Plane Strain Drawing Research Articles

A new method of solving plane-strain boundary value problems for the double slip and rotation model

Abstract A method of solving plane-strain boundary value problems for a reduced version of the double slip and rotation model is developed. It is assumed that the intrinsic spin vanishes. Elastic strains are neglected. The Mohr–Coulomb yield criterion is adopted. An analogy between the solutions for this model and classical rigid plastic solutions of pressure-independent plasticity is revealed. The method is based on introducing auxiliary variables that satisfy the equation of telegraphy in regions where both families of characteristics are curved. Therefore, Riemann's method can conveniently be applied to solving boundary value problems. The method is employed for analyzing the processes of plane-strain drawing and extrusion through a wedge-shaped die. Friction is neglected. The solution is given in terms of ordinary integrals. The effect of the angle of internal friction on processes’ parameters is revealed. The solution reduces to available solutions of pressure-independent plasticity if the angle of internal friction vanishes.

A finite strain framework for steady-state problems: Hyperelasto-viscoplasticity

A numerical framework for analyzing steady-state elastic–plastic material deformation at finite strains is developed and demonstrated in the present work. The framework is an extension of the original method by Dean and Hutchinson (1980) develop to analyze steady-state crack propagation, under a small strain assumption, in which the history-dependent material response is captured through streamline integration. Steady-state problems are encountered in numerous engineering processes, and the original studies of crack growth have already been extended to include rolling and drawing, though within a small strain framework. However, the investigation of such manufacturing processes, where strains greater than 10% easily develop, requires a finite strain formulation to provide accurate results. The framework proposed in the present work offers an efficient and accurate method to extract the steady-state solution at finite strains without encountering the numerical issues related to traditional Lagrangian procedures. Furthermore, the framework also accounts for elastic unloading compared to many existing numerical steady-state schemes as they are often restricted to rigid plasticity. The new numerical framework employs a hyperelastic material model, in terms of a Neo-Hookean material, combined with an isotropic viscoplastic material behavior. However, the framework is not limited to any specific hyperelastic material model, nor any specific model for the plastic behavior. The finite strain steady-state framework has been verified through comparison to a traditional Lagrangian analysis conducted in ANSYS. The benchmark case constitutes a plane strain drawing process where the thickness of the specimen is reduced by drawing it between two circular cylindrical tools.

A Die Profile for Maximum Efficiency in Strip Drawing of Anisotropic Materials

Previous theoretical work has shown that frictionless dies can be designed to produce maximum efficiency in the reduction of isotropic rigid perfectly plastic strip under plane strain conditions. This is accomplished by contouring the die to cause coincidence of trajectories of one of the principal stresses and streamlines throughout the deformation region. Such flows have been named ideal flows. The objective of the present work is to give quite a general ideal flow solution for anisotropic materials. The solution can be adopted to design dies for plane strain drawing. The general structure of the solution in characteristic space shows that the contour of the die consists of two portions. One of these portions may be prescribed whereas the other is determined from the solution. A simple example is provided to illustrate the general solution. This particular solution is practically analytic. Therefore, it can be used to verify the accuracy of numerical codes.

An optimal die profile for plane strain drawing of sheets

An optimal die profile for plane strain drawing of sheets

Modelling of Damage Evolution in the Vicinity of Frictional Interfaces in Metal Forming

Conventional ductile fracture criteria are not applicable in the vicinity of maximum friction surfaces for several rigid plastic material models because the equivalent strain rate (second invariant of the strain rate tensor) approaches infinity near such surfaces. In the present paper, a non-local ductile fracture criterion generalizing the modified Cockroft-Latham ductile fracture criterion is proposed to overcome this difficulty with the use of conventional local ductile fracture criteria. The final form of the new ductile fracture criterion involves the strain rate intensity factor which is the coefficient of the principal singular term in a series expansion of the equivalent strain rate in the vicinity of maximum friction surfaces. When the velocity field is not singular, the new ductile fracture criterion reduces to the modified Cockroft-Latham criterion. The strain rate intensity factor cannot be found by means of commercial finite element packages since the corresponding velocity field is singular. In the present paper, the new fracture criterion is illustrated with the use of an approximate semi-analytical solution for plane strain drawing. It is shown that the prediction is in qualitative agreement with physical expectations.

Design of stationary plane strain drawing based on the ideal flow theory and a fracture criterion

The ideal flow theory provides a tool for designing metal forming processes. However, quite often the design based on this theory is not unique. The purpose of the present work is to investigate a possibility of using an additional design criterion driven by fracture. The fracture criterion adopted is based on the concept of workability diagram for rate-independent materials and its generalization for rate-dependent materials. It is shown that for stationary forming processes of rate-independent materials this criterion does not lead to a design criterion, whereas for stationary forming processes of rate-dependent materials it does. In the latter case, a numerical example for plane-strain drawing is given.

상계요소법에 의한 판재 인발공정에 관한 연구

For metal forming analysis, upper-bound solution is a practical method because the solution is overestimated. However it is not easy to determine the stresses on dies by using upper-bound solution. In this study, new scheme to calculate the stresses on dies based on upper bound solution is proposed. In the velocity fields, imaginary velocity is adapted to analyze the normal pressure on die surfaces. To verify the proposed scheme. plane strain drawing has been considered. The stresses on dies obtained by the proposed scheme are compared with the results of rigid plastic FEM and the experimental results. In the experiments, pressure film is used to measure the normal pressure on dies.

Strain distributions in plane strain deformation through wedge-shaped dies

Abstract In this paper, a theoretical model for plane strain deformation based on a slip line field (SLF) solution is developed to provide the strain distribution across a section of a product in plane strain drawing. The results obtained from the SLF solutions were compared to those of the finite element method. From both solutions, it is shown that the shear strain at a position close to the contact area is greater than that of the centre of the product. Ratios of strain at the surface to strain at the centre against the ratio of the semi-height to the contact length were obtained.

Effects of yield surface shape on sheet metal forming simulations

Three phenomenological yield criteria are adopted to describe the plastic behaviour of sheet metals with normal plastic anisotropy. The sheet metals are assumed to be elastic-plastic, rate-sensitive and incompressible. A rate-sensitive thin shell nite element formulation based on the virtual work principle is derived for the three yield criteria. The eects of the yield surface shapes based on the three yield criteria with the same value of the plastic anisotropy parameter R on the strain distribution and localization are investigated under a hemispherical punch stretching operation and a plane strain drawing operation. The results of the simulations show that the yield surface shape, in addition to the plastic anisotropy parameter R, controls the punch force, strain distribution and strain localization for the punch stretching operation. However, the yield surface shape does not aect the punch force and the strain distribution signicantly for the plane strain drawing operation. ? 1998 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng., 41, 559{584 (1998)

Effects of yield surface shape on sheet metal forming simulations

Three phenomenological yield criteria are adopted to describe the plastic behaviour of sheet metals with normal plastic anisotropy. The sheet metals are assumed to be elastic-plastic, rate-sensitive and incompressible. A rate-sensitive thin shell finite element formulation based on the virtual work principle is derived for the three yield criteria. The effects of the yield surface shapes based on the three yield criteria with the same value of the plastic anisotropy parameter R on the strain distribution and localization are investigated under a hemispherical punch stretching operation and a plane strain rawing operation. The results of the simulations show that the yield surface shape, in addition to the plastic anisotropy parameter R, controls the punch force, strain distribution and strain localization for the punch stretching operation. However, the yield surface shape does not affect the punch force and the strain distribution significantly for the plane strain drawing operation. © 1998 John Wiley & Sons, Ltd.

Behaviour of initial texture components during the plane strain drawing of f.c.c. sheet metals

The behaviour of individual rolling and recrystallization texture components was investigated under idealized cup drawing (plane strain drawing) conditions. Closed-form analytical and numerical solutions were obtained using a rate-sensitive crystal plasticity model together with the full-constraint Taylor theory. Stress responses, slip distributions and lattice rotation fields at and in the vicinity of the ideal orientations are predicted, together with their trajectories in Euler space. The limiting solutions of the rate-sensitive responses are derived; each of these stress states corresponds to the average of all the stress vertices associated with the corresponding Bishop and Hill solution. The slip distributions over the active slip systems are symmetrical, with 4 equal or 2 × 2 equal shear rates. None of the initial texture components is observed to be stable in plane strain drawing, although locations close to the Goss rotate relatively slowly. The only stable end orientation is P ({0 11}〈8 11 11 〉). The influence of the initial texture on grain rotation during plane strain drawing is discussed.

Analysis of friction and pressure distributions along the die during plane-strain drawing

Using the model of the deforming asperities, analyzed by the upper-bound approach, an expression for the global friction factor m as a function of the power consumption J ∗ , of the processing parameters, and of pressure, is provided. Further, using the slug-equilibrium approach, as presented by Sachs, an expression for the required pressure and power of the deformation as a function of the global friction factor is found. Using an iterative procedure, the above two expressions have been solved for the interdependence between the pressure and global friction, during drawing in plane strain, thus satisfying both approaches. The global friction factor, as well as the pressure between the workpiece and the die, is plotted as a function of the location of the asperity along the die and of other independent parameters such as the die angle, the wedge angle, the local friction factor and the reduction of area during the process.

Finite deformations of anisotropic polymers

A model for the thermomechanical behavior of anisotropic polymers under finite deformations below the glass-transition temperature is developed. The anisotropy is deformation-induced and the principal axes of the plastic stretch tensor are used to define the local axes of orthotropy at each material point of the deforming polymer. The formulation is based on a multiplicative decomposition of the deformation gradient tensor into elastic and plastic parts. An expression for the plastic spin, which is the average spin of the continuum as seen by an observer spinning with the axes of orthotropy, is derived. The numerical implementation of the model in a finite-element program is discussed and an algorithm for the numerical integration of the elastoplastic equations is developed. The problems of plane strain drawing and extrusion of polymeric sheets are solved using the finite-element method.

The integral as a function of the upper limit and an analytical solution to plane strain drawing and extrusion

A kinematically admissible velocity field which is different from Avitzur's is established in Cartesian Coordinates. An upper-bound analytical solution to strip drawing and extrusion is obtained by using the integral as a function of the upper limit in this paper.

On the evaluation of the upper-bound approach to problems of rigid-plastic deformation under non-standard conditions

The upper-bound theorem constitutes for rigid-plastic materials an important tool for setting up analytical or numerical solutions to technological problems, the finite element method being one special application. It is known that neither the shape of the free surfaces of the considered body nor the values of the yield limit may be varied when the free parameters of the solution, e.g. the nodal velocities, are being optimized. In the present paper the variation of discontinuity interfaces of the assumed velocity field or of the field of strain rates is investigated. It is then shown that additional terms appear in the derivatives of the upper-bound functional. For illustration, and as an example of application, the free surface of the strip in the plane strain drawing process is calculated behind and in front of the dies, assuming an elementary velocity field.

Finite Element Modeling of Plane-Strain Strip Drawing With Interface Friction

Numerical results of plane-strain strip drawing are presented emphasizing the interfacial friction conditions between workmaterial and drawing die. The results are based on a large-strain elasto-plastic finite element code with a special boundary condition capability to treat interface friction. Two different modeling approaches are developed. In the interface velocity model, finite element solutions are based on the velocity boundary conditions along the die-work interface measured through transparent sapphire dies. In the interface traction model, a technique called “General Linear Boundary Condition,” is developed to incorporate many different existing friction models. Predicted values of drawing force, die separating force, global friction, and residual stresses from both models are favorably compared with the corresponding experimental measurements.

Approximate methods for analysing the cold drawing of polymeric fibres and films

AbstractApproximate methods for analysing steady‐state cold drawing processes in polymer forming are discussed. These methods are applied to both axisymmetric and plane‐strain drawing. The cold drawing process is associated with steady‐state neck propagation along the specimen. Modifications are introduced in the approximate steady‐state analysis which reflect the neck initiation and localization phases prior to steady‐state drawing. This is accomplished without resorting to an analysis of the entire load–deformation history of the process. Finite element results are also presented and compared with those of the proposed approximate analyses. These comparisons show the approximate analysis to be an extremely efficient and quite accurate computational scheme for the analysis of steady‐state cold drawing processes.

Analysis of Central Bursting Defects in Plane Strain Drawing and Extrusion

Based on the upper-bound theorem in limit analysis, the central bursting defect in plane strain drawing and extrusion is analyzed by comparing the proportional flow with the central bursting flow for the metal with voids at the center. A criterion for the unique conditions that promote this defect has been derived. The metal with voids may flow in the identical manner to that of solid strip with no voids to form a sound flow, deterring central bursting. A solid strip, on the other hand, or a material with voids, may flow in a manner so as to produce central bursting defects. A major conclusion of the study is that, for a range of combinations of inclined angle of the die, reduction, and friction, central bursting is expected whether or not the material originally had any voids. On the other hand, central bursting can be prevented even if the original rod contains small-size voids.

Plane strain drawing of a strain hardening material through circular dies

A method is presented for the generation of slipline fields and their hodographs for the problem of plane strain drawing a strain hardening material through circular dies. Experimentally obtained flow fields are used to provide the distribution of maximum shear strain and hence shear flow stress throughout the assumed plastic region and then using the strain hardening form of the stress equilibrium equations the slipline field for the process is constructed. A description is given of the experimental work carried out to obtain the flow fields. These show the existence of deformation and a stress free surface in the work material prior to die entry. It will be seen that this free surface caused a number of problems when attempting to construct the slipline fields.

A theoretical analysis of spread and stress when drawing rectangular sections through wedge-shaped dies

This paper gives an upper bound solution to the problem of drawing rectangular sections through wedge shaped dies. The drawing material is assumed to be rigid perfectly plastic and a constant frictional stress is assumed on the entire die-material interface. The solution predicts spread and the corresponding drawing stress. The drawing stress is different from that for plane strain drawing for sections having low value of width/thickness ratio. As this ratio increases the process approaches the plane strain case. The paper also gives optimum values of wedge angle for minimum drawing stress.