Page 11 - Mechanics of Asphalt Microstructure and Micromechanics

4 Ch a p t e r O n e

molecular dynamics, and numerical implementations in Finite Element Method (FEM),
Discrete Element Method (DEM) and Boundary Element Method (BEM). In the interest
of being concise, only a brief description of these fundamentals is included in Chapter 1,
Chapter 5, Chapter 6 , Chapter 8, Chapter 9 and Chapter 13 for completeness. Readers
are assumed to have a background in elasticity and tensor analysis and the required
mathematics background (such as Laplace Transform) to read these textbooks if need-
ed. With this understanding, the book is organized into six sections with a total of
13 chapters with both straightforward fundamentals and applications so that practicing
engineers and graduate students can learn this topic with convenience.
The first section has two chapters (Chapters 1 and 2) to discuss the fundamental
properties of AC, fundamentals of tensor analysis and continuum mechanics, and the
constituent properties and characterization methods. The second section has three
chapters (Chapters 3, 4, and 5) focusing on microstructure characterization, strain
characterization at the microscopic level, and mixture theory and micromechanics
applications. The third section has two chapters (Chapters 6 and 7) to discuss macro-
continuum types of theories and viscoplasticity models developed for AC. The fourth
section has two chapters (Chapters 8 and 9) to mainly focus on three major numerical
methods including FEM, BEM, and DEM. Section five has three chapters (Chapters 10,
11, and 12) to discuss topics of active research areas with applications of the funda-
mental theories and techniques. Section six has one chapter (Chapter 13) on multi-
scale modeling and topics of current interest and less matured. Appendices docu-
ment the Laplace Transform, Eshelby Tensors for elliptic inclusions, and fundamental
solutions for BEM.
A list of the books recommended for further reading is presented at the end of this
chapter. Again, the philosophy is not to perform a literature review or evaluation of
these books. Instead, the goal is to assist the readers of this book to pick up the back-
grounds quickly, so only a few books in each category are listed. One of the most impor-
tant selection criteria for these books was simplicity, conciseness, and description of the
fundamental physical mechanisms.

1.5 Fundamentals of Mathematics

1.5.1 Scalar
A scalar represents physical or geometrical quantities that require only a magnitude to
represent. Quantities such as time, temperature, density, and distance can be repre-
sented by scalars. It is also called zeroth-order-tensor, requiring 3 = 1 basis (indepen-
0
dent direction). In this chapter, lower-case Greek letters such as a, b and g are used to
represent scalars. In fact, one may judge whether a quantity in an equation is a scalar, a
vector, or a tensor from its physical meaning and dimensional analysis. The algebra
operations such as + and – can be applied to scalars.
Geometrically, scalars are one-dimensional (1D) quantities that have zero as a refer-
ence point and positive and negative “directions” along only one axis.

1.5.2 Vector
Physical quantities such as force and displacement require both a magnitude and a
direction to represent. These quantities can be represented as vectors, generally hav-
ing three independent bases to represent in the three-dimensional (3D) space. If one