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From AN INTRODUCTION TO CONTINUUM MECHANICS by J.N.REDDY

A rigid-body motion is one in which all material particles of the continuum $\mathcal{B}$ undergo the same linear and angular displacements. However, a deformable body is one in which the material particles can move relative to each other. Then the deformation of a continuum can be determined only by considering the change of distance between any two arbitrary but infinitesimally close points of the continuum.

Why can the deformation of a continuum be determined only by considering the change of distance between any two arbitrary but infinitesimally close points of the continuum?

I think one can determine the deformation of a continuum by (for example) directly measuring change of distance between any two arbitrary points of the continuum.

Although, I know that we will use infinitesimally close points for defining the deformation gradient tensor, IMHO, the sentence "Then the deformation of a continuum can be determined only by considering the change of distance between any two arbitrary but infinitesimally close points of the continuum" is not correct.

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    $\begingroup$ I don't know continuum mechanics at all but it is a common occurrence that "pathologically behaving" stuff behave nicely "infinitesimally". For example, transformations that form a smooth group might not commute, but "infinitesimal" transformations do. Parallel transport provided by a connection is path-dependent, but "infinitesimal" parallel transport isn't. If you consider how two distant points get displaced, it gives no idea about what deformation happens, since many deformations can give the same displacement. "Infinitesimally" separated points are in a sense.... $\endgroup$ Feb 23, 2017 at 18:26
  • $\begingroup$ ... adjacent, so this ambiguity is removed. $\endgroup$ Feb 23, 2017 at 18:27
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    $\begingroup$ I think "only" here actually means that "you don't have to consider points separated by a large distance". In other words, I guess he means that "infinitesimal are enough"! $\endgroup$
    – Peaceful
    Feb 23, 2017 at 18:36
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    $\begingroup$ The relative position of two points a finite distance apart may remain the same, even if the material "in between the points" deforms. And don't forget pathological cases like a body whose deformation $u(x)$ close to $x = 0$ follow a pattern like $u(x) = x \sin (1/x)$. $\endgroup$
    – alephzero
    Feb 23, 2017 at 21:16

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It is a calculus requirement.

To solve such problems, you have to consider strain, and stress, which are related to the derivative of the displacement.

To be infinitesimally close means close enough that the change in distance, divided by the original distance, equals the derivative of the displacement field.

That is a really great text, by the way.

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Okay I'm going to get a lot of hate for this answer, by both physicists and mathematicians, but for the sake of keeping this simple and easy to understand intuitively, I'm going to use a very crude example:

Think of this: Rigid object = a block of wood Flexible object ("deformed") = a block of goo

You push the wood and what happens? Every part of it moves together.

You push the goo and what happens? Parts of the goo move fast and change direction, whilst parts of it stay fairly still.

Now there's a risk that if we picked a random 2 points on the block of goo (let's say the very top and the very bottom), by complete coincidence, they might just happen to move at the exact same speed and direction. Does that make the goo rigid? No, it means you got lucky.

To get rid of this "luck" element, we have to pick 2 points that are infinitesimally close to each other. Because we'd expect all the points in the immediate surroundings to be moving at a different speed and direction, if the object is truly flexible all around.

I hope this helps you to understand this theory better.

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It looks like your interpretarion of the word ‘only’ differs from that intended by the author. This is because he uses ‘only’ as in ‘merely’, not as in ‘only if’. That is, he is talking about a sufficient condition for the detection of deformation, not a necessary one.

A more straightforward definition of ‘deformable body’ would certaibly consider arbitrary pairs of points, not solely infinitesimally close pairs. But clearly such a definition would be more difficult to apply, if only because one would not be able to take advantage of the simplifications associated with taking the limits of the expressions involved.

Fortunately, the author remarks that it sufficies to consider arbitrarily close pairs of points: if the distances between them remain constant, the body does not deform; i.e, the distances between an arbitrary pair of points does not change either.

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