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motion of continuum

(Definition)

Definition.

Let $\mathscr{B}$ be a body embedded in an Euclidean vector space $(\mathbb{R}^3,\lVert \cdot \rVert)$ and let $\chi$ be a smooth homeomorphism describing a family of mappings

$\displaystyle \chi\colon \mathscr{B} \to \Re, \qquad \Re=\chi(\mathscr{B}),$

such that every material particle $P\in\mathscr{B}$ is carried over a place¹ $\mathbf{x}\in\Re$ of the space in the course of time, i.e.

$\displaystyle \chi\colon P\mapsto \mathbf{x}, \qquad \mathbf{x}(t):=\chi(P,t).$

These mappings are called configurations, the body being represented as a continuum $\Re$ in the Euclidean space. Such a description we shall call it motion of continuum.

Particle position vector, velocity and acceleration.

Choose an arbitrary system of coordinates. Fix any particle $P\in\mathscr{B}$ . At any instant whatever, once the motion it has took place, for $\mathbf{x}\in\Re$ we define

$\displaystyle \mathbf{x}=\chi(P,t),$

(1)

$\displaystyle \mathbf{v}=\mathbf{\dot{x}}=\frac{d}{dt}\chi(P,t),$

(2)

$\displaystyle \mathbf{a}=\mathbf{\ddot{x}}=\frac{d^2}{dt^2}\chi(P,t).$

(3)

In the course of motion, sometimes it is important to consider any previous instant $\tau,\; t_0 \leq \tau \leq t$ , where

is the initial instant and

the “present” one. Hence, we may restate Eqs. (1)-(3) substituting

by $\tau$ 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 $\mathbf{x}=\mathbf{x}(t)$ , $\mathbf{v}=\mathbf{v}(t)$ and $\mathbf{a}=\mathbf{a}(t)$ , as $\tau=t$ . 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 $P\in\mathscr{B}$ 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 ² of a region $\Re$ of the space. Since the body adopts distinct configurations through the motion $\mathbf{x}(t)$ , we need to precise the concept of reference configuration. So, we shall call a reference configuration to the homeomorphism mapping $\varkappa$ given by

$\displaystyle \mathbf{X}=\varkappa(P), \qquad P=\varkappa^{-1}(\mathbf{X}),$

(4)

where $\mathbf{X}$ is the point of material coordinates

identifying any particle $P\in\mathscr{B}$ in the considered configuration. Such a representation satisfy two necessary requisites: first, a distinction between the motion $\mathbf{x}(\tau)$ and its material description at the considered reference configuration $\chi_\varkappa$ 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 $t=\tau$ , we get

$\displaystyle \mathbf{x}(\tau)=\chi(P,\tau)=\chi(\varkappa^{-1}(\mathbf{X}),\tau)\equiv\chi_\varkappa(\mathbf{X},\tau).$

(5)

Furthermore, if $\overline{\varkappa}$ represents another reference configuration (i.e. another system of coordinates $\overline{X}_i$ ), then from Eq.(4), $\overline{\mathbf{X}}=\overline{\varkappa}(P)$ , $P=\overline{\varkappa}^{-1}(\overline{\mathbf{X}})$ , and so from Eq.(5), $\mathbf{x}(\tau)=\chi_{\overline{\varkappa}}(\overline{\mathbf{X}}, \tau)$ . Since we are speaking about the same motion at some $\tau\in[t_0,t]$ , we have

$\displaystyle \mathbf{x}(\tau)=\chi_{\varkappa}(\mathbf{X},\tau)=\chi_{\overline{\varkappa}}(\mathbf{\overline{X}},\tau),$

(6)

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

$\displaystyle \mathbf{x}(\tau)=\chi(\mathbf{X},\tau),$

(7)

$\displaystyle \mathbf{v}(\tau)=\mathbf{\dot{x}}(\mathbf{X},\tau),$

(8)

$\displaystyle \mathbf{a}(\tau)=\mathbf{\ddot{x}}(\mathbf{X},\tau).$

(9)

Since $\chi$ and its inverse $\chi^{-1}$ are injective and differentiable, we get as a corollary the principle of impenetrability:

$\displaystyle \lq\lq an\, element\, of\, matter\, can\, never\, penetrate\, another\, element\, of\, matter''.$

That principle implies that the Jacobian of the transformation between material and spatial coordinates it must be positive i.e.,

$\displaystyle J= \bigg\vert\frac{\partial{x_i}}{\partial{X_j}}\bigg\vert > 0,$

what means that (from

) a volume element cannot be reduced into a point once the deformation takes place.

Spatial description of motion.

In a spatial description of motion, we do not relate directly to particles but uniquely to their velocities.³So we shall deal with the field of particle velocities $\mathbf{v}(\mathbf{x},t)$ , $\mathbf{x}\in \Re$ , because the “history” about a specific particle $\mathbf{X}\in\mathscr{B}$ 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 $\mathbf{x}$ is the dependent variable, and $\mathbf{X},t$ the independent variables, whereas in the latter $\mathbf{v}$ is the dependent variable, and $\mathbf{x},t$ the independent ones. Let us define the time material derivative or the material rate ⁴ like which it is found by an “observer” moving with a particle , i.e., with material coordinates held constant. Likewise, the local rate $\partial_t$ is found by an “observer” situated at a fixed point $\mathbf{x}$ , i.e., spatial coordinates held constant. Therefore, from those definitions we get

$\displaystyle D_t\mathbf{X}\equiv\frac{D\mathbf{X}}{Dt}\bigg\vert _{\mathbf{X}=... ...rac{\partial\mathbf{x}}{\partial{t}}\bigg\vert _{\mathbf{x}=const.}=\mathbf{0}.$

Thus, we may find out the aceleration field $\mathbf{a}(\mathbf{x},t)$ by calculating the material rate of the velocity field $\mathbf{v}(\mathbf{x},t)$ ,

$\displaystyle \mathbf{v}=\mathbf{v}(\mathbf{x},t), \quad \mathbf{a}(\mathbf{x},t)\equiv\mathbf{\dot{v}}=\partial_t{\mathbf{v}}+(\mathbf{v}\cdot\nabla)\mathbf{v},$

(10)

by application of the chain rule, and where $\mathbf{v}=\mathbf{\dot{x}}(\mathbf{x},t)$ by definition. We can also to express Eq.(10) in a rectangular cartesian system,

$\displaystyle v_i=\dot{x}_i(x_k,t), \qquad a_i=\dot{v}_i(x_k,t)=\partial_t{v_i}+v_jv_{i,j},$

(11)

where we have used the conventional index summation. Eqs.(10)-(11) correspond to the spatial description of motion. In fluid mechanics language, the term $(\mathbf{v}\cdot\nabla)\mathbf{v}$ or equivalently, $v_jv_{i,j}$ , is called the convected rate because the particle-bound “observer” moves, by virtue of its instantaneous velocity $\mathbf{v}$ , 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.

Some applications.

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 $\Re_0$ , 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 $\mathbf{u}$ , as the difference between, the position $\mathbf{x}\in\Re$ of an arbitrary particle of the deformable solid in any distorsioned configuration, and the position $\mathbf{X}\in\Re_0$ of the same particle in the initial undistorsioned configuration of the body, both with respect to a rectangular cartesian system arbitrarily chosen. That is,

$\displaystyle \mathbf{u}=\mathbf{x}-\mathbf{X}.$

We apply now the operator $\nabla_{\mathbf{X}}$ (respect to material particle $\mathbf{X}$ ), to obtain

$\displaystyle \nabla_{\mathbf{X}}\mathbf{u}=\nabla_{\mathbf{X}}\mathbf{x}-\mathbf{1},$

where, $\nabla_{\mathbf{X}}\mathbf{X}=\mathbf{1}$ is the unit tensor. By taking the transpose, we have

$\displaystyle \nabla_{\mathbf{X}}\mathbf{x}=\mathbf{1}+\nabla_{\mathbf{X}}\math... ... \qquad \mathbf{x}\nabla_{\mathbf{X}}=\mathbf{1}+\mathbf{u}\nabla_{\mathbf{X}}.$

Next we consider the square line element

$\displaystyle ds^2=d\mathbf{x}\cdot{d\mathbf{x}}=(d\mathbf{X}\cdot\nabla_{\math... ...bf{X}})\cdot{d\mathbf{X}}\equiv {d\mathbf{X}}\cdot\mathbf{C}\cdot{d\mathbf{X}},$

where $\mathbf{C}$ is the Cauchy-Green deformation tensor[5]. Finally, we define the finite strain as

$\displaystyle \mathbf{E}\equiv\frac{1}{2}(\mathbf{C}-\mathbf{1}),$

in which, by replacing $\mathbf{C}$ , we get

$\displaystyle \mathbf{E}=\frac{1}{2}(\nabla_{\mathbf{X}}\mathbf{u}+\mathbf{u}\n... ..._{\mathbf{X}}+\nabla_{\mathbf{X}}\mathbf{u}\cdot\mathbf{u}\nabla_{\mathbf{X}}),$

or equivalently,

$\displaystyle E_{ij}=\frac{1}{2}(u_{i,j}+u_{j,i}+u_{i,k}u_{k,j}),$

a symmetric cartesian tensor of second order(subscript comma denoting differentiation).

The theory of finite strain is the creation of Cauchy[6],[7].

2. Rate of deformation. Let $X^\alpha\in\mathscr{B}$ be material coordinates describing particles at the sine of a viscous fluid, and let $\dot{x}^i(x^k(X^\alpha,t),t)$ be the velocity field who is tangent to the fluid flow streamlines crossing the region $\Re$ of the space. Since the material derivatives $\dot{\overline{dX^\alpha}}=0$ and $\dot{\overline{x^i\,_{,\alpha}}}=\dot{x}^i_{\,,\alpha}$ , we have (by applying the chain rule)

$\displaystyle \dot{\overline{dx^i}}=\dot{\overline{x^i_{\,,\alpha}dX^\alpha}}= ... ...pha=\dot{x}^i_{\,,j}x^j_{\,,\alpha}d X^\alpha =\dot{x}^i_{\,,j}dx^j. \qquad (a)$

We find now the material rate of $x^i_{\,,\beta}X^\beta_{\,,j}=\delta^i_j$ and then we multiply by $X^\alpha_{\,,i}$ , to obtain

$\displaystyle \dot{\overline{X^\alpha_{\,,j}}}=-\dot{x}^i_{\,,\beta}X^\beta_{\,,j}X^\alpha_{\,,i}=-\dot{x}^i_{\,,j}X^\alpha_{\,,i}.$

Next we evaluate the material derivative of a square line element, with metric tensor $g_{ij}$ ⁵ located in $\Re$ ,

$\displaystyle \dot{\overline{ds^2}}=\dot{\overline{g_{ij}dx^idx^j}}=g_{ij}(\dot{\overline{dx^i}}dx^j+dx^i\dot{\overline{dx^j}}),$

whence, by Eq.(a), we arrive to the Beltrami equation[8]

$\displaystyle \dot{\overline{ds^2}}=2d_{ij}dx^idx^j, \qquad d_{ij}\equiv\frac{1}{2}(\dot{x}_{i,j}+\dot{x}_{j,i}),$

where $d_{ij}$ is a symmetric tensor of second order which represents the required rate of deformation.

Bibliography

1: A.S. Lodge, On the use of convected coordinate systems in the mechanics of continuous media, Proc. Cambr. Phil. Soc. ${\bf 47,}$ 575-584, 1951.
2: C. Truesdell, R.A. Toupin, The Classical Field Theories Enc. of Physics, ${\bf III/1,}$ 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. ${\bf 8}$ (1844-1849), 287-319 = papers ${\bf 1,}$ 75-129, 1845.
5: G. Green, On the propagation of light in crystallized media (1839), Trans. Cambr. Phil. Soc. ${\bf 7}$ (1839-1842), 121-140 = Papers, 293-311, 1841.
6: A.-L. Cauchy Sur le condensation et la dilatation des corps solides, Ex. de Math. ${\bf 2}$ = Oeuvres (2) ${\bf 7,}$ 82-83, 1827.
7: A.-L. Cauchy Mémoire sur les dilatations, les condensations et les rotations produits par un changement de forme dans un système de points materiéls, Ex. d'An. Phys. Math. ${\bf 2}$ = Oeuvres (2) ${\bf 12,}$ 343-377, 1841.
8: E. Beltrami, Sui principi fondamentali della idrodinamica, Mem. Acad. Sci. Bologna (3) ${\bf 1,}$ 431-476, ${\bf 2}$ (1872), 381-437, ${\bf 3}$ (1873), 349-407, ${\bf 5}$ (1874), 443-484 = Richerche sulla cinematica dei fluidi, Opere ${\bf 2,}$ 202-379, 1871.

Footnotes

...¹: This very convenient term was introduced by Lodge[1]
...²: Cf. CFT, pp.507,508 [2]
...³: The velocity was introduced as a primitive concept by D'Alembert[3]
...⁴: a notation introduced by Stokes[4]
...⁵: If the spatial coordinate system be instantaneously stationary, then $\partial_tg_{ij}=0.$ 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, $\delta^i_j\big\vert _k=g_{ij}\big\vert _k=0.$ Therefore, we have $\dot{g}_{ij}=\partial_tg_{ij}+g_{ij}\big\vert _k\dot{x}^k=0.$

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This is version 12 of motion of continuum, born on 2006-05-01, modified 2006-06-16.
Object id is 7889, canonical name is MotionOfContinuum.
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AMS MSC:

53A45 (Differential geometry :: Classical differential geometry :: Vector and tensor analysis)

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