The eigen-value problem associated with the general problem of structural sta

bility. W e consider a given configuration of a deform able bo d y a n d an equilibrium system of body and surface stresses w hich is given to w ith in an a r b itra ry fa cto r X. I f X is sufficiently small, this equilibrium configuration will be stable; we ask fo r that value o f \ fo r which it becomes indifferent, assuming that the additional stresses which are produced by infinitesimal displacements fr o m the given equilibrium configuration are linearly re­

lated to the corresponding infinitesimal strains. T h is critical v alu e of X will be called th e safety factor of th e considered equilibrium configuration. W ith resp ect to a system of re c ta n g u la r C artesian coordinates Xi, le t us d en o te th e co m p on ents of th e given stresses b y Xtr,-,- an d th e com ponents of an infinitesim al d isp lacem en t from th e given equilibrium configuration by If th e u n it vecto r along th e o u tw ard n orm al to the surface is deno ted b y m , th e surface stresses are

\ T j = XcT ijiii. (1 )

T h e q u a n titie s a n m u st satisfy the equilibrium conditions

an.i = 0, (2)

w here th e su b scrip t i a fte r th e com m a denotes differen tiatio n w ith resp ect to Xi, an d th e usual su m m atio n convention regarding re p eated su b scrip ts is ad o p ted .

T h e infinitesim al stra in associated w ith the d isplacem ents is given by

Cij ~ 2 (2L,; T Ujti) . (3)

Since th e relation betw een th is strain an d th e corresponding ad d itio n al stress r,-,- is assum ed to be linear, we have

Tii — Cijkitki, (4)

w here Cijki is a fo u rth ord er tenso r w hich is sy m m etric w ith resp ect to i an d j an d w ith respect to k and I. If, in p artic u la r, m an d e,-,- are assum ed to be related to each o th er b y th e generalized law of H ooke, we have

Cijki = 2Go^S{k5ki — — OijSkiJ^, (5a)

where G o denotes th e m odulus of rig id ity , v Poisson’s ratio , an d 8 a is th e K ro n eck er delta. If th e body u n d er consideration can be expected to behav e like an iso tro pic elastic solid for an infinitesim al d isp lacem ent from th e given equilibrium co nfig ura­

tio n ,2 i.e. if th e stresses Xtr,-,- do now here exceed th e elastic lim it of th e m aterial, th e expression (5a) m ay be used in connection w ith th e stress-strain relation (4). On th e o th e r han d , w here th e stresses Xc,-,- exceed th e elastic lim it, different expressions m u st be used for Cijki according to w h eth er th e stresses r <}- associated w ith th e stra in s e,-,- c o n stitu te “loading” o r “u nloading.” We reserve th e com plete discussion of su ita b le stress-strain relations beyond th e elastic lim it for a n o th e r p ap e r an d give b u t one ex­

am ple here. Defining th e stress deviation as Sij = o n — \a k k8ij a n d its intensity as 5 = \sijSn, we set

2 M . A. B io t [J. Appl. P hys., 10, 860-8 6 4 (1939)] and, m ore recently, F. D . M urnaghan [Proc. N at.

Acad. Sci., 30, 244-247 (1944)] have pointed o u t th at an elastic solid under in itial stress can be strictly isotropic o n ly if the initial stress is of the nature of a h yd rostatic pressure. For th e con ven tion al structural m aterials, how ever, this small anisotropy caused b y the initial stress can be disregarded as lon g as the initial stress does n ot exceed the elastic lim it.

380 W IL L IA M P R A G E R [Vol. IV , N o. 4

(3) th e term x{" , finally, depends on th e ro ta tio n Wj,-; it rep resen ts th e change of stress, w ith respect to th e jfixed coordinate axes, w hich is produced b y th is ro tatio n . sis, th e ord er in which the d eform ation e,-,- and th e ro ta tio n wt)- are applied is im m a­

terial.

T h e an tisy m m e tric ten so r r'tJ depends only on T o find its m a th em atica l ex­

pression, it is therefore sufficient to consider a pure homogeneous deformation, i.e., a d eform ation for w hich m,.,- is in d ep en d en t of the coo rdin ates and «»,) = « /,» = e«/. On a c co u n t of (9), th e equ atio n s of equilibrium for th e deform ed bod y are

Since these eq u atio n s m u st hold n o t only for th e en tire body, b u t also for an a r b itra ry portion of it, we m u st have

Using th e sy m m etry of th e tensors a,-,- an d m in ad d itio n to th e Eqs. (10), (11), (2), and neglecting higher ord er term s, we m ay therefore w rite (12) in th e form

T h e ten so r d epends only on T o find its m ath em atica l expression, it is suffi­

cien t to consider a rigid bo d y ro ta tio n , i.e., a system of displacem ents iii w hich d e­

pend linearly on th e coordinates Xi an d satisfy B y th is ro ta tio n th e com ponents of th e infinitesim al force tra n s m itte d across a given surface elem en t are transform ed according to

F o r the considered rigid body ro ta tio n T ,-,= r<'y = 0. Using (8), (9), and (10), we m ay therefore w rite (14) in th e form

R e tu rn in g now to the consideration of a rb itra ry infinitesim al d isp lacem ents u it we w rite in accordance w ith (10), (13), and (15):

Since only first order term s in and need be considered in th e following

analy-or

(dijXk dikxfj'i 0. (12)

(13)

d j i — (Si, + Ui , j ) d f j = (8ij + o}ij)dfj — d f i + Uijdfj. (14)

(15)

\ < T i j = Xcr.-y + T , j + — O j k ( k i ) — Xcr ^{a-O Jfcy.}

On acco u n t of (2), th e equilibrium condition (11) furnishes therefore

[ r ,- y + ~ ~ j k ( k i ) — = 0,

and the condition d j i = d fi furnishes

[r,-,- + iX(<Tik(ki — O jk tu ) ~ X oikO>kj\n> = 0.

(16)

(17)

(18)

382 W IL L IA M P R A G E R [Vol. IV, N o. 4

th e class of a d m itte d fu nctions is re stric ted an d th e v aria tio n a l problem sim plified.

func-384 W IL L IA M P R A G E R

tion w is odd in x3 as well as in x 3. T ak in g ac co u n t of th is fact, and keeping in m ind th a t an is odd in x2 and even in *3, we find th a t

j'

a u ( u k i u k\ — «ki(ki)dv = — 2c

J

^u'6'xldv = — 2c l3

j*

^{u'6'd xu (26)}

w here I3 denotes th e m om en t of in e rtia of th e cross section w ith respect to th e a;3-axis.

W e now proceed to th e ev alu atio n of term Cpqr,epqers in (19). W ith ^ = 0, Eq. (5a) tak es th e form £;,-*( = 2Go§ii§j! and the' stress-strain relation (4) reduces to

n , = 26V,7. (27)

In ap plying this, we shall replace 2G0 b y £ 0 w henever i —j . In view of (25), we have Cpqrst pqtrj = Tpqepq = Eo(xzu" + •w8")~ + 4Go(ti2 + eh), (28) w here ti2 and ei3 depend on the tw ist 6r and on th e w arping w per u n it tw ist in p re­

cisely th e sam e m an n er as in th e case of p ure torsion. In th is case, how ever, th e in­

tegral of 4Go(e?2+ i ? 3) o ver th e cross section equals.

G

q

C6'2,

w here

G0C

d en o tes th e to r­

sional stiffness of th e beam . A dop ting th e w arping w p er u n it tw ist found in th e case of p ure torsion, and setting*

r = J w2dA, (29)

w here dA denotes the area clem en t of th e cross section, we o btain

f

Cpq„epqer.dv = £0/2

f

u 'n dxi + £ 0r

f

^{+ G0C}

f

^{6n d x u (30)}

J J 0 0 ” 0

where / 2 is the m om en t of in ertia of the cross section w ith resp ect to th e a:3-axis.

S u b stitu tin g the expressions (26) and (30) into (19), we o b tain

£o/2miv + \ci3e" =

⁰

, £»reIV - G0ce" +

\ c i3u "

= o (

^{3 1}

)

as the E u ler eq u atio n s for o u r problem , and

6" — 0 for Xi = 0 and a,'i = I (32) as the n a tu ra l b o u n d ary conditions. In ad d itio n to these n a tu ra l b o u n d a ry conditions, we have th e im posed b o u n d ary conditions

8 — u — u' = 0 a t Xi = 0 and Xi = I. (33) T h e safety fa c to r X is found as th e low est eigen-value of th e problem fo rm u lated by Eqs. (31), (32) and (33).

* N o te th a t for the doubly sym m etric section considered here th e point x u 0, 0 is the shear center of th e cross section *j. Sin ce w is odd w ith respect to xt and x 3, w e have «> = 0 a t th is poin t. T h ese remarks identify the definition (29) w ith th a t given b y J. N . Goodier, E ng. E xp . S tation , Cornell U n iversity, B u lle­

tin N o . 27 (1941), p. 9.

U N S T A B L E S O L U T IO N S O F A C L A S S O F H IL L D IF F E R E N T IA L E Q U A T IO N S *

B Y

G A B R IE L H O R V A Y M cDonnell A ircra ft Corporation

1. In tro d u ctio n . L in ear d ifferen tial eq u a tio n s w ith periodic coefficients p lay an im ­ p o rta n t role in problem s of engineering an d physics. T h e best-kn ow n of these e q u a ­ tions is M a th ie u ’s eq u a tio n . A som ew hat m ore com plicated e q u a tio n is

d*v 1- \9 -ie-u + + d -p r'* + e0 + 0ie{* + d o e ^ v = 0 (la )

# 2

which reduces to M a th ie u ’s eq u a tio n for

do 1

6

o,

⁶

—i =

6

—i =

⁶

\,

⁶

— 2 = 02 = 0,

w here the asterisk is used to den o te th e co n ju g ate com plex q u a n tity . T h is p ap e r is concerned w ith th e d e te rm in a tio n of the solutions

= <?* £ c * lk* (2)

—co of E q. ( la ) su b je c t to th e re stric tio n s

00 = 00, 0-1 = 01, 0-2 = 02 (lb )

and

0 i = 0 ( m ) , 0 2 = O( f f ) , ( 3 )

w here | i i s a sm all positive q u a n tity . I t will be seen th a t solution of th e problem in ­ volves th e d e te rm in a tio n of the “c h a ra c te ristic ex p o n e n t” cr from th e eq u atio n

sin iica = \ / © sin tt\/0o( (4)

w here © d en otes th e expansion

© = 1 + C 55 + C.€ + CnV + CSS* + C !t5e + • ■ • (5) in th e th ree real com b in atio n s

5 = 0_i0i, £ = 0-2^2, v — 2(616-2 + 6-162) (6a) of th e fo u r q u a n titie s, real an d im ag in ary p a rts of 0i a n d 02. © is a pow er series in ju2 since

5 = 0(m2), £ = 0(m4), V = 0(m4). (6b) The coefficients C of the series depend on do alone.

T h e num erical ev a lu a tio n of th e coefficients of th e ex pan sio n is th e p rincipal aim of th is p ap er. T h is is b est accom plished by first re-expressing th e “d o u b ly in fin ite”

I l i l l determinant D in term s of its “sim ply in fin ite” p rin cip al su b d e te rm in a n ts D n,

* Received June 13, 1946.

386 G A B R IE L H O R V A Y [Vol. IV, N o. 4

k ——l: 0 2 C _ l + 0 l C o + [ ( f l r + i ) 2 + ö o ] c i + ö _ l C 2 + 0 _ 2 C 3 = 0 ,

k — —2: diCo-\-diCi~\- [(cr+ 2 i)2+0o]c2+0_iC3+0_2C4= 0,

T h e eq u a tio n s are co n sisten t if th e ir d e te rm in a n t, A(o-), vanishes. T h e consistency criterio n

A (<r) = 0 can be expressed in the m uch sim pler form 3''1

sin i i r a — ± %/D sin t \ /⁶o,

w here

0 m A ( 0 ) =

(¹⁰)

(4)

1 d -\yi d-oy-i 0 0

dll’l 1 d-iyi d-2>’i 0

d^yo ffiyo 1 d-iyo 0-2>’O

0 62yi öiyi 1 d -iy i

0 0 02>'2 d iy2 1

(11a)

is th e d e te rm in a n t of system (9) for cr = 0 w hen each e q u a tio n is d iv id ed b y th e coeffi­

c ie n t of th e diago nal term , an d

L _ . a m

0 is e ith e r positive or n egative, a n d so is do■ T h u s th e q u a n tity y / v s' n ny/do >s e ith e r real or p ure im aginary. In th e first case se t

q = \Æ T sin ir\/do = — \ / ~ - 0 sinh v y / — do. (12a) In th e second case set

q' = q / i = — V — 0 sin

x\/

0

o

= — \ / 0 sinh x \ / — 90. (12b) T h en the solution of th e tra n sc e n d e n ta l e q u a tio n (4) is given by

1

+ a = — log ( q / i + V 1 — ç2) + m i

X

- ?

(m — 0, + 1, + 2, • ■ • ),

+ m i for — 1 ^ q 1, (13a)

log (? + V7?2 “ 1) + (w - 5') 2 for ? g — 1, q ^ 1, (13b) IT

— — log (<?' + \ / g ' 2 + 1) + m i for q im aginary. (13c) x

= — arc tan _____

22 y l - ?2 l

3 W hittaker and W atson , A course of modern a n alysis, Cam bridge, 1927, p. 416.

* M . J. O. S tru tt, LanUsche, M athieusche und verwandle Funktionen in P h y sik und Technik, Springer 1932 (Edw ard B ros., 1944), p. 22.

388 G A B R IE L H O R V A Y [Vol. IV , N o. 4

solutions are u su ally of g re a te st in te re st, because th ey se p a ra te th e /¿-regions of s ta ­

Since, b y v irtu e of (lb )

E n = Z>* = D n, (21a)

T n = 5* ( ^ S„, when $* dh 0* ^ 0s) (21b)

an d by v irtu e of (2) an d (19b)

S nTm = 0 ( M2), (21c)

a re p lacem en t of b y D n a n d a re p e a te d insertio n of (19b) in to (20) g ra d u a lly elim i­

n a te s all b u t th e D„ ty p e of d e te rm in a n ts from th e expression for D . I t is also found t h a t 0i, 0_i, 02, 0-2 a p p e a r o nly in th e co m b in atio n s 5, e, 17 given in (6a). T h u s, b y v irtu e of (6b), th e expansion of D progresses in pow ers of /12. U sing th e n o ta tio n

2

yon2 = >o>i>2, (22a)

one finds t h a t to ni0 term s

© = D0D1 — SyoiDiD2 — t(yo2D\Dz + yuD 2) + ⁷?(²yoii^{© 2} + ’¿yauD 1D3)

— 5 i ( 4 y o n 2 © 2 © 3 + 2 y o i 2 3 © i© 4 ) + e2(> o i2 3 © i© 4 + 2 y a n t D 2D z) 2

+ ejj(4yoii23©2©4 + 2 y n » * P J h + 2yoim©3)

— 5e2(4yoii234©2©5 + Ayo\ i22zDzDi -f- 2yoi2345©i©6) + - - - . (23)

T h e sam e process can also be ca rried o u t for th e sim ply in finite d e te rm in a n t D n.

D isregarding th e exceptional case 6o = k2 (y* = <x>), one finds th a t to /x10 term s Do — Di — 5yoiD2 + ( — «>02 + 2r)yoi2)©3 + ( — 23e + e2)yoi23©4

-f- 2€77yoi234©5 — 25e2yoi2343D6 + • ■ ■ , (24a)

a n d D„ is o b ta in e d from D0 b y increasing th e su b sc rip ts in th e l a t t e r ’s expression b y n.

(24b) I t will be c o n v e n ie n t to in tro d u c e a t th is p o in t th e n o ta tio n

Z

^{T3S6 = y 3 5 6 + 3'467 + >578} + ‘ - , (22b)

Z >12 ¿ C >356 = >12 Z >356 + >23 Z > « 7 + >’34 Z > 5 7 ? - » - ■ • • . ( 2 2 c )

N o tin g th a t

lim Dn = 1, (25)

n—»«

one o b tain s, b y re p e a te d ap p lica tio n of (24),

Do = (1 — 5>’oi)Z?2 + ( — Syi2 — «>02 + 2ijyoi2)©3 d* • ■ ■

= [f — (>oi + >1 2 ) 5 — y 02« + 2yoi27i]£)3 + • • • - (26a)

= 1 + A°s5 + A°,e + A lv + ■ ■■ , (26b)

w here

0 ^ ^ 0 ^ 0 %1 0

d j = — Z >01^{, A , = — Z >}02^, d , = 2 Z >012^, d g 1 = Z >01^{Z >}23, • • ■ . ( 2 7 )

390 G A B R IE L H O R V A Y [Vol. IV, N o . 4

T h e coefficient of a general term , like ôe2, is o b ta in e d as follows: E q u a tio n (24a) gives rise to th e following sym bolic p ro d u c ts co n tain in g 5e2:

[ - 5y0i] [ - eyw\ [ ~ «ryoa], (a)

[ — <5yoi] [«2yoi23], (b)

[ — eyo2][— 25ey»oi23 J. (c)

[l] [ — 25e2yoi2346]. (d)

I t is found t h a t (a) c o n trib u te s

— 22 3« 22 324 22 y ^ ~ 22 yo222 iy3422 357 ~ 22 30222 335 22 3«? (28a) to A ° (,. (N o te t h a t no su b scrip ts can be re p eated , no r can a n y be skipped as one passes from one 22 to th e n e x t 2 2 ; fu rth e rm o re (a) gives rise to th ree d is tin c t su m m a ­ tion expressions, because yi-.r+i can a p p e a r in th e first place, in th e second place, an d in th e th ird place.) T h e relatio n (b) yields

— 22 yoiE y^a — 22

^{^0123}

22 y*^ (

^{2 8}

b)

which (c) yields

2 22 N02 22 3*3456 + 2 X) y„m £ 3*46 (28c) a n d (d) yields

— 2 ^ 3 3012345- (28d)

T h e c o n trib u tio n s (28a, b, c, d) sum up to {0, Se2}. B y increasing th e su b scrip ts of th e expressions (28) b y 2, one o b tain s

A}S = - 22 3*23 22 346 22 379 - 22 3*24 22 3*56 22 379 - 22 324 22 367 22 389

— 22 3*23 22 3*4667 — 22 3*2345 22 3*67 + 2 ^ 3*24 2 2 3*6678

+ 2 ^ ) 32346 2 2 3*68 — 2 2 2 3*234567. (29)

T h e d e te rm in a tio n of the o th e r [n, S V + 'j is sim ilar.

T h e nu m erical values of th e coefficients {11, S'Vr)k ) are given in T a b le I to 5 decim al places, for 0O ranging from + 0 .9 to —1.0 (th e in te re stin g range in h elico pter th eo ry ). In the ev a lu a tio n of {», ô'Vrç*} th e first 51 y m w ere ta k e n in to acco u n t.

(T he a c cu ra cy o b tain ab le is th u s e q u iv a le n t to th e use of a 101-row a p p ro x im a n t to 0 . ) y o to 320 w ere co m p u ted in som e in stan ces to 6, in som e in stan ces to 7 decim al places; 321 to 350 w ere c o m p u ted to 7 decim al places. I t is exp ected t h a t th e e n trie s of T a b le I are in erro r by n o t m ore th a n 2 u n its in th e fifth decim al p lace .6

I t is read ily seen t h a t th e p re se n t m eth o d is n o t lim ited to E q. (1), b u t can be ex te n d e d to th e general H ill d ifferential e q u a tio n w here 8meim* form a co n v e rg en t series.

F o r th e special case of M a th ie u ’s e q u a tio n (e = 7; = 0), one finds b y (23) an d (27) th a t

0 = D^Do - h y M = 1 - 2 5 £ — ~ - - i — — + 0 (5 2)

Ar— 0 0 0 — V o — ( k - f - \ ) £

7T COt 7V\/Oo

= 1 - 2 5 — = + 0 (5 2). (30)

(4 0 o - l ) V 0 o

6 A n experienced com puter can calculate a colum n of T ab le I in som ew h at less than a day.

T h is fo rm u la w as used b y H . B rem ekam p in 1926 in a s tu d y of th e flow of electro ns in m e ta ls.6

4. E xam ple (a). G iven 0 = 0.2, 6i = 0.19685+ 0.3 34 65 f, 02 = 0.03875-0 .1 0 2 5 8 L D eterm in e i>i, v<t. One finds 5 = 0.15074, « = 0.01202, 77= —0.01635, an d , b y T a b le I, A . = 1.8422, A = 0.9434, A = 0.9934, D 3 = 0.9980, A = 0.999, A = A = 1.000. L ike­

wise, b y T a b le II, 0 = 2.3291. T herefore, g = - \ / 0 sin ir^/do— 1.5052 a n d by (13b) (Ti = 0 .3 0 7 8 2 + 7 /2 , cr2 = —0.30782 —i/ 2 . T h e associated fu n ctio n s i'iC/0 a n d Vt(jp) are d eterm in e d from the eq u a tio n system (9). N orm alizin g to c0= 1, a n d using th e e q u a ­ tions k = —4 to k = + 4 , one o b ta in s7

(

^{'P 'P}

= ( - 0.1898 + 1.9817i)fi+0-3078t< — 0.0958 cos — + s i n b 0.2076 cos —

I 2 2 2

3ip 5ip 5 \p 7\p

- 0.0083 s i n 0.0125 cos — - 0.0038 s i n b 0.0008 cos —

2 2 2 2

392 G A B R IE L H O R V A Y [Vol. IV, N o. 4

7 <p + 0.0019 sin — +

2

}

c \p \p

vt = (1.6652 + 0.7467f)e-°-307^ < c o s b 0.4484 s i n b 0.2107 cos —

I 2 2 2

3xP 5<P 5\p 7\p

-b 0.0188 s i n b 0.0041 c o s b 0.0101 sin — + 0.0005 cos —

2 2 2 2

+ 0.0020 sin — +

2 }■

5. E xam ple (b). G iven 0o = O, 01 = O.37249-bO.63323f, 02 = 0.13875-0 .3 6 7 2 8 L D e­

term ine <r. One finds

q = x\/D 0o-sin x \/0 o /x \/0 o = x \ / 0.4392 = 2.082, (t 1 — — <r2 = 0.4339 -b i /2 .

8 M . J. O. S tru tt, loc. cil., p. 26.

7 N o te th a t th e use of <n = 0 .3 0 7 8 2 — i / 2 lead s t o th e ab ove expression o f fi w hen C\ is norm alized to 1, and to the con ju gate com plex of the a b o v e when c0 is norm alized to 1.

T a b l e I.* N um erical v alu es of \n , b't’i f }.

394 G A B R IE L H O R V A Y

T a b l e II. Expansion of 5D (row 1 X row 2 X r o w 3, see E q . 23).

397

O N T H E M E C H A N IC A L B E H A V IO U R O F M E T A L S IN T H E S T R A IN -H A R D E N IN G R A N G E *

BY

G. H . H A N D E L M A N , C. C. L I N a n d W . P R A G E R Brovin U niversity

1. In tro d u c tio n . T h e p re se n t p a p e r is concerned w ith ce rtain stre ss-stra in re la ­ tio n s p u rp o rtin g to describe th e m echanical b eh a v io u r of qu asi-iso tro p ic m e ta ls in th e stra in -h a rd e n in g range. As a p re p a ra tio n for a m ore precise ch a ra c te riz a tio n of these relations, le t us consider th e tension te s t of a m etal like copper or alu m in u m w hich does not flow under a constant stress, but exhibits strain hardening. If th e te s t involves loading only, i.e., if th e reduced tensile stress1 <r o r th e tensile stra in e in ­ crease th ro u g h o u t th e te s t, th e re su ltin g diag ram of redu ced stress versus s tra in will h av e th e general ap p e ara n ce of th e curve OPQ in Fig. 1. On th e o th e r h an d , if th e te s t specim en is unloaded a fte r a cer­

tain p o in t, such as P , has been reached along th is curve, th e stre ss-stra in d ia ­ gram for u nloading is found to be v ery n early a s tra ig h t line P A w hich is p a r­

allel to th e ta n g e n t of th e curve OPQ a t 0 . A fte r com plete unloading, th e specim en shows a p e rm a n e n t extension w hich corresponds to th e p e rm a n e n t stra in re p resen te d b y OA.

T o sim plify th e discussion, le t us assum e a t p re se n t t h a t th e m ateria l is Q' incom pressible. A longitudinal extension

e of th e isotropic specim en is th en ac- Fig. 1. T ypical curve of reduced stress v s. strain,

com panied b y a uniform la te ra l con­

tra c tio n of th e m a g n itu d e e/2. If th e discussion is re stric te d to s ta te s of stress an d stra in w hich can be reached b y a single loading followed b y one com plete or p a rtia l unloading a t th e m ost, th e m echanical b eh a v io u r of th e m a te ria l in sim ple tension is th ere fo re com p letely defined b y th e cu rv e OPQ. I t will b e assu m ed in th e following t h a t for th e m ateria ls u n d e r consideratio n th e stre ss-stra in d iag ram in sim ple com ­ pression (O P 'Q ' in Fig. 1) is o b tain ed b y reflecting th e cu rv e OPQ w ith re sp e c t to th e origin 0 , an d t h a t th e p ra ctically im p o rta n t po rtio n of th e cu rv e Q'OQ, i.e., th e portion corresp ond ing to sm all an d m o d era te strain s, is re p resen ted w ith sufficient accu racy b y a d ev elo p m en t of th e form

« = <7 + a3<r3 + a6iP + • • • , (1) w here a3, as, ■ ■ ■ are c o n stan ts. (T he coefficient of th e lin ear term on th e rig h t-h a n d side of (1) m u s t be u n ity since a is th e reduced stress. N o even pow ers of a can occur

* R eceived Septem ber 17, 1946.

1 T h e reduced stress is defined as th e q u o tien t of th e stress b y Y ou n g’s m odulus.

on th e rig h t-h a n d side of (1), because th e stre ss-stra in d iag ra m s for tension an d com ­

194-7] M E C H A N IC A L B E H A V IO U R OF M E T A L S 399

w here 5,-y is th e K ronecker d e lta . S im ilarly, th e reduced mean normal stress s a n d th e deviation 5tJ- of the reduced stress are defined as

s = j <ru (7)

and

si; cT\j ” sSij. ( S )

A ccording to the definitions of th e d ev iatio n s e,j an d S u , we have

eu = 0, sa = 0. (9)

T h e ta sk of generalizing th e finite stress-stra in relatio n (1) is sim plified b y th e re m a rk t h a t th e first term on th e rig h t-h a n d side re p resen ts th a t p a r t of th e to ta l stra in e w hich is recovered upon complete unloading. T h e rem ainin g term s on th e rig h t- h an d side of (1) accordingly re p resen t th e p e rm a n e n t stra in . In Fig. 1 the total s tra in is rep resen te d b y th e segm ent OB, th e recoverable s tra in b y A B , a n d th e permanent stra in by OA.

S e ttin g

U j = 4 + t'/ j, ( 1 0 )

w here e(, d en o tes th e recoverable an d ef/ the p e rm a n e n t stra in , we m ay assum e th a t th e recoverable stra in is related to th e reduced stress b y m eans of th e generalized law of H ooke:

4 = ( 1 + v ) s a + ( 1 - 2v)s5u. ( U )

H ere v d en o tes P o isson's ra tio . W e are th en left w ith th e ta s k of su p p lem en tin g (11) b y a re latio n w hich expresses th e p e rm a n e n t s tra in o ccurring d u rin g th e first loading in term s of th e reduced stress. F o r an isotropic m aterial, th is relatio n can only con ­ tain scalar c o n sta n ts in a d d itio n to th e ten so rs e[J, a i}- an d 5¿¿, a n d th e ir in v a ria n ts.

F u rth e rm o re , th e principal axes of e f an d a n m u st coincide. U n d er th e pressures com ­ m only en co u n tered in th e te stin g of m aterials, no p e rm a n e n t change of volum e is observed, i.e., « «'= 0 an d e!f = e'f. A s ta te of h y d ro sta tic pressure th erefo re does n o t produce a n y p e rm a n e n t stra in , a n d tw o s ta te s of stress w hich differ only b y a s ta te of h y d ro sta tic pressure m ay be expected to produce id entical p erm a n en t strain s. T h e p e rm a n e n t s tra in (¡J is th u s in d e p e n d e n t of s an d dep en d s only on th e d ev iatio n s,j.

F u rth e rm o re , if th e stress-stra in d iag ra m s for sim ple tension an d sim ple com pression are co n g ru en t, a reversal of th e signs of all stresses m ay be ex pected to prod uce a m ere reversal of th e signs of all principal strain s. F inally , if th e ra tio s of th e principal stresses are k e p t c o n sta n t d u rin g the loading process, th e ra tio s of th e p rincipal p e r­

m a n e n t strain s, too, can be expected to rem ain co n stan t.

In a re cen t p ap e r,2 W. P ra g e r established th e m o st general stress-stra in re latio n w hich is co m patible w ith th e preceding p o stu lates. W ith th e n o ta tio n s

J 2 ^ \ S i j S j i , J 3 = 3S i j S j k S k i , ( 1 7 )

and

t i j — Si k S l c j 3-7 26 i j, ( 1 3 )

P ra g e r’s stress-stra in relatio n can be w ritte n in th e form

* W . Prager, Strain-hardening under combined stresses, J. Appl. P h ys. 16, 837-840 (1945).

u'i = F {J2, A ) [ P ( / 2, + Q(J2, A ) Jz hi], (14) w here P an d Q m u st be hom ogeneous in th e co m p o n en ts of th e stress d ev iatio n , th e degree of P exceeding t h a t of Q b y 4. T h e expressions (12) a re second a n d th ird ord er in v a ria n ts of th e stress d ev iatio n Sij (th e first o rd e r in v a ria n t Su van ishes). T h e te n so r

u'i = F {J2, A ) [ P ( / 2, + Q(J2, A ) Jz hi], (14) w here P an d Q m u st be hom ogeneous in th e co m p o n en ts of th e stress d ev iatio n , th e degree of P exceeding t h a t of Q b y 4. T h e expressions (12) a re second a n d th ird ord er in v a ria n ts of th e stress d ev iatio n Sij (th e first o rd e r in v a ria n t Su van ishes). T h e te n so r

W dokumencie The Quarterly of Applied Mathematics. Vol. 4, Nr 4 (Stron 72-100)