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A variational approach to the Cosserat-like continuum
ht. J. Engng Sci.
Vol. 31. No. Printed in Great Britain
11, pp. 1475-1483. 1YY3
A VARIATIONAL
COZO-7225/Y3$6.00 + 0.00 Pergamon Press Ltd
APPROACH TO THE COSSERAT-LIKE CONTINUUM JAN SACZUK
Institute of Fluid-Flow Machinery, Polish Academy of Sciences, 80-952 Gdalisk, ul. J. Fiszera 14, Poland (Communicated
by J. T.
ODEN)
Abstract-The exposition given here is intended to show an equivalent variational approach to formulation of the virtual work principle for the Cosserat-like continuum. Stationarity conditions of an action integral lead to the Euler-Lagrange equations identified with the balance equations for stresses and couple-stresses within micropolar and micromorphic continua. Vector fields as independent variables are taken so as to satisfy the known Stokes’ decomposition. Based on the standard variational arguments, for a given Lagrangian function and an assumed l-parameter family of transformations of both the independent and dependent variables, the fundamental variational formula identified with the virtual work principle of the Cosserat-like continuum is obtained. To determine the immediate relations between the geometric variation of the boundary and the variation of the field variables the transversality conditions are used. A notion of an independent integral is used to define invariance conditions of the integral in question which is invariant under an action of an r-parameter
Lie group.
1. INTRODUCTION
The importance of variational statements of various problems of continuum mechanics has been confirmed many times. It is also the most general starting point for the discussion of any mechanical system. The reasons why the variational technique is so useful are the following [3,61: (1) It provides tools for studying the existence of solutions of the problem under consideration. (2) It encompasses in a concise statement the collection of boundary, initial and jump conditions of the problem in question. (3) It naturally includes into the system the effects of constraints. (4) It leads to information on bounds to the solution of integro-differential equations. (5) It leads to a very efficient method of constructing approximate solutions. The “trivial” embedding of a given system of the Euler-Lagrange equations (say, the balance equations for stresses and couple-stresses) into a variational statement via the Lagrange multipliers is handy but not, in general, satisfactory. An unsuccessful example of the Lagrange multiplier method takes place already in the linear elasticity, where one cannot derive an unconditional principle from the dual energy principle known as the Hellinger-Reissner one. The reasons for wanting to embed a given system of the Euler-Lagrange equations into a variational statement and their limitations are explained in [3]. Hence it appeared to me to be of importance to investigate circumstances under which the principle of virtual work for the Cosserat-like continuum [2], say, micropolar and micromorphic continua [5], can be obtained from variational arguments, i.e. from a consideration of a given Lagrangian and an assumed l-parameter family of transformations of both its independent and dependent variables. The purpose of this article is to show an equivalent approach to the one used very often in practice, i.e. to the Lagrange multiplier technique. 2.
PRELIMINARY
RESULTS
AND
VARIATIONS
We consider an interacting mechanical continuum in static equilibrium. It is assumed that the (elastic) continuum % is defined by a (connected) geometric configuration space with the 1475
J. SACZUK
1476
structure of a differential manifold G. In turn, by a continuum one can understand a mechanical system which interacts with an external mechanism (the one which forces the system to assume a configuration compatible with constraints) with body forces f and body moments g. It is also assumed that the (virtual) work induced by mechanical reaction is measured. At this stage we assume no constraints to the behaviour of the static continuum. The geometric state of a deformed (elastic) continuum will be described, from assumption, by N + M independent fields (P(ti),
@YQ’))= (u’(t’), . . . ) Lyt’),
@‘(f’), . . . ) rpyt’))
where (t’) = (t’, . . . , t”) are the independent coordinate variables. In the following we will confine ourselves to N = M = n = 3, but results can be extended to more degrees of freedom. The independent fields u a and Gy within the Cosserat continuum [2,5] can be identified with the translation vector field and the rotation vector field, respectively. We assume that the behaviour of our static continuum may be described by the action integral over G I@“, u a> $J‘) = I, L( P, ua(fm), $‘(t”),
Xp(t”),
Y;(P))
dV
(I)
where L is a Lagrangian density function. The method of considering the stationarity, in general, conditions of I is to apply standard techniques of the calculus of variations. One obvious way of extending this theory is to consider the variation of an integral as a consequence of changes in both the independent and dependent variables (cf. Edelen [4]). We consider a set of equations of the type s’(n) =ff(t”,
UP, $“; A),
u”(n) =fU”,(t”, .B; n) +f;&m, $‘(A) =f#“,
@b; A),
4)“; A)
(2)
which represents a l-parameter family of transformations. are of class C* and the matrices of the elements
au“(t”
xg(t”)
=
dt’
Y/(P)
=
dt’
It is assumed that the functions (2)
)
and
a#v7
have rank 3. The variables t”’ can be regarded as the independent Eulerian (xi) or Lagrangian (Xi) ones. REMARK.
coordinate
variables either
Clearly, it is very tempting to extend (2) to the general case, i.e. P(A) =ff(P,
UP, +*; 3L),
ti”(J”) =fu”(P, .@, @b; A), d’(A) =f$(t”,
UP, $P; A);
this case one can easily infer from consideration in [3]. A few words are necessary to connect our formulation with what one knows from literature. The expressions (2)* and (2)3 can be motivated as follows. Let us start from Stokes decomposition of an arbitrary smooth vector field, say, w (Bowen and Wang [l], 328). It means that there exist the Stokes potentials (not unique) h and $J such that following mapping can be represented locally by w-grad/r
+curl#
the the p. the
A variational approach to the Cosserat-like continuum
1477
Or
w~u+curl@;
(3)
here u = grad h, h is the scalar potential and @ is the vector potential. Because the Stokes potentials h and Cpare not unique this mapping is not injective. We can now look at the situation from the standpoint of the Cosserat-like continuum. It suggests to change the expression on the right of (3) by W*(u
+curl@,
#)
(4)
or, more general, w++(f(u,
4J), 0).
(5)
According to (4) the function fEz in the transformation
(2)2 can be written as follows
fEz = (curl rv(tm, @“; A))4
(6)
If we denote
and X = (P, a@, f#P) = (P, f.P + VP, $“) then the coordinate
transformation
(2) takes the form n=T(x;A)
(7)
or, if T(x; J.) satisfies the condition T(x; 0) =x, R =x + nv(x) + a(L),
(8)
where u(x) =
(“‘g *‘),=(; f(O) =x,
u(*) = (d(*), C,(*) + q9,
u&(9).
The variations of X, denoted by 6x, are a simple consequence
of a definition
& = hrn f(A) -x(O) &PO )c * As a result we have 6x = (&i, 6u” + svq S$Y) = (tJf(*), V;,(.) + V,s(*), V$(*)) = U, where, in the last step, we have made use of the above notation. Based on (9) we can obtain 6X: of X7 and SYY of Yiy by a differentiation the variables (t’, u a, @“, X7, Y,? are taken as independent of each other. As a consequence we obtain a&4
sxy = ai&4 ff + -XT+
a
a&-+
where a,(*) = a(.)/&’
a&b”+
process in which
+asr’xI+$ Y$
y-$Yf-x:(a,st*
dUY
sv;=
(9)
(10)
dUY
as@=
(
-Yyi'--Y," aistk+-
w
and vu= (curl ~(t*, #‘))S
astk a@
y,"+-
astk XV
ad
1>’
(11)
.I. SACZUK
1478 3.
LAGRANGIAN
Suppose that we are given a function L = L(t’, ua, C#J y, X7, Y/‘) of class C2 in its 3 + 3 + 3 + 3 x 3 + 3 x 3 arguments, which is defined as a function of the t’ over each subspace C3 of the type U* = u@(t’),
@’ = #‘(t’).
We can form the integral
where G denotes a fixed simply-connected domain in the 3-dimensional space of t’, bounded by a surface dG. In order to derive the fundamental variational formulation we must compare the value I(.) of the action integral at the state (u, $) with the value I(.) of the one at a neighbouring state. Since the variations (9), (10) and (11) entail contribution both to the variation of the integrand and to the variation of the boundary of the domain of integration, one can take it into account as follows. Let us denote by Z(0) the value of the integral (12) over G with respect to C,(O) and bY
Z(A)= I,,,L(t’, ua(fi,
A), @‘(t’, A), Xp(r’, A), Yj’(t’, A)) dV
I
its value at a neighbouring
(13)
state. Here C,(O) is the element of a family of subspaces C,(A) of
the type c4a = Uor(fi, A),
@’ = f$Y(t’, A).
Thus, the first variation of (1) defined as 6Z= Z(A) - Z(0) = Z(ti + A&‘, u”+ MU”, q5’+ A&$Y) - I(& ua, qJ’)
(14)
reduces to 6Z= il
JG((fm +DiTa)8~~+(8, + DiMa)8#n+ T~(a;6~.~+$$Yf) +y) dV
+A
I(
((Lb', + X,“T, + Y,6Mg)6tk - T’,6u” - MiSGb)ni
ac
+ ((LX;
+ LY,y + X;Xj’Ti,
+ Y;XYM;
0th) - XrT’,Gua- YiyMf& “)n, + T
+ X;Y;T, dS
>
+ Y;Y;Mi,)dt”
,
(15)
8+y = S#Y - YySti,
(16)
where &AOz= 6u Ly- X$&‘, dT’
dM’ DiM’, = a;M; + Y” 2. a@ ’
(17)
(18) T;=-&, I
(19)
A variational
approach
to the Cosserat-like
continuum
1479
denote the components of the generalized body forces, the generalized body moments, the generalized force-stresses and generalized couple-stresses, respectively. To obtain (15) we have used standard operations, like differentiation, the application of the divergence theorem and variation definitions (9), (10) and (ll), which lead, in effect, to the Euler-Lagrange equations of the assumed functional. Nevertheless, one moment is worth additional explanation, namely, the one connected with the term in (15)
where vn= (curl o(P, G6))4 If we assume a generally accepted in practice case v”=(tXf$)W where the symbol X denotes usual vector multiplication,
then (20) takes the form
laij T”&#In as it is required. In the above Ed’ is + 1 or -1 according as cu,j, k is an even or odd permutation of 1,2,3 and is zero otherwise. Then, the first integral in (15) can be written as
I
G ((f~ + DiT’,)~u”+
(ga + DiM’,+
E,ijTii)$~n)
dV.
(21)
When terms given in the parentheses are equated to be zero they define the static forms of the balance of momentum and the balance of moment of momentum for the micropolar continuum (Eringen and Kadafar [5], p. 15). Let us now return to the general case, i.e. vn= (curl ~(t”‘, $“))Y From the fact that d.6V I
a = Ejka
diajdwk
it turns out that @vn are components of a spin matrix, i.e. a matrix of the type (dV)y, where d = grad(t X V), V = (a,(*), a,(.), a,(.)), t = (t’, t*, t3). The decomposition of the spin matrix with components a,&, into its symmetric and skew-symmetric parts respectively enables us to obtain the terms E,ijTiiG~ ~
where
here it is used that any skew-symmetric matrix can be put into the one-to-one with its axial vector (with its real eigenvalue), and
correspondence
T’,D~(S@)
where 6D *, = Ejk’cadi,djswk
define microdeformations of material points within the micromorphic expression (21) can be supplemented as follows
I
c ((fa + DiTi,>6Ua + (8, + DiM’, + E,ijTy)A$n
continuum
+ T’,hDy(@))
dV.
[5]. Thus,
(22)
In the light of the form of the balance of momenta for micromorphic continua ([5], p. 43) to seek some convenient representation for disu” would seem a worthy task.
1480
J. SACZUK
When there are no variations of the independent (16) and (21), (15) is reduced to
-A
f(
variables, i.e. 6t’ = 0, then, according to
(T’,du a+ M@#J “)ni + (X;T’,Gu cx+ Y,YM@#+z,
+ y)
dS.
(23)
(2), is taken to be independent
of the un
ac
If, moreover, and @’ then
the function ff in the transformation
6I= A
(fn + DiT$W
+ (gm + D,M’, + ~,;,T~)h#f
(T’,du”+ M&%#?)ni + $))
-A
+ $))
dV
dS
is the classical principle of virtual work for the micropolar continuum. It is seen that by this assumption the variation of our integral is immensely simplified. The fundamental lemma of the calculus of variations implies the set of equations fe+ DiTn=O,
(25)
g, +DiM',+ E,~~T~=O
(26)
known as the balance equation for stresses and for couple-stresses, respectively (Eringen and Kadafar [5], p. 16). The variational principle 61= 0 under conditions (25) and (26) leads to the natural boundary conditions. Details, as standard, are omitted here (cf. Edelen [4]).
4.
TRANSVERSALITY
CONDITIONS
In geometrical language the transversality conditions express the fact that any displacement of a family of geodesically equidistant (M, suq S@Y) is tangential to a hypersurface hypersurfaces with respect to the Lagrangian L(e) (cf. Rund [6], p. 25). To obtain this relation we use (15). In a subset of all variations
{(a;, &.da, S$l'): 6t\a,=O, dUP,,=O,Sf$ra,=O} a necessary condition for the stationary of the (15), under satisfaction of (21), is that 1 ((fa + DiT’,)&“+
(ga + DiM,+
~,,T”)h#f)
dV =0
(27)
G
for all ($ug $4”) that vanish on aG. The lemma of the calculus of variations then gives (25) and (26) at all interior points of G. In turn, under satisfaction of (25) and (26), expression (15) together with (21) is reduced to
I JG
X;Ti,+ YfM',)b?- T',W-M;6@')n;
((@A;+ +
((XJ’+ Yy)(L&+
X;T,+ Y;M;)6tk-X:T,cka- YiyM&%#~~)n~) dS=O.
(28)
If we denote by Hi = Lb; + X;Ti,
+ YZM;
(29)
A
variational
approach
to the Cosserat-like
continuum
the moment-energy complex (cf. Edelen [4]) for the Cosserat-like (28) takes the form
I
ac
(HiM(ni
continuum,
1481
then the equality
+ (XT + Yj’)n,) - T’,6un(ni + Xiyn,) - M$GQ,“(ni + YTn,,)) dS = 0.
(30)
It is said that a set of variations (sti, Sum, SQy) satisfies the frunsversafity conditions on dG if and only if Hi6tk(nj
+ (XT + Yy)n,) = T’,6U”(nj + Xiyn,) + M’,S$,G(nj + Y,‘n,)
(31)
for any point of dG. The above conditions establish the relation between the geometrical variation 6~’ and the variations in the fields 8~” and I%$“. Their importance is evident to problems in which internal boundaries are present, as in the case of cracks within the material. The equality (31) written for a boundary of any region G as follows
I ac
Hiht’(ni
+ (XT + YY)n,) dS = j
ac;
(7”,6U a(ni + Xiyn,) + MbSQ, “(% + Y,‘nr)) dS
can be also regarded, following Edelen [4], as a definition of boundary virtual work on variable boundaries of the system in its motion.
5. INVARIANCE
CONDITIONS
For the purposes of the present discussion we shall assume the case when (2) is expressible in the form
~=(a) =fZ,V,
us; ;ls) +f:*(tm, @; J”,),
@W
9? A,),
=f$(f”,
(32)
where the function ftz is defined by (6). It means that we shall consider an r-parameter Lie group with parameters A,, (s = 1, . . . , r). The identity transformation of the group is obtained under assumption that the values & = 0. The infinitesimal transformations corresponding to (32) are then given by (cf. (9)) 6t’ = v&,&,, ad”= au” + &J” = (tJ,a,(@+ v,q,,)&, WY= “W,
(33)
under summation over s and where (see Section 2)
According to (16)
(34) in which
In order I to be invariant under the infinitesimal transformation (33) up to an independent integral we will try to put the action integral itself (1) into better perspective. For our purposes
J. SACZUK
1482
it is enough by an independent integral to understand the integral defined integrands consisting of divergences. We assume that L is such that under (33) the integral (1) transforms as
I
L(S’, ii”, @‘, Xp, Yy) dV = G(A)
by means of
(L(t', ua,q5”, Xp, Yy)+ @) dV,
(35)
where @ = @(,)(t’, uQ, $“, X,?. YI)A, (cf. Rund [6], Theorem on pp. 257-258) is the integrand of the independent integral. The stationary conditions of (35) (cf. (15) together with (21)), with regard to (34) can be expressed as follows I
c ((fa + DiT’,)UG,, + (g, + D;Mc + EyijT')@&,)h, dV -
&:,s, - T,v,“,(,, - M;I&,) + Gfs,AsdV = 0. I(G iaar(H )
(36)
By definition of Qcs, and a divergence form of the integrand in the second integral it follows that the integrand of the second integral is an independent integral. Then the invariance condition deduced from (36) is reduced to 6I=
IG
@&, dV
(37)
and together with (36) we have therefore
If Qts, = 0 and the integral is invariant under an action of (33), then (37), for instance, implies (23), if there are no variations of the independent variables. From the fact the As are independent parameters of an r-parameter Lie group we infer (cf. Rund [6]): If the Lagrangian L satisfies (35) under an action of an infinitesimal r-parameter Lie group (33) there exist 2r distinct linear combinations of equations
fa+ DiT',=O, g,+DiM:+ E,UT~=O each pair of which is the integrand of an independent integral. The above conclusion is of central importance in a study of conservation micropolar continua [5].
6.
(1) In
laws within the
CONCLUSIONS
the case of absence of the field 4’ the presented exposition coincides with that of Edelen [4]. (2) The variational form of the virtual work for the Cosserat-like continuum (24) stands as one of many possible (cf. (15)). (3) The transversality conditions (31) are obtained as a result of assumed conditions on variations both of the independent coordinate variables and the independent fields of the Stokes’ type. The invariance conditions (38) are defined in the frame of the independent integral to (4) be invariant under an action of an r-parameter Lie group. (5) The presented exposition explains what continuity (regularity) conditions of the Lagrangian function should be assumed as primary.
A variational approach to the Cosserat-like continuum
1483
REFERENCES [I] [2] [3] [4] [5]
R. M. BOWEN and C.-C. WANG, introducfion to Vectors and Tensors, Vol. 2. Plenum Press, New York (lY76). E. COSSERAT and F. COSSERAT, 7Morie des Corps Deformable. Hermann, Paris (1909). D. G. B. EDELEN, Nonlocal Variations and Local Invariance of Fields. American Elsevier, New York (1969). D. G. B. EDELEN, ht. /. Solids Struct., 17, 729-740 (1981). A. C. ERINGEN and C. B. KADAFAR, In Continuum Physics, Vol. IV, pp. l-73 (Edited by A. C. ERINGEN). Academic Press, New York (1976). 161 H. RUND, The H~~~to~-~ocobi Theory in the C&xdus of Variations. Van Nostrand, London (1966). (Received 27 October 1992; accepted 18 December 1992)
Vol. 31. No. Printed in Great Britain
11, pp. 1475-1483. 1YY3
A VARIATIONAL
COZO-7225/Y3$6.00 + 0.00 Pergamon Press Ltd
APPROACH TO THE COSSERAT-LIKE CONTINUUM JAN SACZUK
Institute of Fluid-Flow Machinery, Polish Academy of Sciences, 80-952 Gdalisk, ul. J. Fiszera 14, Poland (Communicated
by J. T.
ODEN)
Abstract-The exposition given here is intended to show an equivalent variational approach to formulation of the virtual work principle for the Cosserat-like continuum. Stationarity conditions of an action integral lead to the Euler-Lagrange equations identified with the balance equations for stresses and couple-stresses within micropolar and micromorphic continua. Vector fields as independent variables are taken so as to satisfy the known Stokes’ decomposition. Based on the standard variational arguments, for a given Lagrangian function and an assumed l-parameter family of transformations of both the independent and dependent variables, the fundamental variational formula identified with the virtual work principle of the Cosserat-like continuum is obtained. To determine the immediate relations between the geometric variation of the boundary and the variation of the field variables the transversality conditions are used. A notion of an independent integral is used to define invariance conditions of the integral in question which is invariant under an action of an r-parameter
Lie group.
1. INTRODUCTION
The importance of variational statements of various problems of continuum mechanics has been confirmed many times. It is also the most general starting point for the discussion of any mechanical system. The reasons why the variational technique is so useful are the following [3,61: (1) It provides tools for studying the existence of solutions of the problem under consideration. (2) It encompasses in a concise statement the collection of boundary, initial and jump conditions of the problem in question. (3) It naturally includes into the system the effects of constraints. (4) It leads to information on bounds to the solution of integro-differential equations. (5) It leads to a very efficient method of constructing approximate solutions. The “trivial” embedding of a given system of the Euler-Lagrange equations (say, the balance equations for stresses and couple-stresses) into a variational statement via the Lagrange multipliers is handy but not, in general, satisfactory. An unsuccessful example of the Lagrange multiplier method takes place already in the linear elasticity, where one cannot derive an unconditional principle from the dual energy principle known as the Hellinger-Reissner one. The reasons for wanting to embed a given system of the Euler-Lagrange equations into a variational statement and their limitations are explained in [3]. Hence it appeared to me to be of importance to investigate circumstances under which the principle of virtual work for the Cosserat-like continuum [2], say, micropolar and micromorphic continua [5], can be obtained from variational arguments, i.e. from a consideration of a given Lagrangian and an assumed l-parameter family of transformations of both its independent and dependent variables. The purpose of this article is to show an equivalent approach to the one used very often in practice, i.e. to the Lagrange multiplier technique. 2.
PRELIMINARY
RESULTS
AND
VARIATIONS
We consider an interacting mechanical continuum in static equilibrium. It is assumed that the (elastic) continuum % is defined by a (connected) geometric configuration space with the 1475
J. SACZUK
1476
structure of a differential manifold G. In turn, by a continuum one can understand a mechanical system which interacts with an external mechanism (the one which forces the system to assume a configuration compatible with constraints) with body forces f and body moments g. It is also assumed that the (virtual) work induced by mechanical reaction is measured. At this stage we assume no constraints to the behaviour of the static continuum. The geometric state of a deformed (elastic) continuum will be described, from assumption, by N + M independent fields (P(ti),
@YQ’))= (u’(t’), . . . ) Lyt’),
@‘(f’), . . . ) rpyt’))
where (t’) = (t’, . . . , t”) are the independent coordinate variables. In the following we will confine ourselves to N = M = n = 3, but results can be extended to more degrees of freedom. The independent fields u a and Gy within the Cosserat continuum [2,5] can be identified with the translation vector field and the rotation vector field, respectively. We assume that the behaviour of our static continuum may be described by the action integral over G I@“, u a> $J‘) = I, L( P, ua(fm), $‘(t”),
Xp(t”),
Y;(P))
dV
(I)
where L is a Lagrangian density function. The method of considering the stationarity, in general, conditions of I is to apply standard techniques of the calculus of variations. One obvious way of extending this theory is to consider the variation of an integral as a consequence of changes in both the independent and dependent variables (cf. Edelen [4]). We consider a set of equations of the type s’(n) =ff(t”,
UP, $“; A),
u”(n) =fU”,(t”, .B; n) +f;&m, $‘(A) =f#“,
@b; A),
4)“; A)
(2)
which represents a l-parameter family of transformations. are of class C* and the matrices of the elements
au“(t”
xg(t”)
=
dt’
Y/(P)
=
dt’
It is assumed that the functions (2)
)
and
a#v7
have rank 3. The variables t”’ can be regarded as the independent Eulerian (xi) or Lagrangian (Xi) ones. REMARK.
coordinate
variables either
Clearly, it is very tempting to extend (2) to the general case, i.e. P(A) =ff(P,
UP, +*; 3L),
ti”(J”) =fu”(P, .@, @b; A), d’(A) =f$(t”,
UP, $P; A);
this case one can easily infer from consideration in [3]. A few words are necessary to connect our formulation with what one knows from literature. The expressions (2)* and (2)3 can be motivated as follows. Let us start from Stokes decomposition of an arbitrary smooth vector field, say, w (Bowen and Wang [l], 328). It means that there exist the Stokes potentials (not unique) h and $J such that following mapping can be represented locally by w-grad/r
+curl#
the the p. the
A variational approach to the Cosserat-like continuum
1477
Or
w~u+curl@;
(3)
here u = grad h, h is the scalar potential and @ is the vector potential. Because the Stokes potentials h and Cpare not unique this mapping is not injective. We can now look at the situation from the standpoint of the Cosserat-like continuum. It suggests to change the expression on the right of (3) by W*(u
+curl@,
#)
(4)
or, more general, w++(f(u,
4J), 0).
(5)
According to (4) the function fEz in the transformation
(2)2 can be written as follows
fEz = (curl rv(tm, @“; A))4
(6)
If we denote
and X = (P, a@, f#P) = (P, f.P + VP, $“) then the coordinate
transformation
(2) takes the form n=T(x;A)
(7)
or, if T(x; J.) satisfies the condition T(x; 0) =x, R =x + nv(x) + a(L),
(8)
where u(x) =
(“‘g *‘),=(; f(O) =x,
u(*) = (d(*), C,(*) + q9,
u&(9).
The variations of X, denoted by 6x, are a simple consequence
of a definition
& = hrn f(A) -x(O) &PO )c * As a result we have 6x = (&i, 6u” + svq S$Y) = (tJf(*), V;,(.) + V,s(*), V$(*)) = U, where, in the last step, we have made use of the above notation. Based on (9) we can obtain 6X: of X7 and SYY of Yiy by a differentiation the variables (t’, u a, @“, X7, Y,? are taken as independent of each other. As a consequence we obtain a&4
sxy = ai&4 ff + -XT+
a
a&-+
where a,(*) = a(.)/&’
a&b”+
process in which
+asr’xI+$ Y$
y-$Yf-x:(a,st*
dUY
sv;=
(9)
(10)
dUY
as@=
(
-Yyi'--Y," aistk+-
w
and vu= (curl ~(t*, #‘))S
astk a@
y,"+-
astk XV
ad
1>’
(11)
.I. SACZUK
1478 3.
LAGRANGIAN
Suppose that we are given a function L = L(t’, ua, C#J y, X7, Y/‘) of class C2 in its 3 + 3 + 3 + 3 x 3 + 3 x 3 arguments, which is defined as a function of the t’ over each subspace C3 of the type U* = u@(t’),
@’ = #‘(t’).
We can form the integral
where G denotes a fixed simply-connected domain in the 3-dimensional space of t’, bounded by a surface dG. In order to derive the fundamental variational formulation we must compare the value I(.) of the action integral at the state (u, $) with the value I(.) of the one at a neighbouring state. Since the variations (9), (10) and (11) entail contribution both to the variation of the integrand and to the variation of the boundary of the domain of integration, one can take it into account as follows. Let us denote by Z(0) the value of the integral (12) over G with respect to C,(O) and bY
Z(A)= I,,,L(t’, ua(fi,
A), @‘(t’, A), Xp(r’, A), Yj’(t’, A)) dV
I
its value at a neighbouring
(13)
state. Here C,(O) is the element of a family of subspaces C,(A) of
the type c4a = Uor(fi, A),
@’ = f$Y(t’, A).
Thus, the first variation of (1) defined as 6Z= Z(A) - Z(0) = Z(ti + A&‘, u”+ MU”, q5’+ A&$Y) - I(& ua, qJ’)
(14)
reduces to 6Z= il
JG((fm +DiTa)8~~+(8, + DiMa)8#n+ T~(a;6~.~+$$Yf) +y) dV
+A
I(
((Lb', + X,“T, + Y,6Mg)6tk - T’,6u” - MiSGb)ni
ac
+ ((LX;
+ LY,y + X;Xj’Ti,
+ Y;XYM;
0th) - XrT’,Gua- YiyMf& “)n, + T
+ X;Y;T, dS
>
+ Y;Y;Mi,)dt”
,
(15)
8+y = S#Y - YySti,
(16)
where &AOz= 6u Ly- X$&‘, dT’
dM’ DiM’, = a;M; + Y” 2. a@ ’
(17)
(18) T;=-&, I
(19)
A variational
approach
to the Cosserat-like
continuum
1479
denote the components of the generalized body forces, the generalized body moments, the generalized force-stresses and generalized couple-stresses, respectively. To obtain (15) we have used standard operations, like differentiation, the application of the divergence theorem and variation definitions (9), (10) and (ll), which lead, in effect, to the Euler-Lagrange equations of the assumed functional. Nevertheless, one moment is worth additional explanation, namely, the one connected with the term in (15)
where vn= (curl o(P, G6))4 If we assume a generally accepted in practice case v”=(tXf$)W where the symbol X denotes usual vector multiplication,
then (20) takes the form
laij T”&#In as it is required. In the above Ed’ is + 1 or -1 according as cu,j, k is an even or odd permutation of 1,2,3 and is zero otherwise. Then, the first integral in (15) can be written as
I
G ((f~ + DiT’,)~u”+
(ga + DiM’,+
E,ijTii)$~n)
dV.
(21)
When terms given in the parentheses are equated to be zero they define the static forms of the balance of momentum and the balance of moment of momentum for the micropolar continuum (Eringen and Kadafar [5], p. 15). Let us now return to the general case, i.e. vn= (curl ~(t”‘, $“))Y From the fact that d.6V I
a = Ejka
diajdwk
it turns out that @vn are components of a spin matrix, i.e. a matrix of the type (dV)y, where d = grad(t X V), V = (a,(*), a,(.), a,(.)), t = (t’, t*, t3). The decomposition of the spin matrix with components a,&, into its symmetric and skew-symmetric parts respectively enables us to obtain the terms E,ijTiiG~ ~
where
here it is used that any skew-symmetric matrix can be put into the one-to-one with its axial vector (with its real eigenvalue), and
correspondence
T’,D~(S@)
where 6D *, = Ejk’cadi,djswk
define microdeformations of material points within the micromorphic expression (21) can be supplemented as follows
I
c ((fa + DiTi,>6Ua + (8, + DiM’, + E,ijTy)A$n
continuum
+ T’,hDy(@))
dV.
[5]. Thus,
(22)
In the light of the form of the balance of momenta for micromorphic continua ([5], p. 43) to seek some convenient representation for disu” would seem a worthy task.
1480
J. SACZUK
When there are no variations of the independent (16) and (21), (15) is reduced to
-A
f(
variables, i.e. 6t’ = 0, then, according to
(T’,du a+ M@#J “)ni + (X;T’,Gu cx+ Y,YM@#+z,
+ y)
dS.
(23)
(2), is taken to be independent
of the un
ac
If, moreover, and @’ then
the function ff in the transformation
6I= A
(fn + DiT$W
+ (gm + D,M’, + ~,;,T~)h#f
(T’,du”+ M&%#?)ni + $))
-A
+ $))
dV
dS
is the classical principle of virtual work for the micropolar continuum. It is seen that by this assumption the variation of our integral is immensely simplified. The fundamental lemma of the calculus of variations implies the set of equations fe+ DiTn=O,
(25)
g, +DiM',+ E,~~T~=O
(26)
known as the balance equation for stresses and for couple-stresses, respectively (Eringen and Kadafar [5], p. 16). The variational principle 61= 0 under conditions (25) and (26) leads to the natural boundary conditions. Details, as standard, are omitted here (cf. Edelen [4]).
4.
TRANSVERSALITY
CONDITIONS
In geometrical language the transversality conditions express the fact that any displacement of a family of geodesically equidistant (M, suq S@Y) is tangential to a hypersurface hypersurfaces with respect to the Lagrangian L(e) (cf. Rund [6], p. 25). To obtain this relation we use (15). In a subset of all variations
{(a;, &.da, S$l'): 6t\a,=O, dUP,,=O,Sf$ra,=O} a necessary condition for the stationary of the (15), under satisfaction of (21), is that 1 ((fa + DiT’,)&“+
(ga + DiM,+
~,,T”)h#f)
dV =0
(27)
G
for all ($ug $4”) that vanish on aG. The lemma of the calculus of variations then gives (25) and (26) at all interior points of G. In turn, under satisfaction of (25) and (26), expression (15) together with (21) is reduced to
I JG
X;Ti,+ YfM',)b?- T',W-M;6@')n;
((@A;+ +
((XJ’+ Yy)(L&+
X;T,+ Y;M;)6tk-X:T,cka- YiyM&%#~~)n~) dS=O.
(28)
If we denote by Hi = Lb; + X;Ti,
+ YZM;
(29)
A
variational
approach
to the Cosserat-like
continuum
the moment-energy complex (cf. Edelen [4]) for the Cosserat-like (28) takes the form
I
ac
(HiM(ni
continuum,
1481
then the equality
+ (XT + Yj’)n,) - T’,6un(ni + Xiyn,) - M$GQ,“(ni + YTn,,)) dS = 0.
(30)
It is said that a set of variations (sti, Sum, SQy) satisfies the frunsversafity conditions on dG if and only if Hi6tk(nj
+ (XT + Yy)n,) = T’,6U”(nj + Xiyn,) + M’,S$,G(nj + Y,‘n,)
(31)
for any point of dG. The above conditions establish the relation between the geometrical variation 6~’ and the variations in the fields 8~” and I%$“. Their importance is evident to problems in which internal boundaries are present, as in the case of cracks within the material. The equality (31) written for a boundary of any region G as follows
I ac
Hiht’(ni
+ (XT + YY)n,) dS = j
ac;
(7”,6U a(ni + Xiyn,) + MbSQ, “(% + Y,‘nr)) dS
can be also regarded, following Edelen [4], as a definition of boundary virtual work on variable boundaries of the system in its motion.
5. INVARIANCE
CONDITIONS
For the purposes of the present discussion we shall assume the case when (2) is expressible in the form
~=(a) =fZ,V,
us; ;ls) +f:*(tm, @; J”,),
@W
9? A,),
=f$(f”,
(32)
where the function ftz is defined by (6). It means that we shall consider an r-parameter Lie group with parameters A,, (s = 1, . . . , r). The identity transformation of the group is obtained under assumption that the values & = 0. The infinitesimal transformations corresponding to (32) are then given by (cf. (9)) 6t’ = v&,&,, ad”= au” + &J” = (tJ,a,(@+ v,q,,)&, WY= “W,
(33)
under summation over s and where (see Section 2)
According to (16)
(34) in which
In order I to be invariant under the infinitesimal transformation (33) up to an independent integral we will try to put the action integral itself (1) into better perspective. For our purposes
J. SACZUK
1482
it is enough by an independent integral to understand the integral defined integrands consisting of divergences. We assume that L is such that under (33) the integral (1) transforms as
I
L(S’, ii”, @‘, Xp, Yy) dV = G(A)
by means of
(L(t', ua,q5”, Xp, Yy)+ @) dV,
(35)
where @ = @(,)(t’, uQ, $“, X,?. YI)A, (cf. Rund [6], Theorem on pp. 257-258) is the integrand of the independent integral. The stationary conditions of (35) (cf. (15) together with (21)), with regard to (34) can be expressed as follows I
c ((fa + DiT’,)UG,, + (g, + D;Mc + EyijT')@&,)h, dV -
&:,s, - T,v,“,(,, - M;I&,) + Gfs,AsdV = 0. I(G iaar(H )
(36)
By definition of Qcs, and a divergence form of the integrand in the second integral it follows that the integrand of the second integral is an independent integral. Then the invariance condition deduced from (36) is reduced to 6I=
IG
@&, dV
(37)
and together with (36) we have therefore
If Qts, = 0 and the integral is invariant under an action of (33), then (37), for instance, implies (23), if there are no variations of the independent variables. From the fact the As are independent parameters of an r-parameter Lie group we infer (cf. Rund [6]): If the Lagrangian L satisfies (35) under an action of an infinitesimal r-parameter Lie group (33) there exist 2r distinct linear combinations of equations
fa+ DiT',=O, g,+DiM:+ E,UT~=O each pair of which is the integrand of an independent integral. The above conclusion is of central importance in a study of conservation micropolar continua [5].
6.
(1) In
laws within the
CONCLUSIONS
the case of absence of the field 4’ the presented exposition coincides with that of Edelen [4]. (2) The variational form of the virtual work for the Cosserat-like continuum (24) stands as one of many possible (cf. (15)). (3) The transversality conditions (31) are obtained as a result of assumed conditions on variations both of the independent coordinate variables and the independent fields of the Stokes’ type. The invariance conditions (38) are defined in the frame of the independent integral to (4) be invariant under an action of an r-parameter Lie group. (5) The presented exposition explains what continuity (regularity) conditions of the Lagrangian function should be assumed as primary.
A variational approach to the Cosserat-like continuum
1483
REFERENCES [I] [2] [3] [4] [5]
R. M. BOWEN and C.-C. WANG, introducfion to Vectors and Tensors, Vol. 2. Plenum Press, New York (lY76). E. COSSERAT and F. COSSERAT, 7Morie des Corps Deformable. Hermann, Paris (1909). D. G. B. EDELEN, Nonlocal Variations and Local Invariance of Fields. American Elsevier, New York (1969). D. G. B. EDELEN, ht. /. Solids Struct., 17, 729-740 (1981). A. C. ERINGEN and C. B. KADAFAR, In Continuum Physics, Vol. IV, pp. l-73 (Edited by A. C. ERINGEN). Academic Press, New York (1976). 161 H. RUND, The H~~~to~-~ocobi Theory in the C&xdus of Variations. Van Nostrand, London (1966). (Received 27 October 1992; accepted 18 December 1992)