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2019 | Buch

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Principles of Continuum Mechanics

A Basic Course for Physicists

verfasst von: Prof. Zdeněk Martinec

Verlag: Springer International Publishing

Buchreihe : Nečas Center Series

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book addresses the basic concepts of continuum mechanics, that is, the classical field theory of deformable bodies. The theory is systematically developed, from the kinematics to the balance equations, the material theory, and the entropy principles. In turn, the linear-elastic solids, the ideal liquid and the Newtonian liquid are presented in detail as concrete applications. The book concludes by covering the theory of small motions in a medium with a finite prestress. In general, the emphasis is on presenting the content in a clear and straightforward way that requires only an elementary grasp of calculus, linear algebra, and Newtonian mechanics. The book is intended for students of physics, mechanics, engineering and the geosciences, as well as applied mathematics, with a year or more of college calculus behind them.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Geometry of Deformation

Abstract

A deformation, in general, is a change in the shape or size of a body caused by the application of a force. Within the context of continuum mechanics, deformation is understood as the transformation of a body from one configuration to another. In the first two chapters, we will study deformation in terms of a pure geometrical description of changes in the shape of a body, regardless of the forces causing it. In this chapter, we confine our attention to two configurations without any regard for the sequence by which the second configuration is reached from the first. Both configurations are kept fixed in time and their geometrical relationship is described as a three-dimensional spatial mapping between them.

Zdeněk Martinec

Chapter 2. Basic Kinematics

Abstract

In the preceding chapter, we discussed the geometric properties of the present configuration κ _t under the assumption that the parameter t describing temporal changes of the body is kept fixed. That is, the motion function \(\vec \chi \) is considered to be a deformation map \(\vec \chi (\cdot ,t)\). Now, we turn our attention to the problem in the case of motion, that is, where the function \(\vec \chi \) is treated as the map \(\vec \chi (\vec X,\cdot )\) for a fixed point \(\vec X\). We assume that the mapping \(\vec \chi (\vec X,\cdot )\) is twice differentiable.

Zdeněk Martinec

Chapter 3. Measures of Stress

Abstract

In this chapter, we abandon the purely kinematic aspects of the motion of bodies and will focus on the actual forces causing the motion. Two distinct types of forces are recognised in continuum mechanics: body forces are conceived as acting on the particles of a body, and the surface (or contact) forces as arising from the action of one part of a body on an adjacent part across a separating surface. Just as there are many different strain measures, there are several different measures of the surface forces. We will see that the surface forces can be described as a second-order tensor. Thus, the surface forces can always be quantified by a set of nine numbers, and the various different definitions are all equivalent.

Zdeněk Martinec

Chapter 4. Fundamental Conservation Principles

Abstract

The fundamental principles of continuum mechanics are those that deal with the conservation of some physical quantity. Conservation means that these quantities cannot change over time, unless an external force intervenes. These conservation principles, as they are often called, are postulated for all material continua, irrespective of the material’s constitution and geometry, and result in equations that must always be satisfied. They deal with mass, linear and angular momenta, energy and entropy. The conservation principles are also valid for all bodies subject to thermomechanical effects.

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Chapter 5. Moving Reference Frames

Abstract

The frame-indifference principle, one of the fundamental concepts of continuum mechanics, means that a physical process remains unchanged when it is observed by different observers.

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Chapter 6. Constitutive Equations

Abstract

The equations listed in the preceding chapters apply to any material body that deforms under the action of external forces. Mathematically, they do not, by themselves, have a unique solution because the deformations are not related to internal contact forces.

Zdeněk Martinec

Chapter 7. Entropy Principles

Abstract

The constitutive theory of materials cannot be complete without thermodynamic considerations. In thermodynamics, two concepts are essential: energy and entropy.

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Chapter 8. Classical Linear Elasticity

Abstract

The classical linearised theory of elasticity provides a model that is useful for studying infinitesimal deformations of an elastic material. In this chapter, we briefly remark on the relationship between this linearised theory for infinitesimal deformations and the exact theory of elastic simple materials.

Zdeněk Martinec

Chapter 9. Infinitesimal Deformation of a Body with a Finite Pre-stress

Abstract

We devote this chapter to deriving the linearised equations and jump conditions that govern infinitesimal elastic deformations of a body initially in static equilibrium. The property of elasticity refers only to incremental deformations, while the state of initial stress may be due to any physical cause. Here, we consider that the initial static stress is caused by self-gravitation, that is, the gravitational forces acting between the material particles of a body. The principal assumption is that the initial static stress caused by self-gravitation is accounted for in a self-consistent treatment of a gravitating body with a three-dimensional density structure.

Zdeněk Martinec

Backmatter

Titel: Principles of Continuum Mechanics
verfasst von: Prof. Zdeněk Martinec
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-05390-1
Print ISBN: 978-3-030-05389-5
DOI: https://doi.org/10.1007/978-3-030-05390-1

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