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Recently, several applications, primarily driven
by microtechnology, have emerged where the use of materials with
tailored electromagnetic (dielectric) properties are necessary for a successful overall design. The ``tailored'' aggregate properties are
achieved by combining an easily moldable base matrix with particles
having dielectric properties that are chosen to deliver (desired) effective properties.
In many cases, the analysis of such materials requires the simulation of the macroscopic and microscopic electromagnetic response, as well as its resulting coupled thermal response, which can be important to determine possible failures in ``hot spots.'' This necessitates
a stress analysis. Furthermore, because, oftentimes, such processes initiate degratory chemical processes, it can be necessary to also include models for these processes as well.
A central objective of this work is to provide basic models and numerical solution strategies to analyze the coupled response of
such materials by direct simulation using standard laptop/desktop equipment. Accordingly, this monograph covers:

(1) The foundations of Maxwell's equations,

(2) Basic homogenization theory,

(3) Coupled systems (electromagnetic, thermal, mechanical and chemical),

(4) Numerical methods and

(5) An introduction to select biological problems.

The text can be viewed as a research monograph suitable for use in an upper-division undergraduate or first year graduate course geared towards students in the applied sciences, mechanics and mathematics that have an interest in the analysis of particulate materials.



Multiphase Continua, an Introduction

Most modern electromagnetic devices owe a significant amount of their success to the tailored electromagnetic material behavior of the components that comprise them. A relatively inexpensive way to obtain macroscopically desired responses is to enhance an easy-to-form matrix material’s properties by introducing microscale particles possessing different electromagnetic, thermal and mechanical properties. (Figure 1.1). The particles are chosen to produce an overall desired electromagnetic effect. The aggregate response of the material is an outcome of the interaction between the smaller-scale (microstructure) constituents that comprise the “effective” material. In the construction of such materials, the basic philosophy is to select matrix/particle material combinations in order to produce desired aggregate responses. For example, in electromagnetic engineering applications, the classical choice is to add a particulate phase with suitable dielectric constants in order to modify the overall properties of an easily moldable base matrix material.
Tarek I. Zohdi

Elementary Notation and Mathematical Operations

In this work, boldface symbols imply vectors or tensors. A fixed Cartesian coordinate system will be used throughout this monograph. The unit vectors for such a system are given by the (fixed) mutually orthogonal triad ( e 1, e 2, e 3). For the inner product of two vectors, u and v, in three dimensions we have
\(\bf u \cdot \bf v=\sum_{i=1}^3v_iu_i=u_1v_1+u_2v_2+u_3v_3=||\bf u|||\bf v||cos \theta,\) (2.1)
\(||\bf u||=\sqrt{u_1^2+u_2^2+u_3^2}\) (2.2)
represents the Euclidean norm in \(I\!\!R^3\) and θ is the angle between them. We recall that a norm has three main characteristics for any two bounded vectors u and v (||u|| < ∞ and ||v|| < ∞):
  • ||u|| ≥ 0, ||u|| = 0 if and only if u = 0,
  • ||u + v|| ≤ ||u|| + ||v|| and
  • ||γ u|| = |γ|||u||, where γ is a scalar.
Tarek I. Zohdi

Governing Electromagnetics: Maxwell’s Equations

Some fundamental definitions and observations in conjunction with electromagnetic phenomena are:
  • If a point charge q experiences a force \({\emph \bf{\Psi}}^e\), the electric field, \({\emph \bf E}\), at a position of the charge is defined by \({\emph \bf{\Psi}}^e=q {\emph \bf E}\).
  • If the charge is moving, another force may arise, \({\emph \bf{\Psi}}^m\), which is proportional to its velocity \({\emph \bf{v}}\). This other field is denoted as the “magnetic induction” (induced force field) or just the “magnetic field,” \({\emph \bf{B}}\), such that \({\emph \bf{\Psi}}^m = {\it qv} \times{\emph \bf{B}}\).
  • If the forces occur concurrently (the charge is moving through the region possessing both electric and magnetic fields), then \({\emph \bf{\Psi}}^{em}=q{\emph \bf{E}} + {\it qv} \times{\emph \bf{B}}\).
  • Electric fields are produced by both electric charges and time-varying magnetic fields. Similarly, magnetic fields are induced by electric currents and by time-varying electric fields.
Tarek I. Zohdi

Classical Linear Constitutive Behavior

We now consider idealized linear material behavior.
The starting point to develop a constitutive theory is to assume that an energy function per unit volume exists, a nonnegative function denoted W. A linear constitutive relation can be derived from
\({\bf D}=\frac{\partial{W}}{\partial{\bf E}}\) (4.1)
\({W}\approx c_0+{\bf c}_1\cdot{\bf E}+ \frac{1}{2}{\bf E} \cdot{\bf \epsilon} \cdot{\bf E}+...\) (4.2)
which implies
D ≈ c 1 + ε·E + ... (4.3)
Tarek I. Zohdi

Extraction of Macroscopic Effective Properties

In order to introduce fundamental concepts pertaining to effective properties of electromagnetic media, we initially start with static, lossless, conditions. Afterwards, we consider more general, thermally-sensitive, time-transient scenarios and the corresponding numerical methods.
Tarek I. Zohdi

Coupled Effects: Joule-Heating

Heterogeneous microstructures lead to a distortion of the electrical and current field within the material mixture. This leads to the fields becoming amplified within the material, which can lead to a variety of detrimental effects. An important quantity of interest is the amount of heat generated from an electrical field. The interconversions of various forms of energy (electromagnetic, thermal, etc.) in a system are governed by the first law of thermodynamics (which will be derived in detail shortly),
\(\rho \dot{w}-{\bf T}:\nabla \dot{{\bf u}}+\nabla \cdot{\bf q}-H=0,\) (6.1)
where ρ is the mass density, w is the stored energy per unit mass, T is Cauchy stress, u is the displacement field, q is heat flux, and H is the rate of electromagnetic energy absorbed due to Joule-heating (a source term)
\(H=a\left(\bf J \cdot \bf E\right),\) (6.2)
where 0 ≤ a ≤ 1 is an absorption constant. This standard form of Joule-heating is derived in the next chapter. Thus, a material designer must seek ways by which to modify a base material in order to deliver a specified overall conductivity (for example, by employing particulate additives), while simultaneously avoiding overheating.
Tarek I. Zohdi

Some Basic Principles of Continuum Mechanics

In order to properly consider multifield coupling effects, we will need to draw on some of the tools of classical continuum mechanics.
The term deformation refers to a change in the shape of the continuum between a reference configuration and current configuration. In the reference configuration, a representative particle of the continuum occupies a point p in space and has the position vector
$$ {\bf X}=X_1{\bf e}_1+X_2{\bf e}_2+X_3{\bf e}_3\,$$
where e 1, e 2, e 3 is a Cartesian reference triad, and X 1,X 2,X 3 (with center O) can be thought of as labels for a point. Sometimes, the coordinates or labels (X 1,X 2,X 3,t) are called the referential coordinates. In the current configuration, the particle originally located at point P is located at point P′, and can also be expressed in terms of another position vector x, with the coordinates (x 1,x 2,x 3,t). These are called the current coordinates. It is obvious with this arrangement that the displacement is u = x\({\emph \bf X}\) for a point originally at \({\emph \bf X}\) and with final coordinates x.
Tarek I. Zohdi

Basic Time-Stepping Schemes

Generally, methods for the time integration of differential equations falls within two broad categories: (1) implicit and (2) explicit. In order to clearly delineate between the two approaches, we first study a generic equation of the form (\({\emph \bf U}\)=unknown variable)
\({\cal M}\dot{{\bf U}}(t)={\bf F}({\bf U},t) \Rightarrow \dot{{\bf U}}(t)={\pmb{\cal G}}({\bf U},t).\) (8.1)
Tarek I. Zohdi

A Model Problem: Dielectrics Undergoing Multifield Processes

An important issue that this monograph addresses is the modeling and simulation of strongly coupled electromagnetic and thermodynamic fields that arise in particulate-doped dielectrics using an adaptive staggered FDTD (Finite Difference Time Domain) method. Of particular interest is to provide a straightforward modular approach to finding the effective dielectric (electromagnetic) response of a material, incorporating thermal effects arising from Joule heating which alter the pointwise dielectric properties such as the electric permittivity, magnetic permeability, and electric conductivity. This is important for “thermal (damage) management” of materials used in electromagnetic applications. Because multiple field coupling is present, a staggered, temporally-adaptive scheme is developed to resolve the internal microstructural electric, magnetic and thermal fields, accounting for the simultaneous pointwise changes in the material properties. Numerical examples are provided to illustrate the approach. Extensions to coupled chemical and mechanical fields are also provided. Here, we follow an approach found in Zohdi [139].
Tarek I. Zohdi

Concluding Remarks and Emerging Applications in the Biological Sciences

In closing, we discuss emerging applications in bioelectromagnetics to Red Blood Cells (RBCs), which are responsible for the transport of oxygen and carbon dioxide, and are the most prevalent type of cells in human blood. The average cellular volume of each cell (82-96 femtoliters) is occupied by a high concentration of the oxygen carrying protein hemoglobin at a concentration of 30-36 %. The lifespan of the human RBCs is approximately 120 days after they are released from the bone marrow as reticulocytes. Typically 4-6 million RBCs per cubic millimeter occupy 41-52 % of blood volume (hematocrit). The typical biconcave shape of RBCs endows the cell with ideal deformability characteristics. This allows RBCs to efficiently perform their function in small capillaries. Alterations in RBC properties, including shape, volume and membrane characteristics will lead to a decreased lifespan and when not compensated by increased production, a lower volume and anemia. Genetic disorders in cytoskeletal proteins (the cell wall “scaffolding”) results in RBC pathologies, such as hereditary spherocytosis and hereditary elliptocytosis (Eber and Lux [22], and Gallagher, [31] and [32]). Deviations in cytosolic and membrane proteins may affect the state of hydration of the cell and thereby its characteristics. Figure 10.1 illustrates some examples of unhealthy cell morphologies. In normal blood, echinocytes and stomatocytes can be observed, in addition to discocytes. Acanthocytes are observed in acquired hepatic syndromes, codocytes are found in thalassemia. In sickle cell disease, hemoglobin polymers will distort the shape of the cell and drepanocytes are observed. Elliptocytes are the result of membrane disorders where interactions in the horizontal direction (e.g., spectrin-spectin interactions) are disrupted, and spherocytes are observed in membrane disorders where the interaction between the lipid bilayer and the underlying membrane skeleton is dysfunctional. The number of humans that are affected by Sickle cell disease, Thalassemia and other hemoglobinopathies, runs in the millions (Forget and Cohen [29] and Steinberg et al. [97]). Such disorders lead to altered hemoglobin and result in changes in RBC properties, which is related to blood pathology, including anemia.
Tarek I. Zohdi


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