Traditional definition of continuous medium also makes three assumptions
(Malvern 1969):
Continuity: A material is continuous if it completely fills the space that it occupies,
leaving no pores or empty spaces, and if furthermore its properties are describable
by continuous functions.
Homogeneity: A homogeneous material has identical properties at all points. The
size is larger than representative volume element (RVE) of the material. RVE is
the smallest volume over which a property measurement can be made that will
yield a value representative of the whole (Hill 1965).
Isotropy: A material is isotropic with respect to certain properties if these properties
are the same in all directions in space.
Only the first assumption is needed to define concepts of stress and strain. The last
two assumptions are needed when we introduce stress-strain relations (constitutive
relation) into continuum mechanics equations.
Of course, anisotropic, inhomogeneous, and discontinues systems can also be
analyzed by continuum mechanics by representing them as separate pieces, as it is
done in computational mechanics, such as finite element method.
Chapter 2 of this book covers only the general principles of continuum mechanics, and associated definitions, such as stress and strain. The fundamentals of
continuum mechanics, presented in the Chap. 2 has been well established by Isaac
Newton (1687); Leonhard Euler (1736); Cauchy, Green, Truesdell, and Toupin
(1960); Trusdell and Noll (1965); Truesdell (1965, 1966); and others. Maugin
(2016) published a comprehensive account of the foundations of continuum mechanics as well as a complete list of references of earlier work. Therefore, there is no
attempt in this book to cover earlier fundamental developments.
Continuum mechanics cannot be discussed without including material constitutive equations. While material behavior modeling is a very important topic of major
current research efforts, it is not the focus of this book. It is covered in the
applications of the unified mechanics theory section, Chaps. 5–8.
Chapter 3 of the book covers basics of thermodynamics. This chapter starts with a
comprehensive literature survey of the use of thermodynamics in continuum
mechanics in the last 150 years. While we do not claim to have included every
paper published on the topic, the survey covers all significant developments. Degradation of materials and structures is discussed in the context of thermodynamics.
Since not all engineering students are required to take the course in thermodynamics,
the reader is assumed to be a beginner; as such, an elementary information is
included.
Chapter 4 presents the formulation of the unified mechanics theory. Great effort
has been made to include all the details of the formulation for mechanical, thermomechanical, and electro-thermo-mechanical loading and for both metal- and particlefilled composite materials. The last part of Chap. 4 is the finite element method
implementation of the unified mechanics theory. Chapter 4 also discusses motivation
behind the development of unified mechanics theory. Essentially, the present-day
mechanics equations are all based on three laws of motion of Isaac Newton.
2
1 Introduction
(Malvern 1969):
Continuity: A material is continuous if it completely fills the space that it occupies,
leaving no pores or empty spaces, and if furthermore its properties are describable
by continuous functions.
Homogeneity: A homogeneous material has identical properties at all points. The
size is larger than representative volume element (RVE) of the material. RVE is
the smallest volume over which a property measurement can be made that will
yield a value representative of the whole (Hill 1965).
Isotropy: A material is isotropic with respect to certain properties if these properties
are the same in all directions in space.
Only the first assumption is needed to define concepts of stress and strain. The last
two assumptions are needed when we introduce stress-strain relations (constitutive
relation) into continuum mechanics equations.
Of course, anisotropic, inhomogeneous, and discontinues systems can also be
analyzed by continuum mechanics by representing them as separate pieces, as it is
done in computational mechanics, such as finite element method.
Chapter 2 of this book covers only the general principles of continuum mechanics, and associated definitions, such as stress and strain. The fundamentals of
continuum mechanics, presented in the Chap. 2 has been well established by Isaac
Newton (1687); Leonhard Euler (1736); Cauchy, Green, Truesdell, and Toupin
(1960); Trusdell and Noll (1965); Truesdell (1965, 1966); and others. Maugin
(2016) published a comprehensive account of the foundations of continuum mechanics as well as a complete list of references of earlier work. Therefore, there is no
attempt in this book to cover earlier fundamental developments.
Continuum mechanics cannot be discussed without including material constitutive equations. While material behavior modeling is a very important topic of major
current research efforts, it is not the focus of this book. It is covered in the
applications of the unified mechanics theory section, Chaps. 5–8.
Chapter 3 of the book covers basics of thermodynamics. This chapter starts with a
comprehensive literature survey of the use of thermodynamics in continuum
mechanics in the last 150 years. While we do not claim to have included every
paper published on the topic, the survey covers all significant developments. Degradation of materials and structures is discussed in the context of thermodynamics.
Since not all engineering students are required to take the course in thermodynamics,
the reader is assumed to be a beginner; as such, an elementary information is
included.
Chapter 4 presents the formulation of the unified mechanics theory. Great effort
has been made to include all the details of the formulation for mechanical, thermomechanical, and electro-thermo-mechanical loading and for both metal- and particlefilled composite materials. The last part of Chap. 4 is the finite element method
implementation of the unified mechanics theory. Chapter 4 also discusses motivation
behind the development of unified mechanics theory. Essentially, the present-day
mechanics equations are all based on three laws of motion of Isaac Newton.
2
1 Introduction