Introduction

Conventional ocean modeling consists of solving the model equations as accurately as possible, and then comparing the results with observations. While encouraging levels of qualitative agreement have been obtained, as a rule there is significant quantitative disagreement owing to many sources of error: model formulation, model inputs, computation and the data themselves. Computational errors aside, the errors made both in formulating the model and in specifying its inputs usually exceed the errors in the data. Thus it is unsatisfactory to have a model solution which is uninfluenced by the data. In the spirit of the inverse methods in Chapter 1, the approach which is developed here finds the ocean circulation providing the best fit simultaneously to the model equations and to the data. The best fit is defined in a weighted least-squares sense, with weights reflecting prior estimates of the various standard errors. Once unknown errors are explicitly included in the model equations and the data, the problem of finding the circulation is underdetermined, and so the least-squares fit may be regarded as a generalized inverse of the combined dynamics and observing system.

Finding the generalized inverse may also be regarded as a smoothing problem. The smoothing norm involves the differential operators for the model equations as well as derivatives of, or covariances for, the errors or residuals in the equations. Indeed, it is shown that the generalized inverse is equivalent to Gauss-Markov smoothing in space and time, based on the space-time covariance of model solutions forced by random fields having prescribed covariances.

The general circulation

Introduction

The general circulation of the oceans is an essential component of the thermodynamic system which determines global climate. The contributions of the oceans to the poleward fluxes of heat and water, for example, are clearly significant if not yet reliably known (Lorenz, 1967). It is widely accepted that modeling has improved our understanding of the general ocean circulation, but the objectives of ocean modeling are evolving along with the models themselves. Goal 1 of the World Ocean Circulation Experiment (WOCE) restates the grand objective of physical oceanography:

(WOCE, 1988). So far models have been developed by exploring the consequences of adding ever more physics and ever more detail. The earliest developments showed that westward intensification in subtropical gyres could be attributed to a combination of the β-effect with flow at high Reynolds number (Stommel, 1948; Munk, 1950). The nonlinear effect of vorticity advection was shown by Bryan (1963) to lead to time-dependent flow on the §-plane at high Reynolds number, even if the flow is steadily forced. An imposed mean density stratification allows the Lorenz cycle of energy exchange between mean and eddy forms of available potential energy and kinetic energy (Holland & Lin, 1975a,b). Stratification, determined internally by thermodynamics, is found to develop plausible thermohaline structure in response to reasonably representative mean surface fluxes of heat, salt and momentum (Bryan & Cox, 1968a,b; Bryan & Lewis, 1979).

The object of the appendix It is clear that elastoplasticity problems are not easily amenable to analytical methods, but for a few exceptions as in the case of the spherical envelope in Chapter 6. In particular, the elastoplastic borderline separating the region where the material still behaves elastically and the already plasticized region is an unknown in such problems. For complex geometries then a numerical implementation seems necessary (Chapter 11). However, the few cases that admit analytical solutions are typical of a methodology of which any student and practitioner of elastoplasticity must be aware. We have thus selected four examples, the first in plane strain (the wedge problem), the second in torsion, the third exhibiting a complex loading and the fourth accounting for anisotropy in a composite material.

Elastoplastic loading of a wedge

General equations

A wedge of angle β < π/2 is made of an isotropic elastoplastic material, satisfying Hooke's law in the elastic regime and Tresca's criterion without hardening at the yield limit. On its upper face it is subjected to a pressure p which increases with time (Fig. A3.1). We look first for the fully elastic solution and then for the elastoplastic solution in which the plasticized zone progresses until the whole wedge has become plastic. The solution of this problem in the elastoplastic framework is due to Naghdi (1957) – see also Murch and Naghdi (1958) and Calcotte (1968, pp. 158–64).

The object of the chapter In the absence of plastic strain, the problem of brittle fracture by extension of cracks can be presented in a thermodynamic framework, analogous to that of elastoplasticity. This means that the fracture criterion (or the criterion of crack propagation) replaces the plasticity criterion. One important notion is the notion of mechanical field singularity (displacement, stresses).

Introduction and elementary notions

We are interested in the problem of fracture, a phenomenon that occurs, more or less violently, under monotonic loading (whereas fatigue concerns cyclical loading). More specifically, we are interested in the problem of cracking, that is, the progagation of macroscopic cracks (of size of the order of one millimetre), whereas the beginning of cracking belongs to the microscopic and to the metal analyses which will not be examined here. (Microscopic cracks are one cause of damage – see Chapter 10.) The aim of this study is to arrive at a formulation of the crack-propagation laws, based upon fracture criteria and the definition of the conditions that may insure resistance to this fracture. We are certainly aware of the interest that such a subject implies for industry; it suffices to think about aeronautical engines and nuclear installations. Actually, our main interest is brittle fracture, that is, the kind that occurs without considerable plastic strain (i.e. the separation mechanism of crystallographic facets through cleavage), whereas ductile rupture is produced by different mechanisms accompanied by great plastic strains).

The object of the chapter This chapter provides a short introduction to the notion of homogenization (i.e., determining the parameters of a unique fictitious material that ‘best’ represents the real heterogeneous material or composite) and then, at some length, its application to the case where all or some of the constitutive components have an elastoplastic behaviour. The essential notions are those of representative volume element, procedure of localization, and the representation of some microscopic effects by means of internal variables. Composites with unidirectional fibres, polycrystals and cracked media provide examples of application.

Notion of homogenization

Homogenization is the modelling of a heterogeneous medium by means of a unique continuous medium. A heterogeneous medium is a medium of which material properties (e.g., elasticity coefficients) vary pointwise in a continuous or discontinuous manner, in a periodic or nonperiodic way, deterministically or randomly. While, obviously, homogenization is a modelling technique that applies to all fields of macroscopic physics governed by nice partial differential equations, we focus more particularly on the mechanics of deformable bodies with a special emphasis on composite materials (as used in aeronautics) and polycrystals (representing many alloys.) Most of the composite materials developed during the past three decades present a brittle, rather than ductile behaviour. As emphasized in Chapter 7, the elastic behaviour then prevails and there is no need to consider the homogenization of an dastoplastic behaviour.

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).