Model of the solid (mechanical)

The model of the solid is frequently used in Mécanique of the systems of material points. It is about a idealization of the usual concept of body (with the state) solid, regarded as absolutely rigid .

Definition

A solid (S) is a whole of material points {M_i} whose mutual distances remain constant during time.
It should be noted that this definition is very restrictive and does not have to be confused with that " of solid state " , for which one takes account of the possibility of (weak) Déformation S.

Models discrete and continuous

As for the material systems in general (deformable or not), one can adopt is a mode of description discontinuous (or discrete), or continuous.

Discrete model

The model easiest to use is to regard the solid (or the material system) as a whole of an infinity of material points M_i of masses m_i .
One defines the total Masse then solid by $M=\; \backslash \; sum\_\; \{I\}\; m\_\; \{I\}\; \backslash \; qquad$, being within the framework of a traditional model with additivity of the masses. All the other extensive physical sizes are formally defined by discrete sums: for example, the Quantité of movement is expressed by $\backslash \; vec\; \{P\}\; =\; \backslash \; sum\_\; \{I\}\; m\_\; \{I\}\; \backslash \; vec\; \{v\_\; \{I\}\}$, etc

Continuous model

The matter has a discrete microscopic structure of course, consisting of atoms, themselves having besides an internal structure. However if one considers an element of " volume; mésoscopique" solid, this one is likely to contain a very great number of atoms. For example an iron cube of 1 μm on side (very small on a macroscopic scale) contains close to 8,5.10¹³ atoms. One can thus treat the solid like a continuous medium. More precisely, a medium will be known as Continu if the number of particles contained in an elementary volume dτ is sufficiently large so that one can neglect his fluctuations. It should be noted that this mode of description is not specific to the bodies " solide" but is also appropriate very well for the Fluide s.

On the assumption of the continuous mediums, one can describe the solid (or also a material system like a fluid) not by a discrete whole of material points, but by a voluminal distribution of mass ρ (M) , defined in any point M of the field of space (V) occupied by the solid.

More precisely one can define the density ρ (M) by considering an element of volume δτ centered in the point M , of mass δm , by $\backslash \; lim\_\; \{\backslash \; delta\; \backslash \; tau\; \backslash \; to\; 0\}\; \backslash \; frac\; \{\backslash \; delta\; m\}\; \{\backslash \; delta\; \backslash \; tau\}\; =\; \backslash \; rho\; (M)$.

Note:: the solid is known as homogeneous if ρ (M) = cte for any point M of the solid (or of the system).
The majority of the kinetic elements of the material systems (and thus, of the solids) can be defined by replacing the summation $\backslash \; sum\_\; \{I\}\; \backslash \; quad$ on the indices of the material points of the discrete model by a voluminal integration on the field (V) , for example:

• Mass of the solid: $M=\; \backslash \; iiint\_\; \{M\; \backslash \; in\; (V)\}\; \backslash \; rho\; (M)\; D\; \backslash \; tau$;

• Center of inertia G: $M\; \backslash \; vec\; \{OG\}\; =\; \backslash \; iiint\_\; \{M\; \backslash \; in\; (V)\}\; \backslash \; rho\; (M)\; \backslash \; vec\; \{OM\}\; D\; \backslash \; tau$;
• Momentum $\backslash \; vec\; \{P\}\; =\; \backslash \; iiint\_\; \{M\; \backslash \; in\; (V)\}\; \backslash \; rho\; (M)\; \backslash \; vec\; \{v\_\; \{M\}\}\; D\; \backslash \; tau$;
• kinetic Moment out of O $\backslash \; vec\; \{L\_\; \{O\}\}\; =\; \backslash \; iiint\_\; \{M\; \backslash \; in\; (V)\}\; \backslash \; left\; (\backslash \; vec\; \{OM\}\; \backslash \; times\; \backslash \; rho\; (M)\; \backslash \; vec\; \{v\_\; \{M\}\}\; \backslash \; right)\; D\; \backslash \; tau$;
• kinetic Energy $E\_\; \{K\}\; =\; \backslash \; iiint\_\; \{M\; \backslash \; in\; (V)\}\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash \; rho\; (M)\; v\_\; \{M\}\; ^\; \{2\}\; D\; \backslash \; tau$;

For a solid, the continuous model is generally regarded as heavier with utliser, also often prefers him one the discrete model, in particular in the demonstrations of the various theorems. Nevertheless, it makes it possible most rigorously to define the concepts of mechanical symmetry and to carry out calculations of moments of inertia, not very easy to carry out by summation.

Mechanical symmetry

It is obvious of share the definition of a mechanical solid that this one has a clean geometrical form - it is moreover about an elementary macroscopic characteristic of the solid state. It is frequent to consider solids of geometrical forms simple (e.g sphere, cubic, cylinder…) presenting elements of geometrical Symmetry (e.g center, Symmetry plane Axis or ) given. However the distribution of the masses within such a solid does not have necessarily the same elements of Symétrie.

One speaks about mechanical Symétrie associated a geometrical Symétrie so for any couple with points (M, Me) of the solid counterparts in the geometrical Symétrie one has ρ (M) =ρ (Me) . The existence for a solid of mechanical symmetries largely simplifies the determination of the position of the Center of inertia G , of the main axes of inertia, calculations of the Moment of inertia.

Movements of a solid

The general study of the movement of a solid is complex in the general case, although the condition of rigidity of the model simplifies the problem largely. indeed, instead of an infinity (or of a very great number) of degrees of freedom a solid has only 6 degrees of freedom:

• 3 correspondent with the coordinates of sound Center of inertia G ;

• and 3 angles (Angles of Euler for example) allowing to describe its rotation " propre" in the barycentric reference frame.

It is interesting to consider some particular cases.

Field of the Speed S of a solid

For two unspecified points M and P of a solid, by assumption one PM = cte . Consequently $\backslash \; frac\; \{D\; (PM^\; \{2\})\}\{dt\}\; =\; \backslash \; frac\; \{D\; \backslash \; left\; (\backslash \; vec\; \{PM\}\; ^\; \{2\}\; \backslash \; right)\}\{dt\}\; =0$, is also $\backslash \; vec\; \{PM\}\; \backslash \; cdot\; \backslash \; frac\; \{D\; \backslash \; left\; (\backslash \; vec\; \{PM\}\; \backslash \; right)\}\{dt\}\; =\; \backslash \; vec\; \{PM\}\; \backslash \; cdot\; \backslash \; left\; (\backslash \; vec\; \{v\_\; \{M\}\}\; -\; \backslash \; vec\; \{v\_\; \{P\}\}\; \backslash \; right)\; =\; \backslash \; vec\; \{0\}$ .
On finds a property of equiprojectivity of the field speeds of the solid, consequently:
$\backslash \; vec\; \{v\_\; \{M\}\}\; =\; \backslash \; vec\; \{v\_\; \{P\}\}\; +\; \backslash \; vec\; \{\backslash \; Omega\}\; \backslash \; times\; \backslash \; vec\; \{PM\}$, (1) , with $\backslash \; vec\; \{\backslash \; Omega\}$ vector rotation of the solid in the Reference frame of study (R) (or the barycentric reference frame (R^*) associated, because both are in translation). This vector with for value the angular Velocity instantaneous of the solid, for direction the instantaneous axis of rotation of the solid.
This relation can be written while utilizing the Center of inertia G of the solid: indeed it is obvious that for any point M of the solid that GM = cte , therefore the formula (1) remains applicable, from where: $\backslash \; vec\; \{v\_\; \{M\}\}\; =\; \backslash \; vec\; \{v\_\; \{G\}\}\; +\; \backslash \; vec\; \{\backslash \; Omega\}\; \backslash \; times\; \backslash \; vec\; \{GM\}$, which makes it possible to show that the movement of an unspecified point M of a solid breaks up into a movement " of translation" and another of " rotation propre".

Translatory movement of a solid

Rotation movement around an axis

See too

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