Chapter 1
Introduction
1.1 What Is Mechanics of Continuous Medium?
Probably one of the best definitions of mechanics of a continuous medium is
provided by Malvern [1969] in his seminal work. Continuum mechanics is a branch
of mechanics concerned with the deformation or flow of solids, liquids, and gases. In
continuum mechanics, the molecular structure and electronics structure of the
material are ignored. When molecular structure must be taken into account, molecular dynamics is used, and when electronic structure must be considered, quantum
mechanics is used. In continuum mechanics, it is assumed that the material is
continuous without empty spaces. It is also assumed that all differential equations
governing the continuous medium are continuous functions, except at boundaries
between continuous regions. It is also assumed that the derivatives of the differential
equations are continuous as well. A material satisfying these requirements is considered a continuous medium. It is important to point out that “empty space” does
not mean there cannot be atomic vacancies. There will always be atomic vacancies.
It just means that there cannot be an empty space proportional to the size of the
object being studied.
The concept of a continuous medium allows us to define stress and strain at an
“imaginary point,” a geometric point in space assumed to be occupying no volume.
When we say a point in continuum mechanics, we do not mean an atomic point. It is
an imaginary point with no volume. Continuum stresses and strain are defined at this
point. Atomic stresses are defined differently, which is outside the scope of this
book. This approach allows us to use differential calculus to study nonuniform
distributions of strain. This assumption of continuous medium allows us to study
deformation in most engineering problems but not all. When continuous medium
assumption is not satisfied or molecular dynamics [movement of atoms] or electronics structure influences the mechanical behavior, then continuum mechanics cannot
be used. In these latter instances, a multi-scale mechanics analysis becomes
essential.
© Springer Nature Switzerland AG 2021
C. Basaran, Introduction to Unified Mechanics Theory with Applications,
https://doi.org/10.1007/978-3-030-57772-8_1
1
Introduction
1.1 What Is Mechanics of Continuous Medium?
Probably one of the best definitions of mechanics of a continuous medium is
provided by Malvern [1969] in his seminal work. Continuum mechanics is a branch
of mechanics concerned with the deformation or flow of solids, liquids, and gases. In
continuum mechanics, the molecular structure and electronics structure of the
material are ignored. When molecular structure must be taken into account, molecular dynamics is used, and when electronic structure must be considered, quantum
mechanics is used. In continuum mechanics, it is assumed that the material is
continuous without empty spaces. It is also assumed that all differential equations
governing the continuous medium are continuous functions, except at boundaries
between continuous regions. It is also assumed that the derivatives of the differential
equations are continuous as well. A material satisfying these requirements is considered a continuous medium. It is important to point out that “empty space” does
not mean there cannot be atomic vacancies. There will always be atomic vacancies.
It just means that there cannot be an empty space proportional to the size of the
object being studied.
The concept of a continuous medium allows us to define stress and strain at an
“imaginary point,” a geometric point in space assumed to be occupying no volume.
When we say a point in continuum mechanics, we do not mean an atomic point. It is
an imaginary point with no volume. Continuum stresses and strain are defined at this
point. Atomic stresses are defined differently, which is outside the scope of this
book. This approach allows us to use differential calculus to study nonuniform
distributions of strain. This assumption of continuous medium allows us to study
deformation in most engineering problems but not all. When continuous medium
assumption is not satisfied or molecular dynamics [movement of atoms] or electronics structure influences the mechanical behavior, then continuum mechanics cannot
be used. In these latter instances, a multi-scale mechanics analysis becomes
essential.
© Springer Nature Switzerland AG 2021
C. Basaran, Introduction to Unified Mechanics Theory with Applications,
https://doi.org/10.1007/978-3-030-57772-8_1
1