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2023 | Book

Kinematics Read chapter Read first chapter

Mathematical Modelling of Continuum Physics

Authors: Angelo Morro, Claudio Giorgi

Publisher: Springer International Publishing

Book Series : Modeling and Simulation in Science, Engineering and Technology

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

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About this book

This monograph provides a comprehensive and self-contained treatment of continuum physics, illustrating a systematic approach to the constitutive equations for wide-ranging classes of materials. Derivations of results are detailed through careful proofs, and the contents have been developed to ensure a self-contained and consistent presentation.
Part I reviews the kinematics of continuous bodies and illustrates the general setting of balance laws. Essential preliminaries to continuum physics – such as reference and current configurations, transport relations, singular surfaces, objectivity, and objective time derivatives – are covered in detail. A chapter on balance equations then develops the balance laws of mass, linear momentum, angular momentum, energy, and entropy, as well as the balance laws in electromagnetism.
Part II is devoted to the general requirements on constitutive models, emphasizing the application of objectivity and consistency with the second law of thermodynamics. Common models of simple materials are then reviewed, and in this framework, detailed descriptions are given of solids (thermoelastic, elastic, and dissipative) and fluids (elastic, thermoelastic, viscous, and Newtonian).
A wide of variety of constitutive models are investigated in Part III, which consists of separate chapters focused on several types of non-simple materials: materials with memory, aging and higher-order grade materials, mixtures, micropolar media, and porous materials. The interaction of the electromagnetic field with deformation is also examined within electroelasticity, magnetoelasticity, and plasma theory.
Hysteretic effects and phase transitions are considered in Part IV. A new approach is established by treating entropy production as a constitutive function in itself, as is the case for entropy and entropy flux. This proves to be conceptually and practically advantageous in the modelling of nonlinear phenomena, such as those occurring in hysteretic continua (e.g., plasticity, electromagnetism, and the physics of shape memory alloys).
Mathematical Modelling of Continuum Physics will be an important reference for mathematicians, engineers, physicists, and other scientists interested in research or applications of continuum mechanics.

Table of Contents

Frontmatter

Basic Principles and Balance Equations

Frontmatter
Chapter 1. Kinematics
Abstract
This chapter deals with the kinematics of deformable bodies. Both deformation and motion of a body are developed by using the reference configuration. Attention is addressed to the topics of objectivity and objective time derivatives, thus establishing a general framework that proves remarkable in the description of material properties in terms of time derivatives. Transport relations are obtained for convecting (or non-convecting) sets and hence basic properties are established for the derivation of local balance equations and jump conditions for discontinuous fields.
Angelo Morro, Claudio Giorgi
Chapter 2. Balance Equations
Abstract
Borrowing from the mechanics of systems of particles, the balance equations for mass, linear momentum, and angular momentum are established. Next, the balance of energy is obtained after having realized that energy in general requires a more general framework including quantities of non-mechanical character. The balance of entropy is made formal by assuming that the entropy rate, deprived of the divergence of an entropy flux and of an entropy supply, is non-negative. Next, balance equations are established for electromagnetism in deformable bodies. Jump relations are derived for balance properties across singular surfaces.
Angelo Morro, Claudio Giorgi

Constitutive Models of Simple Materials

Frontmatter
Chapter 3. Generalities on Constitutive Models
Abstract
The balance equations of a body form an under-determined differential system, insufficient to yield specific results unless further relations are supplied. The balance equations for a continuum (free from internal structures) are the continuity equation, the equation of motion, and the balance of energy, namely five equations, for the fourteen unknowns (mass density, velocity, symmetric stress, energy density, heat flux) in the pertinent space-time domain. The insufficiency of the balance equations to solve a dynamic problem is conceptual and is consistent with the fact that different material properties are expected to provide different responses. Mathematically the material properties of the body are expressed by constitutive equations, or constitutive assumptions, which provide a model of the material behaviour. Constitutive equations are not a mere mathematical model. They have to be physically admissible, and this is ascertained through the compatibility with the objectivity principle and the second law of thermodynamics.
Angelo Morro, Claudio Giorgi
Chapter 4. Solids
Abstract
Solids are described as simple materials namely materials modelled by first-order gradients of deformation, velocity, and temperature. Hence the chapter develops linear and non-linear thermoelastic solids along with corresponding discontinuity waves and time-harmonic waves, thus arriving at the Cristoffel equation. Attention is addressed to models of hyperelastic and rubber-like materials. The concept of phonon is introduced and shown numerically by means of vibrating strings as elastic arrangements of atoms in solids.
Angelo Morro, Claudio Giorgi
Chapter 5. Fluids
Abstract
Fluids include liquid and gas phases. Here fluids are characterized by the constitutive properties. Some models are examined in detail: elastic fluids (along with water wave theories), thermoelastic fluids, the ideal gas and some real gases, heat-conducting viscous fluids, Newtonian fluids, Stokesian fluids, generalized Newtonian fluids, viscoplastic and viscoelastic fluids, and turbulent flows.
Angelo Morro, Claudio Giorgi

Non-simple Materials

Frontmatter
Chapter 6. Rate-Type Models
Abstract
Spatial interaction and memory effects are among the possible features of non-simple materials. As to memory effects in fluids, the inadequacy of the classical Navier-Stokes theory to describe rheological complex fluids, such as geological materials, liquid foams, and polymeric fluids, has led to the development of several theories of non-Newtonian fluids. In this chapter, memory effects are described by rate-type equations: among others, the elastic-plastic and the Kelvin-Voigt solids and the Bingham, the Maxwell-Wiechert, and the Jeffreys fluids are shown to follow from rheological models. Spatial interaction is modelled through materials of higher-order grade as, for example, the Oldroyd-B fluid and the White-Metzner fluid.
Angelo Morro, Claudio Giorgi
Chapter 7. Materials with Memory

Real materials exhibit a continuing strain rate(creep) under persistent constant stress and decay of stress (relaxation) under constant strain. The analogue occurs for electric current or electric polarization and electric field. In this connection, here the mathematical framework of materials with fading memory is developed along with the thermodynamic consistency and the wave propagation properties. First, thermoelastic materials are modelled through histories and summed histories. Next, the wide topic of viscoelasticity is developed; the linear viscoelastic solid with emphasis on free energies; viscoelastic solids with unbounded relaxation functions; viscoelastic fluids. Attention is then addressed to electromagnetism in deformable bodies: dielectrics with memory, magneto-viscoelastic materials.

Angelo Morro, Claudio Giorgi
Chapter 8. Aging and Higher-Order Grade Materials
Abstract
This chapter investigates some aspects of non-simple materials. First, aging materials are examined. Aging is a general problem of the modelling of materials. Mathematically the problem is made formal by letting constitutive parameters depend on time. It is pointed out that the second law of thermodynamics might select physically admissible evolutions of the parameters. Next constitutive equations are considered which involve higher-order derivatives, in time or in space (materials of differential type). The dependence on time derivatives is an alternative to the functionals of histories when memory effects are of interest. Yet the time derivative is not objective and hence recourse is needed to objective time derivatives.
Angelo Morro, Claudio Giorgi
Chapter 9. Mixtures
Abstract
Mixtures consist of different substances which keep their own properties. The individual molecules enjoy being near to each other so that, in the mathematical model, each point in the mixture may be occupied by all of the substances simultaneously. If the chemical structure of a substance changes when it enters the mixture then the mixture is said to be chemically reacting.
Angelo Morro, Claudio Giorgi
Chapter 10. Micropolar Media
Abstract
There are materials with internal structure (e.g. composites, polymers, liquid crystals, soil, and bone) for which a reasonable model should view the points no longer as purely geometric in character but as rigid-body particles or even deformable continua. Models accounting for rigid-body particles are within the theories of micropolar media. In this chapter, the balance laws of mass, linear momentum, angular momentum, and energy are derived by allowing for an orientational momentum, a body couple density, and a surface couple density. As a relevant topic, the micropolar model is applied to the description of liquid crystals. Mixtures of micropolar constituents are considered. As an application of the mixture theory, the model for nanofluids is established.
Angelo Morro, Claudio Giorgi
Chapter 11. Porous Materials
Abstract
A porous material is a substance that contains pores, or spaces, between solid materials through which liquid or gas can flow. Examples of naturally occurring porous media include sand, soil, and some types of stone. The present approach to the modelling of porous media is based directly on the theory of mixtures. Emphasis is given to the model consisting of a solid (skeletal material) and the remaining constituents as fluids. Next Darcy’s law is shown to follow by thermodynamic considerations, if stationary conditions hold, and the Darcy-Forchheimer equation is reviewed. Special models are established: materials with voids, materials with double porosity.
Angelo Morro, Claudio Giorgi
Chapter 12. Electromagnetism of Continuous Media
Abstract
This chapter is devoted to electromagnetism in deformable media. As with any model of material behaviour, the constitutive laws are required to be consistent with the second law of thermodynamics and to comply with the objectivity requirements. The models developed in this chapter describe various material properties and applied electromagnetic fields. The general subject of electroelasticity allows the investigation of piezoelectric materials. Ferrofluids are modelled as a mixture where the ferromagnetic particles are a micropolar constituent. Plasmas, e.g. ionized gases with free electrons and ions, are viewed as both binary mixtures and as a single fluid thus leading to magneto-hydrodynamic equations. Wave propagation of electromagnetic fields in chiral media and in ferrites is investigated.
Angelo Morro, Claudio Giorgi

Hysteresis and Phase Transitions

Frontmatter
Chapter 13. Plasticity
Abstract
Plasticity describes the non-reversible deformation of a material in response to applied forces. The physical mechanisms that cause plastic, that is non-reversible, deformation can vary widely. Consistent with the involved nature of plasticity, this chapter exhibits well-known concepts (yield criteria) and classical models (gradient theories, Kröner decomposition). Next, a decomposition-free approach to the modelling of plastic materials is developed through the essential role of the entropy production as a constitutive function. This allows a general scheme where the hyperelastic regime, the hypoelastic regime, and the hysteretic regime occur depending on the free energy and the entropy production. Polymeric foams are also modelled as hysteretic materials.
Angelo Morro, Claudio Giorgi
Chapter 14. Superconductivity and Superfluidity
Abstract
Superconductivity and superfluidity have a common origin due to the discovery of perfect conduction in mercury and superfluidity in helium. The effects show on a large spatial scale so that we can describe them as macroscopic phenomena. In this chapter, superconductivity and superfluidity are developed in a continuum framework within the theory of mixtures. In superconductivity, the fluid is viewed as consisting of two constituents: normal electrons (satisfying Ohm’s law) and superconducting electrons. In superfluidity, a normal fluid and a superfluid are viewed as a mixture of chemically reacting fluids. The conjecture that second sound is generated by the normal heat flux is made formal by investigating the propagation of discontinuity waves through a Maxwell-Cattaneo objective equation.
Angelo Morro, Claudio Giorgi
Chapter 15. Ferroics
Abstract
Ferroics is the general framework for ferroelectricity and ferromagnetism. Ferroelectrics are materials where the polarization exhibits nonlinear behaviour and hysteresis. They may have a strong permanent polarization but only if the temperature lies below a characteristic temperature, called the Curie temperature. In this chapter, a nonlinear approach to ferroelectrics is investigated and then a general setting is established for the modelling of hysteretic effects. The leading idea is that the entropy production is, per se, a non-negative function here involving the time derivative of the electric field. Also, the GLD theory is examined to describe the transition between linear paraelectric properties and hysteretic behaviours. Further, a nonlinear theory is developed for nonlinear behaviour and hysteresis in ferromagnetic materials.
Angelo Morro, Claudio Giorgi
Chapter 16. Phase Transitions
Abstract
Phase transitions denote transformations of the substance in one phase to another phase. The description of phase transitions is realized essentially within two schemes: a sharp interface across which the constitutive properties suffer a sudden change or a diffuse space region where the pertinent fields vary continuously. First, the jump conditions are derived for the interface among two phases. Next details are determined for the liquid-vapour transition, thus finding the continuity of the Gibbs free energy at the interface and Clapeyron’s equation. The brine channels formation is also developed and the solidification-melting of binary alloys is described. Phase transitions in shape-memory alloys are modelled.
Angelo Morro, Claudio Giorgi
Backmatter
Metadata
Title
Mathematical Modelling of Continuum Physics
Authors
Angelo Morro
Claudio Giorgi
Copyright Year
2023
Electronic ISBN
978-3-031-20814-0
Print ISBN
978-3-031-20813-3
DOI
https://doi.org/10.1007/978-3-031-20814-0

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