motion of continuum
such that every material particle is carried over a place11This very convenient term was introduced by Lodge of the space in the course of time, i.e.
Choose an arbitrary system of coordinates. Fix any particle . At any instant whatever, once the motion it has took place, for we define
In the course of motion, sometimes it is important to consider any previous instant , where is the initial instant and the “present” one. Hence, we may restate Eqs. (1)-(3) substituting by and so they may be refered to as histories of position, velocity and acceleration, respectively. Also it is usual to denote those equations with the more compact notation , and , as . It is important to notice that such equations do not give us a complete description of motion, because they say not the manner about how to identify every and each of the particles conforming the body. So, in order to get that complete identification, we shall now introduce the material and spatial descriptions of continuum.
Material description of motion.
Let an arbitrary point of the body indentified by its material coordinates with respect to a cartesian rectangular system arbitrarily chocen. Those coordinates “map”each point of the body in certain configuration of reference in the same way that spatial coordinates provide a scheme 22Cf. CFT, pp.507,508  of a region of the space. Since the body adopts distinct configurations through the motion , we need to precise the concept of reference configuration. So, we shall call a reference configuration to the homeomorphism mapping given by
where is the point of material coordinates identifying any particle in the considered configuration. Such a representation satisfy two necessary requisites: first, a distinction between the motion and its material description at the considered reference configuration and second, invariance under a change of reference configuration, so being well-defined the motion of continuum, through a material description. In fact, from Eq.(1) and Eq.(4) with , we get
Furthermore, if represents another reference configuration (i.e. another system of coordinates ), then from Eq.(4), , , and so from Eq.(5), . Since we are speaking about the same motion at some , we have
so showing the required invariance. That equation indeed represents a transformation of mappings under changes of reference configuration, therefore we conclude that an invariance condition indicating changes of representation, leads to a transformation relation. In those cases where the configuration is intended, we shall write
That principle implies that the Jacobian of the transformation between material and spatial coordinates it must be positive i.e.,
Spatial description of motion.
In a spatial description of motion, we do not relate directly to particles but uniquely to their velocities.33The velocity was introduced as a primitive concept by D’AlembertSo we shall deal with the field of particle velocities , , because the “history” about a specific particle is not needed. Such a description is a suitable choice in fluid mechanics, for instance. The distinction between material and spatial descriptions is quite basic: in the former is the dependent variable, and the independent variables, whereas in the latter is the dependent variable, and the independent ones. Let us define the time material derivative or the material rate 44a notation introduced by Stokes like which it is found by an “observer” moving with a particle , i.e., with material coordinates held constant. Likewise, the local rate is found by an “observer” situated at a fixed point , i.e., spatial coordinates held constant. Therefore, from those definitions we get
Thus, we may find out the aceleration field by calculating the material rate of the velocity field ,
by application of the chain rule, and where by definition. We can also to express Eq.(10) in a rectangular cartesian system,
where we have used the conventional index summation. Eqs.(10)-(11) correspond to the spatial description of motion. In fluid mechanics language, the term or equivalently, , is called the convected rate because the particle-bound “observer” moves, by virtue of its instantaneous velocity , into regions of different local field values. Furthermore, we call a steady field if the local rate vanishes and a uniform field if the convected rate vanishes.
In continuum mechanics there are two classical field theories: elasticity theory and hydrodymamics.Continuum kinematics it is largely determined by the kind of mechanical response (the cause being, any field forces or impulses solicitation) that is being described. Although we can use in those theories either of the descriptions of motion, from the point of view of mechanical response alone the material description comes to be adequate for elasticity theory, whereas the spatial description (velocity field) is the natural way for hydrodynamics. Basically, a deformable solid possesses a “kind of memory” from an initial undistorsioned configuration , so it becomes necessary to somehow “label” the body particles at that undistorsioned initial configuration. Thus, material coordinates it will be a suitable way in order to make the proper description of the solid in the subsequent distorsioned configurations. On the other hand, a viscous fluid (liquid, gas or plasm) presents a completely different mechanical behavior because its response is determined solely by the instantaneous values of the time rates of deformation. So in general, viscous fluids have no memory at all, not existing past configurations that are special in any way. For that reason, it is natural to use spatial description for viscous fluids.
As an illustration, we shall do a briefly discussion on finite strain in elasticity theory and on rate of deformation which corresponds to hydrodynamics (viscous fluids) kinematics.
1. Finite strain. We define particle displacement , as the difference between, the position of an arbitrary particle of the deformable solid in any distorsioned configuration, and the position of the same particle in the initial undistorsioned configuration of the body, both with respect to a rectangular cartesian system arbitrarily chosen. That is,
We apply now the operator (respect to material particle ), to obtain
where, is the unit tensor. By taking the transpose, we have
Next we consider the square line element
where is the Cauchy-Green deformation tensor. Finally, we define the finite strain as
in which, by replacing , we get
2. Rate of deformation. Let be material coordinates describing particles at the sine of a viscous fluid, and let be the velocity field who is tangent to the fluid flow streamlines crossing the region of the space. Since the material derivatives and , we have (by applying the chain rule)
We find now the material rate of and then we multiply by , to obtain
Next we evaluate the material derivative of a square line element, with metric tensor 55If the spatial coordinate system be instantaneously stationary, then Furthermore, since the components of the metric tensor are constant in orthogonal cartesian coordinates the corresponding Christoffel symbols are zero, and the covariant derivatives of the metric tensor vanish in all coordinate systems. That is, Therefore, we have located in ,
whence, by Eq.(a), we arrive to the Beltrami equation
where is a symmetric tensor of second order which represents the required rate of deformation.
- 1 A.S. Lodge, On the use of convected coordinate systems in the mechanics of continuous media, Proc. Cambr. Phil. Soc. 575-584, 1951.
- 2 C. Truesdell, R.A. Toupin, The Classical Field Theories Enc. of Physics, Springer Verlag, New York, 1960.
- 3 J. L. D’Alembert, Essai d’une Nouvelle Théorie de la Resistance des Fluides,, Paris, 1752.
- 4 G.G. Stokes, On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids, Trans. Cambr. Phil. Soc. (1844-1849), 287-319 = papers 75-129, 1845.
- 5 G. Green, On the propagation of light in crystallized media (1839), Trans. Cambr. Phil. Soc. (1839-1842), 121-140 = Papers, 293-311, 1841.
- 6 A.-L. Cauchy Sur le condensation et la dilatation des corps solides, Ex. de Math. = Oeuvres (2) 82-83, 1827.
- 7 A.-L. Cauchy MÃÂ©moire sur les dilatations, les condensations et les rotations produits par un changement de forme dans un système de points materiÃÂ©ls, Ex. d’An. Phys. Math. = Oeuvres (2) 343-377, 1841.
- 8 E. Beltrami, Sui principi fondamentali della idrodinamica, Mem. Acad. Sci. Bologna (3) 431-476, (1872), 381-437, (1873), 349-407, (1874), 443-484 = Richerche sulla cinematica dei fluidi, Opere 202-379, 1871.
|Title||motion of continuum|
|Date of creation||2013-03-22 15:53:16|
|Last modified on||2013-03-22 15:53:16|
|Last modified by||perucho (2192)|