3.2 Motion of a Continuum
53
3.1 Basic Strategy
To help understand the main ideas, it is instructive to first derive the fundamental
equations in one dimension and sometimes two dimensions, before moving on to
three dimensions. This strategy is used throughout this chapter. It is important
to note that, rather than considering a line as a 1D continuum and a plane as a
2D continuum, all continua are considered three dimensional. An n-dimensional
analysis is characterized by the following:
• All dependent variables depend on n spatial coordinates. For example, the scalar
variable φ is written φ(x, t), φ(x, y, t), and φ(x, y, z, t) in 1D, 2D, and 3D,
respectively.
• Vectors have n scalar components, and second-order tensors can be represented
by n × n matrices.
For simplicity and clarity, 1D and 2D derivations are generally done using scalars
and sometimes vectors in Cartesian coordinates (x, y), while 3D derivations are
done using tensors in general orthogonal coordinates (x i , i = 1, 2, 3). For specific
curvilinear coordinate systems, standard symbols are used regardless of the number
of dimensions, e.g., (r, θ, z) in cylindrical coordinates and (r, θ, φ) in spherical
coordinates.
A Few Words About Dimensions Checking dimensions and their units of measure
often can prevent mistakes during computation. In mechanics, the fundamental
dimensions are force, length, and time. Although correct dimensions are crucial in
the real world, this book generally emphasizes qualitative behavior. Thus, many of
the problems considered in this book do not specify units, and results are presented
in terms of nondimensional quantities or with units ignored entirely. Therefore,
assume this is the case unless stated otherwise. Of course, physically meaningful
numbers always can be obtained by adding the proper units.
3.2 Motion of a Continuum
The volume in space occupied by all the particles making up a continuum at any
instant in time t is called a configuration. The configuration at t = 0 is called
the initial configuration, while the collective motions of these particles define the
configuration at all other times t > 0. We also can define a separate reference
configuration at some time t = t 0 . In this book, the default is to take the reference
and initial configurations to be identical, i.e., t 0 = 0. In some instances, however,
we will find it necessary to define multiple reference configurations for the same
problem. For example, as new fibers appear during tissue remodeling, the reference
configuration for each fiber will be defined by the time at which it is created.
53
3.1 Basic Strategy
To help understand the main ideas, it is instructive to first derive the fundamental
equations in one dimension and sometimes two dimensions, before moving on to
three dimensions. This strategy is used throughout this chapter. It is important
to note that, rather than considering a line as a 1D continuum and a plane as a
2D continuum, all continua are considered three dimensional. An n-dimensional
analysis is characterized by the following:
• All dependent variables depend on n spatial coordinates. For example, the scalar
variable φ is written φ(x, t), φ(x, y, t), and φ(x, y, z, t) in 1D, 2D, and 3D,
respectively.
• Vectors have n scalar components, and second-order tensors can be represented
by n × n matrices.
For simplicity and clarity, 1D and 2D derivations are generally done using scalars
and sometimes vectors in Cartesian coordinates (x, y), while 3D derivations are
done using tensors in general orthogonal coordinates (x i , i = 1, 2, 3). For specific
curvilinear coordinate systems, standard symbols are used regardless of the number
of dimensions, e.g., (r, θ, z) in cylindrical coordinates and (r, θ, φ) in spherical
coordinates.
A Few Words About Dimensions Checking dimensions and their units of measure
often can prevent mistakes during computation. In mechanics, the fundamental
dimensions are force, length, and time. Although correct dimensions are crucial in
the real world, this book generally emphasizes qualitative behavior. Thus, many of
the problems considered in this book do not specify units, and results are presented
in terms of nondimensional quantities or with units ignored entirely. Therefore,
assume this is the case unless stated otherwise. Of course, physically meaningful
numbers always can be obtained by adding the proper units.
3.2 Motion of a Continuum
The volume in space occupied by all the particles making up a continuum at any
instant in time t is called a configuration. The configuration at t = 0 is called
the initial configuration, while the collective motions of these particles define the
configuration at all other times t > 0. We also can define a separate reference
configuration at some time t = t 0 . In this book, the default is to take the reference
and initial configurations to be identical, i.e., t 0 = 0. In some instances, however,
we will find it necessary to define multiple reference configurations for the same
problem. For example, as new fibers appear during tissue remodeling, the reference
configuration for each fiber will be defined by the time at which it is created.