Generalized solutions in elasticity of micropolar bodies with voids

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Rev.Acad.Canar.Cienc., VIII (Núm. 1), 101-106 (1996)
Generalizated solutions in Elasticity
of micropolar bodies with voids
Marin Marin, Faculty of Mathematics, Unive1·sity of lJra.<>ov
Str. Juliu Maniu, 50, 2200 Brasov, Rornania
Abstract. This pa.per a.pplies the general results from the theory of eliptic equa­hons
in order to ~bta.iu the existence and uuiqueness of .the genera.lizated solutions for
the bounda.ry value problems in Ela.sticity of micropola.r materia.Is wit.h voids.
l. Introduction
The theories of the bodies with voids do not represent a material length sea.le, but are
quite sufficient for a large number of the solid mecha.nics a.pplica.tions. Our present
paper is dedica.ted to the beha.vior of the porous solids in which the ma.trix material
is elastic and the interstices are voids of material. The intended applications of thiB
theory a.re to the geological materia.Is, like rocks and soils and to the manufacturcd
porous matcrials. First, we write clown thc ha.sic equ.ations and conditions of the mixed
boundary value problem whitin the context of linear theory of micropolar bodies with
voids, as in the paper [4]. Next we use sorne general results from paper [2], in order to
obt.ain the existence and uniqueness of a weak solution of the problem. For convenience,
the not.ations and termonology chosen are almost identical to those of [3], [4]. Let B
be an open region of Euclidian three-dimensional spacc and boundecl by t.he piece-wise
smoot.h surface 8 B. A fixed system of rectangular Cartesian axes is used and we adopt.
the Cartesian tensor notation. The points in B are denoted by ( x;) or ( 3: ). The variable t
is the t.iiue ancl t E (O, tq). Wc Hball employ t.he usual summatio11 over repeat.ed subHcripA
while subscripts preceded by a comma denote tLe partial differentation with respect to
the spatial argumcnt.. We also u8e a superposcd dot to denote the ~artial diffcrent.at.ion
with respect t.o t. The Laf.in indices are unclerst.ood to range over t.He integen; (1 , 2, :l).
The behavior of micropolar bodies with voids is chara.cterized by the following kincmatic
variables :
u;= 11,(x, t ), cp; = cp;(x , t) , a = a(x, t ), (x, t) E B X [O, lo ). (1)
Our paper is concerned witlt a.u anisotropic and 11onhomogeneo11s material. We restricte
om ronsiderat.ions t.o t.hc t<:last.osta.t.ics i::o tha.t. t.he ha..;ic equations becomc
-t.hc equat.ions of equilibrium
t ,J,J + 11 f i =o,
m;1,j + e,1kt,k + e<l; = O;
101
(2 )
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- t.l1e l_i;;.lann· uf 1.1i .. •·q111li brat.cd force~
h,,, +y + eL = O;
-1.hc const.it.ul.ivc equati1>n8
l. , , = !l,;mn€mn + R,;mn'Ymn + B;; fJ + J),_¡'kU,k,
rn.,J == Hnuu1tmn + (J,j .,nn"Ymn + C,3a + l:J,1ka,k,
hi = J)tm1.i.lmn + f.,1.,nni.'Ymn + d¡,a + Á i;·a,j,
g = -- B,1 e,1 -- C,1 -y,1 - d,u,, - ¿a¡
-t.hc gcometrica.I equations
In thcsc cquat.ions we have used the followiug notations
(4)
(5)
e-t.he consta.u ma.~s densit.y¡ u,- t.he components of the displacement field; <p;-the compo­nents
of the microrot.atia. vect.or¡ v-the vohm1e distribution function which in t.he refer­ence
state i.~ vo; a-a measure of the volume change of.the bulk material which resultll from
void compaction or distension; F;-the components ofthe body forces; G;-the components
of t.hc body couple; L-the extrinsec echilibrated body force; g-the intrinsec equilibrated
force; l;1-the components of t.he stress tensor; m;,-the components of the couple stress
tensor; h,-the component.s of the equilibrated stress; e,¡ , 'Yi¡-kinematic characteristic of
the strain; e,¡k-the alternating symbol; A,, ,.,,,., A;1 , B;1"'"' B,,, C;1mn• C;¡, E,,,., D,,,., d;, e­t.
he cha:racteristic prescribed functions of material and they obey the following syrrune­tries
(6)
The physical significa.ne.es of t.hc funct.ions L and h, a.re presented in the works of Good­man
and Cowin, [1], and Nunziato and Cowi.n, [5].
2. Existence and uniqueness theorems
In t.his sed.ion we use sorne result.s from the theory of eliptic equations in order to derive
tlie existence and uniqueness of a wcak solution of the mixed boundary-value problem
in the context of micropolar hodies with voids. Throughout this section we aasume that
B is a Lipschitz region of the Euclidian three-dimensional space. We use the notations:
(7)
with the com.:ention that A 7 = Ax Ax Ax Ax Ax Ax A and where W"•"' are the familiar
Sobolev spaces. With other words, W is defined as the spaces of ali u = (u¡, cp;, a), where
H;, cp¡, a E W1•2(B) with the norm
3
iultv = lalfvu(B) + L (iu;lfv1.>(B) + l'P1lfv1.>(B)) · (8)
i=l
102
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L\'I. dB = SuUS'iUC be a.disjoint decomposition of oB where C is a sei ofsurface measure
a.nd 5,. and Sta.re either empty or open in 8B .. The following bounda.ry conditions are
used:
U.i = ii.;, 'Pi = rpi, a = a on S,.,
t, = t,;n; = i,, mi:::: mi;n; = m,, h:::: h,ni = h on St,
where ií.;, <{;;, if E W1•2(S .. ) and i,, ·m;, h E L2(St).
(9)
Also we define V as a su bspace of W of all u = ( Ui, 'Pi, a) which satisfy the boundary
conditions :
Ui = O, 'Pi = O, a = O on Su. (10)
Ou W X W we define the bilinear form A( v; u) by
A(v, u)= l {A.;mnEmn(u)e;j(v) + Bijmn[t:i;(v)¡m,.(u) + ei;(uhnm(v)] +
+Cijmn'Yi; (v)¡,,.,.(u) + B;J(e;J(v)a + Eij(u)¡] +
+G11b 11(v)s + 'Y11(u)¡) + D1j1.[t:11(v)a,11: + e;;(uh,11:] +
+ E,1,.b•1(v)a,1< + 'Y•;(uh,k] + d,[a'Y,i +-ya,,]+ A'iª.i'Y.i + ea-r}dV, (11)
where u = (u¡, 'Pi, a), v = (Vi, 1/Ji, -y), Eij(u) = Uj,i + Ejik'Pk, t:¡;(v) = Vj,i + e;;T.1/Jk, 'Yij(u) =
'Pi,i> 'Y;,¡(v) = 1/Ji,i·
We assume that the constitutive coefficients a.re bounded measurable functions in B
which ~atisfy (6). From (11) and (6) we deduce
and
A(v, u)= A(u, v),
A(u, u)= L (Ai;mnEij (u)t:mn(u) + 2B;jm.nEij(u)Tmn(u) +
+C,1mn'Yij(u)Tmn(u) + 2Bi1e,1(n)a + 2Cij'Yi1(u)a +
(12)
+ 2D;1e•.i(u)a,11: + 2E;,n¡1(n)a,,. + 2d;aa,; + A;J'7,it7..i + ea2 ]dV, (13)
and thus :
A(u, u)= 2 fa U dV, (14)
whert' U = ge is the i11ternal euergy density' associated to u and suppose that U is a
poRitive defiuitc quadratic form, i.e. thcre exists a positive constant e such that:
A;jmnXijXmn + 2B,JmriXijYmn + C;1mnYijYmn + 2H;jXi1Z +
+2C;jYijZ + '2D;jkXi111ik + 2E';jkYiiWk + AijWiWj + 2diZW¡ + ez2 2:
2: c(x;1 Xi; + YiiYi; + z2 + w;wi), (15)
103
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for ali x ,.7> Yi;, z a.nd tu,.
Now, wc introduce t.hc functionals
f(v) = { e(F:u, + G,if., t- L-r)<JX , JfJ
y(v) =J. ({.-11, + rii,.11/.i. + fq)1!11,
"'
where v = (v;, 'f,, "Y) E W and {},F., G,, LE L2( H).
Ld ü = ( ii;, cp;, a) E W be su ch that ii.., ,¡?.,a .011 Su ma.y be obtained by 111ca 11~ of t lie
embcdding of t.he W1•2 (B) int.o L2 (S.,).
The element u = (u;, cp;, 11) E W is called weak ( or generaliz1ited) soiulic0ri of f.l1t·
boundary value problem, if
u-: ü E V,
and
A(v, u)= f(v) + g(v)
holds for each v E V . It follows from (15) and ( 13) that
A(v, v) ~ 2c L[e;3(v)e;3(v) + "Y•3(v)"Y;3(v) + "Y2 + "Y,i"Y,;]dV,
for any v = (v;, t/J;, 1) E W.
We consider the operators N1cv,k = 1,2, .. .,22, mapping W into L2 (B)
N;v = ei;(v), N3+;v = e2;(v), Ns+;v = e3,(v),
N9+;V = "Yli(v), N12+;v = "Y2;(v), N1a;V = "YJi(v),
N1s+1V = a,;, N22V = a, ( i = J, 2, 3).
It easy to see that, in fact, the N1cv operators defined above have the general form
n
N1cv = L L n1crc.D"'v., p = ¡ti'¡,
r=l p;Slc,
where n1cra are bounded and measurable on B and D"' is
The N1c v operators form a coercive system on W if, for each v E W, we have
22 7 L IN1cvli.ce) + L lvrli.ce) ~ c1!vl~, c1 >O,
lc=l r=l
104
( 17)
(18)
( 19)
(20)
(21)
(22)
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where c1 does not depend on v. Also, ¡.¡L., j.jw denote the usua.l norms in L2(B) and
W, respllctively. We ha.ve the following theorem, [2],
Theorem l. Let np•a be constants for jaj = le,. Then the N1cv system is coercive
on W if and only if the rank of the matrix
( NP,e) = ( L np.aea) ,
lal=k,
(23)
is equal to m for each e E 01, e :/= O, where 01 denotes the complex three-dimensional
space and ea = ~1 e;· e;·.
We a.BBUme furthermore tha.t for ea.ch v E W
22
A(v, v) ~ c2 L IN1cvll,(B)> c2 >O, (24)
le=~
where c2 does not depend on v . Let
22
P = {v E V, L IN1cvll,(B) =O}. (25)
lc:l
We denote by V /P the factor spa.ce of cla.sses v = { v + p, v E V, p E P} with the norm
1-Vlv¡p = inf !v + Plw.
pEP
From [2] we deduce the following theorem
Theorem 2. Assume that A(v, u)= [v, ñ) define.9 a bilinear form far each v, ñ from
W /P, u E v, v E v. Further we suppose that (22) and (24) hold. Then a necessary and
sufficient condition for. the existence of a weak solution of the boundary-value problem is
p E P ~ f(p) + g(p) =O. (26)
Further,
(27)
for every v E W /P.
N o~v 1 we sha.11 apply ~he a.hove results to prove the existence and mi'iqueness of a weak
solut.ion of our bounda.ry-value problem. Clea.rly, from (19) a.nd (20), we obtain (24).
The matrix (23) has the rank 7 for ea.ch e E 0 3 , ( :/= O. Thus by Theorem 1 we condude
that the system of N, operators, defined in (20), is coercive on W. According to the
definition (25) of ·p, for each v E 'P, we ha.ve
é¡)(v) = º· ri;(v) =o,'= o,
105
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P :=: {v = ! ,.,, , ·::., :)E V, v, =a,_+ e,,d,_, x,, ·:j;, == h,, 'Y== O}, ('.l.>S)
wlwr<' "•· ¡rnd .'., a.r<' a.rl iit.rn.ry C'o 11;;t.;i.nt.r;.
Firnt., wc a s~ 11111i' t.ha t .'i',, is a 11011- r mply ~d. 'l'hen we> rkcluce t.hat.
P = {OL
ami ( '.lt)) is r;a.l.i~h('d . 111 vi,·w of Tl1Pore111 '2 il. folluws lhal.
Ld llS considN t.he CilS(' whf'n s· .. is empt.y, Then p is giV('ll by ('.28), where (l., amf
h, are arbil.rary co11sl.a11ts. WP <Hf' fpcf t.o t.he following t.heorem
Theon,n1 4. '/'{¿ ,; nccessary anti sujfir;1:enl cnn,Liíinns / 01· lhc ea·islcncc of •L w,;cik ~ofol. ion
u E W of thc t r11 r: t.u>n p1'0h/cm circ gú1cn hp
/ e,f~dV + / t",dA =O,
JH JéJH
r (!é j ,kXJ(Jk + Ch)dV + / E;ikXj(ik + mk)dA =O. JB Jou
REFERENCES
(!] GOODMAN, M.A., COWIN ,S.C., A continuum theory far granular materi­als,
Arch. Ra.t.ional Mech. AnaJ.,44(1972), 249-266
(2] HLAVACEK, l., NECAS, .1., Un inequa.lilies of K orn's type, Arch. Rational
Mcch. Anal., 36( J\HO), :105-:111
[3] l\1Ail IN, M., Sw· t't:r.istence da ns la thtrmoela.~ t ic ite des milieux mic1'0polaires, C.
IL i\ca.d . Sci. Pa.ris, t.321, Serie 11 b, H)!)fi, 175-480
[4] MARIN, M., Some basic íheo rems in elastostatics of micropolar materials with
vnids, .J. Comp. AppL Mat.h., 67(1996), 229-242
[5] NUNZIATO, J.W., COWIN, S.C., Linear elastic materials with voids,
J Ela.st.icit.y, 13'(1981), 125-147
Recibido: 12 Abril 1996
106