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is a
part of the body. We assign to each body a
reference configuration, which associates with the body a region
in three‐dimensional Euclidean space. In the reference configuration, each particle in the body has a position
, unique to that particle, as shown in
Figure 2.1. The vector
serves as a label, called the
referential or
Lagrangian coordinates of the particle. As with a person's home address, from a strictly logical point of view the particle need not ever occupy the point
. That said, in some applications it is useful to choose the reference configuration in a way that associates each particle with a position that it occupies at some prescribed time, for example
.
![Geometric representation of a reference configuration of a body, showing the referential coordinates X used to label a particle according to its position in the reference configuration.]()
Figure 2.1 A reference configuration of a body, showing the referential coordinates ![bold upper X]()
used to label a particle according to its position in the reference configuration.
The central aim of kinematics is to describe the trajectories of particles, that is, to determine the position ![bold x]()
in three‐dimensional Euclidean space that each particle ![bold upper X]()
occupies at every time ![t]()
. For this purpose we assume that there exists a one‐parameter family ![bold-italic chi left-parenthesis bold upper X comma t right-parenthesis]()
of vector‐valued functions, time ![t]()
being the parameter, that has the following properties.
1 The vector , having dimension L, gives the spatial position of the particle at time .
2 At each time , the function of the referential coordinates is one‐to‐one, onto, and continuously differentiable with respect to .
3 Also at each fixed time , has a continuously differentiable inverse such that . That is, tells us which particle occupies the spatial position at time .
4 For each value of the coordinate , the function is twice continuously differentiable with respect to .
The function ![bold-italic chi]()
is the deformation of the body, illustrated in Figure 2.2. We call the vector ![bold x equals bold-italic chi left-parenthesis bold upper X comma t right-parenthesis]()
the spatial or Eulerian coordinates of the particle ![bold upper X]()
at time ![t]()
.
![Geometric representation of the deformation mapping the reference configuration R onto the body's configuration at time t.]()
Figure 2.2 The deformation mapping the reference configuration
![t]()
.
![Geometric representation of the regions R and S occupied by a body in two reference configurations, along with the corresponding deformations χ and ψ that map a given particle onto a position vector x at time t.]()
Figure 2.3 Regions
and occupied by a body in two reference configurations, along with the corresponding deformations ![bold-italic chi]()
and ![bold-italic psi]()
that map a given particle onto a position vector ![bold x]()
at time ![t]()
.
Exercise 2.1 Let ![script upper R]()
and ![script upper S]()
be the regions occupied by a body in two different reference configurations, giving the referential coordinates of a certain particle as ![bold upper X]()
and ![bold upper Y]()
, respectively, as illustrated in Figure 2.3. Let ![bold-italic chi]()
and ![bold-italic psi]()
, respectively, denote the deformations associated with these two reference configurations. Thus the spatial position of the particle at time ![t]()
is ![bold-italic chi left-parenthesis bold upper X comma t right-parenthesis equals bold x equals bold-italic psi left-parenthesis bold upper Y comma t right-parenthesis]()
. Justify the relationship ![bold upper Y equals bold-italic psi Superscript negative 1 Baseline left-parenthesis bold-italic chi left-parenthesis bold upper X comma t right-parenthesis comma t right-parenthesis]()
. This relationship makes it possible to reconcile the analyses of motion by observers who choose different reference configurations.
2.1.2 Velocity and the Material Derivative
In classical mechanics, it is straightforward to calculate a particle's velocity: Differentiate the particle's spatial position with respect to time. Continuum mechanics employs the same concept. The velocity of particle ![bold upper X]()
is the time derivative of its position:
(2.1)![]()
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