Hitoshi Moritoki

The criterion for central bursting and its occurrence in drawing and extrusion under plane strain

Central bursting is dealt with, with respect to the collapse of the unique solution. In general, there are two cases bringing about multiplicity: one is a statical multiplicity and the other is a kinematical one. The latter corresponds to localized necking. These two multiplicities occur simultaneously in plane strain processes. In general, it is possible for localized necking to take only two modes, but only one mode can be realized in a plane strain process. The criterion for central bursting is given as (ϵ1/n)(p/2k) = −12, with p as average pressure, k as shear strength. ϵ1 as the tensile strain in forming direction, and n as work hardening exponent. The influences of die geometry and manufacturing conditions on the central bursting are examined.

Free surface ductility in upsetting

Abstract Cold upsetting of a circular cylinder is often used to evaluate the cold forgeability of forming materials. The index of the forgeability is the critical strain at which a crack occurs on the expanding free surface of the upsetting cylinder. The forming limit due to the cracking is analyzed by a method based on the sufficient condition for the collapse of the unique solution. The modes of cracking are determined by comparing the stabilities of deformation from the forming limit to the two modes able to permit strain rate discontinuityon the plane of localized necking. Numerous experiments published in the literature have shown a linear relationship with a slope of −0.5 in circumferential strain vs axial strain at the forming limit. The forming limit and the mode predicted are in very good agreement with the experimental results.

Criterion and mode of the forming limit in sheet forming

Abstract In this paper, the forming limit of sheet metals is examined, with respect to the collapse of the unique solution. In general, there are two cases bringing about multiplicity: one is statically-admissible multiplicity and the other is kinematically-admissible multiplicity. The multiplicities are considered to correspond to plastic instability. After the appearance of statical instability, the strain path cannot be controlled as freely as would be preferred. Ultimately, localized necking occurs when the process satisfies the condition of kinematical instability, the mode of which is consistent with one of the two modes required from the condition permitting strain-rate discontinuity under the continuity of velocity. These instabilities are compared with the diffuse necking of Swift and with the localized necking of Hill from the standpoints of criterion and mode. The dependency of formabilities on the strain path is discussed in terms of several assumed strain paths.

Prediction of central bursting in extrusion

Abstract Central burst defects which are also called chevron cracks, in extruded products are analyzed. The criterion based on plastic instability was used in order to predict central bursting. It satisfies sufficient condition s ˙ i ɛ ˙ i = 0 for the multiplicity of the solution, which is derived from the necessary condition, Δ s ˙ i Δ ɛ ˙ i = 0 where s ˙ i denotes the nominal stress rate, and Δ the difference of any two multiple solutions. The criterion gives the limit strain identical to that derived from Swifts criterion (diffuse necking) under plane stress condition, but conceptually it is entirely different from Swifts. Stress and strain on the specimen were evaluated using the finite element method (FEM). We proposed a measure y M which represents the risk probability of cracking and calculated it using the stress and strain on the central axis of the product. When y M ≤0, bursting occurs. The value of y M is shown in the range of combinations of the die half angle and the reduction area. Boundary curve that separates the defect zone from the non-defect zone in the relationship with respect to the above combinations takes a nose-like shape, which is quite different from that based on Avitzurs prediction.

Intrinsic criterion of plastic instability

Abstract Hitherto the approaches on plastic instability have been carried out mainly in evaluating sheet formability. The necking in press forming is a phenomenon relating to plastic instability. It is classified into diffuse and localized neckings. Swift [1] proposed the criterion of diffuse necking, and for localized necking Storen and Rice [3] presented the modified criterion of Hills [2] in order to make it possible to predict the formability in biaxial stretching. Since there is some imperfections in these criteria and they do not provide sufficient conditions for the emergence of necking phenomena, they must be improved. This paper presents comprehensive criteria based on the multiplicity of solution.

Criterion of plastic instability and its application for predicting ductility in metal forming

Abstract General criteria for ductility based on plastic instability are presented, and unreasonable treatments in the criteria proposed by Swift [1] and by Storen and Rice [2] are pointed out. It is shown that the correction of these imperfections makes their criterion consistent with general criteria proposed here. Then, the applications of the criteria to the prediction of cracking in several metal formings are demonstrated: the formability under linear strain paths in sheet metal forming, the free surface ductility and cracking mode in upsetting, and the central bursting (chevron crack) in drawing and extrusion. The analytical prediction in a wide range of processes are found to be in very good agreement with experimental results.

Influence of Deviatoric Yield Curve upon Plastic Deformation Behaviour in Soil

So far, as a yield criterion on a deviatoric plane, Mises criterion has been widely used in soil as well as in metal in the prediction of plastic deformation behaviour. However, the criterion similar to Mohr-Coulomb fits well in soil rather than Mises criterion. Then, in our simulation of deformation behaviour, we took up the criterion obtained from the stress state at fracture and referred to the experimental results given by Lade & Duncan to estimate the deformation behaviour. In their experiment, compression tests were performed within the range from one dimensional compression to equi-two dimensional compression. On the criterion, J2 is represented as a function of J3 where J2 and J3 are the second and the third invariants of deviatoric stress respectively. In this paper, first, the method for including the criterion into analysis is introduced. Second, the strain developments in these compression processes are simulated and compared with the experimental values. The results in simulation are not always agree with experimental ones. However, stable limits in these compressions are situated nearly on a straight line (a critical line) in the yield plane composing of hydrostatic pressure and equivalent deviatoric stress. Third, we examine what criterion gives good correlations between simulation and experiment. The dependence of J2 on J3 in the criterion is similar on the whole to that obtained from the fracture state, but in detail, the difference is admitted between them.

Shape of Yield Curves on a Deviatoric Plane in Soil

The failure criterion examined using fracture state says that Mohr-Coulomb criterion is more appropriate to soil than Mises criterion. The important aspect which must be confirmed in the examination of the criterion is that the stress state at fracture are situated on a critical straight line in primary yield relation showing the dependency between equivalent deviatoric stress and hydrostatic pressure. Moreover, the equivalent deviatoric stress must be representative of the magnitude and shape of the yield curve on a deviatoric plane. Therefore, the critical straight line is also affected by failure criterion itself. Such mutual dependency makes it complicated to find out the relevant yield criterion. For the discussion of the yield criterion, we used the data obtained in the compression tests performed by Reades & Green and by Lade & Duncan. First, on the assumption that fracture stress points are situated on the critical line in the primary yield relation, the shape of yield curve on the deviatoric plane is examined. Then, the curve shape is represented with the relation between J2 and J3 where J2 and J3 are the second and the third invariant of deviatoric stress respectively. Second, the equivalent stress is defined using the curve shape relation. Third, it is confirmed that fracture points in primary yield relation are located on the straight line when based on the equivalent stress defined

A Proposal on Plastic Constitutive Representation Incorporating Stress Rate Dependence

A constitutive equation incorporating stress rate dependence which has been used widely is shown to have a limit for its application, especially, in the case that strain rate changes abruptly its direction. Therefore, the constitutive equation does not have the potential for representing such plastic behaviour that stress vector largely delays relative to strain rate vector. Before proposing our constitutive equation, the basic correlation in a five dimensional stress space is dealt with between local geometries of associated paths of stress and strain under plastic deformations obeying a general constitutive equation incorporating stress rate dependence. Using the basic correlation, we proposed a constitutive equation with flexibility applicable for stress delay angle observed experimentally between stress and strain rate directions in many strain trajectories.

Yielding Behaviour in Strain Controlled Compactions of Metal Powder Assembles.

Metal powder assembles show some volumetric change under irreversible deformation, and their yield function (yield curve) is represented in the space of equivalent deviatoric stress q and hydrostatic pressure p. In the mechanics of metal powder assembles yield curves are assumed to be contours of constant density. However, there is something questionable in this concept. We already examined the consistency of the concept referring to the experimental data of tri-axial compression tests carried out in the past by other investigators, who compressed cylindrical specimens subjected to constant lateral pressure. The results showed that the concept of constant density was inconsistent with the actual constitutive behaviour under the assumption that associated flow rule is applicable and yield curve expands in a similar shape with increasing of compressive deformation. Here, we newly referred to the data obtained from compaction tests by Brown and A-Chedid who collected them using uni- and tri-axial strain control systems. In these compaction tests the direction of stress vector (p, q) do not change largely, and so whole view of yield curve is not obtainable. But the correspondence between directions of stress and strain rate vectors can be determined in the compactions. The comparison of the results with those in compression tests does not show good agreement. That suggests the necessity of inspection on the suitability of constitutive model used in the mechanics of metal powder assembles. Hardening strain rate ey is defined here which is composed of volumetric strain rate ep and equivalent deviatoric one eq accompanying with constant parameter. Finally, it is found that this definition is valid from the view point of inherent characteristic of hardening.