The object of the chapter The technical difficulties faced in solving plasticity problems, which are free-boundary problems, are such that sooner or later one has to use a numerical implementation. While works fully devoted to numerical methods in solid mechanics give general solution techniques, here we focus on the specificity of the incremental or evolutionary nature of elastoplasticity problems and on Moreau's implicit scheme which is particularly well suited to this.

Introduction

Save for a few exceptions (see Appendix 3) the analytic solution of a problem of elastoplasticity is a formidable task since it involves a free boundary which is none other than the border between elastic and plastic domains, in general an unknown in the problem. In addition, by the very nature of elastoplasticity, the corresponding problems are nonlinear and the nature of certain plasticity criteria does not improve the situation. The relevant question at this point is: what is the quasi-static evolution of an elastoplastic structural member? The very nature of elastoplasticity and the corresponding incremental formulation are well suited to the study of general features of such a mechanical behaviour (see Chapters 4 to 6) and, indeed, via both spatial and temporal discretizations, to a numerical solution for real problems that involve complex geometries, somewhat elaborated plasticity criteria, and complex loading paths (including both loading and unloading). The most appropriate method for the spatial problem obviously is the one of finite elements (for short FEM).

The present book is an outgrowth of my lecture notes for a graduate course on ‘Plasticity and fracture’ delivered for the past five years to students in Theoretical Mechanics and Applied Mathematics at the Pierre-et-Marie Curie University in Paris. It also corresponds to notes prepared for an intensive course in modern plasticity to be included in a European graduate curriculum in Mechanics. It bears the imprint of a theoretician, but it should be of equal interest to practitioners willing to make an effort on the mathematical side. The prerequisites are standard and include classical (undergraduate) courses in applied analysis and Cartesian tensors, a basic course in continuum mechanics (elasticity and fluid mechanics), and some knowledge of the strength of materials (for exercises with a practical touch), of numerical methods, and of elementary thermodynamics. More sophisticated thermodynamics and elements of convex analysis, needed for a good understanding of the contents of the book, are recalled in Appendices.

The book deals specifically with what has become known as the mathematical theory of plasticity and fracture as (unduly) opposed to the physical theory of these fields. The first expression is reserved for qualifying the macroscopic, phenomenological approach which proposes equations abstracted from generally accepted experimental facts, studies the adequacy of the consequences drawn from these equations to those facts, cares for the mathematical soundness of these equations (do they have nice properties?), and then, with some confidence, provides useful tools to designers and engineers.

The object of the chapter In this chapter we are interested in the ruin of perfectly-plastic–elastic structures and we introduce the notions of limit load and of maximum admissible load, the determination of which constitutes the essential object of every engineer's office computations. We shall only attempt an introduction to this type of calculation, which will be illustrated by two examples. Certain minimum principles apply to velocities and to stresses. The static and dynamic methods in the determination of the maximum admissible load are only given in a rough draft.

The notion of limit load

The object of our attention is the notion of limit load and the collapse of perfectly plastic structures under unrestrained plastic strains. What do we mean by that? As certain deformable structures evolve, we observe that the elastoplastic response is produced in three stages. The first phase is elastic, the material being elastic everywhere. This phase lasts until the appearance of the first yielding. But the fact that the criterion of plasticity f(σ) = 0 is reached at one point does not necessarily mean that there is collapse. If the strain rate is still controlled, the plastic strain rate is not unlimited, since it is expressed in terms of έ (Section 4.2). We say then that the plastic strain is still controlled. The second phase of the response corresponds to the appearance and the extension of one or more regions of the structure, generally called plastic zones; in these, the plasticity criterion is satisfied at all the points.

The thrust F produced by the engines is of great importance in almost every phase of flight because it counteracts the drag and enables the aircraft to climb if required. The maximum available thrust Fm depends on the height and speed of the aircraft and is limited by the approved ‘rating’ for the appropriate phase of the flight. The three ratings that are important in relation to aircraft performance calculations are those specified for take-off, climb and cruise, and the rated thrust for each of these is the maximum available. In any phase of flight the thrust can of course be reduced by the pilot below the rated value, usually by moving a single control lever which is commonly known as a ‘throttle’ lever, even though it may act on a complex engine control system.

For almost all aspects of aircraft performance calculation it is necessary to know how Fm varies with the speed and height of the aircraft. In addition, for calculations of range, endurance and operating cost, a knowledge is required of the rate of consumption of fuel and the way in which this varies with flight speed, height and engine throttle setting. In this chapter the principles governing these variations will be discussed and approximate equations will be introduced for representing the variations in calculations of aircraft performance. For this purpose the rate of consumption of fuel will be expressed as the ratio of the rate of consumption to either the thrust or the shaft power of the engine.

The performance of an aircraft is essentially a statement of its capabilities and a different selection of these will normally be specified for the various categories such as transport, military and light aircraft, even though several common performance factors will feature in every such selection. For the engineer involved in the creation of a new design, these performance features serve as design criteria or at least desirable objectives, whereas late in the design and development stages the sales staff will quote the performance features as the basis for the commercial strength of the emerging aircraft. For either reason the performance will be stated in terms of quantities such as direct operating cost (DOC), maximum range for various payloads and fuel loads, cruising speed and airport requirements for landing and take-off. While the sales and design attitudes will be distinct, although related, this book addresses the early stages of the design process which must also bear considerable allegiance to performance as viewed by a potential customer.

The estimation of performance proceeds in stages, starting with parametric studies based on simple assumptions and progressing to more refined calculations as the main features of the design become established and the confidence in data grows. Estimation techniques are important not only because they allow the engineering team to proceed while data are crude or speculative, but also because construction of the new aeroplane will begin well in advance of the engineering refinements, and if there is accuracy in the early estimations this will be rewarded by a reduction in modifications as the fabrication and assembly effort progresses toward regular production.

In the design of a civil aircraft the condition of steady level cruise is of prime importance because improved fuel economy in this flight regime makes a direct and valuable contribution to the reduction of operating costs. Performance in the climb is often less important, but it cannot be ignored because a climb is always needed to reach the required cruising height after take-off and Air Traffic Control may also require the aircraft to change height during the cruise. For military aircraft, performance in the climb may be a primary design requirement because there is often a need to reach a specified height and speed in the shortest possible time, either from take-off or from some other prescribed initial conditions of height and speed.

The quantities that are of most interest in calculations of climbing performance are the rate of climb VC = V sin γ and the time required and fuel used in climbing from one specified height to another. In many cases there is a change of speed during the climb, so that the aircraft is accelerating, but it will be shown that a correction can easily be made for the effect of the acceleration on the rate of climb. The angle of climb is also of some interest, although it is important mainly at low altitudes where there may be obstacles to be cleared or where a large angle of climb may be required for reasons of noise abatement.

In earlier chapters, except in the discussions of the landing flare and the take-off transition immediately after lift-off, it has been assumed that the flight path is straight, so that there is no component of acceleration normal to the flight path. In this chapter flight in a curved path will be considered, concentrating on the usual form of banked turn as shown in Figure 8.1, in which the angle of bank is adjusted so that there is no sideslip and therefore no component of aerodynamic force normal to the plane of symmetry of the aircraft. In such a turn the required lift is greater than the weight, thus CL is greater than it would be in straight and level flight at the same speed and consequently the drag is also greater. This raises the requirement for thrust, even to maintain level flight, and thus the rate of climb obtainable with the maximum available thrust is reduced and may become negative. As the turn becomes tighter and the normal acceleration V2/R is increased, due to either a high speed or a small turn radius, or both, there will be increased demands for CL and for thrust to maintain height, with the consequence that limitations may be imposed by stalling or buffeting or by the engine rating.

This chapter addresses the interdependences among speed, rate of turn, rate of climb and additional ‘g-load’ on the pilot, as well as the limitations on one or other of these when some are fixed.