Balance laws of continuum physics

BALANCE LAWSOF CONX%UtlM WYSICS

FacuTty

of Science,

I,T,U,,

of Applied Mathematics,

Istanbul and Department MSPRI, Turkey

&racterpistic master ‘Lalaxce 1aws of cIassica1 eor;tin~m physics are formlated for general tensor fields, Local field equations and jump conditions are provided Parmulations are given in both spatial ar;d w&u localization is permissible. material descriptions.

Balance

as integral

equations

laws obeyed by continuous where the integrated field

equations

the a@roprtitc

iisually

be ciassif

density

over

by rhecirculat~on When

surfaces

these manifolds

of

be taken inta

a material

;tlaaes

consideration

suffer

where b is the unit L unit tangent vector respectively,

defined in 12.;f

of the

V, u having

present,

note

-.n open surface fii~x

on

is

the

due ta

contributions

laws in same&at

we explore

tens9K

of generality

and cLb To this jump discontinuities

30 = S!i?s,

the

implications

! in the region

$J thKoU&

the

end we shall across

that

umtion

coZlect

the

under

as follows

3’f

and the

supply

of $I with&i

”

CI for the tensor

?J~.X=O on 6. The second . V of

the surface

various

uitj.

te;ms in

make use OF the veil-known the

its

b” = .,t x ?. where t is ths I to G. The tensor densities 2 aTId $ ”

through au and within

to

cc on which

i-e.,

normal vector

we can assume

now

curve

can now be formulated

density

following rry

a singular

with a velocity

equation

quantities

corresponding

Ue

aver

carrying

is moving

to an and go is the unit

deriwtive

f&ids

the supply in the bc&

across

ai balance

briei

tensor

hy the body at time t with the boundary 3Y and

c(t)

normal to the curve

t’ne time

tensor

are

may

forms,

in (Z.13 are self-expianatory, integrals

lines

Xn this

volume V occupied

surface

density

The discusstin

I21.

The master balance

c on o. Without loss denotes

with singular

of discontinuity

the effIux

A whereas X and w are

flux an the boundary and

account.

or’,

physics

along the closed boundary 2nd a supply

discontinuitim

fFig.1).

the jump conditions

the time rate of ctisngc? of a total

flux of a tensor

in IEi ar.d

laws in most general

We consider

are,

5ither

the total

can be found

they yield

IX the regions

laws lead to the local

E&lance laws of classical

CG’&ditiO~S.

of a tensor

we assume that a surface tangent

smoath balance

of discontinuity

should

forms

balance

~th~at~ca~

sufficiently

of discontinuity

are physica?

framework.

cast into a

a body is balanced by a surface

surfncc.

surfaces

are

ied in two categories,

balance

volumes and/or

are

bouitdary

o? the time ra:e of chanpz of

restricted

which

quantities

on surfaces

whereas

particularly,

media

field

farms on mat&al

surface

CT[II

density operator

Other symbols

(2.1)

into

relaiLons

for

q

w _

ERD~ANS. ~~HUBI

284

PIG.l.

Material

body with a surface

I_%. + 9.(v@t)ldv at * *

dv=j v-u

of discontinuity.

- /r U J, I da 8= **

the jump of a field quantity v is the velocity field of the material body, 5 * ] denotes I W of u and@implies the tensor acr’oss 0, U =u n -v n = (:-;) .r& is the speed of displacement

where

product,

In addition

to

(2.2)

we also

da = f 9.i S on a open

Let

us

moving (2.4)

surface

with

the

a velocity that

we nave

da i= 1 n.(%+v SW at

expression Q ?.$ ) da - i dt.(v as

(2.3)

x t)ds

field

$ =r~s @w on the surface u on which a singular * w. x by u as the velocity of the surface o -, Replacing

curve in

co is

(2.3)

and

“jw

n,(n@o)*=-=+ _I wnere

the

S where

now consider

we notice

need

Fs.(uOw) -*

St

introduced

the

surface

y, = Q-n 0 I$” = yp,,

gradient

operator (2.5)

Cl*=“8 Y

I

and

the

displacement

derivative

6W ati . -=--..* tu, &t at Then we can easily

-9 the

(2.6)

vnc

obtain

that

1

%+p,.(u I 0~) da- _f t.[n _.,_ C6t c

,da=j u-co

dt u-cd

If

[?I

parametrization

of

a

is

such

that

x (w-u) OWlds *-

u u = u, eo then

one can

readily

(2.7) show that (2.8)

where

Q is Finally

the mean curvature the

third

integral

of the

surface

in the

II and

right-hand

the side

relation

T.~o=-20

of

can be written

(2.1)

is

employed. as

Balance laws of c~~Ii~u~ by generalizing have

a well-known

supposed

2,:

formula

= (?. Hence

of

the

surface

285

physics

theory

[4,p,2071,

Let

us recall

that

we

we have (2.9)

(2.10)

If

there

are

several

interpreted

as

Whenever

the

Local

in

field

; where

b is again tra,~

open surface the

surface

%

where

Since Hence

where

U=U-v and .A s ”

* i is -,

x $)da L

defined

in ?2,4),

n&v

x y-r;)da -*

+

*-

c is a curve

on o its

tangent

can be written

we employed

the

relation

c +

should

be

assumed

to be continuous

the boundary

8.

The moving

category as

balance

laws,

discontinuity

which

arise

follows (2.12)

c

and

without

loss

of generality

we assume

relations

‘* $1 ds

+ / t.[$lds c * ..** we can

J t.(fu x t-xi-n

J s-c”

(2,15)

-

da + L’+!”

s-c-

are

n.E da t 1 b_.t ds

the well-known

# t.-$ ds = j n.(y as-c”

S with

of c in o

normal

Q_S?.E da =Si,“.i

and co in (2.10)

as

can be formulated

s-c”

= 0 on C. By employing P.

lines,o

integrands

c. The second

curve

interactions,

as-c

the

and the

and jump conditions

da = r# ,E.X ds + i

SQJ*c

that

equations

S through

electromagnetic

and singular

unions,

is permissible

now a material

L1 intersects

surface

set

localization

We consider

mostly

respective

the

we can deduce

of discontinuity

S~~ZXXS

their

-+

1..

easily

transform

x v)ds=O

t can be expressed I

(2.12)

into (2.14)

.,

in terms

of a vector

h as t=h U .V*

x n. Y

as

n .g = 0. 10” *

(2.15)

leads

now the

local

field

equations

and jump

conditions:

(2.16)

3, BA.l,AXE LAW

In many problems the material. of the field

some properties of

description

equations.

proves

Therefore,

to be more advantageous

in this

laws of continuum physics.

the master balance

description

IN MATERIALDESCRIPTIDB

section,

in deducing

we give the material

The maving material

form

volume V(r) in spatial.

is the image

at time t of a vo1.m~ U at rest through the ~~pin~ x=%($t) where - . The moving spatial discontinuity xY denote material and spatial places of a particle, o{ f), on the other hand, is mapped into a moving surface I(t) in the reference

5 and surface

configuration,

One can easily

show that

the speed of displacement

Uw of E(t) is given by 131

-1 l_$=Gn U

13,r1

where C =(n c-‘nfS n-z‘.+, and

= (KC-‘Nj-’ *1*

the normals z and z at respective ? =CI;,F-” *

,

gradient

-1 c-, =FFT ..*‘x

law {2,1)

can

tensor

nc~

by (3.3)

xk,~ and (3.4)

C we deduce that [U] = UN[ Cn I and the uniqueness -LIu *I x to a unique normal. ,” to 0 is secured if E $ 1 = 2. be

transfarmrd

I

da=8 da, * W

into material

of a normal.

form by a simple change of

Bence considering &=J N

dv =J &, we immediately

d.$.t

bk =&lmrL

&I=! dA, da=JCn-I d‘4 *

and B is the surface normal to Cc in F. Here only the * side of f3,4) involves a cmsor ii whid? is 110t yet defined. The

ofineeds nm and

a closer

where f is the unit hk$**‘ds ..*

scrutiny

of the pertaining

=)_ Xk$ T” dS,

that

tangent along LIZ , Therefore =C,

ger

E ‘~~~~,~~,~~~~~

: 1

s in 1 _KLM XK Eklm ’ *LX 4 ,k =;?r”

bk hi”‘& ...

We kmv

Rk = ~~~~,k~~

Making use of tbe relation

we abta.in

integral.

along as

t kds = dxk = xk ,$I#=

On the other

(3.5)

image af co

term in the right-hand

determination

-I

obtain

where Cc is the inverse

in (3,lf

are related

no jump on

The balance

third

of ci and E,respectively,

-1 -1 -1 T C =FIX F

f3 to C corresponding

variable.

points

“TCn_l t$E

Here E is the defo~at~o~

Since UN suffers

(3,2>

C _”

hand T* = ,@Yi B KN

cMPe;‘( C TR ““dS ,P QR NM?‘I&*

dS

j3,7)

wttere

A

(N>

is

the stretch

in the direction

E. It is obvious

that

(3.9)

does not determine

in the form Ii@;. since we can always add to Aan expression A uniqulely in terms of ; 4 5 . I of such terms does not affect the analysis and we However one can easily show that addition Call always assume

that !,?

=O q on C, now the following material

Let us define

field

quantities (3.12)

Bence the balance

and ; by (3.11).

a -

~w+f-~

Qdh=

at v-c” Recalling

that

gradient

(3-33)

operator of

becomes

dA+ j

i dVt @

v-c

”

ax-c;

;;dA B.IZ dS + j z-c i* ”

(3.13)

as

into

of

Z, is

the surface

in material

coordinates

is the surface

and

gradient

the displacerrent

operator

derivative

on L, _

following

is

the

the

L is &T *

at2

6t

at

-=-=.

We also

fJj

Can now be written

where U is the velocity

motion

4;

(3,6)

av-1

dt C-C,-

(2,2)1

we can transform

equation

+ U N.:n N

assumed a stationary

For sufficiectly

(3,151

* -* singular

smooth fields

curve Cc with respect (3,141

leats

to

local

to C for field

the sake of equations

simplicity.

and jump

conditions:

aY _..zL”~,

I:

at * -

Similarly and

-

;=o I

in v-r:

”

the balance

law (2.12)

can be tranformed

inca a material

form by using

(3,s)

the relation t ds =T,FT dS * ., *

between

the

tangent

vector

of

a curve and that of

its

image:

(3,171 where we defined p, = J&l, * ” where

5 = -

_I Jp

”

I’=FTy ****

5,

K=KT is given by (3.11). * *

, M=KFTv -*_-

We thus obtain

(3.181

288

ERDOijANS.$JHUBI

or (3.19)

For sufficiently smooth fields (3.19) leads to

the local field equations and jump conditions:

a0 -CL_

at

xrI? "

NC“LIJ”L

:=O

:

-

I-]

.

in

V (3.20)

+M=O

I..

_

on C _

Appropriately selecting the tensor fields material and spatial forms of field equations of continuous media can deduced

from

various known

(2.11), (2.16), (3.16) and (3.20).

REFERENCES 111.

~RING~N, h.C,, Continuum Physics, Vol.II - Continuum Mechanics of Single-Substance Bodies, Part I, Ch.2, New York, Academic Press, 1975.

[21.

ERINGEN, A.C., Mechanics Company, 1980.

of Continua, Huntington, New York, Robert E. Krieger Publishing

[31. ERINGEN, A.C., and SLJHUBI,E.S., Elastodynamics, Academic Press, 1974.

141. MCCONNELL, A.J., Applications 1957.

of Tensor Analysis,

Vol.1 - Finite Motions, New York,

New York, Dover Publications, Inc.,