Inertia in simple dipolar continua

MECHANICS RESEARCH COMMUNICATIONS

0093-6413/83

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Voi.i0(5),273-278, 1983, Printed in USA. Copyright (c) 1984 Pergamon Press Ltd.

INERTIA IN SIMPLE DIPOLAR CONTINUA J. Lubliner Department of Civil Engineering, University of California Berkeley, California 94720, USA.

(Received 11 February 1983; accepted for print 9 September 1983)

Introduction Using the author's [2] axiomatic d e f i n i t i o n of i n e r t i a in simple body models, the most general form (subject to some elementary r e s t r i c t i o n s , namely l o c a l i t y and mass continuity) of the i n e r t i a in a simple dipolar continuum is derived. I t is shown that in a simple continuum with couple stresses, the dipglar i n e r t i a must vanish. Background The purpose of this note is to present the general form of simple dipolar i n e r t i a . This form w i l l

be derived, not postulated, as i t was, for example, in [ I ] .

The

basis is the axiomatic d e f i n i t i o n of i n e r t i a as given in [ 2 ] , where i t was shown that in a simple monopolar continuum - i.e.

one in which the only possible force

systems are measures - the only possible i n e r t i a is the monopolar one. We begin by reviewing some concepts from [2]. quadruple ( 3 ,

2,

F, u ), where ( I )

9,

A simple body model is the ordered

the material set or body manifold, is a

set with the structure of a manifold (or a manifold with boundary) of dimension at most three; (2)

:}13, the space of configurations (mappings of ~o°

dimensional Euclidean affine space ~ ) , sional) modeled on a topological

is a manifold ( f i n i t e -

into the three-

or infinite-dimen-

vector space K whose elements are E-valued func-

tions on ~ (E being the translation space of E, i.e. the three-dimensional Euclidean vector space, and ~ the closure o f ~ ) ; (3) F = ~ F is the forceK~ K system bundle, each f i b e r F< being a subspace of K* (the algebraic dual of K) whose elements (force systems) are E-valued d i s t r i b u t i o n s on

~;

and (4) ~, the i n e r t i a , :

is a functional defined on ~CCxV×V, where V is a vector space containing the velocity-field

spaces V<, < ~ ] C , as subsets. 273

The space V< is essentially the tangent

J. LUBLINER

274

space to ~

at K w i t h respect to the weak topology induced by FK; t h a t i s ,

it

is ,

the subspace of F*~ (the algebraic dual of FK) generated by the velocity, f i e l d s ~Lt: (also E-valued d i s t r i b u t i o n s ) × ( t ) d~f ×t = '<

in all

admissible motions x. [a,b],[YC

at some time t in the i n t e r v a l

[a,b].

such t h a t

The v e l o c i t y f i e l d s

::tk are

defined by the l i m i t s h lim ÷0+

h1 < F,~ Xt±h - x t >

= ~ _

(I

f o r every f E F~, where < . , - > denotes the d u a l i t y p a i r i n g on F~ x F~, ×

is admissible i f

the l i m i t s

:~t+ and : t -

e x i s t at every t # [ a , b )

and a mot on and t E ( a , b ] ,

respectively. I f the dual of V~. is defined as V< on VK which thus becomes a l o c a l l y

FK/V <

then o(V,, V#) is a Hausdorff topology

convex t o p o l o g i c a l

vector space [ 3 ] .

The axiomatic properties of the i n e r t i a

functional

II

u(~.~,-,-) def ~ < ( . , - )

(basic d e f i n i t i o n ) .

For any

Ke ~ ,

on V, and f o r any u ~. V , the r e s t r i c t i o n 12 (smoothness).

u are:

of u<(u,-)

is an inner product

to V~. belongs to V,L.

At any (K,u,v,)E~/d'~ x V x V there e x i s t s a uF(u,v) e V

such t h a t

4Uxt(g,y)It± =
(2)

>.

I f K and K* are two c o n f i g u r a t i o n s

related by a r i g i d - b o d y

i.e., ~*(p)=xo+c+g

f o r some XoC E ,

c ~ E and

(~(p)-xo)

V

p E~

Q~ ~ (~ is the r o t a t i o n group on E), then

(3) f o r any u, y # V . (Note:

Qu__is defined by <¢, Qu > =
E-valued t e s t f u n c t i o n . ) 14-15 ( l i n e a r momentum, mass).

I f ci denotes a uniform t r a n s l a t i o n a l ~

f i e l d w i t h value c, then f o r any K E ~

velocity

and u £ V ,

u~(Cl, U ) = C. / u(p) m(dp),

(4)

where m is a p o s i t i v e measure, called mass, on a a-algebra ~ of subsets o f ~ (subbodies).

In [2] the existence of mass was not assumed but derived from more

INERTIA

IN SIMPLE D I P O L A R

CONTINUA

basic assumptions, namely the configuration-independence (4) and the p o s i t i v i t y

275

of the left-hand side of

of the r e s u l t i n g l i n e a r f u n c t i o n a l .

Dipolar I n e r t i a We define a simple d i p o l a r continuum as a simple body model in which the force systems are E-valued d i s t r i b u t i o n s

of f i r s t

order.

Thus, i f we w r i t e , as in [ I ] ,

T f o r the tensor space L(E,E), then a force system f E F~ w i l l pair of measures f(1) and f(2) on ~ ( ~ ) , E-valued and the l a t t e r

be given by a

the former (the monopolar force measure)

(the d i p o l a r force measure) T-valued, such that

<~, V > = f

[vifi(1)(dx)

+ v i , j fi(j2)(dx)] •

I t follows from II that the i n e r t i a must be given by p<(u,y) : # ( ~ ) [ u i v j p ~ j ( d x ) where po,

+ (uivj, k + viuj,k)Pljk(dX)

pl and o2 are tensor-valued measures.

+ ui,jVk,LP#jkz(dx)],

By 14-15 we see that

P ° i j ( d x ) = aijm(dp) and PijkZ = O. In f a c t , we shall assume that the i n e r t i a of any subbody vanishes with its mass so that ~2 is mass continous, i . e . p 2lJ . . k t ( d x ) = 7r.. 13k~ = ~k~lO lJk~ m( dp) ' with ~'" .. and ( A~ , B ) ~ i j k ~ A i j B k ~ semidefinite to s a t i s f y I I . We next use 13 to r e s t r i c t

the configuration

dependence of ~.

positive

Let F = VoK,

where Vo denotes the gradient with respect to a reference c o n f i g u r a t i o n ,

and

C = [TF. I f we define ~ by ~ijk~FilFjjFkKFLLHIJKL , then we can show by standard arguments that ~ can depend on the configuration only through the C f i e l d . Assuming this dependence to be l o c a l , we have ~ = ~(C,VoC, ~oVoC. . . .

), and

consequently u<(u,v) f - : ~

[u ivi + Fi IFkK~IJKL(~C, . . . )u i , j V k , L ] m(dp) ,

where we used the chain rule

ui, J = F j j u i , j .

F i n a l l y we use 12.

xt+ = ~t- = xt at time t

Assuming

and l e t t i n g

×t = <'

276

J. LUBLINER

we have d-t-d uxt(u'v)~ =~3/ [(#iIFkK + FiIFkK)'rIJKL + FiIFkK(3C N~IJKL CMN Note that Fil

=

Fml #i,m and

;~CMN,p ;~IJKL CMN,P)]ui ,jVk, Lm(dp) " CMN = FmMFnN(Xm,n

+ ~ n ,m )

"

But CMN,P = (FnN FmM,P + FnMFmN,P + FmMFnN,P + FmN FnM,P)Xm,n + Fpp (FmM FnN + FInNFnm) Xm, np Consequently the presence of a nonvanishing 9~IJKL/JCMN,P would imply that u'(u,v) is a d i s t r i b u t i o n of at least second order, contrary to assumption 12 for a dipolar continuum. Finally, then, u~ (u,v) r [uiv i * FilFkK i1IJKL (C) ~ ~ = #9 - u i ,jVk,L ]m(dp)

(5)

is the most general form of simple dipolar inertia. This is more general than usually presented; for example, Green and Naghdi [ l ] consider only the special case corresponding to i T;IjKL = CIKMjL or equivalently ~ijk:! = ~ikm'~ "JL The i n e r t i a l

force system at time t , f ~ , is defined [2] by = ~d- ~

= ~xt (zt,w) _

xt

( ~t w) ,~

I/2<~, x t ( x t , x t ) ,w >

+ ~t,w)~ , ~ - 1/2- .u;
For a dipolar continuum we write

for any v#6V zt

i

< f t ' '# > = ~ (XkWk + "rk~Wk,L )m(dp)'

INERTIA IN SIMPLE DIPOLAR CONTINUA

where # is the dipolar i n e r t i a l and (5) to obtain I

<~<(u,v),w> ~

+

force density.

277

To determine ~ we use (2)

= ~,~~ [TIIJKL(FkKWi, I + FilWk, K)

~

~IJKL - ~CMN FmMFnN Fil FkK (Wm,n + wn,m )] u i , j V k , L m( dp) (~

•

njk~. ~im

+

7..

ion~ ~km + 2~ijk~mn)Ui, j Vk, wm,n m(dp)

where ~'.. ijk;~mn ~HIJKL ~MN

(Note that

=

F

i l FjjFkK

F

~LFmNFnN

)~IJKL ~CMN

~HIJKL .) ~CNM

_

Consequently Yk~ = ~'" 1jk~ xi ,j

+ (~nj k~
I (~.jmn ~ik + ~.. - 2_-lj~n ~km + 2~'.. Ijmnk;, ) xi ,j =

~.. ljk~ (x i ,j

+

" Xm,i

A m ,j ) + ( ~ . .ljm~.

-

7 .1j . m)xi

i

+ (2~ijk~mn - ~ijmnk~ )~
,j

m,n

m,n Xk

,m (6)

The Couple-Stress Case A frequently considered special case of a dipolar continuum is the continuum with couple stresses, in which the measure f(2) is antisymmetric. In such a body we would need ~ijk~ = with

eijm ek~n Xmn

X symmetric, or, equivalently, ~IJI(L = elJM eKLN AMN

with AMN = (det

F) ~ -2 FmMFnN Xmn

278

,].

LUBLINER

However, the gyroscopic terms in (6) w i l l In p a r t i c u l a r , A = constant.

not, in general, be antisymmetric.

' ~ijmnk; Z
'

'

mn

~, = r), hence

'

and ~k:: = e( ~(k,

- S~ ,k ) * 2C#k, i(~

- ~,

)

which is not antisymmetric f o r a r b i t r a r y motions unless c = O.

Thus in a

simple continuum w i t h couple stresses the i n e r t i a

is monopolar, as in [ 4 ] . An

inertial

(Cosserat) continuum with

body couple arises only in a generalized

couple stresses. References I.

A.E. Green & P.M. Naghdi,

2.

J. L u b l i n e r , Mech. Res. Comm. lO, I - 7 (19°3)

Q. Appl. Math. 28, 458-460 (1970).

3.

J. Horvath, Topological !,Jesl ey, 1966).

4.

C. Truesdell & R.A. Toupin, The Classical F~eld Theories, Encyclopedia of Physics (ed. S. Flugge) I I - I / l (Springer, B e r l i n , 1960).

Vector Spaces and D i s t r i b u t i o n s ,

I (Addison-