet metals under plane-stress conditions

The paper presents a new yield criterion for orthotropic sheet metals under plane-stress conditions. The criterion is derived from the one proposed by Barlat and Lian (Int. J. Plasticity 5 (1989) 51). Three additional coefficients have been introduced in order to allow a better representation of the plastic behaviour of the sheet metals. The predictions of the new yield criterion are compared with the experimental data for an aluminium alloy sheet and a steel sheet.

PC-based blank design system for deep-drawing irregularly shaped prismatic shells with arbitrarily shape flange

Abstract A method for determining optimum blank shapes for the production of irregularly shaped prismatic shells with an arbitrarily shaped flange and stepped bottoms is proposed. All of the procedures for calculation are incorporated into a CAD system that can output the optimum blank shape in a few seconds using a personal computer. The calculation method is based on the slip-line field theory, a kinematically admissible velocity field for the flange region is determined from the theory and the calculated velocity field is used to conduct a backward calculation of the metal flow, from the designed flange shape to the initial blank shape. In the calculation it is assumed that the material is isotropic and that the thickness of the blank does not change during the deep-drawing process. A deep-drawing experiment is conducted for a square shell with six different flange shapes, using aluminum alloy sheet A5182-0 1.0mm thick and cold-rolled steel sheet SPCE 0.5mm thick. Experimental flange shapes are in good agreement with the designed (predetermined) ones. The influence of material characteristics on the blank design is found to be relatively small.

Shear–band development in polycrystalline metal with strength–differential effect and plastic volume expansion

Experimental studies have shown that the flow stress of some metals is clearly influenced by superimposed hydrostatic pressure. Flow stress increases with the hydrostatic pressure, and consequently it is often observed that the flow stress is clearly larger in a uniaxial compression test than in a uniaxial tension test. This phenomenon is known as the strength–differential effect (SDE). In addition, metals undergoing uniaxial tension and compression show a permanent volume expansion (dilatancy) which is insensitive to the sign of the hydrostatic pressure. In this paper, shear–band development in polycrystalline metal with the SDE and dilatancy is studied, using a rate–dependent crystal–plasticity model with a full three–dimensional, body–centred–cubic, slip–system structure. It is postulated that the appearances of the SDE and dilatancy are consequences of non–Schmid effects existing in each slip system. In the finite–element analyses performed here, each Gaussian integration point in a finite element represents a polycrystal consisting of many of crystal grains having different orientations. As a boundary–value problem, a rectangular specimen subjected to plane–strain tension is considered. Although the finite–element geometry is chosen to be two dimensional, the constitutive model incorporates the full three–dimensional slip–system structure. As a polycrystal model to be embedded in each integration point, an extended Taylor model is employed. Thus, macroscopic manifestations of non–Schmid effects are studied. The influences of hydrostatic stress, internal friction, plastic volume expansion and strain–rate sensitivity on macroscopic shear–band formation in polycrystals are investigated. Results are directly compared with predictions from a classical Schmid–law–based crystal–plasticity theory.

Anisotropy and Formability

The chapter presents synthetically the most recent models of the anisotropic plastic behavior. The first section gives an overview of the classical models, In the next step, the discussion is focused on the anisotropic formulations developed on the basis of the theories of linear transformations and tensor representations, respectively. Those models are applied to different types of materials: body centered, faced centered and hexagonal-close packed metals. A brief review of the experimental methods used for observing and modeling the anisotropic plastic behavior of metallic sheets and tubes under biaxial loading is presented together with the models and methods developed for predicting and establishing the limit strains. The capabilities of some commercial programs specially designed for the computation of forming limit curves (FLC) are also analyzed.

Biaxial Stress Testing Methods for Sheet Metals

A constitutive model capable of accurately reproducing the work-hardening behavior of sheet metals is fundamental to the creation and implementation of accurate numerical simulations of sheet metal forming. This chapter introduces the material testing methods that aid the quantification of the deformation behavior of sheet metals under biaxial compression, biaxial tension, combined tension–compression, and in-plane stress reversal. Examples of material testing and modeling results are also presented.

Advances of Plasticity Experiments on Metal Sheets and Tubes and Their Applications to Constitutive Modeling

This paper provides a review of experimental techniques which are effective in observing and modeling the anisotropic plastic behavior of metal sheets and tubes under a variety of loading paths. These include biaxial compression tests, biaxial stress tests for metal sheets (cruciform specimens) and tubes using closed‐loop electrohydraulic testing machines, an abrupt strain path change method for detecting a yield vertex and subsequent yield loci without unloading, in‐plane compression or stress reversal tests for metal sheets and multistage tension tests. Comparison of observed material response with those predicted using phenomenological plasticity models are presented, where possible. Special attention is focus on the verification of the validity of conventional anisotropic yield criteria and its associated flow rule. The effects of the anisotropic yield criteria on the accuracy of forming simulations, such as springback and forming limit strains and stresses, are also discussed.

Forming Limit Stresses of Sheet Metal under Proportional and Combined Loadings

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.