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Über dieses Buch

Mechanical Engineering, an engineering discipline borne of the needs of the in­ dustrial revolution, is once again asked to do its substantial share in the call for industrial renewal. The general call is urgent as we face profound issues of pro­ ductivity and competitiveness that require engineering solutions, among others. The Mechanical Engineering Series features graduate texts and research mono­ graphs intended to address the need for information in contemporary areas ofme­ chanical engineering. The series is conceived as a comprehensive one that covers a broad range of concentrations important to mechanical engineering graduate education and re­ search. We are fortunate to have a distinguished roster ofconsulting editors on the advisory board, each an expert in one ofthe areas ofconcentration. The names of the consulting editors are listed on the next page of this volume. The areas of concentration are applied mechanics, biomechanics, computational mechanics, dynamic systems and control, energetics, mechanics ofmaterials, processing, ther­ mal science, and tribology. Frederick A. Leckie,the series editor for applied mechanics, and I are pleased to presentthis volume in the Series: Nonlinear Computational Structural Mechan­ ics: New Approaches and Non-Incremental Methods of Calculation, by Pierre Ladeveze. The selection of this volume underscores again the interest of the Me­ chanical Engineering series to provide our readers with topical monographs as well as graduate texts in a wide variety of fields.

Inhaltsverzeichnis

Frontmatter

Chapter 1. The Reference Problem for Small Disturbances

Abstract
This chapter introduces the main notation. In the case of small disturbances, the basic (or reference) problem is specified which describes the motion of a medium (usually a solid), subject to given (usually time-dependent) external conditions. We finish by stating two uniqueness results that, in a sense, are a synthesis of various classical results.
Pierre Ladevèze

Chapter 2. Material Models

Abstract
The modeling of the mechanical behavior of materials nowadays nearly always uses the internal variable approach; the functional formulation is used rarely at the theoretical or numerical level.
Pierre Ladevèze

Chapter 3. Solution Methods for Nonlinear Evolution Problems

Abstract
Nearly all the methods one encounters in mechanics for solving nonlinear evolution problems are incremental methods. The load, or rather the interval of time [0, T] considered, is decomposed into a succession of (generally) small intervals. The history of different quantities being known until the present instant t, one studies a new interval of time [ t, t + Δt] where Δt is the increment. The problem is then to determine the history on the interval [t, t + Δt]. In assuming, for example, a linear history on [t, t + Δt] that is, a history that depends only on values at the instant t + Δt we are led to a classical nonlinear problem where the time does not enter. This problem, which determines various quantities at time t + Δt is generally treated by a Newton-type method. We note that the only mechanics property used is the principle of causality.
Pierre Ladevèze

Chapter 4. Principles of the Method of Large Time Increments

Abstract
The large time increment method (acronym: LATIN) was introduced by Ladevèze [1985a, b]. It represents a break with classical incremental methods in the sense that it is not built on the notion of small increments; the interval of time studied, [0, T] does not have to be partitioned into small pieces. It is an iterative method that sometimes starts with a relative gross approximation (generally coming from an elastic analysis) for displacements, strains, and stresses at each point M belonging to the domain Ω and for all t belonging to [0, T]. At each iteration, an improvement is always made to these different quantities for all t ∈ [0, T] and for all M ∈ Ω. For the interval of study, [0, T] the method is built on three principles:
  • P1, separation of the difficulties—partition of the equations into two groups:
    • a group of equations local in space and time, possibly nonlinear
    • a group of linear equations, possibly global in the spatial variable.
  • P2,a two-step iterative approach where, at each iteration, one constructs, alternatively, a solution to the first group of equations and then a solution to the second group. The first problem is local in the spatial variable, perhaps nonlinear, and the second is linear but generally global
  • P3,use of an ad hoc space-time approximation based on mechanics for the treatment of the global problem defined on Ω × [0,T].
Pierre Ladevèze

Chapter 5. A Preliminary Example: A Beam in Traction

Abstract
A beam in traction serves as our first example illustrating how the method of large time increments works. The simplicity of such a model permits us to appreciate only the first two principles P1 and P2 of the method. No conclusion can be drawn about the principle P3 which consists of exploiting some space-time approximations having physical content. The different steps of the method and the results obtained at each step are detailed for two types of materials: viscoplastic and (hyper)elastic. The analysis is quasi-static.
Pierre Ladevèze

Chapter 6. A “Mechanics” Approximation and Numerical Implementation

Applications and analysis of performance
Abstract
This chapter is concerned, above all, with approximations on Ω × [0,T] having a strong mechanics content that are introduced to solve the global step in the method of large time increments (Principle P3). The linear global step is, even for sophisticated models of behavior, by far the costliest. For small-disturbance static problems, the approximation used at each iteration is an extension of the “radial loading” approximation, which works remarkably well in a number of plastic and viscoplastic problems. This consists of approximating a function defined on Ω × [0,T] by a sum of products, each product being formed by a scalar function of the time variable multiplied by a function of the space variable. A sum of n products defines an approximation of order n. This nonclassical mode of approximation is studied as such in this chapter, where the best approximation to order n is characterized and some general convergence properties are given. The numerical treatment of different steps is also detailed. The discretizations used here in space as well as in time are classical.
Pierre Ladevèze

Chapter 7. Modeling and Calculation for Structures under Cyclic Loads

Abstract
The method of large time increments is not a fixed framework for constructing approximations to nonlinear evolution problems of mechanics—many approaches are possible. Through means of an example, we describe in this chapter a way of treating cyclic phenomena.
Pierre Ladevèze

Chapter 8. Formulation and “Parallel” Strategies in Mechanics

Abstract
This chapter develops a formulation of, and some computational strategies for, structural mechanics adapted to computers with parallel architecture. This approach has much in common with methods of domain decomposition.
Pierre Ladevèze

Chapter 9. Modeling and Computation for Large Deformations

Abstract
The modeling and computation of the large deformation of bodies has been of interest for the last 20 years. Despite many efforts, substantial progress remains to be accomplished in the industrial domain on problems such as the modeling and computation of the stamping of sheet metal. On the other hand, progress in other areas, such as postbuckling analysis, is quite advanced.
Pierre Ladevèze

Backmatter

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