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Foundations of Micropolar Mechanics

Authors: Victor A. Eremeyev, Leonid P Lebedev, Holm Altenbach

Publisher: Springer Berlin Heidelberg

Book Series : SpringerBriefs in Applied Sciences and Technology

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

The book presents foundations of the micropolar continuum mechanics including a short but comprehensive introduction of stress and strain measures, derivation of motion equations and discussion of the difference between Cosserat and classical (Cauchy) continua, and the discussion of more specific problems related to the constitutive modeling, i.e. constitutive inequalities, symmetry groups, acceleration waves, etc.

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Table of Contents

Frontmatter

Chapter 1. Introduction

Abstract

Within the Cosserat continuum theory, many problems were successfully solved. They demonstrate qualitative and quantitative difference between the solutions by micropolar and classic elasticity models. One of the principal difficulties of any micropolar theory is to establish its constitutive equations. This question was not discussed in the Cosserats’ original monograph [1], and it was a reason why the ideas of micropolar continuum were not recognized by many researchers. But even if for a material the constitutional equations are formulated we are faced with a new hard problem: the identification of the material parameters. For a homogeneous material in classical elasticity we should know two Lamé moduli whereas in the linear micropolar elasticity of isotropic solids the material is determined by six material parameters.

Victor A. Eremeyev, Leonid P. Lebedev, Holm Altenbach

Chapter 2. Kinematics of Micropolar Continuum

Abstract

In this chapter we briefly recall general kinematical relations for a micropolar continuum.

Victor A. Eremeyev, Leonid P. Lebedev, Holm Altenbach

Chapter 3. Forces and Couples, Stress and Couple Stress Tensors in Micropolar Continua

Abstract

In this chapter using the balance of momentum and balance of moment of momentum (Euler’s laws of motion) we introduce the stress and couple stress tensors. Then we derive the motion equations of the micropolar continuum which contains the motion equations of simple (non-polar) continuum as a special case.

Victor A. Eremeyev, Leonid P. Lebedev, Holm Altenbach

Chapter 4. Constitutive Equations

Abstract

For an arbitrary part of the body, Eqs. (3.30) and (3.31) express the balance equations for the moment and the moment of momentum. These six scalar equations contain 18 unknown quantities that are the components of tensors \(\mathbf{ T} \) and \(\mathbf{ M} \). The dependence of \(\mathbf{ T} \) and \(\mathbf{ M} \) on medium deformations is determined by the constitutive equations or constitutive relations that depend on the material properties. They are determined experimentally. The constitutive equations must obey some principles that restrict their form, see [1].

Victor A. Eremeyev, Leonid P. Lebedev, Holm Altenbach

Chapter 5. Strong Ellipticity and Acceleration Waves in Micropolar Continuum

Abstract

In this chapter we will consider acceleration waves in nonlinear thermoelastic micropolar continua. We will establish kinematic and dynamic compatibility relations for a singular surface of second order in the media. We also will derive an analogue to the Fresnel–Hadamard–Duhem theorem and an expression for the acoustic tensor. The condition for acceleration wave’s propagation is formulated as an algebraic spectral problem. It is shown that the condition coincides with the strong ellipticity of equilibrium equations. As an example, a quadratic form for the specific free energy is considered and solutions to the corresponding spectral problem are presented.

Victor A. Eremeyev, Leonid P. Lebedev, Holm Altenbach

Backmatter

Title: Foundations of Micropolar Mechanics
Authors: Victor A. Eremeyev
Leonid P Lebedev
Holm Altenbach
Copyright Year: 2013
Publisher: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-28353-6
Print ISBN: 978-3-642-28352-9
DOI: https://doi.org/10.1007/978-3-642-28353-6

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