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2001 | Buch

Modelling the Continuum Kapitel lesen Erstes Kapitel lesen

Handbook of Continuum Mechanics

General Concepts Thermoelasticity

verfasst von: Professor Jean Salençon

Verlag: Springer Berlin Heidelberg

Enthalten in: Professional Book Archive

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Über dieses Buch

The scale that concerns the practitioner in mechanics is usually qualified as macroscopic. Indeed, applications are rarely much below the human scale, and in order to be relevant models must be constructed on a similar scale, several orders of magnitude greater than the objects that are normally attributed to the physicist's sphere of interest. The mechanicist is therefore aware of the limits of these models, no matter how elegant their mathematical formulation may be, when the time comes far experimental validation. The mechanicist has a deep concern for the microscopic phenomena at the heart of what is being modelled, exposed by the physicist's research, which can today explain a wide range of material behaviour. The aim of this book is to present the general ideas behind continuum mechanics, thermoelasticity and one-dimensional media. Our approach to constructing mechanical models and modelling forces is based upon the principle oi virtual work. There are several advantages to this method. To begin with, it clearly emphasises the key role played by geometrical modelling, leading to mechanically consistent presentations in a systematic way. In addition, by requiring rigorous thought and clear formulation of hypotheses, it identifies the inductive steps and emphasises the need for validation, despite its axiomatic appearance. Moreover, once mastered, it will serve as a productive tool in the reader's later research career. This duality is used in the chapter devoted to variational methods for the solution of thermoelastic problems.

Inhaltsverzeichnis

Frontmatter
Chapter I. Modelling the Continuum
Abstract
The notion of a deformable continuous medium comes to mind when we observe the kind of solid deformations shown in Figs. 1 and 2, during cold or hot forming processes, or again the flow of a liquid, or the expansion and compression of a gas. From this kind of experience, the observer extracts the idea that certain problems can be treated on a macroscopic scale by assimilating the material to a ‘continuous’ medium, without in any way contradicting the assumptions of microscopic physics.
Jean Salençon
Chapter II. Deformation
Abstract
Figure 1 illustrates, for the die stamping process already shown in Fig. 2, Chap. I, two stages in the evolution of the block as it undergoes such an operation:
  • in its original state (configuration k 0) the block is rectangular and covered by a square grid,
  • in its current state (configuration k t ), application of the die has forced material down to the base and into the upper part as required for the forming process, and the grid has been deformed.
Jean Salençon
Chapter III. Kinematics
Abstract
In the previous chapter our investigations were based on a comparison between the current configuration and the reference configuration, being concerned essentially with the geometrical point of view, without considering the intermediate states of the system. As stated at the time, the argument t served merely as a parameter to index the current configuration.
Jean Salençon
Chapter IV. The Virtual Work Approach to the Modelling of Forces
Abstract
The last three chapters have been devoted to geometrical modelling of the deformable continuum on the basis of our experimental intuition. In order to model the mechanics of the continuum, we must now introduce the idea of forces and establish laws governing the motion and equilibrium of a system in this model. It should be emphasised that, as in all classical textbooks, the word ‘forces’ is used here and in all that follows as a generic term. It does not imply that the corresponding actions are intended to be modelled as concentrated or distributed forces. It is the very aim of the analysis to investigate what force model is relevant.
Jean Salençon
Chapter V. Modelling Forces in Continuum Mechanics
Abstract
In Sect. 6 of the previous chapter we set out some general results which follow from the principle of virtual work, expressed in terms of wrenches. These require, for any model:
• The law of mutual actions, equivalent to the statement (4.1) in Chap. IV, expressing the fact that the virtual rate of work by internal forces must be zero in any rigid body motion
$$\left\{ {\begin{array}{*{20}{c}} {forS,\left[ {{{F}_{i}}} \right] = 0,} \\ {\forall S',\left[ {{{F}_{i}}} \right] = 0.} \\ \end{array} } \right.$$
(1.1)
The fundamental law of dynamics
$$\left\{ {\begin{array}{*{20}{c}} {{\text{in a Galilean frame }}\mathcal{R},} \hfill \\ {{\text{for }}S,\left[ {{{\mathcal{F}}_{e}}} \right] = \left[ {\mathcal{M}a} \right],} \hfill \\ {\forall S',\left[ {{{{\mathcal{F}'}}_{e}}} \right] = \left[ {\mathcal{M}a'} \right].} \hfill \\ \end{array} } \right.$$
(1.2)
Jean Salençon
Chapter VI. Local Analysis of Stresses
Abstract
The aim of Chap. V was to show how the virtual work method could be used to set up a representation of internal forces in the classical 3-dimensional continuum. We thus arrived at a symmetric second rank tensor field defined on the current configuration k t , which we called the Cauchy stress tensor field.
Jean Salençon
Chapter VII. Thermoelasticity
Abstract
The thermoelastic behaviour of materials is revealed by everyday experience in the form of linear or bulk expansions and contractions under the effect of temperature changes, applications of the elastic properties of metals and even polymers to the production of springs, pins, clips and the like. From a phenomenological point of view, thermoelasticity is thereby linked to a notion of reversibility. The material responds to mechanical or thermal excitation in an instantaneous manner and, when the excitation is removed, returns to its initial state without showing any memory of the recent changes.
Jean Salençon
Chapter VIII. Thermoelastic Processes and Equilibrium
Abstract
As already observed, the equation of motion (1.1) results in a system of three first order partial differential equations for the tensor field σ Type="Italic">, that is, for the six scalar fields that are the components of the symmetric tensor σ, functions of the three space variables at each time t:
$$\left\{ {\begin{array}{*{20}{c}} {{\text{in a Galilean frame }},} \hfill \\ {{\text{div}}\underline{\underline \sigma } \left( {\underline x ,t} \right) + \rho \left( {\underline x ,t} \right)\left( {\underline F \left( {\underline x ,t} \right) - \underline a \left( {\underline x ,t} \right)} \right) = 0{\text{ on}}{{\Omega }_{{t,}}}} \hfill \\ {\left[\kern-0.15em\left[ {\underline{\underline \sigma } \left( {\underline x ,t} \right)} \right]\kern-0.15em\right] \cdot \underline n \left( {\underline x } \right) = 0{\text{ on }}{{\Sigma }_{{\underline{\underline {\sigma \cdot }} }}}^{1}} \hfill \\ \end{array} } \right.$$
(1.1)
Jean Salençon
Chapter IX. Classic Topics in Three-Dimensional Elasticity
Abstract
The aim in this chapter is to present several classic problems in the study of linearised isothermal elastic equilibrium for homogeneous, isotropic materials. The initial state of the system under zero loading, taken as the reference state, is always assumed to be natural. The following points will be emphasised:
  • statement of the problem (in particular, boundary conditions),
  • form of the solution,
  • remarks (e.g., other formulations of the problem),
  • practical applications (e.g., yield point of the system).
Jean Salençon
Chapter X. Variational Methods in Linearised Thermoelasticity
Abstract
In Sect. 2 of Chap. VIII, we stated the small perturbation hypothesis (S.P.H.). This allows a physical, then geometrical linearisation of the constitutive law and leads to a linearised expression for the equations governing a quasi-static thermoelastic process relative to the known reference configuration. The equations obtained in this way (Chap. VIII, Sect. 2.3) show that the thermal problem decouples (so that the temperature change field becomes one of the known fields), and define at each instant of time a thermoelastic equilibrium problem that depends only on the current excitations and the initial state.
Jean Salençon
Chapter XI. Statics of One-Dimensional Media
Abstract
We shall now present two approaches to one-dimensional modelling of the continuum. The starting point for this theory of one-dimensional media must clearly be geometrical, based on the observation that many solids used as structural elements in constructions (civil or industrial engineering, ship building, aeronautics, etc.) have a slender shape (Fig. 1). This suggests that it should be possible to carry out mechanical studies on the one-dimensional geometry defined by a director curve.
Jean Salençon
Chapter XII. Thermoelastic Structural Analysis
Abstract
Despite the considerable simplification of the one-dimensional model constructed in the last chapter (Sect. 3), the static analysis of a structure made from one-dimensional elements generally leads to a statically indeterminate problem that can only be solved by introducing the constitutive law for the one-dimensional medium. The latter is the one-dimensional counterpart, for this type of medium, of the constitutive laws discussed in Chap. VII for the classical three-dimensional continuum, and it obeys the same general principles.
Jean Salençon
Backmatter
Metadaten
Titel
Handbook of Continuum Mechanics
verfasst von
Professor Jean Salençon
Copyright-Jahr
2001
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-56542-7
Print ISBN
978-3-642-62556-5
DOI
https://doi.org/10.1007/978-3-642-56542-7