Basic definitions in Kinematics

Observers/frames of reference

We will now introduce the crucial notion of an observer, or, equivalently, a frame of reference. For our purposes, a frame of reference is defined as a map that assigns to each event $\mathfrak{e}$ in spacetime a unique time and space coordinates: $$\mathfrak{e} \mapsto (\mathsf{x}, t) \in \mathbb{R}^3\times\mathbb{R}.$$ Here $\mathsf{x} \in \mathbb{R}^3$ and $t \in \mathbb{R}$ are the position and time coordinates of the event $\mathsf{e}$ according to the observer, or, equivalently, according to the frame of reference. Suppose that we have a second observer who assigns coordinates $\mathsf{x}^+,t^+$ to the same event $\mathfrak{e}$, then the observer transformation map, or, equivalently, the change of reference frame map is computed as $$(\mathsf{x},t) \in \mathbb{R}^3\times\mathbb{R} \mapsto (\mathsf{x}^+,t^+) \in \mathbb{R}^3 \times \mathbb{R}.$$ We will assume that this is a diffeomorphism on $\mathbb{R}^3\times\mathbb{R}$. Recall that a diffeomorphism is an invertible map such that both it and its inverse are continuously differentiable.

Remark

The space in which the collection of all events $\mathfrak{e}$ reside is not defined here. What is actually behind this definition is the assumption of a flat spacetime, in line with the Newtonian view of absolute space and absolute time. Rather than getting into the details of this definition, it is sufficient to focus only on the observer transformation/change of reference frame maps, which are diffeomorphisms on $\mathbb{R}^3\times\mathbb{R}$ since that is what we will need in this course.

The eventual theory that we will develop in this course will be invariant with respect to choices made by the observer. To begin with, we will focus on the case of a single observer. Once we develop the appropriate descriptions of motion, deformation and the laws of thermodynamics with respect to one observer, we will revisit this notion and study the implications that observer consensus places on the theory.

Bodies, configurations and deformations

A body $\mathcal{B}$ is defined as an abstract collection of material points. This is a useful abstraction for the development of continuum mechanics. In practice, what we are interested is how the body $\mathcal{B}$ moves and deforms in space over a period of time. Towards this end, it is helpful to introduce the notion of a configuration of the body in space. A configuration of the body $\mathcal{B}$ is a map of the form $$\chi:\mathcal{B} \to \mathbb{R}^3,$$ that assigns coordinates $\chi(p) \in \mathbb{R}^3$ to every material point $p \in \mathcal{B}$.

Remark

Some authors prefer to define a configuration as a map from the body $\mathcal{B}$ into the ambient three dimensional space $\mathcal{S}$. A choice of a coordinate system on $\mathcal{S}$ subsequently yields the foregoing description of a configuration as a map from $\mathcal{B}$ into $\mathbb{R}^3$. This two-step procedure is conceptually better, but we adopt the simpler approach here.

The image $B = \chi(\mathcal{B}) \subseteq \mathbb{R}^3$ of the body $\mathcal{B}$ under $\chi$ is called the placement of the body $\mathcal{B}$ according to the configuration $\chi$. We will typically assume that $B$ is an open subset of $\mathbb{R}^3$. Further, we will assume that the map $\chi:\mathcal{B}\to\mathbb{R}^3$ is injective, or, equivalently, that $\chi:\mathcal{B} \to B$ is bijective.

Remark

It is important to distinguish between the configuration and placement of a body. For instance, a homogeneous cylinder, and a twisted version of the cylinder both occupy the same region of space, but amount to distinct configurations.

Note

The assumption that $B = \chi(\mathcal{B}) \subseteq \mathbb{R}^3$ is open in $\mathbb{R}^3$ is quite a strong one. For instance, this precludes bodies that occupy a two dimensional surface or one dimensional curves in $\mathbb{R}^3$. Such special bodies are important for certain applications; appropriate extensions of the theories developed here using the tools of differential geometry need to be developed to handle those cases. We focus on the simple case mentioned here in the interest of keeping the mathematical development simple.

Suppose now that $\chi_1$ and $\chi_2$ are two configurations of $\mathcal{B}$, the deformation of the body with respect to these configurations is the map $$\varphi:B_1 \subseteq \mathbb{R}^3 \to B_2 \subseteq \mathbb{R}^3,$$ where $B_1 = \chi_1(\mathcal{B})$ and $B_2 = \chi_2(\mathcal{B})$. Notice that the deformation of a body is a map between open subsets of $\mathbb{R}^3$, which is much easier to handle than abstract configuration maps. For this reason, we will work almost entirely with deformation maps from now on. We will further assume that all the deformation maps we encounter in these notes are diffeomorphisms: this means that they are invertible, and both the deformation and its inverse are smoothly differentiable.

Note

The assumption that the deformation map is a diffeomorphism is also a strong one. There are important applications where this assumption fails, like shocks, interfaces in solids, etc. We will nevertheless adopt the stronger assumptions since this makes the development of the theory clearer. Appropriate corrections to the theory we develop here can be used to handle the special cases where these assumptions fail.

Motion of the body in space

Let us now look at how a body is described by an observer, or, equivalently, how a body is represented in a frame of reference. The observer assigns, for every instant of time $t \in \mathbb{R}$ according to her clock, a configuration $\chi_t:\mathcal{B} \to \mathbb{R}^3$ such that the body occupies the region $B_t = \chi_t(\mathcal{B})$. Note that this description is with respect to the specific coordinate system attached to the observer.

The sequence of configurations $(\chi_t)_{t \in I}$, where $I \subseteq \mathbb{R}$ is a time interval of interest, of the body $\mathcal{B}$ according to the observer is called a motion of $\mathcal{B}$ in $\mathbb{R}^3$. The one-parameter family of deformation maps $\varphi_t:B_0 \to B_t$, where $t \in I$, is defined as follows (note that we are implicitly assuming $0 \in I$): for any $\mathsf{X} \in B_0 = \chi_0(\mathcal{B})$, $$\varphi_t(\mathsf{X}) = \chi_t \circ \chi_0^{-1}(\mathsf{X}).$$ It is useful to introduce the following related variants of the deformation map: $\varphi:B_0 \times I \to \mathbb{R}^3$ defined as $\varphi(\mathsf{X}, t) = \varphi_t(\mathsf{X})$, and $\varphi_{\mathsf{X}}:I \to \mathbb{R}^3$ defined as $\varphi_{\mathsf{X}}(t) = \varphi(\mathsf{X},t)$.

A few important terminologies that will be used repeatedly in our development of continuum mechanics are introduced now. In the description of the motion and deformation of a body in space, it is often possible to single out a special configuration $\chi_R:\mathcal{B} \to \mathbb{R}^3$ called the reference configuration. The corresponding image $B_R = \chi_R(\mathcal{B}) \subseteq \mathbb{R}^3$ is called the reference placement of the body $\mathcal{B}$. It is important to note that a reference configuration need not necessarily be a configuration of the body during the motion of the body as observed by an observer. However, it is convenient and typical to identify the reference configuration with the configuration of the body at the initial instant of time, as observered by an observer. Without loss of generality, we can assume the initial instant of time to correspond to $t = 0 \in I$. The configuration at $t = 0$, $\chi_0:\mathcal{B} \to \mathbb{R}^3$, is called the initial configuration. The corresponding image $B_0 = \chi_0(\mathcal{B}) \subseteq \mathbb{R}^3$ is called the initial placement of the body $\mathcal{B}$. The configuration of the body at the current instant of time, $\chi_t:\mathcal{B}\to\mathbb{R}^3$ is called the current configuration of the body $\mathcal{B}$, and the corresponding image $B_t = \chi_t(\mathcal{B}) \subseteq \mathbb{R}^3$ is called its current placement.

Note

For the sake of simplicity, we will identify the reference configuration with the initial configuration throughout these notes. These two terms will therefore be used interchangeably. It should however be kept in mind that it is perfectly well admissible to have distinct initial and reference configurations. It is also possible to choose a reference configuration that is not actually occupied by the body during its motion.

Lagrangian vs Eulerian viewpoints

Consider a motion $\chi_t:\mathcal{B} \to \mathbb{R}^3$ of a body $\mathcal{B}$ in three dimensional space $\mathbb{R}^3$ according to an observer. Let $\chi_0$ and $\chi_t$ be the initial and current configurations, respectively, of the body. Let $B_0$ and $B_t$ be the initial and current placements, respectively, of the body in $\mathbb{R}^3$. Let us now focus on a real valued physical property of the body. We will now see how the same observer can talk about this physical quantity in many different ways, depending on the point of view she adopts.

For concreteness, let us suppose we are interested in assigning a temperature to every point of the body $\mathcal{B}$ at the current time $t$. This amounts to a map of the form $$\theta_t:B_t\to\mathbb{R}$$ that assigns a temperature $\theta_t(\mathsf{x})$ to every point $\mathsf{x} \in B_t$. This is called the Eulerian, or, spatial description of the temperature field. We will also write $\theta(\mathsf{x},t)$ for $\theta_t(\mathsf{x})$. Thus, $\theta(\mathsf{x},t)$ denotes the temperature at the current time $t$ of the material point $\chi_t^{-1}(\mathsf{x}) \in \mathcal{B}$.

We can alternatively describe the temperature at the current instant of time in terms of either the initial/reference configuration or the material body as follows. If $\mathsf{X} = \varphi_t^{-1}(\mathsf{x}) \in B_0$ is the position occupied by the material point $\chi_t^{-1}(\mathsf{x})$ at time $t = 0$, then we can define the Lagrangian, or, referential, or initial description of the current temperature using the map $$\Theta_t:B_0 \to \mathbb{R}, \qquad \mathsf{X} \mapsto \Theta_t(\mathsf{X}) = \theta_t(\varphi_t(\mathsf{X})).$$ Note again that this is the temperature at the current time $t$ of the particle that initially occupied the position $\mathsf{X} \in B_0$.

Finally, we can also talk about the current temperature in terms of the material points using the maps $$\hat{\Theta}_t:\mathcal{B} \to \mathbb{R}, \qquad p \mapsto \hat{\Theta}_t(p) = \theta_t(\chi_t(p)).$$ This is called the material description of the current temperature of the material particle $p$.

Note

Some authors use the terms material and Lagrangian/initial interchangeably. If a particular reference configuration that is distinct from the initial configuration is present, then the reference description refers to a description with respect to the reference configuration, and the terms Lagrangian/initial are reserved exclusively for the description with respect to the initial placement. It is always a good idea to be aware of the specific conventions followed in each case.

To reiterate a very important point, note that all these descriptions pertain to the description of the temperature field at the current instant of time $t$. This is summarized in the following equation: $$\theta(\mathsf{x},\mathbf{t}) = \Theta(\mathsf{X},\mathbf{t}) = \hat{\Theta}(p,\mathbf{t}).$$ Here, $\mathsf{X} = \chi_0(p)$ and $\mathsf{x} = \chi_t(p)$.

Remark

The pointwise assignment of temperature at every instant of time as outlined before is an example of a scalar field. The material, Lagrangian and Eulerian descriptions of vector and tensor fields are defined in an analogous manner.