motion of continuum

motion of continuum

Definition.

Let

ℬ

be a body embedded in an

Euclidean vector space

(

ℝ

3

,

∥

⋅

∥

)

and let

χ

be a smooth

homeomorphism

describing a family of

mappings

χ

:

ℬ

→

ℜ

,

ℜ

=

χ

⁢

(

ℬ

)

,

such that every material particle

P

∈

ℬ

is carried over a

place

1

1

This very convenient term was introduced by Lodge

[

1

]

x

∈

ℜ

of the space in the course of time, i.e.

χ

:

P

↦

𝐱

,

𝐱

⁢

(

t

)

:=

χ

⁢

(

P

,

t

)

.

These mappings are called

configurations

, the body being represented as a

continuum

ℜ

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

∈

ℬ

. At any instant

t

whatever, once the motion it has took place, for

𝐱

∈

ℜ

we define

𝐱

=

χ

⁢

(

P

,

t

)

,

(1)

𝐯

=

𝐱

˙

=

d

d

⁢

t

⁢

χ

⁢

(

P

,

t

)

,

(2)

𝐚

=

𝐱

¨

=

d

2

d

⁢

t

2

⁢

χ

⁢

(

P

,

t

)

.

(3)

In the course of motion, sometimes it is important to consider any previous instant

τ

,

t

0

≤

τ

≤

t

,  where

t

0

is the initial instant and

t

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

t

by

τ

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

𝐱

=

𝐱

⁢

(

t

)

,

𝐯

=

𝐯

⁢

(

t

)

and

𝐚

=

𝐚

⁢

(

t

)

, as

τ

=

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

∈

ℬ

an arbitrary point of the body indentified by its material coordinates

X

i

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

x

i

provide a scheme

2

2

Cf.   CFT,   pp.507,508

[

2

]

of a region

ℜ

of the space. Since the body adopts distinct configurations through the motion

𝐱

⁢

(

t

)

, we need to precise the

concept

of

reference configuration

. So, we shall call a reference configuration to the homeomorphism mapping

ϰ

given by

𝐗

=

ϰ

⁢

(

P

)

,

P

=

ϰ

-

1

⁢

(

𝐗

)

,

(4)

where

𝐗

is the point of material coordinates

X

i

identifying any particle

P

∈

ℬ

in the considered configuration.
Such a representation

satisfy

two necessary requisites: first, a distinction between the motion

𝐱

⁢

(

τ

)

and its material description at the considered reference configuration

χ

ϰ

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

=

τ

, we get

𝐱

⁢

(

τ

)

=

χ

⁢

(

P

,

τ

)

=

χ

⁢

(

ϰ

-

1

⁢

(

𝐗

)

,

τ

)

≡

χ

ϰ

⁢

(

𝐗

,

τ

)

.

(5)

Furthermore, if

ϰ

¯

represents another reference configuration (i.e. another system of coordinates

X

¯

i

), then from Eq.(4),

𝐗

¯

=

ϰ

¯

⁢

(

P

)

,

P

=

ϰ

¯

-

1

⁢

(

𝐗

¯

)

, and so from Eq.(5),

𝐱

⁢

(

τ

)

=

χ

ϰ

¯

⁢

(

𝐗

¯

,

τ

)

. Since we are speaking about the same motion at some

τ

∈

[

t

0

,

t

]

, we have

𝐱

⁢

(

τ

)

=

χ

ϰ

⁢

(

𝐗

,

τ

)

=

χ

ϰ

¯

⁢

(

𝐗

¯

,

τ

)

,

(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

𝐱

⁢

(

τ

)

=

χ

⁢

(

𝐗

,

τ

)

,

(7)

𝐯

⁢

(

τ

)

=

𝐱

˙

⁢

(

𝐗

,

τ

)

,

(8)

𝐚

⁢

(

τ

)

=

𝐱

¨

⁢

(

𝐗

,

τ

)

.

(9)

Since

χ

and its

inverse

χ

-

1

are

injective

and

differentiable

, we get as a corollary the

principle of impenetrability:

`

⁢

`

⁢

a

⁢

n

⁢

e

⁢

l

⁢

e

⁢

m

⁢

e

⁢

n

⁢

t

⁢

o

⁢

f

⁢

m

⁢

a

⁢

t

⁢

t

⁢

e

⁢

r

⁢

c

⁢

a

⁢

n

⁢

n

⁢

e

⁢

v

⁢

e

⁢

r

⁢

p

⁢

e

⁢

n

⁢

e

⁢

t

⁢

r

⁢

a

⁢

t

⁢

e

⁢

a

⁢

n

⁢

o

⁢

t

⁢

h

⁢

e

⁢

r

⁢

e

⁢

l

⁢

e

⁢

m

⁢

e

⁢

n

⁢

t

⁢

o

⁢

f

⁢

m

⁢

a

⁢

t

⁢

t

⁢

e

⁢

r

⁢

"

.

That principle implies that the

Jacobian

of the transformation between material and spatial coordinates it must be positive i.e.,

J

=

|

∂

⁡

x

i

∂

⁡

X

j

|

>

0

,

what means that (from

d

⁢

v

=

J

⁢

d

⁢

V

) 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.

3

3

The velocity was introduced as a

primitive

concept by D’Alembert

[

3

]

So we shall deal with the field of particle velocities

𝐯

⁢

(

𝐱

,

t

)

,

𝐱

∈

ℜ

, because the “history” about a specific particle

𝐗

∈

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

𝐱

is the dependent variable, and

𝐗

,

t

the independent variables, whereas in the latter

𝐯

is the dependent variable, and

𝐱

,

t

the

independent

ones. Let us define the

time material derivative

or the

material rate

D

t

4

4

a notation introduced by Stokes

[

4

]

like which it is found by an “observer” moving with a particle

X

, i.e., with material coordinates

X

i

held

constant

. Likewise, the

local rate

∂

t

is found by an “observer” situated at a

fixed point

𝐱

, i.e., spatial coordinates

x

i

held constant. Therefore, from those definitions we get

D

t

⁢

𝐗

≡

D

⁢

𝐗

D

⁢

t

|

𝐗

=

c

o

n

s

t

.

=

𝟎

,

∂

t

⁡

𝐱

≡

∂

⁡

𝐱

∂

⁡

t

|

𝐱

=

c

o

n

s

t

.

=

𝟎

.

Thus, we may find out the aceleration field

𝐚

⁢

(

𝐱

,

t

)

by calculating the material rate of the velocity field

𝐯

⁢

(

𝐱

,

t

)

,

𝐯

=

𝐯

⁢

(

𝐱

,

t

)

,

𝐚

⁢

(

𝐱

,

t

)

≡

𝐯

˙

=

∂

t

⁡

𝐯

+

(

𝐯

⋅

∇

)

⁢

𝐯

,

(10)

by application of the

chain rule

, and where

𝐯

=

𝐱

˙

⁢

(

𝐱

,

t

)

by definition. We can also to express Eq.(10) in a rectangular cartesian system,

v

i

=

x

˙

i

⁢

(

x

k

,

t

)

,

a

i

=

v

˙

i

⁢

(

x

k

,

t

)

=

∂

t

⁡

v

i

+

v

j

⁢

v

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

(

𝐯

⋅

∇

)

⁢

𝐯

or equivalently,

v

j

⁢

v

i

,

j

, is called the

convected rate

because the particle-bound “observer” moves, by virtue of its instantaneous velocity

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

ℜ

0

, so it becomes necessary
to somehow “label” the body particles at that undistorsioned initial configuration. Thus, material coordinates

X

i

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

𝐮

, as the

difference

between, the position

𝐱

∈

ℜ

of an arbitrary particle of the deformable solid in any distorsioned configuration, and the position

𝐗

∈

ℜ

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,

𝐮

=

𝐱

-

𝐗

.

We apply now the

operator

∇

𝐗

(respect to material particle

𝐗

), to obtain

∇

𝐗

⁡

𝐮

=

∇

𝐗

⁡

𝐱

-

𝟏

,

where,

∇

𝐗

⁡

𝐗

=

𝟏

is the unit tensor. By taking the

transpose

, we have

∇

𝐗

⁡

𝐱

=

𝟏

+

∇

𝐗

⁡

𝐮

,

𝐱

⁢

∇

𝐗

=

𝟏

+

𝐮

⁢

∇

𝐗

.

Next we consider the square

line element

d

⁢

s

2

=

d

⁢

𝐱

⋅

d

⁢

𝐱

=

(

d

⁢

𝐗

⋅

∇

𝐗

⁡

𝐱

)

⋅

(

𝐱

⁢

∇

𝐗

⋅

d

⁢

𝐗

)

=

d

⁢

𝐗

⋅

(

𝟏

+

∇

𝐗

⁡

𝐮

)

⋅

(

𝟏

+

𝐮

⁢

∇

𝐗

)

⋅

d

⁢

𝐗

≡

d

⁢

𝐗

⋅

𝐂

⋅

d

⁢

𝐗

,

where

𝐂

is the Cauchy-Green deformation tensor

[

5

]

. Finally, we define the

finite strain

as

𝐄

≡

1

2

⁢

(

𝐂

-

𝟏

)

,

in which, by replacing

𝐂

, we get

𝐄

=

1

2

⁢

(

∇

𝐗

⁡

𝐮

+

𝐮

⁢

∇

𝐗

+

∇

𝐗

⁡

𝐮

⋅

𝐮

⁢

∇

𝐗

)

,

or equivalently,

E

i

⁢

j

=

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

α

∈

ℬ

be material coordinates describing particles at the sine of a viscous fluid, and let

x

˙

i

⁢

(

x

k

⁢

(

X

α

,

t

)

,

t

)

be the velocity field who is

tangent

to the fluid flow streamlines crossing the region

ℜ

of the space. Since the material derivatives

d

⁢

X

α

¯

˙

=

0

and

x

,

α

i

¯

˙

=

x

˙

,

α

i

, we have (by applying the chain rule)

d

⁢

x

i

¯

˙

=

x

,

α

i

⁢

d

⁢

X

α

¯

˙

=

x

˙

,

α

i

d

X

α

=

x

˙

,

j

i

x

,

α

j

d

X

α

=

x

˙

,

j

i

d

x

j

.

(

a

)

We find now the material rate of

x

,

β

i

⁢

X

,

j

β

=

δ

j

i

and then we multiply by

X

,

i

α

, to obtain

X

,

j

α

¯

˙

=

-

x

˙

,

β

i

⁢

X

,

j

β

⁢

X

,

i

α

=

-

x

˙

,

j

i

⁢

X

,

i

α

.

Next we evaluate the material derivative of a square line element, with

metric tensor

g

i

⁢

j

5

5

If the spatial coordinate system be instantaneously

stationary

, then

∂

t

⁡

g

i

⁢

j

=

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,

δ

j

i

|

k

=

g

i

⁢

j

|

k

=

0

.

Therefore, we have

g

˙

i

⁢

j

=

∂

t

⁡

g

i

⁢

j

+

g

i

⁢

j

|

k

⁢

x

˙

k

=

0

.

located in

ℜ

,

d

⁢

s

2

¯

˙

=

g

i

⁢

j

⁢

d

⁢

x

i

⁢

d

⁢

x

j

¯

˙

=

g

i

⁢

j

⁢

(

d

⁢

x

i

¯

˙

⁢

d

⁢

x

j

+

d

⁢

x

i

⁢

d

⁢

x

j

¯

˙

)

,

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

[

8

]

d

⁢

s

2

¯

˙

=

2

⁢

d

i

⁢

j

⁢

d

⁢

x

i

⁢

d

⁢

x

j

,

d

i

⁢

j

≡

1

2

⁢

(

x

˙

i

,

j

+

x

˙

j

,

i

)

,

where

d

i

⁢

j

is a

symmetric tensor

of second order which represents the required

rate of deformation.

References

1

A.S. Lodge,

On the use of convected coordinate systems in the mechanics of

continuous

media,

Proc. Cambr. Phil. Soc.

𝟒𝟕

,

575-584, 1951.

2

C. Truesdell, R.A. Toupin,

The Classical Field Theories

Enc. of Physics,

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.

𝟖

(1844-1849), 287-319 =

papers

𝟏

,

75-129, 1845.

5

G. Green,

On the propagation of light in crystallized media

(1839), Trans. Cambr. Phil. Soc.

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.

𝟐

=

Oeuvres

(2)

𝟕

,

82-83, 1827.

7

A.-L. Cauchy

MÃÂ©moire sur les dilatations, les condensations et les

rotations

produits par un changement de forme dans un système de points materiÃÂ©ls,

Ex. d’An. Phys. Math.

𝟐

=

Oeuvres

(2)

𝟏𝟐

,

343-377, 1841.

8

E. Beltrami,

Sui principi fondamentali della idrodinamica,

Mem. Acad. Sci. Bologna (3)

𝟏

,

431-476,

𝟐

(1872), 381-437,

𝟑

(1873), 349-407,

𝟓

(1874), 443-484 =

Richerche sulla cinematica dei fluidi,

Opere

𝟐

,

202-379, 1871.

Title

motion of continuum

Canonical name

MotionOfContinuum

Date of creation

2013-03-22 15:53:16

Last modified on

2013-03-22 15:53:16

Owner

perucho (2192)

Last modified by

perucho (2192)

Numerical id

15

Author

perucho (2192)

Entry type

Definition

Classification

msc 53A45

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