R. Sowerby

A Theoretical and Experimental Investigation of Limit Strains in Sheet Metal Forming

The paper deals with an analytical technique for predicting FLDs based on linear, bilinear and trilinear straining paths—although any general curvilinear strain path can be handled by the method. The analytical procedure was based on the work of Marciniak and Kuczynski (known as the M-K model). The influence of material properties, such as anisotropy, strain hardening and strain rate sensitivity, as well as the effect of strain path, on the shape and level of the FLD were investigated. Material property data were determined from tests on Interstitial Free (IF) steel sheet. n nAn experimental FLD was constructed for the IF steel sheet in the as-received condition. Additional FLDs were also produced following pre-straining of the as-received sheet. Comparison between the experimentally determined FLDs and those predicted from the theoretical model was favourable. n nConventional FLDs are constructed in strain space with the principal surface strains as coordinate axes. However, they can be plotted in principal stress space, and some investigators have claimed this is a better representation. By knowing the strain path the stress state at the limit strain can be determined, and these limit stresses were plotted in principal stress space in order to construct a Forming Limit Stress Diagram (FLSD). It turned out that regardless of the shape of the FLD and the type of pre-strain imposed, all the FLSDs were almost identical. In contrast when plotted in strain space the FLD was very sensitive to the type of straining path.

The modelling of sheet metal stampings

Abstract A method of evaluating the strains over the deformed surface of a stamping is described. Measurements of the nodal points of a grid marked on the undeformed sheet are analyzed to determine the strain distributions and these are displayed automatically in the form of contours of strain intensity. In addition to this experimental technique, a modelling method is described for predicting in an approximate manner the strains necessary to deform a sheet from the flat to the final shape of the stamping. The technique is intended as the basis of an interactive computer design aid capable of dealing with typical stampings, including deep parts and complex shapes.

The pure plastic bending of laminated sheet metals

Abstract This paper deals with the pure bending of bonded laminated metals under plane strain conditions. Each laminate is classified by its initial yield strength, work-hardening characteristics and its percentage thickness in relation to the whole sheet. Computer programs have been developed which enable the sheet thickness, the magnitude of the bending moment and the distribution of the radial and tangential stress across the sheet to be calculated as a function of the radius of curvature of the bent sheet. Detailed calculations were performed for rigid non-strain-hardening bi- and tri-metals. The results indicated that the relative position of the strong and weak laminates in the sheet determined whether the overall sheet thickness increased or decreased during bending and whether the bending process was stable, i.e. took place under an increasing moment. Incorporating work hardening into the analysis increased the computation time but generally yielded similar results. A few bending experiments on laminated sheet confirmed the theoretical predictions. Detailed calculations were also performed for a rigid-strain-hardening mono-metal adopting a simple Bauschinger effect. These results were compared with the same material without the Bauschinger effect.

The development of ideal blank shapes by the method of plane stress characteristics

Abstract Asymmetrical cup drawing operations have been analysed employing the method of plane stress characteristics. Optimum blank shapes can be developed rapidly and accurately using a personal computer. In the present study the network of characteristics is continuously updated to account for the change in geometry of the outside edge of the blank, as the material is drawn into the die. It was found that the predicted blank shapes were in close agreement with those determined using a finite element analysis. In addition the predicted profiles correspond with experimentally determined optimum blank shapes when deep drawing square cups.

Blank development and the prediction of earing in cup drawing

The article deals with the prediction of blank shapes using the method of plane strain characteristics. In one case the material is assumed to be an incompressible, non-hardening, isotropic solid, and ideal blank shapes are developed when deep drawing prismatic cups. The significance of the study is that the resulting slip line field pattern does not violate the Hencky equations. The earing behaviour when deep drawing cylindrical cups from circular disks has also been predicted based on a particular form of anisotropy which allows for four fold symmetrical earing. The technique permits the blank shape to be calculated throughout the entire drawing operation, and demonstrates how the ears develop.

Probabilistic model of limit strains in sheet metal

Abstract A mathematical model is presented in which the mean value and standard deviation of limit strains are obtained for forming a sample of sheet metal parts in a repeatable stretch forming process. It is assumed that tearing originates in defects in the material which can be characterized by an equivalent population of voids which have an exponential size distribution. The model predicts that for a given material the mean limit strains in forming many pieces will increase with material thickness, severity of overall strain gradient and with decreasing volume fraction of equivalent voids. Numerical results indicate that the relationship between formability and volume fraction is approximately of an inverse logarithmic nature. The behaviour of the model using arbitrary values of parameters describing the equivalent void population is in good agreement with many phenomena observed in sheet metal forming.

Variability of forming limit curves

Summary The forming limit curve indicates the maximum uniform strain which can be achieved in an element of a sheet which is strained in a proportional biaxial straining process; it refers to the strain in a region adjacent to, but not within, any area of localized straining associated with rupture. In this work forming limit curves were determined for four different grades of mild steel; gridded test pieces were stretch formed to failure in elliptical and circular hydrostatic bulge dies. Each test was repeated twenty times for each material and each die, in order to provide data for statistical investigation of scatter in measured forming limits. It is shown that the variation in forming limits is much greater than that due to the experimental error and it is proposed that this scatter reflects an intrinsic property of the material which is important in determining material formability. A three-dimensional forming limit diagram is presented which can be used to determine the probability of failure of an element which is stretch formed to a particular strain level. The experimental data are also employed to determine the number of tests required to determine the mean forming limit curve within a specified accuracy.

An approximate analysis for studying the deformation mechanics of rate sensitive materials

Abstract To provide reliable predictions about a forming process it is necessary to have access to a realistic constitutive equation, a good estimate of the boundary conditions pertaining to the problem and a proven computational technique. This present article deals with this latter aspect and a numerical scheme is proposed for analysing the deformation of strain rate sensitive materials. The method of solution is an approximate one and employs a rate of energy functional which is applicable to a certain class of strain rate sensitive materials. The functional is minimised with respect to a kinematically admissible velocity field and is used in a discretized form in a finite element analysis. The functional is similar to an energy functional proposed by Lee and Kobayashi for a rate insensitive, rigid-plastic material. To test the theoretical technique two simple deformation processes were considered namely, the expansion of a thin walled spherical shell under internal pressure and the uniaxial deformation of a prismatic bar under its own weight. Exact solutions to these problems can be obtained based on a material constitutive equation of the type: σ = ϵ m . The computational technique based on the minimization of the proposed functional gave excellent agreement. To provide a more practical application of the method the following problems were considered. The expansion of a circular hole in a square plate, the deformation in the flange when drawing circular cups from square and octagonal plates and the tube drawing (and sinking) process. The examples serve to illustrate the flexibility of the method and the results of the analysis appear promising. However, since little information exists about the behaviour of superplastic alloys in these processes then it is difficult to infer anything about the “accuracy” of the predictions.

On the bauschinger effect and its influence on U.O.E. pipe making process

Abstract A plenomenological model is proposed which accurately portrays the observed forward and reverse loading characteristics of an elastic-workhardening material during unidirectional straining. The forward and reverse flow behaviour was employed to assess the magnitude of the residual stresses in the wall of a pipe resulting from an assumed version of the actual manufacturing process. The pressure-expansion characteristics of the pipe were then predicted using a finite element analysis. The analysis permits an assessment to be made of the influence of the Bauschinger effect on the pressure-expansion behaviour.

An approximate analysis for studying the plane strain deformation of strain rate sensitive materials

Abstract The paper presents a numerical method for analyzing the plane strain deformation of rate sensitive materials. A rate of energy functional is introduced which is thought to take adequate account of the strain rate sensitivity of the material. In the numerical technique the functional is minimized with respect to a kinematically admissible velocity field and used in a discretized form in a finite element analysis. To serve as an illustration the frictionless, plane-strain, side extrusion process was considered. To simulate actual side extrusion processes friction was incorporated into the analysis by assuming a constant fraction, α, of the current shear stress of the material. Data were available from some preliminary experiments on the side extrusion of a superplastic tin-lead alloy. The theoretically predicted forming pressure, taking α = 0·3 , showed reasonably good agreement with the experimental values.