Wei H. Yang

Stress and deformation analysis of the metal extrusion process

Abstract A complete stress analysis of a metal-forming process is necessary in order to assess the onset of metal-forming defects such as the initiation of internal or surface cracks or the generation of residual stresses. This demands elasticplastic analysis. A program to evaluate complete stress distributions has been developed and applied to the extrusion process. Such solutions have not previously been obtained for general two- and three-dimensional problems encompassing the technologically important steady state processes, although these solutions are essential for the rational assessment of limits on process variables which will ensure a satisfactory metal-forming procedure. The stress fields obtained for the extrusion process exhibit features which are consistent with the known development of extrusion defects, such as the appearance of surface cracks.

A general algorithm for limit solutions of plane stress problems

A computational approach to limit solutions is considered most challenging for two major reasons. A limit solution is likely to be non-smooth such that certain non-differentiable functions are perfectly admissible and make physical and mathematical sense. Moreover, the possibility of non-unique solutions makes it difficult to analyze the convergence of an iterative algorithm or even to define a criterion of convergence. In this paper, we use two mathematical tools to resolve these difficulties. A duality theorem defines convergence from above and from below the exact solution. A combined smoothing and successive approximation applied to the upper bound formulation perturbs the original problem into a smooth one by a small parameter e. As e → 0, the solution of the original problem is recovered. This general computational algorithm is robust such that from any initial trial solution, the first iteration falls into a convex hull that contains the exact solution(s) of the problem. Unlike an incremental method thut invariably renders the limit problem ill-conditioned, the algorithm is numerically stable. Limit analysis itself is a highly efficient concept which bypasses the tedium of the intermediate elastic-plastic deformation and seeks the most important information directly. With the said algorithm, we have produced many limit solutions of plane stress problems. Certain non-smooth characters of the limit solutions are shown in the examples presented. Two well-known as well as one parametric family of yield functions are used to allow comparison with some classical solutions.

Large deformation of structures by sequential limit analysis

Although most structures are designed to function under small deformation, their large deformation behavior can be used to estimate reliability and safety for survivorship from an accident or a natural disaster. Structures such as buildings, bridges, ships, vehicles and machinery are designed with a safety factor to protect certain assets from the unexpected and unknot elements. The large deformation analysis provides the rational basis for a safety factor. In this paper, the method of sequential limit analysis is used to compute large deformation solutions of truss and frame problems. Differing from the incremental method of plasticity, the limit analysis method is numerically stable, more effecient and requires simpler input data. A duality theorem serves as the foundation of an algorithm for computing the complete static and kinematic solutions sim- uhaneously and for establishing their accuracy in each step of a deformation sequence. The phenom- ena encountered in large deformation such as loading-unloading under monotone deformation, bifurcation (more generally, loss of uniqueness) of solutions, and internal contact of structural members are revealed by the sequential limit analysis presented in this paper.

A general algorithm for plastic flow simulation by finite element limit analysis

Abstract Limit analysis has been rendered versatile in many structural and metal forming problems. In metal forming analysis, the slip-line method and the upper bound method have filled the role of limit analysis. As a breakthrough of the previous work, a computational approach to limit solutions is considered as the most challenging area. In the present work, a general algorithm for limit solutions of plastic flow is developed with the use of finite element limit analysis. The algorithm deals with a generalized Holder inequality, a duality theorem, and combined smoothing and successive approximation in addition to a general procedure for finite element analysis. The algorithm is robust such that from any initial trial solution, the first iteration falls into a convex set which contains the exact solution (s) of the problem. The idea of the algorithm for limit solutions is extended from rigid⧹perfectly plastic materials to work-hardening materials by the nature of the limit formulation, which is also robust with numerically stable convergence and highly efficient computing time.

The determination of limiting pressure in simultaneous elongation and inflation of nonlinear elastic tubes

&WI&-A nonlinear elastic tub&r membrane bonded at its ends to rigid plates is subjected to internal pressure and elongation. For a fixed elongation, pressure initially increases with radius, reaches a local ~ximum and then decreases. The purpose of this work is to determine this limiting pressure for each prescribed elongation. The usual boundary value problem formulation is such that a two dimensional search must be conducted. This can be computationally very costly. A method is presented which reduces the determination of the limiting pressure to a one-dimensional search. A numerical example is presented. An interesting p~enomenun in non~n~ elasticity arises in the problem of the inflation of a sph&d me,mbrane by internal pressure. In studying this problem for an incompressible Mooney model, Green and Shield111 showed that the inflating pressure d-need not montonicahy increase with deformed radius r. In particular, p can increase montonically. to a local maximum, decrease to a local minimum and then increase once more. This local maximum represents a limiting pressure. If a higher internal pressure is applied, the membrane will either respond dynamically or assume a much larger equ~b~um state. The same phenomenon appears in other membrane problems, such as the inflation by lateral pressure of a flat circular membrane clamped along a boundary[2] or the inflation of a torus by internal pressure[3]. In each of these cases, the pressure has a local maximum when considered as a function of an appropriate deformation parameter. Again for the reason discussed above, it is useful to know the magnitude and deformation at this local m~imum. In the above examples there is only one load parameter, the internal pressure. Now consider a nonlinear, elastic, cylindrical membrane which is bonded at its ends to rigid plates. The membrane is to be subjected to simultaneous inflation and elongation. Two load parameters must now be specified, the internal pressure and either the elongation or force applied to the end plates. This problem was first solved an~~ic~fy by Kydoniefs and Spencer[4] for the special case of zero end forces. They presented deformed profiles for a number of internal pressures. Their results indicate that the relation between pressure and deformed radius at mid-length has a local maximum. However, this case is governed by only one load parameter, the pressure. The present work is concerned with determining the local maximum in the more general case, when two parameters are varied. For the cases of the pressurized spherical and toroidal membranes, and the clamped membrane, it is possible to construct the pressure-geometry relation by an inverse procedure. The problem formulations are such that the local maximum can be found by a one parameter search. On the other hand, the formulation of the tube problem is such that a two parameter search must be conducted. Since this can be very time consuming, often prohibitive in computing cost, an alternate method is desirable The one presented here is an optimization scheme using a projected gradient. By means of this method, the search for the maximum pressure is confined to a path (a one dimensional space) instead of a two dimensional space, allowing a significant saving in computer time to be realized. It is applied to the tube problem to obtain results for this interesting case as well as to demonstrate the method for more complicated problems.

A variational principle and an algorithm for limit analysis of beams and plates

Abstract A variational principle for limit analysis of beams and plates is developed from a yield function based on the Frobenius matrix norm. The formulation produced a pair of maximization and minimization problems with a duality relation between them. Exact solutions of two simple problems are presented as verification to the validity of the new variational principle. An iterative algorithm is constructed to solve the minimization problem. The algorithm, tested successfully on the two example problems, is intended for beams and plates with general loading and boundary conditions and shapes.

Optimal design by a homotopy method

An optimal design problem is formulated as a system of nonlinear equations rather than the extremum of a functional. Based on a new homotopy method, an algorithm is developed for solving the nonlinear system which is globally convergent with probability one. Since no convexity is required, the nonlinear system may have more than one solution. The algorithm will produce an optimal design solution for a given starting point. For most engineering problems, the initial prototype design is already well conceived and close to the global optimal solution. Such a starting point usually leads to the optimal design by the homotopy method, even though Newtons method may diverge from that starting point. A simple example is given.

A duality theorem for plastic plates

SummaryLimit analysis studies the asymptotic behavior of elastic-plastic materials and structures. The asymptotic material properties exist for a class of ductile metals and are designed into optimal structural members such as I-beams and composite plates. The analysis automatically ignores the relatively small elastic deformations. Classical lower and upper bound theorems in the form of inequalities are mathematically incomplete. A duality theorem equates the greatest lower bound and the least upper bound. Although some general statement has been made on the duality relation of limit analysis, each yield criterion will lead to a specific duality theorem. The duality theorem for a class of plastic plates is established in this paper. The family of β-norms is used to represent the yield functions. Exact solutions for circular plates under a uniform load are obtained for clamped and simply supported boundaries as examples of the specific duality relations. Two classical solutions associated with Tresca and Johansen yield functions are also presented in the spirit of their own duality relations, providing interesting comparison to the new solutions. A class of approximate solutions by a finite element method is presented to show the rapid mesh convergence property of the dual formulation. Complete and general forms of the primal and dual limit analysis problems for the β-family plates are stated in terms of the components of the moment and curvature matrices.

MINIMIZATION APPROACH TO LIMIT SOLUTIONS OF PLATES

Limit analysis which predicts the ultimate load carrying capacity of a structure or machinery provides very useful information especially for designs that must survive accidents and abnormal conditions. In this paper, maximization of lower bound formulation and minimization of upper bound formulation are presented as primal and dual problems respectively. The dual problem is solved by a minimization procedure for circular, triangular and square plates with three types of boundary conditions and two types of loadings. A detailed parametric analysis for a cracked plate is also given to demonstrate the effectiveness of this approach.

METHODS FOR OPTIMAL ENGINEERING DESIGN PROBLEMS BASED ON GLOBALLY CONVERGENT METHODS

Abstract An optimal design problem is formulated as a system of nonlinear equations rather than the extremum of a functional. Based on the Chow-Yorke algorithm, another globally convergent homotopy method, and quasi-Newton methods, two algorithms are developed for solving the nonlinear system. Although the base algorithms are globally convergent (under certain fairly general assumptions), there is no theoretical proof of global convergence for the new methods. Some low dimensional numerical results are given.