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## Über dieses Buch

This textbook explores the theory of Cosserat continuum mechanics, and covers fundamental tools, general laws and major models, as well as applications to the mechanics of granular media. While classical continuum mechanics is based on the axiom that the stress tensor is symmetric, theories such as that expressed in the seminal work of the brothers Eugène and François Cosserat are characterized by a non-symmetric stress tensor. The use of von Mises motor mechanics is introduced, for the compact mathematical description of the mechanics and statics of Cosserat continua, as the Cosserat continuum is a manifold of oriented “rigid particles” with 3 dofs of displacement and 3 dofs of rotation, rather than a manifold of points with 3 dofs of displacement. Here, the analysis is restricted to infinitesimal particle displacements and rotations. This book is intended as a valuable supplement to standard Continuum Mechanics courses, and graduate students as well as researchers in mechanics and applied mathematics will benefit from its self-contained text, which is enriched by numerous examples and exercises.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
This chapter succinctly describes the need for a compact representation in order to describe continua with higher degrees of freedom than the classical translational ones.
Ioannis Vardoulakis

### Chapter 2. Rigid-Body Mechanics and Motors

Abstract
This chapter lays down the fundamental representation concepts that will be used in the book thereafter. It eventually defines the concept of a “von Mises motor”, which is a compound vector including force and moment vectors. This compound representation of forces and moments in turn defines a geometric space/representation, where all the balance laws are going to be formulated upon. It continues by laying the basic theorems that will be used to formulate the Cosserat continuum, together with the appropriate kinematic fields conjugate to the “motor” vectors that are naturally called “kinematic von Mises motors”. Such a kinematic motor is a compound vector including linear velocity and spin (angular velocity), fully describing a rigid body motion in the new reduced geometric representation.
Ioannis Vardoulakis

### Chapter 3. Cosserat Continuum Kinematics

Abstract
This chapter derives the kinematic fields (deformation and deformation rate tensors) in general curvilinear coordinates, before reducing them to the familiar forms of Cosserat continuum in Cartesian coordinates. It showcases this way that the motor calculus approach has the same information as the classical representation as a limiting case, but can be used in a generic framework. It finishes with the integrability, compatibility and discontinuity conditions for the considered representation.
Ioannis Vardoulakis

### Chapter 4. Cosserat Continuum Statics

Abstract
In this chapter conservation considerations are firstly introduced. The virtual work principle, together with equilibrium equations in generalized curvilinear coordinate systems, are briefed under the auspices of the new mathematical representation, with working examples of how they reduce in cartesian, polar cylindrical and spherical coordinate systems. Finally, the definition of the concept of the traction motor—in accordance to the concept of the traction vector through Cauchy’s tetrahedron—is detailed.
Ioannis Vardoulakis

### Chapter 5. Cosserat Continuum Dynamics

Abstract
In this chapter the conservation (balance) laws for mass, linear and angular momentum are presented. It is shown that the Cosserat continuum differs from the Boltzmann continuum only in the angular momentum balance equation. Examples of how momentum balance laws can be applied to cartesian and polar coordinate systems are also presented.
Ioannis Vardoulakis

### Chapter 6. Cosserat Continuum Energetics

Abstract
In this chapter the energy and entropy balance laws for a Cosserat continuum are presented. It is shown that the higher-order continuum introduces additional terms in the local dissipation of a system.
Ioannis Vardoulakis

### Chapter 7. Cosserat-Elastic Bodies

Abstract
This chapter explores certain classes of materials. It begins with linear elastic bodies, laying down the basic concepts for linear isotropic elasticity. Examples are following, including bending of a beam, annular shear and torsion of a sphere of Cosserat elastic material.
Ioannis Vardoulakis

### Chapter 8. Cosserat Fluids

Abstract
This chapter moves from linear elastic solids to Cosserat fluids, showcasing the versatility of the approach. It presents the Navier-Stokes equations, generalized for an incompressible, linear viscous Cosserat fluid. Following this, examples are given, including shear flow and shallow flow slide of a granular fluid.
Ioannis Vardoulakis

### Chapter 9. Mechanics of Discrete Granular Media

Abstract
This chapter links the Cosserat continuum with discrete granular media. Through a discrete modelling approach, it presents a homogenisation method based on intergranular energetics and fabric averaging.
Ioannis Vardoulakis
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