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.

These lectures cover a selection of topics from recent developments in the geometric approach to mechanics and its applications. In particular, we emphasize methods based on symmetry, especially the action of Lie groups, both continuous and discrete, and their associated Noether conserved quantities veiwed in the geometric context of momentum maps. In this setting, relative equilibria, the analogue of fixed points for systems without symmetry are especially interesting. In general, relative equilibria are dynamic orbits that are also group orbits.

In this chapter, we extend the theory of reduction of Hamiltonian systems with symmetry to include systems with a discrete symmetry group acting symplectically. The exposition here is based on the work of Harnad, Hurtubise and Marsden [1991].

For antisymplectic symmetries such as reversibility, this question has been considered by Meyer [1981] and Wan [1990]. However, in this chapter we are concerned with symplectic symmetries. Antisymplectic symmetries are typified by time reversal symmetry, while symplectic symmetries are typified by spatial discrete symmetries of systems like reflection symmetry. Often these are obtained by taking the cotangent lift of a discrete symmetry of configuration space.

There are two main motivations for the study of discrete symmetries. The first is the theory of bifurcation of relative equilibria in mechanical systems with symmetry. The rotating liquid drop is a system with a symmetric relative equilibrium that bifurcates via a discrete symmetry. An initially circular drop (with symmetry group S1) that is rotating rigidly in the plane with constant angular velocity Ω, radius r, and with surface tension τ, is stable if r3Ω2 < 12τ (this is proved by the energy-Casimir or energy-momentum method). Another relative equilibrium (a rigidly rotating solution in this example) branches from this circular solution at the critical point r3Ω2 = 12τ. The new solution has the spatial symmetry of an ellipse; that is, it has the symmetry ℤ2 × ℤ2 (or equivalently, the dihedral group D2).

In this chapter we discuss the cotangent bundle reduction theorem. Versions of this are already given in Smale [1970], but primarily for the abelian case. This was amplified in the work of Satzer [1977] and motivated by this, was extended to the nonabelian case in Abraham and Marsden [1978]. An important formulation of this was given by Kummer [1981] in terms of connections. Building on this, the “bundle picture” was developed by Montgomery, Marsden and Ratiu [1984] and Montgomery [1986].

From the symplectic viewpoint, the principal result is that the reduction of a cotangent bundle T*Q at µ ∈ g* is a bundle over T*(Q/G) with fiber the coadjoint orbit through µ. Here, S = Q/G is called shape space. From the Poisson viewpoint, this reads: (T*Q)/G is a g*-bundle over T*(Q/G), or a Lie-Poisson bundle over the cotangent bundle of shape space. We describe the geometry of this reduction using the mechanical connection and explicate the reduced symplectic structure and the reduced Hamiltonian for simple mechanical systems.

Mechanical G-systems

By a symplectic (resp. Poisson) G-system we mean a symplectic (resp. Poisson) manifold (P, Ω) together with the symplectic action of a Lie group G on P, an equivariant momentum map J : P → g* and a G-invariant Hamiltonian H : P → ℝ.

Following terminology of Smale [1970], we refer to the following special case of a symplectic G-system as a simple mechanical G-system.

In this chapter, we study some examples of bifurcations in the Hamiltonian context. A lot of the ideas from the previous chapters come into this discussion, and links with new ones get established, such as connections with chaotic dynamics and solution spaces in relativistic field theories. Our discussion will be by no means complete; it will focus on certain results of personal interest and results that fit in with the rest of the chapters. Some additional information on bifurcation theory in the Hamiltonian context may be found in the references cited below and in Abraham and Marsden [1978], Arnold [1978], Meyer and Hall [1991] and the references therein.

Some Introductory Examples

Bifurcation theory deals with the changes in the phase portrait structure of a given dynamical system as parameters are varied. One usually begins by focussing on the simplest features of the phase portrait, such as equilibrium points, relative equilibria, periodic orbits, relative periodic orbits, homoclinic orbits, etc., and studies how they change in number and stability characteristics as the system parameters are changed. Often these changes lead to new structures, such as more equilibria, periodic orbits, tori, or chaotic solutions, and the way in which stability or instability is transfered to these new structures from the old ones is of interest.

For conservative mechanical systems with symmetry, it is of interest to develop numerical schemes that preserve this symmetry, so that the associated conserved quantities are preserved exactly by the integration process. One would also like the algorithm to preserve either the Hamiltonian or the symplectic structure — one cannot expect to do both in general, as we shall show below. There is some evidence (such as reported by Chanell and Scovel [1990] and Marsden et al. [1991]) that these mechanical integrators perform especially well for long time integrations, in which chaotic dynamics can be expected. It is well known that, in general, may standard algorithms can introduce spurious effects (such as nonexistent chaos) in long integration runs; see, for example, Reinhall, Caughey, and Storti [1989]. We use the general term mechanical integrator for an algorithm that respects one or more of the fundamental properties of being symplectic, preserving energy, or preserving the momentum map.

Definitions and Examples

By an algorithm on a phase space P we mean a collection of maps Fτ: P → P (depending smoothly, say, on τ ∈ ℝ for small τ and z ∈ P). Sometimes we write zk+1 = Fτ(zk) for the algorithm and we write Δt or h for the step size τ.