Vol. 20, 1969 Nurze Mitteilungen - Brief Reports - Communications br~ves 979

4. Schlussbetrachttmgen

Dank der ausserordent l ich hoben Bandbre i t enkompres s ion ist das thermische Rauschen sehr gering. Die Grenze der Empf ind l i chke i t unseres A b t a s t s y s t e m s scheint, wie aus D i a g r a m m 6 und 7 ers icht l ich ist, eher durch S t6rspannungen gegeben zu sein. Diese werden beispielsweise durch Ref lexionen infolge Feh lanpassung oder durch Ampl i tudenschwankungen der Ab ta s t impu l se erzeugt. Bet Verwendung eines sym- metr i schen Sampl ing Gates wiirden al lerdings diese Ampl i t udenschwankungen nicht mehr auf den Ausgang t~bertragen werden. Man miisste jedoch die Erzeugung eines b ipo la ren Ab ta s t impu l se s in Kauf nehmen.

Ich danke Her rn Dr. A. S~AH ft~r seine wer tvol len Diskuss ionen und Anregungen sowie Her rn Dr. E. BAs ffir die in grosszfigiger Weise zur Verf~lgung gestel l te Fern- fokus-Elek t ronenkanone . Besondern Dank geb/ ihr t Her rn J. CHERIx far die sorg- f/iltige Real is ierung der verschiedenen e lekt ronischen Schal tungen.

L I T E R A T U R V E R Z E I C H N I S

[1] H. RIEDLE, NTZ 3, 135-140 (1957). E2] J. MARCUS, Echantillonnage et q~anti/ication (Ed. Gauthier-Villars, Paris 1965). [3] E. B~s und F. GAYDOU, Z. angew. Phys. 71, 370-375 (1959). [4] J. MOLL, S. IKRAI<AUER und R. SHEN, Proc. I R E 50, 43-53 (1962). [5] H. FELDMANN, Thesis, Berlin 1964.

Rdsumd

Nous d@crivons un syst@me d'6chantillonnage destin6 & reproduire le signal pr61ev@ aux bornes d 'une sonde capacit ive situ6e ~ l'int@rieur d 'un cyclotron isochrone. L'6chantil- lonnage est effectu@ -A l 'a ide d 'une porte Iin4aire qui est contr616e par des impulsions ex t r ;mement 4troites et dont la phase varie r4guli~rement entre deux limites. La fr4quence de ces impulsions n 'es t qu'une fraction de la fr6quence du signal ~ 6chantillonner; cela permet d 'op4rer avec une excursion de phase qui est 6galement une fraction de 2 =. (Eingegangen: 27. Mat 1969.)

Kurze Mitteilungen - Brief Reports - Communications br@ves

Singular Stress Concentrations in Plane Cosserat Elasticity

By STEPHEN C. COWIN, Dept. of Mechanical Engineering, Tulane University, New Orleans, Louisiana, USA

Based on the theory of Cosserat elasticity, closed form expressions are presented here for the singular par t of the stress in the problems of the half plane under concentrated normal and tangential loads and under discontinuous normal and tangential surface tractions. These same problems were considered in the context of the couple stress theory, which is a special case of Cosserat elasticity, by MUKI and STERNBERO [1] and by BOGY and STERNBERG [2]. The formulas for the singular par t of stress given here differ from those given in [1, 2] only by the occurance of a dimensionless number N called the coupling number. N was defined in [3] where i t was shown tha t N must satisfy the inequali ty 0 < N < 1. "When N is set equal to one in the expressions presented here the results of

980 Kurze Mitteilungen - Brief Reports - Communications br~ves ZA~P

[1, 2] are recovered. When N is set equal to zero the same expressions reduce to the corresponding expressions in classical l inear elasticity.

The theory of plane Cosserat e last ic i ty was developed by SCtiAEI~R [4]. In the nota t ion of [3] the cons t i tu t ive equat ions for plane Cosserat e last ic i ty are

3 1 1 = ( ~ + 2/~)s n + z % 2 , 3 2 2 = ( Z + 2 / ~ ) e 2 2 + e u , 1

T 1 2 4 - 612= 2/~e12-- 2 3 ((0[121 @ %0112] ) , /~1[121 = 2 r/ %O[12],1 ' /~g[12] = 2 r] %O[12] '2' I (1)

w h e r e e 1 1 , 822 and e~, are components of the usual inf ini tesimal symmet r i c strain tensor; 31~, 3̀ 2̀ 2 and T12 are components of the usual symmet r ic stress tensor; A and/~ are the usual Lam~ modul i ; w[12] is the component perpendicular to the plane of the usual ro ta t ion vec tor ; %op2j is the componen t perpendicular to the plane of the vec tor represent ing the to ta l ro ta t ion of the Cosserat t r iad ; %2 is the skew symmet r i c par t of the shear stress; #lira] and/~.z[12] are components of the couple stress tensor; ~ is a modulus of ro ta t iona l r igidi ty and ~l is a ro ta t iona l gradient modulus. SCHAEFER [4] int roduces the stress funct ions r and %0,

h i = r - ~ ,1 `2 , 3~`2 = r n + ~ , t '2~ h 2 + %2 = - r + %o,~i, i ! 2#1[1~]= V, 1, 2#~[r2]=%o,2

which, using the equi l ibr ium conditions, can be shown to sat isfy the differential equat ions

( '~ ) ~ r ~ _ N v ~ ~7~%o=0, (3>

where l is a mater ia l p roper ty wi th the dimension of length and N is the dimensionless coupling number ,

l = V ~ ' N ~ y # + - r ~ (4)

Plane Cosserat e last ic i ty wi th N = 1 is ident ical wi th the plane couple stress theory. When N = 1 the stress funct ions r and %O coincide wi th those employed in the couple stress theory and when l = 0 the funct ion r coincides wi th the Airy stress funct ion of classical elasticity.

In a series of papers MUKI and STERNBtgRG [13 and BOGY and STERNBERG [2, 5] under took a p rogram to solve, in the con tex t of the couple stress theory, certain plane strain problems which, according to classical elast ici ty theory, give rise to infinite con- centra t ions of stress. Mux<I and STElaNBE~G [t] de te rmined closed form expressions for the singular pa r t of the stress in the problems of the half p lane under concent ra ted normal and tangent ia l edge loads. By paral le l ing the developments of [1] these problems can also be solved for the Cosserat theory. Le t the concent ra ted load _P be applied at the origin of the half plane del imited by -- co < ;q < oc and 0 < x 2 < co and let 0(1) denote terms t h a t are finite and cont inuous th roughou t the half plane and its bounding plane. The singular par ts of the stresses for these two problems are given by the following equat ions:

Concentrated Normal Load - 2 P x~ z~ [1 - 2 (1 - v) N~]

+ N ~ 0(~), 311 = " "iN r 4 [1 + 2 (1 -- f) N ~]

+ N ~ 0(1), 322= a r 4 [1 + 2 (1 - ~) N 2]

- - 2 P x i [x~ 4- (1 - - 'v) N 2 (.*'12 - - x~>] + N~ o(1), (s) h 2 = ~ r ~ [ 1 + 2 ( 1 - - v ) N ~3

- 2 P x ~ (1 - ~) N ~ -1- N 2 0(1) ,

- - P ( 1 -- v) N =logr 4- Ne 0(1), /~2[~'2J= N=0(1) . /~1[12]= z [ 1 4- 2 (i -- v) N ~]

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Kurze Mitteilungen - Brief Reports - Communications brgves 981

Concentrated Tangential Load

2 B x fIX1 ~ + 2 (1 -- v) N 2x~] N i0(1) ] z n = ~ r 4 [ l + 2 ( 1 - v ) N ~] ~ ' I

2 P x ~ x ~ [ 1 - - 2 ( 1 - - v ) N 2] N ~0(1) ] z'29-= ~ r 4 [1 + 2 (1 -- v) N 2] +

2 P x 2 [x~ + ( I -- v) N 2 (x~ -- x l ) ] N 2 0(1) [ (6) ~12= :~r ~ [ 1 + 2 (1 -- v) N 2] + '

/ - - 2 P x ~ (1 - - v) N 2 = N 2 0 ( 1 ) = N ~ 0(1) r 2 1 1 + 2 ( 1 - - v ) N ~] + N 2 0 ( 1 ) ' #~[1~.] , #~.[12] ' ]

The coordinate sys tem used for expressing the results g iven above is ro ta ted by n ine ty degrees f rom the one employed in [1]. For N = 1 equat ions (5) and (6) reduce to the corresponding equat ions in the couple stress theory, namely equat ions (4.12) and (4.13) of [1], when al lowance is made for the ro ta t ion of the coordinate system. Fur thermore , for N = 0 equat ions (5) and (6) give the analogous equat ions for classical e last ic i ty (cf., for example, equat ions (4.14) and (4.15) of [1]). The s trong dependence of the s t rength of the s ingular i ty upon N is mani fes t in equat ions (5) and (6).

BOGY and STERNBERG [2] considered the p rob lem of the half plane under discontinuous normal and tangent ia l surface t ract ions. A load funct ion p(x~) t h a t was an t i - symmet r ic about the origin and had a finite j u m p d iscont inu i ty a t the origin was considered. Paral- leling the deve lopmen t of [2], a sympto t i c es t imates show t h a t the singular par ts of the stresses for the discont inuous normal load are

4 (1 - ~,) N~ p(0+) ~1~ = ~ = ~i~- ~ (1 - ; ) ~ ) l o g r + 0(~) (7)

while there is only one s ingular i ty in the stress for the discontinuous tangent ia l load, namely

4 p(0+) logr ~ ' = ( 1 + 2 ( 1 ~ ) m ) ~ + ~ (s)

When N = 1 the results (7) and (8) coincide wi th those given in [2] for the corresponding problems in the couple stress theory. For N = 0 the results (7) and (8) reduce to the corresponding results in classical e las t ic i ty ; note t h a t the s ingular i ty (7) in the discon- t inuous normal load vanishes in classical elasticity.

BoGY and STERNBERG [5] consider the plane strain problem of an or thogonal elastic wedge subjected to a rb i t ra ry shearing t rac t ions on one face and no other loading. They consider the p rob lem wi th in the con tex t of both the couple stress theory and classical e las t ic i ty theory. I f the loading fails to vanish a t the apex of the wedge, the classical theory gives rise to singularit ies in the stress and ro ta t ion fields a t the apex. BoGY and STERN- BERG [5] showed tha t these singularit ies will no t occur in the couple stress theory, and the i r resul t can be ex tended to show tha t these singularit ies will also not occur in the Cosserat theory.

An in te res t ing and inconsis tent result uncovered by BoGY and STERNBERG [5] has been here ex tended to the Cosserat theory. The results no ted in the previous paragraphs show t h a t there are problems for which the classical theory produces singularit ies which do not appear in the Cosserat theory. Conversely, the resul t (7) for the stress s ingular i ty in the p rob lem of the discont inuous normal t rac t ion is unbounded in the Cosserat theory ye t the corresponding resul t in classical e las t ic i ty is bounded.

A signif icant feature of the results (5), (6), (7) and (8) is t h a t their re la t ionship to the corresponding results in classical e las t ic i ty is immed ia t e ly apparen t ; one has only to set N = 0 to recover the corresponding resul t in classical elasticity. The relat ionship of the corresponding results g iven for couple stress theory in [1, 2, 5] to classical e las t ic i ty is much more difficult to see because they appear to be independen t of any pa ramete r character iz ing a devia t ion f rom classical elast ici ty, N hav ing been set equal to one.

982 Kurze Mitteilungen - Brief Reports - Communications brgves za,'vlP

R E F E R E N C E S

_FIJ R. MUKI a n d E. STERNBERG, T/ze In/luence o~ Couple-stresses on Singular Stress Conce~ztmtio~s ira Elastic Solids, Z. angew. Math . Phys . 76, 611 (1965).

E21 D. B. BOGY a n d E. STERNBERG, The E//ect o/ Couple-stress on Siragula~4ties Due to Disco~r Loadirags, In t . J. Solids S t ruc t . 3, 757 (1967).

I3J S. C. Cowlx , Stress Furacliom /or Cossemt Elasticity, In t . J. Solids St ruct . , in press. E41 H. SCHAEFER, Vemuch eimr Elastiaitiitstheo,4e des aweidimemional ebenen Cossemt-

Koratirauums, Misz. angew. Mech. (Akademie Verlag, Ber l in 1962), S. 277. [51 D. B. BOGY and E. STERNBERG, The Effect o/Couple Stresses on the Corner Singula~,ity

Due to ar Asymmetric Shear Loading, In t . J. Solids S t ruc t . 4, 159 (1968).

Zusammen/assung

Auf G r u n d der Cosse ra t schen E la s t i z i t~ t s theo r i e werden e x a k t e L6sungen gegeben fiir den s ingu la ren Tell des Spannungs %l des in der H a l b e b e n e u n t e r k o n z e n t r i e r t e n Norma l - u n d T a n g e n t i a l l a s t e n und u n t e r d i skon t inu ie r l i chen n o r m a l e n und t a n g e n t i a l e n Ober- f lgchenbe las tungen .

(Eingegangen: 29. Juli 1969.)

Ins tab i l i t i e s in a Perfec t ly 'Plastic Sol id Cyl inder under Axia l C o m p r e s s i o n

B y HENRY VAUGHAN, Dept . of Mech. Eng. , The U n i v e r s i t y of B r i t i sh Columbia , V a n c o u v e r 8, C a n a d a

Introduction

I n 1947, St-IANLEY EI~ showed how a co lumn m a y buck le u n d e r large axia l compress ive stresses. Bas ica l ly t he ax ia l compress ion causes a d o m i n a n t regular p las t ic flow and b e n d i n g develops d u r i n g th i s f low w i t h o u t caus ing a n y unloading , a t leas t n o t un t i l t h e b e n d i n g is well es tab l i shed . I n some c i r cums tances a s imi la r concep t m a y be used for a b o d y unde rgo ing two or t h r e e - d i m e n s i o n a l d i s tor t ion . The b o d y m a y be cons idered to be d i s to r t ed u n i f o r m l y b y t h e presence of d o m i n a n t u n i f o r m stresses a n d a smal l p e r t u r b a t i o n , a s sumed to be smal l enough t h a t loading r e m a i n s posi t ive, c an be i n t r o d u c e d and e x a m i n e d us ing a l inear t h e o r y supe r imposed on a un i fo rm mot ion . Th i s concep t was used b y GOODIER [23 in an e x a m i n a t i o n of t he buck l ing of t h i c k slabs. Assuming a s t a t e of p lane s t ress he e x a m i n e d the s y m m e t r i c a n d a n t i s y m m e t r i c ins tab i l i t i es t h a t c an occur w h e n a t h i c k s lab is compressed well b e y o n d t he elast ic l imit . A t a b o u t t he same t ime COWPER and ONAT [33_, a s suming a s t a t e of p l a n e s t ra in , e x a m i n e d s imi la r ins tab i l i t i e s in staSs. I n a d d i t i o n t h e y were able to d e m o n s t r a t e t he i n i t i a t i on of neck ing t h a t occurs u n d e r large tensi le loads. I n b o t h of t h e theor ies g iven in [21 a n d [31 a t a n g e n t modu lus was essent ia l for the occur rence of buckl ing .

More r ecen t ly GOODIER [41 ha s cons idered t he f o r m a t i o n of wr ink les in r e c t a n g u l a r p la tes u n d e r sus t a ined axia l compress ive flow. In p a r t i c u l a r t h e b e n d i n g m o m e n t in t he plate , assoc ia ted w i t h a smal l p e r t u r b a t i o n , was s h o w n to be fa i r ly insens i t ive to the h a r d e n i n g modulus . The d o m i n a n t t e rm, wh ich Goodier cal led t he d i rec t iona l m o m e n t , is due to po in t s t h r o u g h t he p la t e t h i ckness co r respond ing to d i f fe ren t po in t s on t he c u r r e n t yield ell ipse a n d is a consequence of cons ider ing a b i -ax ia l s t ress s ta te . The effect of the h a r d e n i n g a n d d i rec t iona l m o m e n t s on t he p las t ic buck l ing of cy l indr ica l shells ha s been cons idered b y VAUG*IAN a n d FLORENCE [51. Fo r a l u m i n u m a n d magnes iuna alloys the direc- t i ona l m o m e n t d o m i n a t e s for b iax ia l buck l ing and t he inc lus ion of a h a r d e n i n g modu lus changes n e i t h e r the p red ic ted mode n u m b e r no r the load requ i red to buckle the shells.