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**Microcontinuum field theories. I. Foundations and solids.**
*(English)*
Zbl 0953.74002

Berlin: Springer. xvi, 325 p. (1999).

This very good book presents, in a unitary manner, the micromorphic, microstretch and micropolar field theories. These theories constitute a generalization of classical field theories (elasticity, fluid dynamics, electromagnetism) to microscopic length and time scales.

A micromorphic continuum is a collection of material particles which deform and undergo classical motion. The deformation of material particles is assumed to be affine, so that any material point in the body possesses twelve degrees of freedom: three for macromotion, and nine for micromotion. Micropolar materials are classical materials with extra independent degrees of freedom for local rotations. These materials respond to spin inertia and to body and surface couples, but they also exhibit certain new static and dynamic effects. The theory of microstretch continua is a generalization of the micropolar theory, and is simultaneously a special case of the micromorphic theory. The material particles of microstretch continua are able to stretch and contract independently of their translations and rotations. For engineering applications, these theories can model liquid crystals, composites with rigid chopped fibers, porous media, elastic solids with rigid granular inclusions, bubbly fluids.

This volume presents the fundamentals of the dynamics of microcontinua. The book consists of eight chapters: 1. Kinematics; 2. Stress; 3. Constitutive equations; 4. Electromagnetic interactions I; 5. Theory of micropolar elasticity; 6. Microstretch elasticity; 7. Micromorphic elasticity; 8. Electromagnetic interactions II. The book also contains a bibliography and an index.

The first chapter gives a treatment of the geometry of deformations and microdeformations, strain measures, deformation-rate tensors, relative strain tensors, compatibility conditions, material time-rates of tensors, objective tensors, mass, inertia, momenta, and kinetic energy. Chapter 2 is devoted to stress tensors and balance laws. This chapter contains also the basic notions of thermodynamics of continuous media. Chapter 3 is concerned with the theory of constitutive equations. Elastic solids, viscous fluids, memory-dependent solids and fluids are discussed here in detail, together with thermodynamic restrictions imposed on the constitutive functionals. Electromagnetic interactions with micromorphic, microstretch, and micropolar continua are studied in chapter 4. Constitutive equations obtained in this chapter are used later (chapter 8) to formulate micropolar piezoelectricity and micropolar magnetoelasticity.

In chapter 5, the author presents the theory of micropolar elasticity. This chapter contains also solutions of many problems, the plate theory, and the discussion of some new physical phenomena predicted by the micropolar theory. Chapter 6 deals with the linear theory of microstretch elastic solids, presenting fundamental solutions, plane harmonic waves, surface waves and a lattice model for microstretch continuum. Chapter 7 is concerned with the linear theory of micromorphic thermoelasticity, which in special cases gives microstretch theory of micropolar theory. The author also discusses restrictions imposed on material constants by the nonnegativity of the strain energy, and plane harmonic waves. The last chapter describes electromagnetic interactions with micropolar media. First, the author establishes here linear constitutive equations for anisotropic and isotropic solids, and then demonstrates applications of field equations and material stability conditions in piezoelectricity and magnetoelasticity.

The book gives a detailed account of important results in the field, to which the author has made fundamental contributions. Most of the formulations are new, and some results are presented for the first time. The book includes also solutions of many applied problems. Summarizing, this monograph is a useful work which can be of interest to research workers, mathematicians, physicists and engineers dealing with microcontinuum theories.

A micromorphic continuum is a collection of material particles which deform and undergo classical motion. The deformation of material particles is assumed to be affine, so that any material point in the body possesses twelve degrees of freedom: three for macromotion, and nine for micromotion. Micropolar materials are classical materials with extra independent degrees of freedom for local rotations. These materials respond to spin inertia and to body and surface couples, but they also exhibit certain new static and dynamic effects. The theory of microstretch continua is a generalization of the micropolar theory, and is simultaneously a special case of the micromorphic theory. The material particles of microstretch continua are able to stretch and contract independently of their translations and rotations. For engineering applications, these theories can model liquid crystals, composites with rigid chopped fibers, porous media, elastic solids with rigid granular inclusions, bubbly fluids.

This volume presents the fundamentals of the dynamics of microcontinua. The book consists of eight chapters: 1. Kinematics; 2. Stress; 3. Constitutive equations; 4. Electromagnetic interactions I; 5. Theory of micropolar elasticity; 6. Microstretch elasticity; 7. Micromorphic elasticity; 8. Electromagnetic interactions II. The book also contains a bibliography and an index.

The first chapter gives a treatment of the geometry of deformations and microdeformations, strain measures, deformation-rate tensors, relative strain tensors, compatibility conditions, material time-rates of tensors, objective tensors, mass, inertia, momenta, and kinetic energy. Chapter 2 is devoted to stress tensors and balance laws. This chapter contains also the basic notions of thermodynamics of continuous media. Chapter 3 is concerned with the theory of constitutive equations. Elastic solids, viscous fluids, memory-dependent solids and fluids are discussed here in detail, together with thermodynamic restrictions imposed on the constitutive functionals. Electromagnetic interactions with micromorphic, microstretch, and micropolar continua are studied in chapter 4. Constitutive equations obtained in this chapter are used later (chapter 8) to formulate micropolar piezoelectricity and micropolar magnetoelasticity.

In chapter 5, the author presents the theory of micropolar elasticity. This chapter contains also solutions of many problems, the plate theory, and the discussion of some new physical phenomena predicted by the micropolar theory. Chapter 6 deals with the linear theory of microstretch elastic solids, presenting fundamental solutions, plane harmonic waves, surface waves and a lattice model for microstretch continuum. Chapter 7 is concerned with the linear theory of micromorphic thermoelasticity, which in special cases gives microstretch theory of micropolar theory. The author also discusses restrictions imposed on material constants by the nonnegativity of the strain energy, and plane harmonic waves. The last chapter describes electromagnetic interactions with micropolar media. First, the author establishes here linear constitutive equations for anisotropic and isotropic solids, and then demonstrates applications of field equations and material stability conditions in piezoelectricity and magnetoelasticity.

The book gives a detailed account of important results in the field, to which the author has made fundamental contributions. Most of the formulations are new, and some results are presented for the first time. The book includes also solutions of many applied problems. Summarizing, this monograph is a useful work which can be of interest to research workers, mathematicians, physicists and engineers dealing with microcontinuum theories.

Reviewer: Dorin Ieşan (Iaşi)

### MSC:

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

74A60 | Micromechanical theories |

74A20 | Theory of constitutive functions in solid mechanics |

74A15 | Thermodynamics in solid mechanics |

74A35 | Polar materials |