Part 3 · Kinematics

3.1 Bodies, placements and deformations

Source: Favata’s notes, Kinematics §1.1–1.2, p. 3–5

Objectives

Place kinematics among the three pillars of continuum mechanics.
Understand a body through its placements, and a deformation as a map between two of them.
Picture: body, two placements, and the deformation that connects them.
Vocabulary: body, material point, placement, reference and current configuration, interpenetration.
Essential vocabulary: deformation.

Three pillars

Modern continuum mechanics keeps three things apart:

kinematics: what the bodies are and how they can move and deform;
the balance laws: forces, and what they must satisfy;
the constitutive relations: what makes rubber different from steel.

The first two are general, the same for every material. The third is particular: it defines one material. This part is kinematics, the first pillar. It describes bodies, their configurations and their admissible changes of shape, and never asks what causes them.

Bodies and placements

Definition 3.1.1Body and placement

A body $B$ is a set of material points $X$ , taken as primitive, like numbers in analysis. A placement (configuration) of $B$ is an injective map onto a region $Ω$ of the Euclidean space $E$ : $κ : B \to Ω \subset E, X \mapsto x = κ (X) .$ $x$ is the position of $X$ in the placement $κ$ ; loosely, $x$ is called a point and one writes $x \in Ω$ .

Injective: each place holds at most one material point. Two pieces of matter never share a place: no interpenetration of matter.
Onto $Ω$ : every point of $Ω$ is $κ (X)$ for some $X$ . The placement creates no new points.

So $κ$ has an inverse $κ^{- 1} : Ω \to B$ , which answers “which material point is here?”.

Truesdell: bodies are available to us only in their configurations, the regions they happen to occupy at some time. Favata’s image is Plato’s cave: the regions are the shadows on the wall, and the theory models the shadows, the only thing we ever see.

The notes assume the body is regular: it can be smoothly identified with a domain of Euclidean space. What a body or a part of it is, in general, turns out to be a hard question.

The deformation

Definition 3.1.2DeformationEssential

For two placements $κ$ and $χ$ of $B$ , the deformation from $κ$ to $χ$ is $f := χ \circ κ^{- 1} : κ (B) \to χ (B) .$

Go back from a position to the material point there, then forward to where that point sits in $χ$ . If $x = κ (X)$ , then $x$ and $f (x)$ are the positions of the same material point before and after.

The body

B

is only seen through its placements.

κ

puts the material point

X

x

χ

puts it at

f (x)

; the deformation

f

goes straight from one to the other.

$f$ is a bijection, with inverse $f^{- 1} = κ \circ χ^{- 1}$ (beyond the notes).
One placement is fixed once and for all as the reference configuration; another is called the current one. Where it matters, each gets its own copy of the spaces: $E_{ref}, V_{ref}$ and $E_{cur}, V_{cur}$ .
The body never appears again: everything is said about $f$ , a map between regions of space, where calculus works.

Key observation 3.1.3A deformation acts on points

$f$ is a vector field on the reference region: it says where each point goes. Whether it is a deformation in the physical sense depends on two things the next section adds: smoothness, and never letting matter pass through itself.

Non-example 3.1.4An overlap is not a placementbeyond the notes

Fold a strip of paper flat onto itself through a crease and squeeze until two layers coincide: two material points occupy the same place. That map is not injective, so the folded state is not a placement, and the map to it not a deformation. The fold in the body demo shows how this announces itself in the derivative.

Exercises

Exercise 3.1.1beyond the notes

Is $f (x) = - x$ a deformation of a ball centred at the origin? Of a ball centred at $3 e_{1}$ with radius $1$ ? Which property, if any, fails?

Solution

It is injective and smooth with a smooth inverse in both cases, so as a map of points it qualifies; for the second ball the image is simply elsewhere. But it reverses orientation, $\nabla f = - I$ with $det (- I) = - 1$ in 3D, and the next section excludes exactly this (Definition 3.2.4). A placement can be mirrored; a real body cannot be turned into its mirror image by a continuous motion.