Thermomechanics of material continua with generalized constraints

MECH. RES. COMM.

THERMOMECHANICS

Vol.3,

331-336,

OF M A T E R I A L

1976.

CONTINUA

Pergamon Press.

WITH

Printed in USA.

GENERALIZED

CONSTRAINTS

Cz. Wo~niak Institute of Mechanics, University of Warsaw, Poland (Received 23 March 1976; accepted as ready for print 24 April 1976)

Introduction The aim of the contribution is to formulate the foundations of thermomecban~cs in which the ,mWnown physical fields (deformations, temperature, stresses etc.) are restricted not only by the constitutive equations but also by certain extra relations. The latter are said to be the generalized constraints and may be interpreted as heuristic hypothesis for special problems of thermomechsnics. The known simple internal constraints also constitute a special case of the generalized constraints. The thermomechanice of continua with generalized constraints enable us to obtain a variety of mathematical models of the real bodies, usefull in such technical applications as thermomecb-nics of thin plates, shells or rods, discretized formulations of thermoelasticity etc. The approach is developed by use of some concepts of the non-standard mathematical ~nalysis, [I] • Analysis Let~be a region in the physical space occupied by the bodj in the reference configuration and let ~ b e its enlargement. Let be given an internal fine partition of.~48Ronto a set of disjoined infinitesimal elements JPX ,-~e~ c ~ . We assume that each such ele. ment is an independent homogeneous thermomech-nical system with the deformation functlon~k~,~),~,~)CZ~_~) and the temperature extension Of

distribution~,~)+9~,~)~Z~-~),~e~,'~e~,~e~.By

the analytical form of the basic laws of thermomechanics we obtain for each infinitesimal system the following relations Scientific Communication

331

332

Cz. WO~.NIAK

and

,

h

, c/

,

Vol.3, No.4

,

,

are

the

=as.

d e n s i t y , t h e e x t e r n a l body l o a d s , t h e h e a t a b s o r p t i o n , t h e f i r s t Piola-Kirchhoff extra-stress tensor,[7~, the heat flux, the specyfic entropy and the specific internal energ.v, respectivel~; subscriptU~denotes t h a t t h e q u a n t i t ~ i s r e l a t e d t o t h e r e f e r e n c e con. figuration. M o r e o v e r , d1 a n d ~ a r e ~minnOWn i n t e r n a l f o r c e s and i n ternal heat densities, respectively, due t o t h e i n t e r r e l a t i o n s among infinitesimal s T s t e m s ~ , ~ J l . The fields ~ , F , ~ , ~ , , ~ , ~ , have to satisfy the suitable known constitutive equations i n which no c o n s t r a i n t s r e s p e n s e i s t a k e n i n t o a c c o u n t and form of which has to turn (1. 5 ) into identit-j. For ~ B ~ Eqs. (1) a r e assumed t o be s t a n d a r d , ~ 1 ~ . The i n t e r r e l a t i o n s among i n f i n i t e s i m a l systems ~ ,~e~, as w e l l as their extra properties let be defined by the following restrictions. :imposed on ~ , F , # , e , i n ~ x ; ~ :

where k~ , ~ , a r e ~nown d ~ f f e r e n t i a b l e f u n c t i o n s and where d o t s d e n o t e t h e h i g h e r d e r i v a t i v e s . We s h a l l a l s o p o s t u l a t e t h e g l o b a l p r i n c i p l e o f c o n s e r v a t i o n o f e n e r g y and t h a t o f d i s s i p a t i o n :

where p~ '-/~8 a r e boundar~ loads and a heat entering i n t o

~(B=, ~;) through the .boundqry, respectively. From (1) and (3) we obtain

0= -~ == . where £~_=-~= a r e i n t e r n a l boundaz~3 f o r c e s . Summing Eqs ( 4 ) t e r m wise.we p.ostulate, that the resultin~ equation has to hold for an~ • , ~ , ~ , ~ aamlssible by Eqs. (2). Putting into resulting , k + b ; r , ~ - ~+~# , ; 3= +/ ~ ,.here/z , ; r , ~ , I ~ are suitable virtual increments, we arrive finall~ at the condition 7E= ~

equation£~b;Z

Vol.3, No.4

THERMOMECHNICS

~

~

OF CONTINUA

~

,

-

333

&

¢50

which has t o h o l d f o r s n ~ ; ~ ,~ , 2 [ ,~8 a d m i s s i b l e by (2) • The equations h =0,R=0 w i l l be c a l l e d the 5 e n e r a l l z e d c o n s t r a i n t s p r o -

~ided t h a t E q s . ( 5 ) a n d ( 1 ) h o l d . The s t a n d a r d p a r t of Eqs.(1 ) , the known c o n s t i t u t i v e e q u a t i o n s , t h e e q u a t i o n s of g e n e r a l i z e d constraints (2) and the "ideality" principle (5) are ~overning relations of thermomechanics with generalized constraints. It must be stressed that no constraints response i s included into constitutive relations. We can also observe that the reference constant ~ h a s to be known only if the kinematic and the thermal fields are c o n s t r a i n e d a t l e a s t "~y one common e q u a t i o n b e l o n g i n g to

(2).

Now we s h a l l a n a l y s e some cases of the g e n e r a l i z e d c o n s t r a i n t s . I f Eqs. (2) have the form;/k,v-~a=O , ~ = - ~ = O (i.e.g~=~),~:~,*;*, , then f r o = ~ ) = d < ~ ) . , obtain

,*,,..;~,~,,.,.>-o.~(x,z*;z, vz,.,;a,e~..,)=o

i s a u n i t v e c t o r normal toaB~. ~8=eonatt h e n E q . ( 6 ) l e a d s t o t h e known p r i n c i p l e of v i r t u a l work. I f ~ depend onl~ on V~, ~ , V~, t h e n we a r r i v e a t the known concept of simple c o n s t r a i n t s , [ 3 - 6 ] . I f Eqs. (2) r e d u c e to

~,k,~t-~=D, ~ -

-~ c~=, =~=,~ ,~, &= ~,=

ea= 0 , t h e n from< 5 ) we o b t a i n , - ~ - ~ , = , t h ~ we conclude that the

i n t r o d u c e d g e n e r a l i z e d c o n s t r a i n t s d e f i n e the c l a s s i c a l m a t e r i a l thermo=e chaz~ c c o n t ~ u u m .

I f the f i x e d components ~ ,

~

~# are a b s e n t from Eqs. (2), t h e n from

fonow, that ~*~0. h~=O ~.~./3 bei~, ei,,~) . Zt is ,~ e ~ l e

of what can be c a l l e d the " s l e n d e r " continuum. I f t h e r e a r e no g e n e r a l i s e d c o n s t r a i n t s , ~ z D , ~ 4 - - D , t h e n from(5 )

,e obt~

~=~.~R-~.

L~-~. ~ = ~ .

~-0.

~=0-

~

t ~ s caso n

have

a r r l v e d a t the contlnuum o f n o l i n t e r r e l a t e d ~ m f i ~ L t e s i m a / 8,~s~ems (using the non-standaz~ t e z m l n o l o ~ ) o r t o an a b s o l u t e l ~ s l e n d e r m a t e r i a l contlmuum. The d e t a i l e d a n a l y s i s i n t ~ s

field

is in progress.

334

oz. WO#.NIAK

Vol.3, No.~

a / t e r n ~ i y , ~,oz',~ o:~ constraints l ~ r s t l y , l e t us assume t h a t i n s t e a d e f E q s . (2) we p e s t u l a t e a t i o n s o~ o o n s t r a i q t s i n t h e form

equ-

R~Ix>~/e,~£,...~ ~ ~#,...):o~

~:s,...,~, x~=, t~, (7) where t h e known d i f f e r e n t i a b l e f u n c t i o n s b v and R# can a l s o depend on ' ~ / ~ , . . . > ~ F , .... , ~ ~:,vZ, .... . respectively. T, this case ,e shalZ asaume that ~ = ~g , where ~ is the reference absolute temperature ( p r o w l e u s l y ~ -~ had a d i m e n s i o n e f time) o F r o m ( l ) s a d ( 3 ) w e obtalm as before

• mminK E q s . ( 8 ) t e r u u ~ s e we p e s t u l a t e , t h a t t h e r e s u l t i n g e q u a t i o n h a s t o h o l d f o r an7 ~ , F , # , Z adncLssible by E q ~ . ( 7 ) . P u t t i n g , ~te the eq~tlon obta~ed from ( 8 ) , we a r r i v e a t t h e c o n d i t i o n w h i c h h a s t h e same form as Eq. ( 5 ) , b u t i n which t h e v i r t u a l i n c r e m e n t s h a v e d i f f e r e n t s e n s e . The e q u a t i o n s (7) , p r e w l d e d t h a t Eqs. ( 1 ) , ( 8 ) h e l d , w i l l be c a l l e d the anholonomic (nonlinear) generalized constraints.

~:->~:*;X, ~ / " , I F , ~,, ~,I'~, ~.~,6~

interactions in the continuum (mind that these fields were inter9meted b e f o r e as c e n s t r a i n t s r e s p o n s e due t o t h e c o n s t r a i n t s (2) o r ( 7 ) ) . I t i m p l i e s t h a t f e r an7 r e g u l a r r e g i o n ~z , ~ c ~ , we have

.he~. mt

~

t = ~ I~ ~ = a o t e ~ , . ~st~t

the interaotle-- b.~,..= I the ~ i o l d s ~ , ' ~ , . . . , ~ by the o e - - t r ~ t e

~,,a ~=-~.

where ]~0 , ~w are known d i f f e r e n t i a ~ l e f u n c t i o n s . ~dm~LnK Eqs. [~ )tezmwlse we shall pestulate new that the resultlu8 equation has to hold for ~ ~,~o..) ~z admissible by Eqa.(9) and (10) .

Vol. 3, I~o .4

h~*~z,

THERMOMECHNICS

we a r r i v e

OF CONTINUA

335

a t the r e l a t i o n

which is assumed to hold for a z ~ , ~ 4 , ~ , ~ ,~'~ , ~ z admissible by ~s. (10] and restricte'd b~ Eqs. gg) • The laws of thermomechamics ~I ) , the constitutive equ~tlons, the equations of constraints (10) and the "idealit-y" condition (11J are governing relations o f t h e r m o m e c h ~ c s w i t h g e n e r a l i z e d c o n s t r a i n t s imposed on t h e i n t e r a c t i o n s i n t h e continuum. F o r such c o n s t r a i n t s t h e r e f e rence constant#~-~ ~ b @ ~ ~ ~I~, I f among E q s . ( l O ) t h e r e onCe3~, t h e n from (11)we o b t a i n

for an~#~, ~

a d m i s s i b l e by t h e o t h e r c o n s t r a i n t s .

no other c o n s t r a i n t s , then Zro= e l m z o n ~ and we a r r i v e

a t the c l a s s i c a l

(11) i s an i d e n t i t y nil interactions.

thermoslastic

that~:

If there are

~ , ~=~

,

continuum. I f Eqs.

and we o b t a i n t h e c o n t i n u ~ a w i t h o u t i n t e r -

The c o n c e p t o f g e n e r a l i z e d c o n s t r a i n t s f o r m o t i o n and t e m p e r a t u r e can be combined w i t h t h a t f o r t h e i n t e r a c t i o n s . To t h i s a i d we i n t r o d u c e t h e d e c o m p o s i t i o n

r.')/hR

where ~ constitute constraints response to the constrai n t (2) and *dRj.°.) # ~ a r e i n t e r n a l i n t e r a c t i o n s s u b j e c t e d t o t h e c o n s t r a i n t s o f t h e f o r m (10) and ~ e s t r i c t i o ~ ( 9 ) (where In this

case both ideality

~ a t ,,e ,',,place in (~J

conditions

~, ~ ,...., 4

b~

~5) and (11) h o l d p r o v i d e a

'~,~,'~>...~%.,

re,,pective~,

At the same time t h e v a l u e s o f ~ ' ~e are assumed t o be d e t e r m i n e d by t h e c l a s s i c a l c o n s t i t u t i v e e q u a t i o n s ( w i t h o u t cons t r a i n t s r e s p o n s e ) . S p e c i a l c a s e s of t h e s e combined c o n s t r a i n t s can be o b t a i n e d as b e f o r e . ~he detailed a ~ s i s

of such cons~-ra~ts will be given separately.

336

cz. WOINIAK

Vol.3, No.4

Concluding Remarks I. The obtained equations define material contlmua which can be, roughly speaking, not only more "rigid" but also more "slender" (including ~ne continuum of non interacting infinitesimal elements as the limit case) then the classical material continuum. That is why restrictions ( 2 ) w e r e

called generalized constraints.

This

concept corresponds to the concept of constraints in [ 2 3 • 2. The known simple thermomechanical

constraints analysed i n [ 3 - ~

are special cases of the generalized constraints. Also simple mechanical constraints,J7], as well as arbitrary non-simple constraints,[8,

9], can be obtained from the generalized constraints as

special cases. 3. By an approach given in[9]for mechanical constraints one can obtain from(1), (2), (5) various discretized formulations of thermomechanics as well as thermomechanics of rods, plates and sh~llso Using an approach given in[qO]for mechsn~cal problems, one can also obtain from (I~,(2) , (5)thermomecb-n~cs of such "slenier" bodies as membranes and c ~ d s . References I.

A. Robinson, Non-Standard Analysis, Amsterdam-London (I974)

2.

M, A..~aK, /-leKnaccu~ec~ue ~o~.~e~ Mer_aHu~ e ~ o t a ~ z .r1~u#~pq~cKo~o ~l~u~el~u'rera, (#q~)

34-

T. Manacorda, Boll. Un. Mat. Ital., No 3, I_~2 (19572 A. E. Green, P. M. Naghdi, J. A. Trapp, Znt.~Engng.Sci., 8 (I 970 ) J. A. Trapp, ibid. ~ , (I971) F. Andreussi, P. Podio Guidugli, Bull. Acad. Polon. Sci., ser. sci. tecb~., No 4, 2_I ,(1973) C. Truesdell, W. Noll, The Non-I~near Field Theories of Mechanics, Encyclopedia of Physics, Vol. III/3, p.69, Springer Verlag, Berlin-Neidelberg-New fork, (1965) Cz • Wo~.niak, Arch. of Mech., No I, Warszawa, 2__6,(I 974 ) Cz. Wo~niak, Bull. Acad. Polon. Sci., set. sci. teobn., N03-4

56. 7. 8. 9-

10. ~ •,(19~3) Wozniak,

ibid., No 4, 2~,(1975)

North-Holland Publ. Cemp. cpe@, U~.