Dualities in the analysis of cosserat plate

MECH. RES. C O M M .

DUALITIES

Vol.3, 399-406, 1976.

IN T H E A N A L Y S I S

Pergamon Press.

OF COSSERAT

M. Farshad, Shiraz, Iran B. Tabarrok Dept. of Mech. Engineering,

Printed in USA.

PLATE

University of Toronto,

Toronto,

Canada

(Received 24 January 1976; accepted as ready for print 4 May 1976)

Introduction H i s t o r i c a l l y the analysis of thin bodies as exemplified by rods, plates and shells has evolved through three stages of development. In the first, the formulation is based, to a large extent, on physical intuition. As examples one may cite the EulerBernoulli beam and Lagrange's plate equations. In the second stage the governing equations are obtained, by imposing certain restrictions on the three dimensional equations of elasticity. Here we may cite the plate theories of Reissner [i] and Mindlin [2] as well as other examples outlined in the recent text by Wempner [3]. The third approach, the so-called direct approach, is based on the concepts of directed curves and surfaces and the Cosserat model of the continuum. A comprehensive treatment of these latter two approaches can be found in reference [4-5] wherein a large bibliography to numerous related works is given. In this paper the governing equations of linear elastic plates are derived by the direct method with an emphasis on the physical meaning of these equations. Subsequently inherent dualities in the governing equations of plates are exposed.

Governing

equations of ~eneralised

Consider an infinitesimal fig.

i.

Cosserat plate

element of a flat plate as shown in

Denoting the unit vectors by

ai,

(i = x, y, z) we may

describe the deformations of the element in terms of three displacement components of vector u = a. u. and three independent --

rotation components

--i

1

forming the vector 8 = a. 8. --

element possesses

--i

six degrees of freedom.

element we assign a s s o c i a t e d f o r c e

vectors

Thus the

1

Along the edges of the denoted by

F --~

Nie

and

M

--~

= --i a . M . IS

as shown in fig.

Scientific Communication

399

1 (~ = i, 2)

= a. --1

400

M.

FARSHAD

and

B.

TABARROK

V o l . 3,

No. 5

Z

g3

a

y

Components

Kinematic strain

the

deformation

vector

displacement

t

( ),i

= a

It

should

t

plate,

( )

denotes t

the

plate

that

of t h e

vectors

as

follows

0

( ),2

-- ~ 3y

outward

, describing in t e r m s

of

the

: 0

K

(I)

( )

unit

normal

equilibrium

equations In the and for

vector

so t h a t

vector

Cosserat u

and

the

model rotation

independent.

may

be o b t a i n e d

absence

along

the

vanishing

I (M

unconstrained

displacement

The

condition

expressed

in t h i s

equations

surface

<

--2

be n o t e d

work.

vector

= a

equilibrium

the

of

be

the

The

the

element

can

and

completely

on

an

curvature

x

are

virtual

0

the

+ t

--2

derivatives

vector

and

= u

and

--I

the

on

e

rotation

- ~ ~x

vector

--I

of

and e

The

force

relations

The

where

of

of

distributed

edges of

of

virtual

6K--~ + --
via

the

the

forces

plate

work

6e--~ )dx dy

theorem

as = 0

and

we m a y

of

couples write

follows (2)

Vol.3,

No.5

DUALITIES

The application faction

of virtual

- rotation

in equation

ment and rotation equation

401

the satis-

To this end we use the

- displacement

relations

in mind that the admissible

fields must be continuous.

given

displace-

Thus we rewrite

(2) as I

On

equations.

and strain

(i) and bear

PLATE

work theorem presupposes

of the compatibility

curvature

OF COSSERAT

{M_~

carrying

conditions

6@ --~

+ --~ F

(6u --~

+ t dy = 0 --~ x 68)}dx --

out the integrations for vanishing

(3)

by parts we obtain the necessary

of virtual

work as

In the domain

Alon$

6u --

:

F ~ (~ --~

= 0

68 --

:

M - -

+ ~t

~

(4)

the boundary Either

and either Equations

M

= 0

or

e

is prescribed

F

= 0

or

u

is

(4) express

In expanded Nxx,x Nxy,x

the conditions

+N

yx,y

+ Nyy,y

M

=0

M xx,x + M yx,y - Q x = 0

By

tions [i,

and the presence

, and bearing

2, 6].

Nowacki

These

of the moment

+Qy=0

(6)

and inplane

of the distributed

in mind the vector

are identical

yy,y

PX,X + Py,y + N xy - N yx

from the non symmetry

tensors

+M

of the plate.

forms

=0

xy,x

equations

couples

sign convention

to those obtained

(5)

prescribed

of equilibrium

form they take the following

Qx,x + Qy ,y = 0 Apart

x --~ F = 0

in classical

have been previously

=

0

force Bx

and

these

equa-

plate theories

derived

by

[12].

The compatibility

equations

We may obtain

the compatibility

complementary

virtual

placements

work.

and rotations

equations

via the theorem

In the absence

of prescribed

on the plate and along

its edges,

of disthe

402

M.

expression

FARSHAD

for the vanishing

a n d B.

TABARROK

condition

Vol. 3, No. 5

of complementary

virtual

work becomes I [< This time the theorem equilibrium function

$

6F ]dx dy = 0

requires

equations.

vectors

+ s

6M

the A priori

To this

and

~

(7)

satisfaction

end we introduce

defined

through

of the

the stress

the following

re-

lationships. F = $ -~ e~B -,6

(~:B = i, 2)

(8)

M : e~B --,8 $ + e~8 ! s x ~ --~

where tions

ee6

is the two dimensional

in equation

solutions checked

of

by a direct

force

£

variables.

force

Equation

~

Returning

vectors

{<-~

( e~B

Carrying the

65-,6

and

$

along

that the

0

and

u

virtual

stress

set

forms

any pres-

of the plate.

to equation

(i).

function

In

vectors

same role as the rotation respectively,

(7) and employing

we may write

selected

satisfy

the edges

similarity

func-

equilibriating

that the

will also

play the

to equation

£

one to obtain

functions

(8),

to provide

(4), as can be readily

we assume

be noted

Rela-

In this way the stress

conditions

in equation

of complementary

In

stress

it should

tion vectors

f

Further

boundary

and displacement (i).

enable

(8) has a remarkable

particular and

~

tensor.

constructed

equations

substitution.

and

of the admissible cribed

(8) are specifically

to the equilibrium

tion vectors

alternating

in equation

the stress

the vanishing

func-

condition

work as

+ e~B !8

out the indicated

x

6~) -

+ -e ~

integrations

e B 6¢, _ 8}dx dy : 0

(9)

one obtains

domain

6~_

:

e B ~,~

6!

:

ec~8(R(~, B + i 8 x ~< ) : 0

Alon E the

: 0

(i0)

boundary

65_

:

e8

~

65_

:

ec~ 8 _e~

: 0 = 0

(ii)

Vol. 3, No. 5

DUALITIES

Since

the

stress

force

boundary

functions

conditons,

(Ii) are of the natural generally

prescribed.

of c o m p a t i b i l i t y noting tions

that (I) if

u

the r e s u l t i n g

conditons

only

Equations

and

i.e.

~

(10)

and

express

and c u r v a t u r e to these

8

403

to satisfy

type

solution

PLATE

are assumed

of strain

the

OF COSSERAT

in e q u a t i o n

¢

are not

the c o n d i t i o n s

fields.

equations

are assumed

any p r e s c r i b e d

It is worth

is p r o v i d e d

as c o n t i n u o u s

by equa-

vector

func-

tions.

Dualities

in various

On c o m p a r i n g

differential

of these

equations

equations

(4) have

We may refer set of dual

theories

the e q u i l i b r i u m

compatibility tures

plate

differential

equations

are

variables

(10) we note

identical.

kinematic

to such paired

equations

Some

counterparts variables

is shown

TABLE

, exy

, ey x

M

Nxx

, Nyy

, Nxy

, Ny x

- Ky x

Kyz

' Kxz

, ~x

'

of flexure form dual inplane

' Cy

to note

variables.

nature

interest

but in a d d i t i o n

analysis

of specific

developed in terms

that

are dual

in

(i0).

variables.

The

u

, ¢

, My x , - Mxy

xx

, - Kyy

' - eyz

' 8x

'

they

for the analysis

XX g

'

(kinematic)

of interest variables

can be used

XZ

quantities

of s t r e t c h i n g

to note

to shear

that

theoretical

a computer

deformations

the

deformations.

to a d v a n t a g e

For instance

variables,

K

z

quantities

of inplane

,

ey

are not only of intrinsic

problems.

of d i s p l a c e m e n t

' ~y

(force)

M

' Qy

~x z

-

, Kxy

the force

It is also

stresses

of this

variables

i.

,

yy

- Qx

and the k i n e m a t i c

couple

Dualities

' ~y

ez

It is of interest

struc-

Bending

, eyy

~z

the

1

exx

, ~x

the

in e q u a t i o n s

Stretching

u x , Uy

that

force

as dual

in Table

(4) with

in the program

of a plate,

can in p r i n c i p l e

be e m p l o y e d

404

M. FARSHAD and B. TABARROK

for the tions Next

flexural

analysis

certain

constraints

let us c o n s t r a i n local rigid

some

Mindlin

the

z

component

body rotations

Cosserat

Further,

~

the c u r v a t u r e

the f o r c e

quantities

of the

(12)

of

u

the d u a l i t y

and e q u i l i b r i u m

equations

functions. in e q u a t i o n

t h a t the r e q u i r e d

this

x directly

)

(12)

of this

type of con-

x,y

the

influ-

[8].

(i) shows

that

under

)

and

K

also b e c o m e yz N o t e h o w e v e r that

y effected

structure

let us i n t r o d u c e

This constraint (12).

equation

by this

kinematic

must

From equation

of the c o m p a t i b i l i t y

a constraint

amongst

be dual to that (8) it is a p p a r e n t

is

(13)

- ~x,y )

(13)

into e q u a t i o n

(8) we note

that

constraint

the c o n s t r a i n t Qx

modification mentioned

<

in the

constraint

Mxy

ponents

+ u

y,x

to

that

concentrations

xz and u

~z = ½(~y,x On s u b s t i t u t i n g

vector

and N r e m a i n i n d e p e n d e n t in spite xy yx t h e i r a s s o c i a t e d s t r a i n s are n o w i d e n t i c a l . In

fact that

indicated

For i n s t a n c e

N

o r d e r to r e t a i n

Again

func-

by i m p o s i n g

variables.

into e q u a t i o n

= ½(u

yx

are not

and h e n c e

stress

x,y

theory

components

of d e r i v a t i v e

constraint

theories

of the r o t a t i o n

- u

y,x

on s t r e s s

= ~

xy

functions

under

of s t r e s s

p l a t e w i t h a v i e w on d e m o n s t r a t i n g

of e q u a t i o n

constraint

plate

we r e q u i r e

the b a s i c

ence of c o u p l e - s t r e s s e s Substitution

i.e.,

= ½(u

z

has d e a l t w i t h

strained

simplified

on some of the b a s i c

0

the

in t e r m s

[7]. let us c o n s i d e r

this

of the p l a t e

Vol.3, No.!i

and

Qy

= - Myx

: ½ (@y ,x + ~ x ,y )

in e q u a t i o n

(13) w i l l

effect

as can be seen f r o m e q u a t i o n

is a n a l o g o u s

to the m o d i f i c a t i o n

of

the

force

(8). Kxz

com-

This and


earlier.

The net r e s u l t

of i n t r o d u c t i o n

tions

(13)

(12) and

of the p l a t e the

of the two c o n s t r i a n t s

is as follows.

strain tensor

For the

becomes

stretching

symmetrical

in e q u a analysis

but the

Vol. 3, No. 5

DUALITIES

force tensor remains stresses,

OF C O S S E R A T

unsymmetrical.

405

PLATE

Further we have two couple

which together with the non symmetric

shear forces,

tend to bring about the local rigid body rotation of the element. This

stretching model of the plate is dual to the following

flexural model of the plate. but the curvature accounts

The moment tensor

tensor remains

for two components

unsymmetrical.

moderately

thick plates

average of Finally

tensor,

K

and

xy

= - u

y

From equation

theory for

yx

e

z,x

x

kinematic = u

constraints (14)

z,y

(i) it is apparent that these latter constraints

suppress the shear deformations

e

xz same time they relate the curvatures the derivatives

of the displacement

way the symmetry of

<

and

<

xy yx dual constraint would then appear as ex : ~z,y

and from equation suppress

to Mindlin's

In his theory Mindlin employs the

let us impose the additional 8

It will be

apart from non symmetry

is identical [2].

<

The model

of shear deformation.

r e c o g n i s e d that this flexural model, of the curvature

is symmetrical

and

e

and at the yz < , K and K to XX yy xy component u In this z

is also established.

The

8y : ~z,x

(15)

(8) is then apparent that this constraint will

the distributed

time it will relate

couples

~x

and

~y

and at the same

, N and N to the derivatives of xx xy yy ~z In this way the symmetry of N and N will be estabxy yx lished. From the foregoing

N

it is apparent

that in this case we have

regained the well known duality between the stretching and flexural analysis

of classical

plate theory.

Southwell

pointed out

this duality by introducing a pair of stress functions our notation functions

and

~y ~z

and

~x

are equivalent

to Southwell's

In

stress

is the Airy stress function.

Finally we point out that when the equations plate are discretised of grid works

[9].

of the Cosserat

they take a form analogous

[i0, ii].

to the equations

Through this analogy the non symmetry

of the force tensor becomes

entirely natural and the m e a n i n g

406

M.

of couples

~x

and

~y

FARSHAD

a n d B. T A B A R R O K

Vol. 3, No. 5

becomes clear.

References i.

E. Reissner, "The Effect of Shear Deformation on the Bending of Elastic Plates", Journal of Appl. Mech. Trans. A.S.M.E., Vol. 87, PA-69, (1945).

2.

R.D. Mindlin, "Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic Elastic Plates", J. Appl. Mech. Trans. A.S.M.E., Vol. 73, pp. 31-38 (1951).

3,

G.A. Wempner, Mechanics of Solids with Applications to Thin Bodies, McGraw Hill, 1973.

4

P.M. Naghdi, "Theory of Shells and Plates", Handbuch der Physik, Vol. 13, S. Fl~gge (editor), Springer-Verlag, 1973. E. Kroner, (editor), I.U.T.A.M. Symposium, "Mechanics of Generalized Continua", Springer-Verlag, 1967. S. Timoshenko, S. Woinowsky Kreiger Theory of Plates and Shells, McGraw Hill, 1959. Z.M. Elias, "Dualities in Finite Element Method", J.Eng'g. Mech. Div., A.S.C.E 94, EM4, pp. 931-946, (1968). R.D. Mindlin, "Influence of Couple-Stresses on Stress Concentrations", Experimental Mechanics, Vol. 8, pp. 1-7, (1963) R.V. Southwell, "On the Analogues Relating Flexure and Extension of Flat Plates", Quart. J. Mech. and Appl. Math., Vol. 3, pp. 257-270, (1950).

i0.

Z.P. Bazant, M. Christensen, "Analogy Between Micropolar Continuum and Grid-Frameworks Under Initial Stresses", Int. J. Solids Structures, Vol. 8, pp. 327-346, (1972).

ii.

M. Farshad and B. Tabarrok, "Dualities in the Analysis of Cosserat Plate", UTME-TP 7601, Tech. Pub. Ser., Dept., of Mech. Eng., Univ. of Toronto, (1976).

12.

W. Nowacki, "Theory of Asymmetric Elasticity PWN, Warszawa, 1970.

(in Polish)