Moriaki Goya

The influence of void distribution on the yielding of an elastic-plastic porous solid

Abstract In a previous study, the authors proposed an anisotropic yield function for a porous metal. The present investigation extends the previous work by deriving a rate-type constitutive law based on the proposed yield function. A comparison is also made between yield loci derived from the yield function and those determined from an FE model. The FE calculations were performed for a flat plate, with a periodic array of cylindrical holes, under biaxial loading; and also for a porous solid with a periodic array of spherical voids under axial load and superimposed hydrostatic pressure. For the plate, the agreement between the yield function and numerical results was good, less so far the three-dimensional case. Good correspondence can be obtained by introducing a scalar multiplier in the pressure-dependent term in the yield function.

Prediction of limit strain in sheet metal-forming processes by 3D analysis of localized necking

Theoretical prediction of forming limit strain of sheet metal is developed in the framework of the three-dimensional general bifurcation theory. The onset of the three-dimensional discontinuous velocity field in the biaxially stretched uniform sheet is predicted. Three fundamental mode vectors, i.e. shear horizontal, shear vertical and normal modes are introduced and it is demonstrated that any bifurcation mode is represented by the linear combination of them. The onset of the bifurcation is numerically analyzed in terms of the modes by the use of the linear comparison solid originally introduced by Hill in 1959. In this study, a linear constitutive relation is adopted for the linear comparison solid, which is developed based on the constitutive theory proposed by Goya and Ito and is capable of incorporating the directional dependence of the plastic strain rate on the stress rate. The numerical results show that forming limit strains predicted by the three-dimensional mode theory is much higher in general than that given by Storen and Rice in 1975. Then, it is revealed from the three-dimensional mode analysis that the bifurcation mode that arises can be changed from one type to another according to the sign of stress ratio. It is also shown that the strain limit predicted by the three-dimensional mode analysis gives upper limit lines for the bifurcation lines proposed in the past for any linear strain-path directions.

An anisotropic yield function for porous metal

Abstract In this paper a new yield function for porous metals is proposed. In a similar manner to the function developed earlier by Gurson, yielding is shown to be dependent upon the hydrostatic pressure and volume fraction of voids. At the same time the present model allows for the effect of the spatial distribution of the voids. The model is developed assuming a three-dimensional regular array of ellipsoidal voids, but allowing for variations in spacing along each of the coordinate axes. Yield loci are calculated for two specific types of loading. The first case examined was that of biaxial stressing, and the analysis was reduced to the yielding of a plate with circular holes. The second stress state considered was uniaxial tension under a superimposed hydrostatic stress. For simplicity the voids were assumed to be spherical, but with variable spacing along each of the coordinate axes. A comparison is made with predictions from the Gurson model for each type of loading.

Directional dependence of plastic strain-rate vector on stress-rate vector: Numerical experiments based on the Lin's polycrystal model

Abstract The response of plastic strain increment to the various directions of stress increment is calculated numerically based on the Lins polycrystal model. The directional dependence of the plastic strain-rate on the stress-rate is formulated in quite simple expressions from the numerical results. Once the “natural direction” is given where the plastic strain-rate has the same direction with the stress-rate, the above expressions can predict the strain-rate direction for arbitrary stress-rate direction. In a polycrystal the directional dependence of plastic strain-rate does not always have to be related to the corner of the yield surface.

A Model to Describe the Yielding of an Elastic-Plastic Porous Solid

In a previous study, the authors proposed an anisotropic yield function for a porous metal. The present investigation extends the previous work by comparing yield loci derived from the yield function and those determined from a FE model. The FE calculations were performed for a flat plate, with a periodic array of cylindrical holes, under biaxial loading; and also for a porous solid with a periodic array of spherical voids under axial load and superimposed hydrostatic pressure.

Mechanism of Ductile Fracture Originating in Surface Defect and Evaluation of Fracture Ductility. Case of Specimen Having a Single Hole.

Tensile fracture tests of specimens having a single blind hole were carried out in order to investigate the mechanism of ductile fracture originating in a surface defect. Although a crack was initiated at the hole edge, the true stress-true strain curve of the holed specimen approximately coincided with that of the plain specimen with no hole untill the true stress reached a maximum value. In the deformation stage of the specimen after this stress value was reached, the growth behavior of the crack was not affected by the deformation of the hole. On the basis of these phenomena, parameter fac was defined by the ratio of crack area to minimum cross section of the specimen at which the true stress reached the maximum value, and the effect of hole size on the fracture ductility was examined using the parameter fac. The ture strain ec at the point of maximum true stress showed good correlation with the parameter fac, and the shear mode growth of the crack started unstably at this point. The relation between the fracture ductility ef and the parameter fαc was predicted from the relation between ec and fac and from the growth behavior of the crack. This relation coincided with the experimental results, and it was found that the fracture ductility ef of the present experiment is determined mainly by the starting conditions of unstable fracture and the growth behavior of the crack.

Validity of Finite Element Unit Cell Model for Studying Yield Condition of Isotropic Porous Materials.

Assuming a spherical void in an infinite rigid plastic material, Gurson proposed a yield function for isotropic porous solids. It is, however, well known that the Gurson model gives harder response than those predicted by experimentation on actual porous solids. In numerical studies to check the validity of Gursons model, most of the past researchers have introduced a cubic unit cell model, in which a spherical void is placed at the center of the cube. The cubic model can be a good approximation of a porous solid, if the void volume fraction is very small. The cubic model, however, may be not appropriate for the study of porous solids with high ratio of void volume fraction since the model automatically introduces an orthotropy effect due to the geometrically repetitive distribution of voids in three orthogonal axis directions. This research will propose a new unit cell model which is appropriate for the study of the yield functions for isotropic porous materials. The model is also favorable for the study of the anisotropic effect due to the void shape because the unit cell includes less of the anisotropy based on the distribution than the cubic model does.

Comparison Between Numerical and Analytical Predictions of Shear Localization of Sheets Subjected to Biaxial Tension

The classical J2-Flow(J2F) rule results in unrealistic critical loads when applied to plastic bifurcation problems in which stress paths change their directions abruptly at critical points. This unrealistic response is due to the property of the J2F rule that the direction of plastic strain increments is independent of the direction of stress increments. Most of constitutive rules proposed in the past are given through an attempt to loosen the severe restriction on the direction of plastic strain increments in the J2F rule[1,2].

F. E. M. Analysis of Shear Localization using a New Constitutive Equation for Isotropic Porous Materials

In sheet metal forming, the failure is caused by the necking deformation called “shear localization” or by the wrinkling. It is well known that the shear localization of sheets subjected to biaxial tension is not observed in Hill’s theoretical frame work based on the use of the classical J2-Flow rule which limits the direction of plastic strain increments to the same one of corresponding total deviatoric stresses: the classical J2-Flow theory gives unrealistic critical values when it is applied to bifurcation problems where the stress increments change their directions at critical points. Several modifications have been introduced to the J2F rule in trying to make the constitutive rule applicable to the bifurcation problems. In the past researches, the modification have been discussed mainly in terms of two different constitutive properties such as “stress increment directional dependence” and “dilatational effects of voids”.

Determination of constitutive parameters by forming-limit tests

Abstract Goya and Ito have developed a constitutive expression of plastic materials introducing two essential parameters μ(α) and β(α) which denote the magnitude and directional angle of the plastic strain increment, respectively, where α denotes the directional angle of the stress increment in the deviatoric stress space. In this report, the authors investigate specific forms of μ(α) and β(α), and determine the values of material constants through the application of the linear-comparison solid theory to the localized necking problem of a rigid-plastic plate subjected to biaxial stretching. The specified forms of the transition parameters are described as follows: β(α)=(β max /α max )∗φ( Θ )α , μ(α)={ cos (α−α)} 1+0.9φ , φ( Θ )=(1−2K 0 )∗(1− Θ ) r +K 0 , where γ and K0 are new material constants, and Θ denotes the deviation of the current stress state from the uniaxial stress state.