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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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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 .

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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"

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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). Exter