Ghostscript wrapper for D:\Digitalizacja\MTS86_t24z1_4_PDF_artyku³y\mts86_t24z3.pdf M E C H A N I K A TEORETYCZNA I STOSOWANA 3, 24 C1986) BALANCE EQUATIONS FOR MIXTURE AND POROUS MEDIA IN THE LIGHT OF NONSTANDARD ANALYSIS KRZYSZTOF NOBIS Instytut Mechaniki UW Introduction The list of papers and monographs concerning mixtures and porous media is very extensive (for example, cf. [1 - 20]). Most of them is based on the rather unphysical postu- late that in every point of the region occupied by the mixture there are all its components. At the same time, also porosity is assigned to every such point. Moreover, in [I - 19], the balance equations for mixture and porous media are introduced in a form of certain a priori assumptions, which take into account the forementioned postulates being not related to the balance equations of the classical continuum mechanics. The aim of the paper is to derive the balance equations for the mixture and porous media directly from the well known balance equations of continuum mechanics. We are to show that it can be done using the methods of the nonstandard analysis. Such approach makes it possible to assign the clear physical interpretation to all terms in the resulting balance equations for mixture and porous media. At the same time, the extra unphysical postulates, such as mentioned in the begining of the Introduction, are avoided. The present paper is self contained but the partial results and ideas concerning the proposed approach were published in [24 - 26], The approach proposed here has certain features common with that of [14], [21 - 22]. Denotations. Throughout the paper n is the fixed positive integer. Subscripts oc, {$ run over the sequence 1,2, . . . , « , subscript y runs over 0, 1,2, . . . , « . Moreover, tQ, tt are the known real numbers, t0 < tt, which determine certain time interval. 1. Physical basis Let 8 be Euclidean real 3-space of places, O be a set of all regular regions in e, V be a translation space for e, Vf be a Euclidean real 3-space of forces, U be the material uni- verse of all 3-dimensional elementary differentiable manifolds (deformable bodies). Define J a s a set of all n-tuples (%i, • • • , Xn) where #a: i2ax [t0, fx] -> e, a = 1, ..., n are deformation functions of n disjointed deformable bodies $a, J"a e U such that (Va, /0(VO((« * ft =*(*«(£«, t)nX^Qfi, t) * 0 ) ) . 282 K., N OBIS L et Q a ,oc= 1, . . . , «, be th e disjoin ted region s in e occupied by bodies Ś S a in th e kn own r e fe r e n c e c o n fi gu r a t i o n s x a : @ a - *e. L e t v a (- ,t)eV, wh e r e v a (x, t) = %{X, t), X = x~ x{x, t),XeQ a ,te [ f0 , fj, be the velocity field defined, for every £, on the region %*(&*, 0 occupied by t h e body Ś S a at the time instant t. Define i2J, a # «(£«, 0 and i2'0 = e\ uQa- By virtue of the well known assumptions of solid mechanics, to every (Xi> • • •> X«) e  - ^  a n ^  t o  every  /J e  O the  following  functions  are  uniquely  assigned: [t 0 ,tx)at- * J Q«(X, t)dv e R + , [t 0 , tj 3 t - * J edx, t)b a (x, t)dv  G Vf, [t o ,tx]at- * J t ar (x,t)daeV f ,  /««(• ) =   0, J T a (x, t)n SA (x)da e V,, J f aP (x,t)daeV f . F unctions  (1.1)  are  assumed  to  be  continuous  for  almost  every t e [t 0) tj].  The  values of  (1.1)  represent  for  any  fixed t,  the following  objects:  mass  of 3$ a inA,  resultant  body forces  acting  at 3$ a  in A,  resultant  of  the boundary  tractions executed  by 3$ v  (for y  #   0) on SS a  in A,  resultant  of  the surface  tractions  on 8A  within  body 3S a (T a (x, t)  is Cauchy stress  tensor), respectively.  M o reo ver./ ^*, t) is the density of the forces  due to the friction between @ a   and @ p ,  which  act  on SS X . H ere/ a / 3(x, t)- j- f^ (x, t)  =   0.  Term ta0  in  (1.1)3 represents boundary tractions due to the external forces  acting at @t a . F or  an  arbitrary A, A eO,  define A*  = e\ A, ri y (A,A e )  s d(Q^ nA)nd(Q' Y nA e ) c 8A, As  it is  known, for  each  ( J 1 !,  ..., @ n ), 3S a   e  C/,  and every  ( ^ ,  ..., %„) e M,  the integrands in  formulas  (1.1) satisfy  the well known  system  of conditions ~  / ' e*(x, t)*>  =   01),  (1.2) Anal l )  Here ~  / Ax, t)dv s  J for  an  arbitrary  function/ (x, 0  on  the  LHS of  Eqs.  (1.2) BALAN CE EQUATIONS FOR M I XTU R E ... 283 DC,, V f ' V f — v a (x, t)Q a (x, t)dv = > fa8(x, 0 ^ «+ > , W ^ , t)da + + J T a (x,t)n SA (x)da+ J b a (x, t)c a (x, t)dv, (1.2) [ cont.] — J (x~X 0 )xv a (x,t)Q a (x,t)dv = 2J J  (*- *O)><''«?( n +   Z J  j  (*- *o)xf«v(xlO< fa+   J  (^- ^o)xya(x,/ )«aX^ +   J  (x- 0̂)xAa(x,  0e«(^> t)dv, which  has  to  hold  for  every A e O  and a.e.  ( 6 [ ( 0 ,  ?J .  H ere x0  is an  arbitrary  point  in e and  4^(- )  =  0, for a =  /3,  while ^ ( x ,  0  = T a (x, t)n SQ *Jx) =  - 7> ( x,  *) »«$( *) + / .*( *, Of a* P,  (1.3) holds for  a.e. x e <%2£n 5 ^ .  Eqs. (1.2) represent th e well known  balance equations  of  mass, momentum  and moment of momentum, respectively,  for the system  of n  non  intersecting deformable  bodies. 2.  N onstandard  definition  of mixtures  and  porous  media Let SO?  be the full  structure in which e,V,V f ,R  are disjointed  relations of the type  (0) (cf.  [23], p.  19). The balance  equations  (1.2), which  hold in 3W,  hold  also  in  *9K as certain internal  relations.  It means  that  for  every  (internal)  f , e * [ / ,  « = 1 , . . . , « ,  for  every internal (xi Xn) e * - M• >  f°r  every  internal A e* O and x 0 e s,  as well  as for  every internal  function  of  the form  (1.1)  (with  domain  *[ć 0, ti]  in *R and with  the  values  in *R and *V fi   respectively),  relations  (1.2)  and  (1.3) hold  in  *2ft. Define  in *50i m a {x,r)= J g a (x,t)dv, where Ś S(x, r) stands  for  a ball  with a center x and the radius r, r e* R + . Let J(°  be a subset  of *Jt which satisfies  the following conditions: 1°  F or  every (xi, • • - ,%„) e ^ °  there is °(x«) = 1  for a =   1,  ..., n where %  is  defor- mation  function  in 9ft,  i.e., %\ QY. \ U, tj] - * e 2 ) .  H ence we  see  that  there  are  regions Qc in s, such  that = Q*  for a =   1,...,«  and  f e [t 0 , h].  (2.1) 2 )  Symbols  "(• ) stands for a stan dard part of function or set,  [23], p . 115. 284 K. N OBI S We also assume that all points belonging to each Q' a are near standard and Q' are regular regions in s. 2° F or every x e S- 'mV*Q' 3 ) , the internal sequences vol [B(x, r m )ni2' a ] jn a (x,  /- „,) vol B(x,rJ '  vol [B(x, rJnQH ' ^ U) /• „, =  - ^-, °(r l )>0, m  = 1 , 2 , . , . , have  F- limits  (cf.  [23], p. 109). By virtue  of the theorem which can be found  in  [23], p. 110, there exists  Ao s *N \ N   such that for  every v e* N \ N  and v < Xo  each  value vo\ [B{x,r v )nQ' a ] m a (x, r v ) vo\ B(x,r v )  '  vol[i?(x, r, )n fla"s  l  j is  the  F- limit  of any  sequence  (2.2). 3°  There  exist  functions Q1 a x- * v a {x, t) e [0, 1], Ql 3 x -> ga(jc, 0  e 7?+ ,  continu- ous  a.e.  on i3* for  I e [( 0, tt\  (where  £«(• , t) is also  differentable)  which  are  the  standard parts  of  functions ^ ^ / ̂ s* [0,1L N ow  we  formulate  the  following. D efinition.  The  system  of  bodies $ a   e *U, a =   1,  ..., n,  for  which (%i, . . . j j e  ../ Z0, will be  called  th e mixture.  Every SS a  is  said  to be the component of the mixture. The  value )'K(x,  / ) will be called  the saturation  of the mixture by  the a- th component. D efine  function i3'3x^ v(x,i)e[0,l], putting v(x,t) m l- If v(x, t) is not  identically  equal  to  zero  and « =   1,  then the mixture  is  called  the  porous medium.  F unction v( • , t) is known  as  a porosity,  and  has  been  derived  here by  the non- standard  approach  from  the real  porous  structure  of  the body.  In the traditional approa- ches,  porosity v{- , t)  and  saturation vj^ - , t) are  postulated  a priori. 3. General form of the balance equation in  9Jt and  *9?l All  balance  equations  (1.2)  can  be  written  down  in  what  is called  the  general  form of  the  balance  equation 3 >  Symbol S- 'mt, where A <= *R"  stands  for  5- interior  of A ,cf. [23], p.  107.  M oreover, for  every A in  SCR,  by *A  we  define  t h e corresponding stan dard entity in 9JŁ *. BALANCE EQUATIONS FOR M I X T U R E . . . 285 DC ' v% r — J Wa(x, t)Qa(x, t)dv = 2J J Gav(x,t)da + n + 2J J Gav{xj)da+ J 0a(x,Ond^x)da+ (3.1) + J £a(x,t)Qa(x,t)dv, 8 " 1 n, where Ga/3( •) = 0 for « = /3 and where Gafi(x, t) = $ a ( x , 0 » a 4 ( * ) =  ­@p(x,  t)naatfi(x) + Fati(x,  t);  a.  ?= /?, (3.2) here Fa/)(jc, f) =  ­F^{x,t) hold for a.e. x  eP^ iń ,  A") c <9zl and ( s f o ^ J . Eqs. (3.1) have to be satisfied by an arbitrary  3§ a e  U, a. = 1, ..., n, by every fe, ..., # „) s  Jt and by every zl 6 0. In Eqs. (3.1), scalar field Q%(- , t) has the same meaning as before, W a { • , t), B a ( • , t) are tensor fields of the /c- th order, defined  a.e. on£}„. Moreover, G ay {- , t) are  tensor fields  of  the £>th order  defined  a.e. on dQf a r\ dQp  and & a { • , t)  are  tensor  fields of  the k + 1- th  order,  defined  a.e.  on Q^ .  H ere 0 a ( - , t)  is  the flux  field,  !?<,(• , t)  is  the internal and G ay { • , t) is the external supply  in the a- th component. In what follows  instead of  (1.2),  (1.3)  we  shall  deal  with  the  general  form  of  the balance  equation  (3.1)  an d  with the  continuity  conditions  (3.2). On  passing  to  enlargement  *9Jl  of  9Jt  we  shall  take  th e  internal  conditions  (3.1)  and (3.2)  as  the  basis  of  the  analysis. 4.  From  micro-   to  macro-  general  balance  equations The  general  balance  equation  in  *S0t  given, by  (3.1),  (3.2),  under  assumptions  th at (Xi >  ••• > Xn)  e ^ °  c **&',  will  be  called  the  general  micro- balance  equation  for  mixtures and porous media. The forementioned micro- balance equation constitutes only the  starting point for  further  considerations, being  the basis for  obtaining in 90? what will  be called  the the general  macro- balance  equation.  In order  to  pass  from  the  micro- balance  equation in  *9K to  the  macro- balance  equation  in  501, a  number  of  the extra  assumptions  has  t o be  introduced. It  must  be  emphasized  that  the macro- balance  equations  exist  only  for rather  special kinds  of  structure  of  mixtures  and porous  media  (which in the n on stan dard sense  were  defined  in  Sec. 2). Let  us  substitute  to  (3.1): A = B(z, r r ),  z e S- int*®', v <  Ao, v  e *N \ N .  Then  for a.e. zeS- mt*Q*, te*[t 0 , tj],  from  (3.1)  we  obtain fJD  1 Dt  volJ?(z,  /• „) (4.1) 286 K. NOBIS [cont.] 1  C _ / i r x , + ,— —__—-  BJx,  t)px(x,  t)dv. vol B(z,  rv)  J S(z,rr)na t a We shall assume that all terms in (4.1) are near standard and that in 9JI there is field 1 f W(x yol[B(z,rv)nQ'a]  J ( a^ ' such that the following formulas hold4) Analogously, we assume that in S0i there are fields 5a( •, f) and Say(  •,  t) such that J B«(̂ , 0 ^ holds in ^-int*^',  t e [/0, ^ ] . Let J1 be an arbitrary smooth surface in  S­int*Q' oriented by the unit normal  nr(z), z  eF. Let the minimum radius  rmin of the curvature of  F is greater than  rv. Define by jff+(z,  r),  B~(z,r),  z eF the semisphers of the ball  B(z,  r), which are situated on the positively and negatively oriented side of F, respectively. Define tfvlB+iz,  r), B­(z, /• )] •  d[B+(z,  r)nQla]^d[B­  (z, ** We write a ~  b, where a, b are finite numbers in  *R, if  \a­b\ is infinitesimal BALANCE EQUATIONS FOR MIXTURE... 287 for  x  — 1, . . , « ;  y = 0 , . . . , « . We shall assume that for every  z  e rn(S­mt*£2'), the internal sequences area [Tnl?(z, ~ have f-limits. Then by virtue of [23], p. 110 there exist  Xx e  *N\N such that for every d e *J/V\iV and  d <  Xx, the following values area [FnB(z,  ra)]  '  ^  '   } are the F-limits. We shall assume that for every pair of surfaces  T, "J1 in S'-int*^', which satisfy con- ditions analogous to those imposed on J1, and for every z e '/TV1/1 there is ^ ^ ^ z ,  rt),B~^y  rs)] area[Tn5(z,ra)] area[TnB(z, ra)] where  Fly(  • ), u / 5 y ( • ) have the same meaning as T ^ ( • ) . Define the system of functions t0> i ] , and assume that there exist functions which are standard parts of functions defined above. We also postulate that there exist functions Q'3x­> ( *  t)W {z / )] -   y  div[fi mv {z,  t)0 ay (z, It X  S ay (z,  t)+Q a (z,  t)v a (z,  t)£ a (z,  t),  (4.3) hold  for  a.e.  z  eQt  and  for  a.e.  t e  R. Let  us  observe,  that  from 0 a (x,t)n r (x)da=  J an d  under  assumption  that  fi a p(x,  t)  =   (ifjjc,  t)  and  i ^ (  • ) s  0  we  obtain  *a f i ( x,  0  = Thus,  for  every  &1  c  Q  and  every  f  e  [*0,  fi],  there  is n —  J  ^a(z, t)va(z, t)W a(z,  i)dv =  2J  J n At  th e  same  time,  assuming  th at  [i a p( • ,  t)  =   ^ j 9 a ( • , / )  we  define Ą ?(^,  0  =   *«u(z, 0  ~  */ J«(2»  0 , as  a  field  determining  the  effect  of  friction.  Condition  (4.4)  will  be  called  the  general integral  macro- balance  equation  for  mixtures  and  porous  media,  while  (4.3)  be  the  local form  of  this  equation. F rom  the  foregoing  consideration  it  follows  that  the  general  macro- balance  equation in th e form  (4.3)  or  (4.4) holds  under rather  strong  regularity  conditions, which  have  been succesively  introduced  in  this  Section.  The  mixtures  and  porous  media  for  which  Eqs. (4.3)  and  Eqs.  (4.4)  take  place, will  be  called  the ideal  mixtures  and  ideal  porous  media, respectively. 5.  Macro- balance  equations  for  ideal  mixtures and  porous  media F rom  the  general  balance  equation  (4.4)  we  shall  obtain  now  the  macro- balance  equ- ation s  of  mass,  momentum  and  moment  of  momentum.  I t  will  be  done  by  the  speci- ficat io n  of  fields  lF a (- , t), G ay {- , t), & a (- , t)  and B a (- , t)  in  (3.1)  and  (3.2). 5.1.  Mass  conservation.  In  order  to  obtain  the  principle  of  mass  conservation  from  the general  balance  equations  (for  a- th  component of  the mixture), we  have  to  substitute % ( • ) • ! • G av (- )~0,  <*>«(• ) =   0,  £«(• )  =   0, BALAN CE EQUATIONS FOR M I XTU R E ... 289 in to E q. (3.1), where  Q a (- , t) stan ds for a mass density of the  a- th  com pon en t  in  con - figuration Q' a .  Takin g  in to  accoun t  the  forementioned  substitution s,  we  o bt ain  fields yya( ' > t),  0 « y ( ' >  0 . S*i>(''  0 i  3 x ( ' .  0 .  S«("»  0   a n d v *(  • , 0  in  501 where  in *ffl *vtf t 7 t \ —  L ^ _ _ _  I m ( Y  f\ x,  _  i y ^ 2 ' t} vo\ [B(z, r v )nQ' a ] J ( * Ą X> t ) m ~ l > *S av (z, t)  =   y o  * f )  J Gav(x, t)da  =   0,  (5.1) an d  where  z  e S- int*.^', v, X o e*N \ N , v < X o .  N ow,  from  (5.1)  and  from  E q .  (4.3) we  obtain ~  (&,(*, t)v a (z,  0 )  =   0,  (5.2) where  ga( z,  / )  an d va(z, t)  satisfy  th e  postulated  regularity  con dition  and (5.3) hold  for  every  z  e ^ - i n t *^ '.  Moreover  if 0>  is  an  arbitrary  regular  subregion  of Q,  then • ̂ J §- (*, 0»«( ̂ 0 * = 0, (5- 4) holds.  The  resulting  equation s  (5.2),  (5.4)  represent  macro- mass  balance  equation s  in  th e local  and  the  integral  form,  respectively.  F ields ql(- ,t), v a (- ,t)  are  n o t  postulated a  priori  (which  takes  place  in  the  known  approaches  t o  mechanics  of  po ro u s  m edia)  bu t are  given  by  formulas  (5.3). 5.2.  Conservation  of  momentum. I n this  case  we  have  to assume  t h at in  the  gen eral  balan ce equation  (3.1) 4  Mech.  Teoret.  i  Stos.  3/86 290 K. N OBIS B a (;t)  =  b a (;t). By virtue of the assumptions formulated in Sec. 4 we have f J  r. ( x, o* W* .  (5.5) and »  • §; [*e«(z, 0*^( z, 0*®.(*» 0 ] , for  z e JS1—int*i3r, Ao, v e *N \ N ,  v < Ao. U sing the procedure analogous to that applied in Sec. 4, instead of (4.3) we obtain - prr  &*(*, 0"«(^» i)v a {z,  0] =  ^ d ivŁ M a v( z, 0 ^ ^ , 01 + Q a (z,  t)v a (z,  t)b~(z,  t ) .  (5.6) r= o The  meaning  of  ,«av(  • , t) has been  explained  in  Sec.  4. H ere 1  C div[*fi av (z,  t)*T ay (z,  / )] a  - —r- Ę r—^ r  t ay (x,  t)}i 8B (x)da. voiayzs rv) j r K J.B+(i,r f ),(B- (i,r f )l If  y«ai(z,  0  =  ... =  ^„ „ (z, t)  then  the following  equalities  hold « n 2 1  d ivK v( z, t)T ay(z,  t)] =  div [/ / ay(z, 0 J]  T aY(z, t)}. P utting we  shall  refer  T a (  • , t) to as a partial  stress  tensor  related  to the  a- th  component of the mixture. At the same  time, for an  arbitrary  regular  subregion  by virtue of (4.4) we obtain BALAN CE EQUATIONS FOR M I XTU R E... 291 D r „ .   .   .   .„  . J  ̂ 0 ̂ j  (5.7) V= 0 ,   0  W z  > O *  • Eqs. (5.6), (5.7) represent the local and global form, respectively, of the  macro- balance equation  of momentum.  The  macro- fields  occuring in th e forementioned  equations  have been  not postulated  a priori  but  are  related  to  the micro- structure of the  body  by means of  Eqs.  (5.5). 5.3.  Conservation of moment of momentum. Applying  the procedure  analogous  to  that  of Sec.  5.1,  5.2, we assume  now  that W Jpc,  t) a  ( X - XO ) X B » ( I ,  t), G av (x, t) m (x- x 0 )  x t av (x, t), 0 a (x, t)ft dA {x) m (x~x o )xT a (x, t)n dń {x), BJx, t) = (x- x Q )xb a (x, t). H ence in 501 there  are  fields v a ( • ,  t), s ay ( • , t),  T ay (  • ,  t) such  that in *30t  the  following relations  hold ( Z - XO ) X *S aJz,  t)  ~  j- 57  r-   I  ( X- XO )  X  *a y( x,  0 ^ «5  f5  Ŝ VO1ZJ(Z,  r y )  J  \ - " v J (z- xo)  x *T ay {z, t)n r {z)  ~  J  .  ^  J  (x- xo)x 3"«(*»  t)n r (?c)da, and  where 4 * 292 K, NOBIS Applying the forementioned relations we arrived iv|>a,,(z,  t)(z­x0) x Tay(z, /)] + (5.9) y=0 O, O(z­^o)xi(z, 0­ Taking the time derivation of Eq. (5.9) and bearing in mind (5.6), we obtain fiav(z,  t)6xwzxTay{z,  t) =  QK(Z,  t)va{z,  t) —  Y.va{z, 0 = 0, u n d e r t h e e x t r a a s s u m p t i o n  fit( • ,  t) «• ... = £ „ ( • ,  t)  — v{­,  t) where Hence, after simple calculation, we arrive at n , 0 Tay(z, 0 = [ J £ ^ ( z , 0 ra),(z, O] 7- (5.10) Defining n fa(z, 0 S  ^^(.Z,  t)Tav{z,  t), y = 0 as a total stress tensor of the a-th component of the mixture, we obtain here the symmetry condition of this tensor. The resulting equation (5.10) represents the macro-balance law of the moment of momentum in its local form. Final remarks The main feature of the resulting macro-balance equations is that they are not postu- lated a priori but are derivied from the balance equations for system of unintersecting and coacting deformable bodies, i.e., from Eqs. (1.2). Such procedure has been realized here by applying the methods of the nonstandard analysis. On this way we^are able to give the exact fenomenological definitions of mixtures, ideal mixtures, porous media, and ideal porous media. The approach used in the paper assignes to every term in the resulting macro-balance equations its physical interpretation in terms occuring in Eqs. (1.2), which have the clear physical meaning. For the particulars and the further analysis of the obtained resultSj the reader is referred to [27]. BALAN CE EQUATIONS FOR M IXTU RE... 293 References 1. M. A. BI OT,  General T heory ofT hree- Dimensional Consolidation, J. Appl. P hys., 12, (1941). 2. M. A. BI OT, T heory of Propagation of Elastic W aves in Fluid Saturated Porous Solid, J. Acoust. Soc. of Amer., 28, 2, (1956). 3. W. D E R SK I , Equation far Motion for a Fluid Saturated Porous Solid., Bull. Pol. Ac. T ech n .: 26, 1, (1978). 4. F . A. L. D U LLI E N , Porous Media Fluid T ransport and Pore Structure, Academic P ress, N ew York, (1979). 5. A. G OOD MAN , E. C O WI N , A Continuum T heory for Granular Material. A. R. M . A., 44, 4, (1972). 6. A. E. G REEN , P. M. N AG H D I , A N ote on Mixtures, I n t. J. Eng. Sci., 6, (1968). 7. A. E . G REEN , P . M . N AG H D I , On Basic Equations for Mixtures, Quar. Jour. M ech. an d Appl. M at h ., v. XXX, 4, (1969). 8. A. E. G R E E N , N . LAWS, Global Properties of Mixtures, A. R. M . A., 43, I , (1971). 9. M . E. G U R T I N , G . M . de la P H E N A, On the T hermodynamics of Mixtures, Mixtures of Rigid Heat Conditions, A. R. M. A., 36, 3, (1970). 10. J. IG N ACZAK, T ensorial Equations of Motion for a Fluid Saturated Porous Media, Bull. P ol. Ac. T ech n .: 26,  8- 9,  (1978), 11.  S. J.  KOWALSKI, W spół rzę dne normalne i warunki brzegowe w teorii mieszanin,  R ozpr.  H abil.,  I P P T, (1980). 12.  J.  K U BI K , Mechanika silnie odksztalcalnych oś rodków o anizotropowejprzepuszczalnoś ci,  I P P T ,  (1981). 13.  V. N . N IKOLAEVSKIJ, On processes of unsteady deformations in water- saturated solids,  Arch. M ech.  Stos. 17,  (1965). 14.  V. N .  N IKOLAEVSKIJ,  K . S.  BASN EV,  A. T.  G ORBU N OV,  G .  A.  ZOTOV, Mechanics of Saturated Porous Media, (in Russian),  M oscow,  (1970). 15.  A. E.  SCH EID EG G ER, T he Physics of Flow T hrough Porous Media,  Toron to,  (1957). 16.  G .  SZEF ER, N onlinear Problems of Consolidation T heory,  P roc.  of  Polish- F rench  Symp.,  Cracow, (1977). 17.  C.  TRU ESD ELL,  R . A.  T O U P I N , T he Classical Field T heories, In H an dbuch der  Physik, Bd I I I / I , Sprin- ger,  (1960). 18.  B.  UziEMBŁO, Podstawy termodynamiki aksjomatycznej wieloskł adnikowych oś rodków cią gł ych,  R ozpr. dokt.,  I P P T,  (1979), 19.  W.  O.  WILLIAM S, On the T heory of Mixtures,  A. R . M . A.,  51, 4, (1973).  *  • 20.  C z.  WOŹ N IAK, Podstawy dynamiki ciał odksztalcalnych,  P WN , Warszawa,  (1969). 21.  C z. WOŹ N IAK,  M . WOŹ N I AK, Effective Balance Equations for Multiconstituent and Porous Media, Bull.  P ol.  Ac. Tech n .:  29, 1- 2,  (1981). 22.  M . WOŹ N I AK, On the Formulation on Conservation L aws in Multiconstituent and Porous Media,  Bull. P ol.  Ac. Techn .:  29,  1- 2,  (1981).  . 23.  A. ROBIN SON , N on- standard Analysis,  N orth- H olland  P ubl.  C om p., Amsterdam, (1966). 24.  K.  N OBI S, An Applications of N onstandard Analysis in Mechanics of Porous Media,  Bull.  P ol. Ac. Techn.:  32, 7 - 8, (1984). 25.  K.  N OBI S, E.  WI E R Z BI C KI ,  C z. WOŹ N I AK, On the Interpretation of N onstandard Method in Mechanics, Bull.  P ol.  Ac. Tech n .:  32, 7 - 8, (1984). 26.  C z. WOŹ N IAK,  K. N OBI S, N onstandard Analysis and Balance Equation in the T heory of Porous Media, Bull.  P ol.  Ac. Tech n .:  29, 11- 12,  (1981). 27.  K.  N OBI S, Formulation of the Balance Equation for Mixtures and Porous Media by the N onstandard Analysis Methods,  (in Polish), D iss., D ept. of M ath., C om p. Sci. and M echanics, U niversity of  Warsaw, to  be prepared. 294 K-   N OBIS P  e  3 io  M e yPABH EH H fl  BAJIAHCA  flJM   CMECEft  H   IIOPH CTH X TEJI B  BH fly  H ECTAH flAPTH OrO  AHAJIH3A H ac T o m n ero  cooSm eH H a  —  STO  noJiyqH TB,  MeTonaiviH   H eoTaH flapTH oro 6an aH ca  «J I H   c wec ea  n  n o p H c n r x  Teji  H3 H3BecTHŁK  ypaBHeHHH   6ajiaH ca  MexaHHKH  cm ioiiiH bix  cpefl.  T a - lo rił   n oflxofl  Aaer  BO3MO>KH OCT&  H CH OH  H H TepnpeTanjtra  Bcex  n o n efi  B  n o JiyieH bix  ypaBHeiCHHX  6an aH ca. S t r e s z c z e n i e R Ó WN AN I A  BILAN SU   D LA  M IESZ AN IN   I  CIAŁ  POROWATYCH W  Ś WIETLE  AN ALI Z Y  N IESTAN D ARD OWEJ Celem pracy jest otrzymanie, za pomocą  metod analizy  niestandardowej,  równań  bilansu  dla mieszanin i  ciał   porowatych  wprost  ze  znanych  równ ań  bilansu  mechaniki  kontinuum.  Podejś cie  takie  umoż liwia jasn ą   interpretację   wszystkich  pól  w  otrzymanych  równaniach  bilansu. Praca  wpł ynę ł a do  Redakcji  dnia  12  kwietnia  1985  roku