time dilatation of a volume element

time dilatation of a volume element

Introduction

The rate of change of a

volume element

constitutes an important subject in certain applications related to the

velocity

field in fluid flow and elasticity,
besides it admits some physical

interpretations

intimately entailed to
tensor

invariants

, as we shall see.

Formulae derivation

Let

X

α

and

x

i

be material and

spatial coordinates

, respectively. Consider

diffeomorphic

the

mapping

1

1

See

motion of continuum

X

α

↦

x

i

⁢

(

X

α

,

τ

)

,

t

0

≤

τ

≤

t

,

which represents the motion of continuum

ℜ

⊂

(

ℝ

3

,

∥

⋅

∥

)

. The

Jacobian

J

=

|

x

,

α

i

|

(comma denoting partial differentiation with respect to the indicated

coordinate

) of coordinate

transformation

is given by

J

=

ϵ

α

⁢

β

⁢

γ

⁢

x

,

α

1

⁢

x

,

β

2

⁢

x

,

γ

3

≡

C

3

γ

⁢

x

,

γ

3

,

ϵ

α

⁢

β

⁢

γ

being the Levi-Civita density,

C

3

γ

the

cofactor

of

x

,

γ

3

in the

determinant

expansion

and it comes expressed as

C

3

γ

=

ϵ

α

⁢

β

⁢

γ

⁢

x

,

α

1

⁢

x

,

β

2

=

1

2

⁢

(

ϵ

α

⁢

β

⁢

γ

⁢

x

,

α

1

⁢

x

,

β

2

+

ϵ

β

⁢

α

⁢

γ

⁢

x

,

β

1

⁢

x

,

α

2

)

=

1

2

⁢

ϵ

α

⁢

β

⁢

γ

⁢

(

x

,

α

1

⁢

x

,

β

2

-

x

,

β

1

⁢

x

,

α

2

)

=

1

2

⁢

ϵ

α

⁢

β

⁢

γ

⁢

ϵ

i

⁢

j

⁢

3

⁢

x

,

α

i

⁢

x

,

β

j

.

So that, for any arbitrary cofactor

C

k

γ

,

C

k

γ

=

1

2

⁢

ϵ

α

⁢

β

⁢

γ

⁢

ϵ

i

⁢

j

⁢

k

⁢

x

,

α

i

⁢

x

,

β

j

.

Let us multiply by

x

,

γ

n

.

C

k

γ

⁢

x

,

γ

n

=

1

2

⁢

ϵ

i

⁢

j

⁢

k

⁢

ϵ

α

⁢

β

⁢

γ

⁢

x

,

α

i

⁢

x

,

β

j

⁢

x

,

γ

n

=

1

2

⁢

|

x

,

α

i

|

⁢

ϵ

i

⁢

j

⁢

k

⁢

ϵ

i

⁢

j

⁢

n

=

J

⁢

δ

k

n

,

C

k

γ

J

⁢

x

,

γ

n

=

δ

k

n

,

(1)

where we have used well-known alternator’s

properties

. Moreover, since the cofactor

C

k

γ

is

independent

on

x

,

γ

k

by its own

definition

,

∂

∂

⁡

x

,

γ

n

⁢

(

J

⁢

δ

k

n

)

=

∂

⁡

J

∂

⁡

x

,

γ

k

=

C

k

γ

.

(2)

But

X

,

k

γ

⁢

x

,

γ

n

=

δ

k

n

,

which it is compared with Eq.(1) to obtain

X

,

k

γ

=

C

k

γ

J

,

C

k

γ

=

J

⁢

X

,

k

γ

.

So from Eq.(2) we get

∂

⁡

J

∂

⁡

x

,

γ

k

=

J

⁢

X

,

k

γ

.

(3)

Let us now consider the

relation

d

⁢

v

=

J

⁢

d

⁢

V

between the spatial and material volume elements, and by taking the material time derivative

d

⁢

v

¯

˙

=

J

˙

⁢

d

⁢

V

,

(4)

because

d

⁢

V

¯

˙

=

0

,

by definition. From Eqs.(3)-(4),

(

d

⁢

v

d

⁢

V

)

¯

˙

=

∂

⁡

J

∂

⁡

x

,

α

i

⁢

x

,

α

i

¯

˙

=

∂

⁡

J

∂

⁡

x

,

α

i

⁢

x

˙

,

α

i

=

J

⁢

X

,

i

α

⁢

v

,

α

i

=

J

⁢

v

,

i

i

where

x

˙

,

α

i

≡

v

,

α

i

are material

gradient

components of velocity field, thus arriving to the result due to Euler

[

1

]

J

˙

J

=

v

,

i

i

≡

∇

x

⋅

𝐯

,

log

⁡

J

¯

˙

=

∇

x

⋅

𝐯

,

(5)

expressing the spatial

divergence

of velocity field.
Also, by substituting

d

⁢

V

=

d

⁢

v

/

J

in Eq.(4) we get

log

⁡

d

⁢

v

¯

˙

=

∇

x

⋅

𝐯

.

(6)

Physical interpretations

•

The time

logarithm

of dilatation and the first invariant

I

∇

x

⁡

𝐯

associate to the tensor of velocity spatial gradient, coincide exactly.

•

If we consider the

Lagrangian strain

tensor

E

i

⁢

j

=

1

/

2

⁢

(

u

i

,

j

+

u

j

,

i

+

u

i

,

k

⁢

u

k

,

j

)

(large strain) for

`

⁢

`

⁢

s

⁢

m

⁢

a

⁢

l

⁢

l

⁢

"

strain, i.e. the initial undistorsioned (material) reference

configuration

χ

ϰ

⁢

(

X

i

,

t

0

)

maps to near distorsioned (spatial) reference configuration

χ

ϰ

+

Δ

⁢

ϰ

⁢

(

X

i

,

τ

)

(as

τ

→

t

0

) during the motion of continuum

ℜ

,

coinciding approximately the spatial coordinates with the material coordinates and therefore, as a

consequence

, the quadratic displacement gradient

u

i

,

k

⁢

u

k

,

j

≈

0

.

2

2

Indeed for small strain is required that

∇

X

⁡

𝐮

⋅

𝐮

⁢

∇

X

≈

𝟎

and

∇

X

⁡

𝐮

⋅

∇

x

⁡

𝐮

≈

𝟎

.

To see this, we consider the coordinates transformation

𝐱

⁢

(

τ

)

=

χ

⁢

(

𝐗

,

τ

)

,

as

τ

→

t

0

.

So

d

⁢

𝐱

=

d

⁢

𝐗

⋅

∇

X

⁡

𝐱

.

But

d

⁢

𝐮

=

d

⁢

𝐗

⋅

∇

X

⁡

𝐮

=

d

⁢

𝐱

⋅

∇

x

⁡

𝐮

,

then by the first equation, we have

d

⁢

𝐮

=

(

d

⁢

𝐗

⋅

∇

X

⁡

𝐱

)

⋅

∇

x

⁡

𝐮

=

d

⁢

𝐗

⋅

(

∇

X

⁡

𝐱

⋅

∇

x

⁡

𝐮

)

,

and since

d

⁢

𝐗

is arbitrary

∇

X

⁡

𝐮

=

∇

X

⁡

𝐱

⋅

∇

x

⁡

𝐮

.

(The

chain rule

!) Recalling now

𝐱

=

𝐗

+

𝐮

,

we get

∇

X

⁡

𝐮

=

[

∇

X

⁡

(

𝐗

+

𝐮

)

]

⋅

∇

x

⁡

𝐮

=

(

𝟏

+

∇

X

⁡

𝐮

)

⋅

∇

x

⁡

𝐮

=

∇

x

⁡

𝐮

+

∇

X

⁡

𝐮

⋅

∇

x

⁡

𝐮

,

which shows that quadratic gradient is approximately equal to

zero

whenever

∇

X

⁡

𝐮

≈

∇

x

⁡

𝐮

,

i.e. the material undistorsioned reference configuration

ϰ

be approximately equal to the spatial distorsioned configuration

ϰ

+

Δ

⁢

ϰ

.

In fact elasticity theory defines

infinitesimal

strain

tensor as

e

i

⁢

j

≡

1

/

2

⁢

(

u

i

,

j

+

u

j

,

i

)

,

i.e.

e

i

⁢

j

≈

E

i

⁢

j

for small strain, but the definition of tensor

𝐞

is exact. So, in the vector displacement

u

i

=

x

i

-

X

i

,

we take the material rate

u

˙

i

=

x

˙

i

≡

v

i

and hence

u

˙

i

,

j

=

v

i

,

j

.

Therefore, according to the mentioned approximation, the material time derivative for tensors

𝐄

and

𝐞

are given by

E

˙

i

⁢

j

≈

e

˙

i

⁢

j

≈

1

/

2

⁢

(

v

i

,

j

+

v

j

,

i

)

.

3

3

The last approximation because the tensors

𝐄

and

𝐞

are usually defined with respect to material coordinates and not with respect to the spatial ones.

Now by contracting

j

=

i

, we get

E

˙

i

⁢

i

≈

e

˙

i

,

i

≈

v

i

,

i

=

∇

⋅

𝐯

.

4

4

Notice that although this is an approximated result, Eqs.(5)-(6) are exact.

•

Considering the infinitesimal strain tensor

𝐞

(or

𝐄

for small strain), we see that sum of

normal strain

e

i

⁢

i

=

e

11

+

e

22

+

e

33

=

u

1

,

1

+

u

2

,

2

+

u

3

,

3

represents the

trace

or first invariant

I

𝐞

. Thus, in the initial undistorsioned reference configuration

ϰ

,

we can use principal centered axes (i.e. along the

eigenvectors

of tensor

𝐞

,

whose representation corresponds to pure normal strains) of a volume element

d

⁢

V

in order to measure the induced

dilatation

(

d

⁢

v

-

d

⁢

V

)

/

d

⁢

V

in the distorsioned reference configuration

ϰ

+

Δ

⁢

ϰ

. So, for an elemental rectangular parallelopiped of volume

d

⁢

V

, we have

(

d

⁢

v

d

⁢

V

)

≈

(

1

+

e

11

)

(

1

+

e

22

)

(

1

+

e

33

)

=

1

+

e

i

⁢

i

+

o

(

e

n

⁢

n

2

)

.

(

n

n

o

t

s

u

m

m

e

d

)

By taking now the material time derivative,

(

d

⁢

v

d

⁢

V

)

¯

˙

≈

e

˙

i

⁢

i

≈

E

˙

i

⁢

i

≈

u

˙

i

,

i

=

x

˙

i

,

i

=

v

i

,

i

=

∇

⋅

𝐯

,

thus completing the aimed physical interpretation.

•

A volume-preserving motion is said to be

isochoric.

Then

J

=

|

∂

⁡

x

i

∂

⁡

X

α

|

=

1

,

div

x

⁡

𝐯

=

0

.

V

is called

material volume

and

v

is called

control volume.

•

Although we have used Cartesian rectangular systems, if we introduce generalizated tensors, all the results obtained are also valid for

curvilinear coordinates

. For

instance

,

(a)

Divergence

∇

x

⋅

𝐯

=

𝐠

i

⋅

∂

∂

⁡

x

i

⁢

(

v

j

⁢

𝐠

j

)

=

𝐠

i

⋅

𝐠

j

⁢

v

j

|

i

=

v

i

|

i

=

v

,

i

i

+

Γ

j

⁢

i

j

⁢

v

i

,

Γ

j

⁢

i

j

=

1

2

⁢

g

⁢

∂

⁡

g

∂

⁡

x

i

,

g

=

|

g

i

⁢

j

|

,

g

j

⁢

k

⁢

g

k

⁢

i

=

δ

j

i

,

where

v

j

|

i

,

𝐠

i

,

𝐠

j

,

stand for

covariant
derivative

and contravariant and covariant spatial base vectors, respectively.

(b)

Gradient

∇

X

⁡

𝐮

=

𝐆

α

⁢

∂

∂

⁡

X

α

⁢

(

u

β

⁢

𝐆

β

)

=

u

β

|

α

⁢

𝐆

α

⁢

𝐆

β

=

(

u

,

α

β

+

Γ

γ

⁢

α

β

⁢

u

γ

)

⁢

𝐆

α

⁢

𝐆

β

,

Γ

γ

⁢

α

β

=

G

β

⁢

δ

⁢

Γ

γ

⁢

α

⁢

δ

,

Γ

γ

⁢

α

⁢

δ

=

1

2

⁢

(

G

γ

⁢

δ

,

α

+

G

α

⁢

δ

,

γ

-

G

γ

⁢

α

,

δ

)

.

𝐮

⁢

∇

X

=

∂

∂

⁡

X

β

⁢

(

u

α

⁢

𝐆

α

)

⁢

𝐆

β

=

u

α

|

β

⁢

𝐆

α

⁢

𝐆

β

=

(

u

,

β

α

+

Γ

γ

⁢

β

α

⁢

u

γ

)

⁢

𝐆

α

⁢

𝐆

β

,

where

𝐆

α

,

𝐆

β

are the covariant and contravariant
material base vectors, respectively.

mutatis mutandis

for spatial gradient
tensors

∇

x

⁡

𝐮

,

𝐮

⁢

∇

x

.

(c)

Material time derivative

𝐯

˙

=

v

i

⁢

𝐠

i

¯

˙

=

∂

⁡

v

i

∂

⁡

t

⁢

𝐠

i

+

v

i

|

j

⁢

v

j

⁢

𝐠

i

=

∂

⁡

v

i

∂

⁡

t

⁢

𝐠

i

+

(

v

,

j

i

⁢

v

j

+

Γ

k

⁢

j

i

⁢

v

k

⁢

v

j

)

⁢

𝐠

i

,

where the local time

derivative

∂

⁡

𝐠

i

/

∂

⁡

t

=

𝟎

,

by definition.

References

1

L. Euler,

Principes gÃÂ©nÃÂ©raux du mouvement des fluides,

Hist. Acad. Berlin

1755,

274-315, 1757.

Title

time dilatation of a volume element

Canonical

name

TimeDilatationOfAVolumeElement

Date of creation

2013-03-22 15:54:28

Last modified on

2013-03-22 15:54:28

Owner

perucho (2192)

Last modified by

perucho (2192)

Numerical id

10

Author

perucho (2192)

Entry type

Definition

Classification

msc 53A45

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