Germain VirtualPower

8/11/2019 Germain VirtualPower

1/21

The Method of Virtual Power in Continuum Mechanics. Part 2: Microstructure

P. Germain

SIAM Journal on Applied Mathematics, Vol. 25, No. 3. (Nov., 1973), pp. 556-575.

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2/21

THE METHOD OF VIRTUAL POWER IN CONTINUUM MECHANICS.

PART

2 :

MICROSTRUCTURE

Abstract. Th e advantages of a systematic application of the m ethod of virtual power are exemplified

by the study of continuous media with microstructure. Results on micromorphic media of order one

are easily found and the equations of motion for the general micromorphic medium are established

for the first time. Various interesting special cases, which have been previously considered in the

li terature, may be derived upon imposing some convenient constraints. The paper ends with some

comparisons with other related work.

1. Introduction.

In a recent note ( Ge rma in (1972)). it was emphasized how

the classical method of virtual work may be used in a more systematic way than

it has been in the past in order to derive the fundamental equations of a given

theory in con tinuu m m echanics. As an exam ple, first and second gradient theories

have been considered with some details in a m ore recent paper (G erm ain (1973a)).

It is not claimed of course that the idea is new. Nevertheless, it may give new

insight into som e questions which have been treated so far by more con ventional

tools. Mo re im portant , i t provides the shortest way to obtain the required equa-

tions and t o avoid any am biguity or er ror . Recent papers from Casal (1972) and

Breune val (1973) give new illu stration s of the relev ance of this prop osition .

The present co ntribu tion is related to co ntinu ous media inside which micro-

structure may be taken into ac cou nt. This topic has been considered by ma ny

auth ors, for instance, Mindlin (1964) an d Eringen and his coworkers in m any

pap ers published since 1964--some of them m ay be found in the References.

These media are called "microm orphic" in Eringen's termino logy. Th e present

paper is quite limited in scope. The main steps of the application of the method

are briefly reviewed in 4 2. Then classical equations for equilibrium in the most

simple case are derived in

3.

The dynamical equations require special attention

( 5 4) .

Generalizations when a more refined description of the kinematics and of

the stresses is desired follow in 5 5. Special cases and comparison with other

theories are finally considered in 5 6 and 7. We do not intend in this paper to

introduce constitutive relations which of course must be taken into account in

order to be able to apply a micromorphic theory to specific physical problems.

Numerous papers in the literature deal with these applications. Our aim here is

just to give a new illustration of the simplicity and usefulness of the method of

virtual power.

Th e aut ho r is specially pleased to be able to publish this work in the volume

dedicated to Professor Prage r a nd t o join his friends and colleagues in this tribute

of adm iration a nd g ratitud e to this great scientist who, in a q uite different context,

has shown on many occasions how d'Alembert 's principle of virtual work is a

very precious tool in various problems of continuum mechanics.

Received by the editors Ma rch 12. 1973.

tuniversity of Paris (VI). Place Jussieu, PARIS,

5 .


3/21

METHOD OF V I R T U A L POWE R

Th e following n otation is used :

i v e l o c i t y (c om p on en ts ).

Di s t r a i n r ate t ensor.

g r a d i e n t of the relat ive veloc it ies-symmetr ic pa r t : micro-

X i j

strain rate tensor, antisymmetric part : microrotation rate

tensor.

V i j

r e l a t i v e mic rove loci ty g rad ien t.

x i jk

7

s e c o n d gradient of the relat ive ve locit ies.

. I J ~

0 = 0

-intrinsic stress tenso r.

1 J J 1

S. .

i n t r i n s i c m ic ro st re ss t en s or .

v . .

L ~ -intrinsic second microstress tensor.

zij aij sij-stress tensor.

f

-volumic force.

y j

-volumic dou ble force.

s u r f a c e trac tio n.

Mi -surface dou ble traction .

i a c c e l e r a t i o n .

' .

1 1 -microacceleration.

2

Description of the method Without going into too many details, i t is

nevertheless worthwhile to review the main ideas involved in the application of

the me thod . Forces an d stresses are not introduce d directly but by the value of

the virtual power they prod uce for a given class of virtual motion s. The n, roughly

speaking, if one extends the class of virtual motions which are considered, one

refines the description of forces and stresses.

M ore precisely, for a given system

S

and at a given time, a virtual motion is

defined by a vector field V on

S : V

is the field of virtual velocities.

A

class of virtual

motions is a vector space

Y

whose elements are

V.

Th e "system of forces" one

wants to consider is defined by a linear continuous application

Y

R

o r

Y real number, is the virtual power produced by "this system of forces" in the

virtual motion V. Obviously, Y must be a topological vector space in order to

be able to define the continuity of 9 ( V ) .

In fact, virtual velocities like real velocities are defined with respect to a

given fram e. Th e velocity fields of th e

same

virtual mo tion in two different frames

differ only by a rigid body velocity field

C

Rigid body velocity fields, sometimes

called distributor define a vector space g It will always be assumed in what

follows tha t

g

is a subspace of Y

Th e various "forces" which act on the mechanical system are divided in a

very classical way into two classes :

externa l forces

which represent the dynamical

effects on

S

due to the interaction with other systems which have no common

part with

S

a n d internal forces which represent the mutual dynamical effects

of subsystems of

S ;

or in stan ce, if

S

and

S

are tw o disjoint subsystems of

S

the

action of S on S represents "external forces" acting on S J but "internal forces"

i on e considers the system

S

itself. Let us stretch the m eanin g of the word "forces"


4/21

558 P.

G E R M A I N

for the time being an d let it be taken in the vague sense it may have in the com m on

language and not in any precise mathematical conception it may receive (and

will eventually receive below).

It is crucia l to notice tha t the definition of the virtual power of intern al

forces is subject to a limita tion which may be expressed as the following axiom .

A X I O MO F POWEROF I N T E R N A L F O R C E S

Th e virtual polr,er of the inter nal

forces acting on a syste m S for a given cirtual mo tion is an objective qu an tit y;

i.e., it has the same value whatecer be th e jkanze in which the m otion is observed.

I t

is obviously equivalent to stat e:

Th e virtual power of the internal forces acting on a syst em

S

is null for an y

dlstributor.

It is quite clear that this axiom is the counterpart in this presentation of

dynam ics of the principle of super pose d rigid body motion or of the principle

of objectivity (material indifference). External forces are no t subjected to a

similar limitation. It is barely necessary to recall d'Alembert's famous statement

:

PRIN IPLE Le t a sys tem be in equilibriunz with respectF

VIRTUALPOWER.

to a given G alilean fi.am e; th en, in any virtual ,notion , the cirtual power of all the

inter nal forces and ex ter na l forces acting on

S

is null.

Looking at the definition of the virtual power, one sees that the natural

mathem atical frame for such a theory would be the frame of generalized functions

or distributions. In fact we will not use it, first for simplicity and second because,

as we are dealing with physical ph eno me na, one may reasonably assume that the

distributions involved are regular enough in orde r to be represented by func-

tions ( or densities ). T o be m ore ex plicit, let us close this section by recalling

briefly the simplest situa tion which leads to th e classical theo ry.

Assume that

S

is a simple connected open do ma in in equilibrium in a given

Galilean frame. The crucial point in order to build a continuum theory is to

decide on the kinematical descr iption which will be used, i.e., to choose and V,

and the linear application

V +

Choose V as a velocity field Ui defined on

S

( U i are the comp one nts of a velocity vector in an ortho norm al C artesian frame)

and state that the definition of

9

will involve th e valu e of U i an d of its first deriva-

tives U i S j .The correspon ding theory is the first gradient theory (G erm ain (1973a)).

Th e internal forces acting on a given subd om ain of S have a virtual power

which may be w ritten1

2 ) i )

S

p d v

with

p

a linear form on Ui and U i S j .But with the axiom , only the symmetric part

of the gradient can be involved and then

(3)

p

= J J

D . .

J = % U . . U . . ), l

g D

I , J

Here a defines a symmetric second order tensor called the

intrinsic stress

tensor.

Follo win g the classical assum ption s, external forces acting on B will

We assume here that p is defined at every point M of S independently of the subdomain 3

This assum ption mean s that internal forces are short range forces and excludes, as is usually

done in continuum mechanics, long range internal forces . This more general case, which may also

be investigated by the present metho d. is not considered in this paper


5/21

559

ETHOD OF

V I R T U A L

POW R

be of two types first long rang e volume forces, prod uce d on by systems located

outside S , giving the virtual power

3

J

( f , u i C i j n i j

OijDij)

d r .

where nijis the antisymm etric part of U i,j.Form ula (4) shows th at f , is the volumic

force, Ci j and O ij the antisym metric and symmetric volumic double force-Cjj

is in fact a volumic couple; usually all these quantities

f , ,

Cij ,

c ij

are considered

as given (some of them may be zero of course). Second, contact forces are con-

sidered, produced on by the parts of

S

located outside , giving the virtual po wer

(it is not necessary to w rite here the con tributio ns of R ij and Dij because it may be

shown that they must be ze ro) ; 7 ; defines the usual traction a cting on the bou nda ry

of Using

2 ,(3), (4) , 5 )

and applying the principle of virtual power, one finds

after some integrations by pa rts that

f , oi j , j C . . O . . ) U i

c

(7,

(o i j

C i j

@ .

.)n

.I

U .

d o

0

I 1 1 3 1

[ I

I

which must be satisfied for any motion U i. It is easy to show that,

i f

one introduces

the stress tensor zij,

one must have

Form ulas (6) an d (7) are the fu ndam ental equations of the first gradient theory,

which reduces to the classical theory if one assum es tha t Cij Q ij

0

3 Equilibrium of a micromorphic continuum of degree 1

The main point in

the application of the method, as reviewed above, is to define the kinematics of

the virtual motions which are being considered. In the classical description, a

con tinuu m is a contin uous distribution of particles, each of them being represented

geonletrically by a point M and characterized kinenlatically by a velocity Ui.

In a theory which takes microstructure into acco unt, from the macroscopic point

of view (which is the point of view of a continuum theory), each particle is still

represented by a point M, but its kinematical properties are defined in a more

refined way. In order to discover how to define the kinematical properties of a

particle, one has to look at it from a microscopic point of view. At this level of

observation, a particle appears itself as a continuum

P

(M ) of small extent. Let

us call M its center of.mass, M a point of

P

(M ), Ui the velocity of M, xi the co-

ordinates of M in a Ca rtesian fram e parallel to the given frame xi with M as the

origin, U: the velocity of M with respect to the given frame, and xi the coordina tes

of M in the given frame . If on e assume s for the mom ent that the xi ar e given,

i.e., if one looks at a given particle, the field U I ap pe ars a s a function of the

x

defined on

P

(M ). As

P

(M ) is of small exten t, it is natu ral t o look at the Tay lor

expansion of U( with respect to the x j an d also, as a first app roxim ation , to s top


6/21

560 P G E R M I N

this expansion with the terms of degree 1. Then, with this assumption, one can

write

(8)

C'i U .

+ ;C..'u

I

1 J . J '

The physical significance of this assu mption is cle ar : one pos tulates that

one can get a sufficient description of the relative motion of the various points

of the particle if one assum es that th is relative motio n is a hom ogen eou s deforma-

tion. With such an assumption, the macroscopic kinematical description of our

continuum will be completely defined if one knows two fields on S, the

velocity

field U , , and the field x of the gradients of the relative velocities of P

M),

also

called the

nticrocelocity

gradient, x obviously being a second order tensor.

The

sym metr ic part of ;c,, will be called the nlicrostrain rate tenso r, the antisym met ric

part-the nlicrorotation rate tensor.

With such a choice of the vector space V of

the virtual motions, we say that we are dealing with a micromorphic continuum

of degree 1 .

The next step is to choose ou r linear application o be specific and to

stick to a simple case, we shall be satisfied with a first gradient theory. Let us

note that

Th e power of the inter nal forces will be written as in (2) with

p

a linear combina-

tion of U i , U i , j , i j x i j kor equivalently of U i , U i , j , i j , i j k :

But as is an arbitrary sub dom ain of

S ,

and

p

is independent of 2-see

footn ote 1-the axio m on intern al forces tells us th at itself mu st remain un-

changed w hen com pute d in a different frame. Ob viou sly, such a change of frame

implies that

U i

is to be replaced by

U i U i O i j x j ,

where

Di

and

O i j

are a

vector and a second ord er tensor respectively, both co ns tan t in S. similar change

is to be applied to U,: ,the velocity field of the subparticle, which leads to the

conclusion that z i j has to be replaced by x i j Ti i j . As a result. qi j defined by 9 )

ren~ uins nvariant

by this change of frame, a qulte natural conclusion because,

obviously, q . . is the

relative microcelocity gradient

an d then is related t o the local

I

relative motion of the particle with respect t o the con tinuu m. Then, one must

have

K i D i o i j T i i j

identically at every point of S , which implies that in l o ) ,K i is zero and o i j is

a

symmetric tensor.

Finally, the virtual power of the internal forces in may be written

s i j x i j v i j k x i j k ) v ,

or, after an integration by parts.

J { o i j s i j ) n j U i s i j k x i j n k ) o .

?B


7/21

METHO OF V I R T U A L POWER

56

The following step is to introduce the virtual power of the external forces

applied to 2 We can generalize what has been reviewed in 2 by considering

long range volumic forces with virtual power

and contact forces with v irtual pow er

Expressions (12) and (13) are no t as general a s a systematic first gradient theory,

but will be quite sufficient for the main applications.

Note that i jis a volumic double force, and M i j a double surface traction,

both of them including in the general case a symmetric and an antisymmetric

par t .

The last step is to apply the principle of virtual power. First assume that

U i and

x j

are chosen in such a way that U i and

x j

are zero outside a compact

subset interior to

2

n the sum

qi qd

q c only the volume integral re-

mains. which m ust be zero for any choice of U iand z i j .As an obvious consequence,

if one assumes that all the quantities which have been written are continuous

in

S

one must have, when putting

the two equa tions of equilibrium

It is natural to call z i j the (Cauchy) stress tensor ai j the intrin sic part of tlze

stress tensor sij the inicrostress tensor vijk he second inicrostress tensor. With this

partial result one may write

and , again. as and

x l

may take arbitrary values in every point of 2 9 . one

obtains the bou ndary conditions

which express the traction

(stress vector) and the d oub le surface traction M i j

as linear functions of the unit n orm al n i the linear ap plication so defined being

the stress tensor and the second microstress tensor respectively.

These results are summ arized in the following theorem .

TH OR M stress in a nzicronzorphic inediunl of degre e is. The s ta te o f

coinpletely defined b y the intrinsic stress tensor aij-a symm etric tensor-the

microstress tensor s i j and th e second nzicrostress tensor t Ji jk . et f and i jbe the

zjolunlic force and dou ble for ce ; and M i j tlze su fa ce tract ion and double surface

traction respectively. Define the stress tensor b y T~~ a ij s i j . Th en t lze equations


8/21

562

P. GERMAIN

in S of equilibriuin are giuen b y 15) and (161, the boundary conditions bjl 17) and

the volunzic energy of stresses by

where U and

x

are the velocity and the gradient of th e relatice velocities of a

particle and x the second gradient of the relatice uelocities.

c

The above results were announced in a previous note Germain 1972)).

Their development here is much easier than the way previously proposed in the

literature.

4

Dynam ical equa tions for a micromorphic medium of order

1 Inertia effects

must now be taken into account. Before giving the convenient definitions for a

micromorphic medium and the generalized statement of the principle of virtual

power, it is worthwhile to derive some preliminary results for a homogeneous

deformation.

4 1 Homogeneous deformation

Let us consider

a

body V-which later on

will represent a particle-and call ~ its mass. p its mass density, the position

of its center of mass M, x/ the position of a point M of V with respect t o its center

of mass M. Then

One also introduces a reduced inertia tensor I i j defined by

In this section, one has to distinguish between real kinematic concepts and their

virtual counterparts; the latter will be denoted with

a

A-for example, U i is the

real velocity of M, Qi is the virtual velocity of

M,

U i is the real velocity of

M ,

;

the virtual velocity of M .

It is assumed that

any tiine

the velocity field on

V

is defined by 81, where

z i j

is independent of the xl Accordingly, the vector space of virtual motions is

defined by a vector and a second order tensor tijn such a way that the virtual

velocity field of the body V is given by

The real) acceleration field of V is given by

where a dot means the time derivative material derivative, more precisely) and

where

y i

ui is the acceleration of the center of mass

M.

But i Is the relative

velocity of a point

M

with respect to the frame of the center of mass M and then,

as i I U j U j one may write

Now, the kinetic energy of

V,

the virtual power of mass-velocity or of momentum


9/21

563E T H O D

OF

V I R T U A L

POW R

of

V,

and the virtual power of mass-acceleration of

V

are given respectively by

Thu s, a straightforward calculation show s the following result.

L E M M A

The quantities

k . j? j?

defined

b y

( 2 3 ) , 2 4 ) ,and (2 5) are gicen

b y

with

It is also easy to prove the next lemm a.

LEMMA. W he n t varies, then

( 3 0 )

' i j

XipIpj XjpIpi.

For , as / i is a constan t, one has by de rivation of ( 2 0 ) :

p ' ( x y ;

~ i j c ; )du'

p ( i l j p x : x b x ~ , x ~ x ~ )u

4 2 Dynamics

of

a micromorphic medium

As in previous sections, but here

with different not atio ns, a virtual mo tion of suc h a me dium will be defined

by the two fields

oi

a n d

j

oth functions of the point x i in

S

at a given time.

Th e inertia properties ar e defined by the m ass density and the field of the reduce d

particle inertia tensor I i j given by ( 2 0 ) .Con serva tion of mass implies, for the real

mo tion (see in particular ( 3 0 ) ):

0 ,

I

j P p l

31)

i J l l p P j ;C I

Note that the mass of a volume A ( M ) around M , small with respect to the

dime nsion s of S , but large with respect to the particle size, is approximately given

by

l'(M)AV Pi,

where Au is the volume of A ( M ) ,and is the mass of the (great numbe r of) par-

ticles included in A ( M ) .With this rem ark, it is clear that one is led to the following

definitions :

Kinetic energy

of S :


10/21

564 P G E R M A I N

Vir tua l power of mass zjelocity of S

(33)

.qK

Jsp i K

da

Jsp ( O i ~ i

dv 2ijiipIpj

Virtu al power of m ass acceleration of S :

(34)

. A =

Js

l a i i i ; . ip~pj

sP ( ~ i 7 i

d v .

Ob viou sly, similar definitions ma y be given for any subd om ain of S. Let us

denote by V the real velocity field (defined by Ui a n d z j on S) and the real

accele ration field (defined by y and y j on S-the accele ration an d microaccelera-

tion) and by p the v irtual velocity fie ld; then .gKs a symmetric bilinear form of

V an d ?-say 2-and on e ma y write

Moreover, it is possible to differentiate

gK

with respect to time, provided we

supplement the definition of virtual motion with the following precision: the

zjirtual moti on of each point of the syst em is a uniform translation. For a micro-

morph ic medium, this means that the mo tion of each point M of any particle

P (M ) is a uniform translation. Thus, one must have

a n d , a s i S

U j x j k x ; , for any point M of P ( M ) ,one m ust write

As a consequence, one can state the following lem ma .

LEMMA.

Diferentiation of

a

virtual veloc ity field must be compute(1 with

the following formulas

With such a result one may relate ,, to K .

LEMMA. If all the quant ities inzjolved are continuously dif ere nti abl e on S,

then

The proof is quite eas y; s tart ing with (33 ), one has just to co mpu te

using (36 ), (30) an d of course the symm etry of I i j .

It is time now to recall the statem ent of the principle of virtual powe r when

dynamical effects have to be taken into a ccount.

Principle of zjirtual power

(dynamics) .

Let a sys tem

S

be in motion with respect

to a gicen Galilean p a m e ; at any time , and for any virtual motion, the material

derivative of the virtual power of mass-zjelocity of S is equal to the zjirtual power

of all th e internal forces and extern al forces actin g on S.

Disregarding first the case when discontinuities exist on S, using Lemma 4

one immediately obtains the following theorem.


11/21

565

ETHOD OF V I R T U A L POWER

THEOREM the dynanzics of inicronlorphic med ium of. T h e equations of a

degree

1

are obtained b y replacing, in tlze equations of the statics (see The orem

I ) ,

f i by

i

pyi Yi b y Yij pyiplPj where

y i

and

y i j

are the acceleration and the

nzicroacceleration of a particle (tlze latter be ing given by

29)).

It is obviously quite easy to supplement Theorem 2 with the statement

giving the jump relations ac ross an y surface of discontinuity. If C is the surface

across which discon tinuities may arise, W the velocity of a point of C, V U W

the relative velocity of the medium with respect to C, one know s tha t

where

v

is the norm al co mpon ent of V with respect to C ),

[ h ]

the jump of

h

across

C

provided that

p

an d are piecewise continuo usly differentiable in

S .

By con-

servation of mass, it is seen that m pv is con tinuo us across C. Then by classical

reasoning, the following theorem is obtained see, for instance, Ge rm ain 1973b)).

TH EO RE M. f

in pv is the mass flux dens ity across an y discontinuity

sullface

C,

then

4 3

Energy

equation The problem is to give significance to the concepts

involved in the classical statement of the first principle of thermodynamics: the

inaterial derizjative of th e total ener gy of a systein is the suin of the power of ext ern al

forces actin g on this systein and of the rate of heat supplied into S-say

Q.

Assuming the total energy to be the sum of the kinetic energy

K

and the

inter nal energ y E-this lat ter being defined by a specific int er na l ene rgy e-one

may write

B + R = q * + q c + Q .

But th e theorem of energy in dym amics , i .e., the statem ent of the principle of

virtual power applied w ith the real mot ion, gives

and then one ob tains

If, as usual, we app ly this result to a subsyste m of S and write

( r

the rate of volumic heat,

q ,

the heat flow ve ctor), we can derive. by the applica-

tion of the divergence theorem and the introduction of the expression of .qi

given in 3,

It is outside the scope of the present paper to put more emphasis on this result

an d to continue the theory to considerations of constitutive relations-for this.


12/21

66

P. G E R M A I N

the reader is referred t o the literature an d, in particular, t o the pape rs of Eringen

an d his cowo rkers. We w ant only to point o ut the advan tage of the present method

which, starting directly with th e expression of the r ate of energy of intern al forces,

leads immediately t o the expression

4 0 )

of the equation of energy

3 9 ) .

5 General micromorphic medium In the last two sections a new method has

been proposed in order to derive quite simply results which were obtained pre-

viously by different and, according to the author's opinion, more complicated

methods. This theory for the time being is rich enough to fit various physical

situations. The principal difficulty indeed is to discover the practical significance

of some of the concepts which have been in trodu ced, to design a metho d in order

to exhibit their physical validity i f any

),

and to measure them in some specific

physical situations. Thus, the generalization proposed in this section is not given

with the idea of providing a new schematization which can be used at the present

time, but just for the sake of showing the possibilities of the method and for

eventual further comparisons with other theories which may be found in the

literature.

5 1 Statics Starting again with the model of a continuum with a micro-

structure. a natural generalization is to describe more accurately the relative

motion of the particle with respect to its center of mass by replacing

(8)

by a

finite or infinite expansion

W e begin here with the formal

the or.^,

when the full infinite expansion is written.

Without loss of generality, the zijk1may be assumed to be fully symmetric with

respect to the indices

j

k,

I

As in

3,

a virtual motion is defined here by a sequence of fields

W riting the power of intern al forces as in

2 ) ,

and restricting ourselves to a

first gradient theor y, on e is led to define p by

where

As above,

ai j

s sy mm etric; without loss of generality, the various tens ors which

have been introduced in order to describe the state of stress may be assumed

to satisfy som e symm etry properties-for instance, sijk vijk are symmetric with

respect to the indices and

k.

Note also that definition

42 )

fulfills the cond ition

arising from the axi.om of virtual power of internal forces.

Similarly, form ulas

12 )

and

13 )

have to be generalized by

43 )

.qu J U i

+ Yijl i j

Yijkzijk

) da


13/21

567

E T H O D

OF V I R T U A L POWER

Note again that, for instance, Y a n d M,,, are symmetric with respect to the

indices j and k. Introducing the stress tensor T , , by 14), one gets by application

of the principle of virtual work the equations of equ ilibrium for the general nzicro-

inorphic medium

5 2

Dynamics

If one wants to take into account the m otion of the system.

one has again to distinguish between the real motion defined by V : U i ,

x i j

an d a virtual motion which will be denoted by

= o i ,Z i j ,

It is not difficult

to generalize the above analysis of

5 ,

even i the final expressions are evidently

more involved.

In order to describe inertia properties of the medium, one must introduce

the generalized reduced inertia ten sors

I i j , i j k ,

j k lully symm etric with respect

to their indices) an d defined for any particle

V =

P M) by form ulas generalizing

20), such that

It is not difficult to generalize 30) also . Th e result may be w ritten

if on e introduce s the following no tatio n :

Note that formulas

19)

are still valid. Accordingly, differentiation with respect

to time gives the following identity becaus e

;

= Ui Uj) which is satisfied by

the real mo tion lr

In wh at follows, it will be assum ed also th at the virtual mo tion satisfies the

same relation ~ 90.

The kinetic energy is still a qu ad ra tic form of V-say 2K

=

2 V ,V t

w he re 2 9 , V ) is the expre ssion of the virtual pow er of mass velocity of S , a

bilinear symmetric form which may be expressed with the inertia components

I,,, Th e inte gran d of 24) is


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568 P

GERMAIN

andthenwemaywrite

The t e rms in side the braces-say ;,-may a l so bewr i tt en a s

r -

with

Th eabo veresultgivesthe generalization of(33)an dconse quen tlyof( 32).

W enow have to dea l with acce le rat ions . F i rs t we ob ta in the general iza t ion

of(22)bydifferentiationof(41)

:

L

3C

. .

X

n l x i j k . . . l ( ? ~ ~ ~

;

=

y i

i i j k . . . I ~ j

-~;)(jk . . . l) ,

m = l i j m= 1

if one u s es no ta t ion (48 ), t ak ing a ccoun t of th e s ymme try w i th r espect to th e

indices j, k. . , I But

x j , . . . s ~ ~

53)

=

u j

- u j =

, , S

rn' = 1

Then,wemaywrite

7,'= 7 7rJk lSJXk . .

X;

(54)

8 ,

n t = 1

with

For ins tance, for the microaccelerationtensorof lowerorderonegets

Thegeneralizationof(28)iseasilywritten

(57)

with

(58)

rip q ijljp...q

j ijkljkp...q

Finallytheformulawhichreplaces(34)is


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569ETHO OF

V I R T U A L

POW R

if we denote by d the set of acceleration fields y

q .

It is essential for

a later application of conservation of energy) to check that

for reul moti ons, the

pon.er of muss uccelerution is tlze time derivutive of tlze kinetic ene rg y:

d

d V ,d

=

- i 2 ~ , ) )

dt

For this purpose it is interesting to give an alternative expression for Tip . . .

Going back to 52), 53) an d introducing

7;

directly in 25), one obtains 57) with

a n d

Ajk . . . l p . . .qn l [x ja l

k . . . l p . . q

](jk...l) .

Th e left-hand side of 60) con tains

a) terms like

whos e sum gives the term s of the right-h and side of 60) when th e coefficients

I in are taken as con stan ts;

b) terms like

But it is easily checked th at, according to 47) an d the p roperties of symmetry

with respect t o th e indices,

an d these term s give the other terms in 60), i .e., those w hich are obtained by

material derivatives of the coefficients I .

A

sinzilur calculution showss thut Lenlmu 4

1s still vulid.

Let the virtual motion of each point of the system be a uniform translation,

on e may first generalize 36). Differentiation of a virtual velocity field must be

comp uted with the following formulas see 56) above)

:

After all these prelimina ries of kinetics, it is easy to get the final result by a pplic a-

tion of the principle of virtual work see the statem ent of 4.2). Th e conclusions

are given in the following theorem.

TH OR M

.

T h e equutions of statics of u general m icromorphic nlediunz ure

given by 45). T h e equations of dynunlics ure obtuined by replacing

f

bjl f

pyi

uizd the

Yi j . , . k

y

Yi j . . .k- p r i j . . , k ,

lzere ma y be expressed in term s of the ucceleru-

tion fields and the velocity jields und its deriv atice s)

by 58)

und

5 6 t o r 55).
http:///reader/full/Ajk...lp..http:///reader/full/ak...lp.,http:///reader/full/ak...lp.,http:///reader/full/ak...lp.,http:///reader/full/Ajk...lp..


16/21

570

P G E R M A I N

This general theory is quite formal. For the applications, expansion (41)

may be reduced to the first

n

+ 1 terms-and the theory obtained in this way

will be relative to nzicronlorphic medium of order n. For instance, for

n

3, is

described by

U i ,

x i j , x i j k ,

i j k l ;

he state of stresses by

a i j , i j , s i j k , i j k l , i j k ,

i j k l ,

vij,,,,: long range forces by f Y i j i j k i j k l;nd generalized surface traction

by T M i j ,M i j , , M i j , , . On ly the first four lines of (45) have to be w ritten for equ i-

librium. As far as inertia forces are concerned, we keep y i , r i p , i p q ,ipqrnly.

It would not be difficult to write the jump relations and the energy equation.

This m atter is left to the reader.

6 Micropolar medium and other particular cases

6 1

Micropolar medium very interesting particular case is obtained when

assuming that the particle

P

( M ) of the continuum moves as a rigid body. Th e

virtual motioils which will be considered will also be assumed to udlzere to this

condition. In this case, the micromorphic medium is necessurily of order 1, und tlze

tensor

z i j

is antisymmet~ic .As a consequence, x i j , is also antisymmetric with

respect to the indices

i

a n d j (see (9)) .

Going back to 4 and putt ing T~~ = o i j

+

s i j ,as in (14 ), (10) may be w ritten

P = K i U i + T ~ ~ U ~ , ~i j x i j+ 11,'11

x

and without loss of generality,

s i j

and

v i j ,

may be assumed to be antisymmetric

with respect to the indices i and j As p is invariant, by superposition of a rigid

body velocity field, one must have for any constant Di and Rij

- a J i :

then

K i

0 and

o i j

is symm etric. As a result,

the tensor s ij is nothing else than the

antisynlnzetric purt of tlze stress teilsor z i j

Similarly, in (12) an d (13),

Y i j

and

M i j

may be assumed, without loss of generality, antisymmetric with respect to i a n d

j :

they define the long range volumic couple distribution and surface couple stress

respectively. Again in .qA-see (34)-one has only to consider the antisym me tric

part of

T i j

=

y i p l P j ,

say

T I i j l .

The equations of motion and boundary conditions

for a micropolar medium may then be w ritten

f + 7 . . =

p q b .

L J J

J . J .

r'

T T ,

.n .

11 =

1 1 k k .

~ijl

+

t ' i j k , k Y i j= p T L i j l ,

M . . n

It is obviously easy t o get a m ore classical description when one introduces

the adjoint tensors with respect to the indices i and j P u t

7 . .

=

- . .

7

I1

, ] ,

,,

M i ,

-

2 i l k

M

,,

Y i j k

-L

i j k

Y

k

(62)

Then,

As only the antisymmetric parts of K i j and T i jplay a role, one may introduce

and then


17/21

METH O D O F V I R TU A L PO WER

if on e introduce s the inertia tensor

Th e interpretation is quite easy :

x

is a vector rotation rate,

M,

and

Y,

are couple

densities,

v i j

the couple stress density, o the spin. The last line of 61) may then

be written

and the volumic kinetic energy is

Then, one recovers the main result given in the paper of Kafadar and Eringen

1971), in which m any q uite interesting interpretations and applications can be

found.

6 2

Second gradient theory The second gradient theory may be obtained

very easily from the theo ry of mic rom orph ic m edia of order 1 when on e assumes

that the particle is subject to the same deform ation a s the general continu um , i.e.,

Let us investigate briefly the consequences of such an assumption. It will be

sufficient to use virtual motions which satisfy the condition 68 s a y 2 i ,:,.

The equa tions of 5

3

and 4 have been obtained by writing, for any subdo main

relation such tha t

is valid whatever be the fields

O

4 . In the present case these two fields are no

longer independent: in order to apply the principle of virtual work, one must

write the left-hand side of 69) in such a way tha t only indepen dent quan tities

related to the virtual motions appear. First, one obtains easily for the volume

integral,

A,s the volum e inte gral mus t be zero w hatever the value of

i

ne gets first the equa -

tion

A

Bij,j

0.

If one assum es equilibrium, this volum e integral comes from

qd

.?,,,

and one has

A = f + o . . + s . .

r 11 11

B . . = y . . + s . . + v i j k , , :

1 11 L

and writing

one gets

2)

i

+ T i j j 0.

Th e term sijdisappears, as is quite clear when one looks at

1

1) taking account of


18/21

(68). Intr insic stresses are defined by the symmetric tensor o i j an d the tensor ui jk,

symmetr ic wi th respect to an d

k.

No w consider the term in th e surface integral of (69) ar ising from

which

is given by (13). Let us introduce, following Mind lin, the n otat ion DK i an d DjKi

for wr it ing the no rmal derivat ive o n 2 9 of the vector f ie ld K i and i t s t angentia l

d er iv at iv es o n 2 9

O n 2 9 ,

i

a n d O^ may be chosen independently , but DjO i may be computed

from th e values of

i

o n 2 9 . Th en

;2.l.o.

J 1 1

( : ~ l , , f r , ) , ~

M , ~ , ~ @ ,

Dj(h l i jO i )

r 2 . 1 i j n j ~ o i ( D ~ M ~ ~ ) ~ , .

The divergence theorem for an open subdom ain C of 2 2 gives

where

t

is the uni t tangent vector to 2C an d v

~ t n

the uni t normal vector

to ZC, tangent to 2 9 . Then

Here,

is a ridge, i.e., a line on

2 9

if any) where the tangent p lane of 2 9 s d is -

continuous. The only tract ion elements which may be defined on 2 9 n terms of

the stress tensors are the n:

(i) a surface traction

r

DjMij (Dkn, )Mi jnj ,

( i i) a norm al dou ble t ract ion Ri Mijn j ,

(iii) eventually a line traction

W i

[[Mij\tj],

a n d

Finally, making a similar transformation for the other terms of the surface

integral of (69) arising from

4

4 .

nd taking into a ccou nt the surface integral

of (70), one gets the boun dary condit ion o n an y surface

2 8 :

which, with the f ield eq uat ion s

(71)

an d (72), form a com plete set of equa tions of

equilibriu m for the second g radien t theo ry (see M indlin an d Eshel (1968)) fully

equivalent to the system given by Germain (1973).


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573ETHO OF V I R T U A L

POWER

he equution of motion ma y be derived b y applicutioil of heo rem

2. This last

statemen t show s the significance of the extra inertia term s on e has to in troduce in

the dynamical equa t ion.

6 3

Indeterminate couple stress theory This model ca n be viewed as a micro-

polar m edium subjected to the constraint

The particle has a rotation motion which is exactly the rotation of the general

cont inuum. In order to obtain the corresponding equat ions, one can star t again

wi th (69) and assume tha t e i ther f i j fi i jwith the same B i j and Fijo r f i j o i , j ,

as in the preceding subsection, but by keeping only the antisymmetric parts of

B i j and Fij This last poin t of view leads immediately to th e results we are loo king

for. Using the not ation (62), (71) an d (72) are replaced by

? . . =

IJ r j i j l Ik .k i j k y k 3

(77)

f + z s 0 .

O n t he o t he r hand , v i j k is ant isymmetric with respect to an d k. Th us (73) reduces to

(78)

;

zijnj i i j lDj ( ~lk nk) .

Th e r ight-hand side of (74) is zero. Then 2Ni -ci jkMknj 0. Th e couple M i

is then nor ma l to an d its intensity is fully indeterm inate: it plays n o role when

one writes .?,,,which reduces to the pow er of the surface tractio ns T-at least

if there is no ridge o n 2 . Th en (77) an d (78) are the m ain eq uations of this theory

which has many applications-see, for instance, Min dlin and Tiersten (1962).

The relation of this theory with the micropolar theory has been previously

emphasized in Eringen (1966a), (1966 b). Of course the theories of 6.1 an d 6.3

originate in the famoh s work of the Cosserat bro thers (1909).

6 4

Microstructure without microstresses Looking back again to a micro-

morphic medium, one may assume that the relat ive motion of the part icles is

possible without an y expense of energy. In fact,

i f

one assumes v j k 0 in (10) an d

Y ij 0 in (12) , 16) shows that also si j 0 an d thus we recover the classical theory

where z i j a i j defines completely the state of stress. Nevertheless, the parameters

which define the o rientation of the particle,

i f

they d o not appea r in the equations

of equilibrium, may be present in the definition of the specific internal energy,

an d consequently in the constitutive equation s. They then play the role of hidden

param eters. Such a m odel h as recently been considered by M and el (1971) in its

general th eor y of viscoplasticity.

7

Comparison with other work We conclude this paper with a few remarks

dealing with the comparison of the present work with other theories presented

in the li terature.

The theory of micromorphic media of order one has been extensively con-

sidered by Eringen an d his cowo rkers. Th e results of

3 and 4 are then no t new,

but the method which is proposed seems to be original and is , in the author 's

opinion , simpler than the me thods which have been prop osed so far. This is why


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574

P

G E R M A I N

it has been possible to develop the general theory of 5 which, to the autho r's

knowledge, is completely new.

Mention must be made at this stage of another generalizaticsn proposed by

Eringen (1968) an d (1970) under the title,

Micronzorplzic coiztinuum

of

higher grades

Eringen has proposed to obtain the results of 4 an d 5 by a meth od of surface

and volume averages on the basic concepts valid at the microscopic level. The

generalization is obtained formally by taking higher moments in this process of

averaging. Th e present auth or must confess that , in the present s i tuation. the

physical significance of such a form ulation is no t clear to him. Th e basic kinematics

of the particle is still given by (8 ); how then is it possible to introduce a description

of stresses which apparen tly is mo re refined th an th e one given in 3 which is

based on the fundamental duali ty between the kinematical description and the

description of stress? What is the significance of the various micro-energies which

are intro duced ? At first glance, at least, they seem to h ave n o direct relation with

the usual stateme nt of the first principle of therm odyn amics. If such a refinement

is worth being considered, what would be i ts coun terpart in the classical continu um

theory? These are som e of the questions which rem ain, until n ow, without answer,

an d which seem necessary to elucidate before such a theory ma y be accepted and

used for the description of physical situations.

We no w w ant t o com men t on a very interesting n ote by G reen (1965) which

intends to establish the connection between the micromorphic theory of order

one and the theories of multipolar continuum introduced by Green and Rivlin

(1964a, b) an d (196 5). Th e startin g poin t of these theories is quite similar to th e

one o n which the present paper is based (desp ite som e difference in the wo rds

used for describing the same concepts) generalized forces ar e introdu ced by their

power for a given kinematical description. Nevertheless these au tho rs d o not

apply th e principle of virtual power, bu t instead , an d m ore simply, the first principle

of thermodynamics or the equation of energy. supplemented by a condition of

invariance under superpo sed rigid body motions. When applied to th e classical

continu um m echanics, this meth od provides at the sam e time the three s tatements

of conservation of mass, mom entum an d energy. Wh en applied to a m ore complex

situation, this method does not give the same results as the method of virtual

po w er: obviously th e num ber of equation s obtained by the former is less tha n the

num ber obtain ed by the latter. Mor e precisely, in case of micromorphic media,

the last condition (17) is included in an oth er on e in which he at flow concep ts are

present as well as double surface traction

M i j

and the microstress tensor v jk

Thi s difference app ears still m ore clearly w hen lookin g, for instance, at the p aper

of Green an d Rivlin (1964a) devote d to w hat is called in th e present terminology

theory of the second an d higher gradients . If long range forces an d generalized

surface tractions are essentially those introduced here, the generalized stresses

are different and consequently the basic dynamics equations are different and

moreover less in num ber. Of course the number of equatio ns which ar e missing are

supplem ented by a greater num ber of constitutiv e equations-for instance. when

the m aterial is assumed t o be elastic. There is no reason to suspect this theory to

be inconsistent, but the present author does not know if statements of existence,

uniqueness an d v ariational formulations can be established a s has been d one for

the second gradient theory (Ge rmain (1973a)).A further study of these questions


21/21

575ETHOD OF V IRTUAL POWER

an da mo reextendedanalysisof thephysical significanceof thetheory of Gree n

andRivlinincomparisonwiththenthgradienttheorywouldbeveryvaluable.

finalremarkontheworkofGreenandRivlinisrelatedtotheiruseofthe

tetra hed ronargumentwhichtheysystematicallyapp lyinord erto relategeneralized

surfacetractionswithgeneralizedstresstensors.Itisnotclearthatsucha nargu-

mentmaybesafelyappliedwhenhighorderderivativesareinvolved,ashasbeen

shownindetailintheabovequotedpaperonsecondgradienttheory.

REFERENCES

P . BREUNEVAL1973),Principe des travaux virtuels et equations des coques, Mecanique , to appear .

P.CASAL1972),

La thdorie du second gradient et la capillarite,

C.R.Acad.Sci. ParisSer.A,274,pp.

1571-1574.

E.COSSERAT (1909), Theorie des corps deformables, Hermann, Par is . ND F COSSERAT

A. C . ERINGEN

ND

E .S. SUHUBI1964),

Nonlinear theory o fsi mp le m icroelastic solids,

Internat.

J

Engrg.Sc i . ,2 , pp. 189-203, 38 94 04 .

A. C . ERINGEN1966a),

Linear theory of micropolar elasticity , J

Math.Mech., 15,pp.909-924.

(1966b), Theory of micropolar f luids, Ibid ., 16,pp. 1-18.

(1968),

Mechanics of micromorphic continua,

Mechanicsof Genera l izedCont inua , IUTAM

Symposium,E.Kro ner, ed .,Springer-V erlag,Berlin,p. 18-35.

(1970a),

Mechanics ofmicropo lar continua,

ContributionstoMechanics,Dabir, ed., Pergamon

Press,Oxford,pp.23-40.

(1970b),Balance laws of micromorphic mechanics, Internat. J Engrg.Sci.,8,pp.819-828.

P. GE RM AIN1972),

Sur I application de la methode des puissances virtuelles en mecanique des milieux

continus, C. R. Acad.Sci. ParisS r . A,274,pp. 1051-1055.

(1973a),

La metho de des puissances virtuelles en mecaniq ue des mili eux continus-1Pre Partie,

Theorie du second gradient,

J

Mecan ique, 12,pp.235-274.

(1973b),

Cours de mecanique des milieux continus,

Tome1,Masson,Paris.

A . E . GREEN

ND

R. S . R IVL IN 1964a), Simple force and stress multipoles, Arch . Ra t i ona l Mech.

Anal.,16,pp.325-353.

(1964b),Multipolar continuum mechanics, Ibid.,17,pp.113-147.

(1965),Multipolar continuum mechanics: unctional theory, Proc . Roy. Soc . Ser . A, 284, pp.

303-324.

A . E. GREEN1965) , Micro-materials and multipolar continuum mechanics, Internat. J Engrg. Sci. ,

3,pp.533-537.

C . B . KAFADAR

ND

A . C . ER INGEN 1971),Micropo lar media-The classical theo ry, Ibid . , 9 , pp.

271-305.

J . MANDEL1971) , Plusticite classique et viscoplasticite, Cou rs a u C . I .S .M . , Udi ne , Ed it io ns du

C.I .S.M.

R .

D.

MI ND LIN1964),Microstructure in linear elasticity, Arch.RationalMech.Anal., 16,pp.51-78.

R . D . M INDL IN ND H . F TIERSTEN1962),Effects of couple-stresses in linear elasticity, Ibid . , 11,

pp . 415448 .

R . D . MINDLIN

ND

N .N .E SHEL1968),

O n j r s t strain gradient theories in linear elasticity,

Internat.

J.

SolidsSt ru ct., 4,pp. 109-124.

R . A . TOUPIN 1964), Theory o f elasticity with couple-stress, Arch . Ra t i onal Mech . Ana l ., 17, pp .

85-1 12.

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