General Planar Ideal Flow Solutions with No Symmetry Axis.

Bulk ideal flows constitute a wide class of solutions in plasticity theory. Ideal flow solutions concern inverse problems. In particular, the solution determines part of the boundary of a region where it is valid. Bulk planar ideal flows exist in the case of (i) isotropic rigid/plastic material obeying an arbitrary pressure-independent yield criterion and its associated flow rule and (ii) the double sliding and rotation model based on the Mohr-Coulomb yield criterion. In the latter case, the intrinsic spin must vanish. Both models are perfectly plastic, and the complete equation systems are hyperbolic. All available specific solutions for both models describe flows with a symmetry axis. The present paper aims at general solutions for flows with no symmetry axis. The general structure of the solutions consists of two rigid regions connected by a plastic region. The characteristic lines between the plastic and rigid regions must be straight, which partly dictates the general structure of the characteristic nets. The solutions employ Riemann's method in regions where the characteristics of both families are curvilinear. Special solutions that do not have such regions are considered separately. In any case, the solutions are practically analytical. A numerical technique is only necessary to evaluate ordinary integrals. The solutions found determine the tool shapes that produce ideal flows. In addition, the distribution of pressure over the tool's surface is calculated, which is important for predicting the wear of tools.

A New Semi-Analytical Solution for an Arbitrary Hardening Law and Its Application to Tube Hydroforming.

The present study consists of two parts. The first part supplies an exact semi-analytical solution for a general model of rigid plastic strain hardening material at large strains. The second part applies this solution to tube hydroforming design. The solution provides stress and velocity fields in a hollow cylinder subject to simultaneous expansion and elongation/contraction. No restriction is imposed on the hardening law. A numerical method is only required to evaluate ordinary integrals. The solution is facilitated using Lagrangian coordinates. The second part of the paper is regarded as an alternative to the finite element design of tube hydroforming processes, restricted to rather simple final shapes. An advantage of this approach is that the hardening law is not required for calculating many process parameters. Therefore, the corresponding design is universally valid for all strain hardening materials if these parameters are of concern. In particular, the prediction of fracture initiation at the outer surface is independent of the hardening law for widely used ductile fracture criteria. The inner pressure is the only essential process parameter whose value is controlled by the hardening law.

Application of the Generalized Method of Moving Coordinates to Calculating Stress Fields near an Elliptical Hole.

The distribution of stresses near holes is of great importance in fracture mechanics and material modeling. The present paper provides a general stress solution near a traction-free surface for an arbitrary piecewise linear yield criterion, assuming plane-strain conditions. The generalized method of moving coordinates is proven efficient in this case. In particular, the solution reduces to evaluating one ordinary integral. The boundary value problem solved is a Cauchy problem for a hyperbolic system of equations. Therefore, the stress solution in the plastic region is independent of other boundary conditions, though the occurrence of plastic yielding at a specific point is path-dependent. The general solution applies to calculating the stress field near an elliptic hole. It is shown that the parameter that controls the pressure-dependency of the yield criterion affects the stress field significantly. The aspect ratio is less significant as compared to that parameter. However, for a given material, the aspect ratio should also be considered to predict the stress field accurately, especially in the near vicinity of the hole. The solution reduces to an available solution for the pressure-independent yield criterion, which is a particular yield criterion of the considered class of yield criteria.

An Upper Bound Solution for the Compression of an Orthotropic Cylinder.

The upper bound theorem is used in conjunction with Hill's quadratic yield criterion for determining the force required to upset a solid cylinder. The kinematically admissible velocity field accounts for the singular behavior of the real velocity field in the vicinity of the friction surface if the maximum friction law is adopted. The regime of sticking is also taken into consideration. The effect of this regime on the upper bound limit load is revealed. In particular, the kinematically admissible velocity field that includes the regime of sticking may result in a lower upper bound than that with no sticking. The boundary value problem is classified by a great number of geometric and material parameters. Therefore, a systematic parametric analysis of the effect of these parameters on the compression force is practically impossible. An advantage of the solution found is that it provides a quick estimate of this force for any given set of parameters.

Finite Plane Strain Bending under Tension of Isotropic and Kinematic Hardening Sheets.

The present paper provides a semianalytic solution for finite plane strain bending under tension of an incompressible elastic/plastic sheet using a material model that combines isotropic and kinematic hardening. A numerical treatment is only necessary to solve transcendental equations and evaluate ordinary integrals. An arbitrary function of the equivalent plastic strain controls isotropic hardening, and Prager's law describes kinematic hardening. In general, the sheet consists of one elastic and two plastic regions. The solution is valid if the size of each plastic region increases. Parameters involved in the constitutive equations determine which of the plastic regions reaches its maximum size. The thickness of the elastic region is quite narrow when the present solution breaks down. Elastic unloading is also considered. A numerical example illustrates the general solution assuming that the tensile force is given, including pure bending as a particular case. This numerical solution demonstrates a significant effect of the parameter involved in Prager's law on the bending moment and the distribution of stresses at loading, but a small effect on the distribution of residual stresses after unloading. This parameter also affects the range of validity of the solution that predicts purely elastic unloading.

Effect of Rotation of the Principal Stress Axes Relative to the Material on the Evolution of Material Properties in Severe Plastic Deformation Processes.

Severe plastic deformation (SPD) processes are widely used for improving material properties. A distinguishing feature of many SPD processes is that the principal axes of the stress tensor intensively rotate relative to the material. Nevertheless, no measure of this rotation is involved in the constitutive equations that predict the evolution of material properties. In particular, a typical way of describing the effect of SPD processes on material properties is to show the dependence of various parameters that characterize these properties on the equivalent strain. However, the same level of the equivalent strain can be achieved in a process in which the principal axes of the stress tensor do not rotate relative to the material. It is, therefore, vital to understand which properties are dependent and which properties are independent of the rotation of the principal axes of the stress tensor relative to the material. In the present paper, a new multistage SPD process is designed such that the principal stress axes do not rotate relative to the material during each stage of the process but the directions of the major and minor principal stresses interchange between two subsequent stages. The process is practically plane strain, and it may be named the process of upsetting by V-shape dies. In addition, axisymmetric compression by Rastegaev's method is conducted. In this case, the principal stress axes are fixed in the material throughout the entire process of deformation. Material properties and microstructure generated in the two processes above are compared to reveal the effect of the rotation of the principal stress axes relative to the material on the evolution of these properties.

Description of Residual Stresses in Autofrettaged Open-Ended Cylinders Made of High-Strength Steel.

The elastic range in loading-unloading processes is often reduced with a Bauschinger effect. This material property may have a high impact on residual stresses and, as a result, on the performance of autofrettaged cylinders under service conditions. The objective of the present paper is to demonstrate this impact using a material model that accounts for the response of typical high-strength steel. The solution is semi-analytic and, therefore, can be used for fast and accurate analysis of the process of autofrettage. A numerical example illustrates the general solution. This example shows that the Bauschinger effect has a significant impact on the residual circumferential stress in the vicinity of the inner radius of the cylinder. This stress is the most significant quantity of autofrettaged cylinders. Therefore, the main result obtained suggests that even a moderate Bauschinger effect should be taken into account in analyses of the process of autofrettage.

A Novel Approach to Predict Wrinkling of Aluminum Alloy During Warm/Hot Sheet Hydroforming Based on an Improved Yoshida Buckling Test.

In order to predict the wrinkling of sheet metal under the influence of fluid pressure and temperature during warm/hot hydroforming, a numerical simulation model for sheet wrinkling prediction was established, taking into account through-thickness normal stress induced by fluid pressure. From simulations using linear and quadratic elements, respectively, it was found that the latter gave results that were much closer to experimental data. A novel experimental method based on an improved Yoshida Buckling Test (YBT) was proposed for testing the wrinkling properties of sheets under the through-thickness normal stress. A wrinkling coefficient suitable for predicting wrinkling was also presented. Based on the numerical simulations, an experimental validation of wrinkling performance was conducted. Ridge-height curves measured along the main diagonal tensile direction of the sheet were presented and showed that the wrinkling prediction criterion provided good discrimination. Furthermore, the wrinkling properties of several different materials were simulated to evaluate the accuracy of the prediction method, and the results revealed that the improved YBT gave good predictions for wrinkling in the conventional sheet metal forming process, while the prediction results for wrinkling in warm/hot sheet hydroforming were also accurate with the fluid pressure of zero.

Solution Behavior near Envelopes of Characteristics for Certain Constitutive Equations Used in the Mechanics of Polymers.

The present paper deals with plane strain deformation of incompressible polymers that obey quite a general pressure-dependent yield criterion. In general, the system of equations can be hyperbolic, parabolic, or elliptic. However, attention is concentrated on the hyperbolic regime and on the behavior of solutions near frictional interfaces, assuming that the regime of sliding occurs only if the friction surface coincides with an envelope of stress characteristics. The main reason for studying the behavior of solutions in the vicinity of envelopes of characteristics is that the solution cannot be extended beyond the envelope. This research is also motivated by available results in metal plasticity that the velocity field is singular near envelopes of characteristics (some space derivatives of velocity components approach infinity). In contrast to metal plasticity, it is shown that in the case of the material models adopted, all derivatives of velocity components are bounded but some derivatives of stress components approach infinity near the envelopes of stress characteristics. The exact asymptotic expansion of stress components is found. It is believed that this result is useful for developing numerical codes that should account for the singular behavior of the stress field.

Description of Residual Stress and Strain Fields in FGM Hollow Disc Subject to External Pressure.

Elastic/plastic stress and strain fields are obtained in a functionally graded annular disc of constant thickness subject to external pressure, followed by unloading. The elastic modulus and tensile yield stress of the disc are assumed to vary along the radius whereas the Poisson's ratio is kept constant. The flow theory of plasticity is employed. However, it is shown that the equations of the associated flow rule, which are originally written in terms of plastic strain rate, can be integrated with respect to the time giving the corresponding equations in terms of plastic strain. This feature of the solution significantly facilitates the solution. The general solution is given for arbitrary variations of the elastic modulus and tensile yield stress along the radial coordinate. However, it is assumed that plastic yielding is initiated at the inner radius of the disc and that no other plastic region appears in the course of deformation. The solution in the plastic region at loading reduces to two ordinary differential equations. These equations are solved one by one. Unloading is assumed to be purely elastic. This assumption should be verified a posteriori. An illustrative example demonstrates the effect of the variation of the elastic modulus and tensile yield stress along the radius on the distribution of stresses and strains at the end of loading and after unloading. In this case, it is assumed that the material properties vary according to power-law functions.

Evolution of internal variables in an expanding hollow cylinder at large plastic strains.

An efficient method for calculating the evolution of internal variables in an expanding hollow cylinder of rigid/plastic material is proposed. The conventional constitutive equations for rigid plastic, hardening material are supplemented with quite an arbitrary set of evolution laws for internal variables assuming that the material is incompressible. No restriction is imposed on the hardening law. The problem is solved in Lagrangian coordinates. This significantly facilitates a numerical treatment of the problem. In particular, the initial/boundary value problem is reduced to a system of equations in characteristic coordinates. A finite difference scheme is used for solving these equations. An illustrative example is presented assuming that the internal variables are the equivalent plastic strain and a damage parameter.

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In theories with N=2 supersymmetry on R^{3,1}, supersymmetric bound states can decay across walls of marginal stability in the space of Coulomb branch parameters, leading to discontinuities in the BPS indices Ω(γ,u). We consider a supersymmetric index I which receives contributions from 1/2-BPS states, generalizing the familiar Witten index Tr(-1)^{F}e^{-ßH}. We expect I to be smooth away from loci where massless particles appear, thanks to contributions from the continuum of multiparticle states. Taking inspiration from a similar phenomenon in the hypermultiplet moduli space of N=2 string vacua, we conjecture a formula expressing I in terms of the BPS indices Ω(γ,u), which is continuous across the walls and exhibits the expected contributions from single particle states at large ß. This gives a universal prediction for the contributions of multiparticle states to the index I. This index is naturally a function on the moduli space after reduction on a circle, closely related to the canonical hyperkähler metric and hyperholomorphic connection on this space.

An accurate upper bound solution for plane strain extrusion through a wedge-shaped die.

An upper bound method for the process of plane strain extrusion through a wedge-shaped die is derived. A technique for constructing a kinematically admissible velocity field satisfying the exact asymptotic singular behavior of real velocity fields in the vicinity of maximum friction surfaces (the friction stress at sliding is equal to the shear yield stress on such surfaces) is described. Two specific upper bound solutions are found using the method derived. The solutions are compared to an accurate slip-line solution and it is shown that the accuracy of the new method is very high.