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An elastic-plastic plane stress solution using the incremental theory
Int. J . mech. Sc/. P e r g a m o n Press. 1971. Vol. 13, pp. 97-106. P r i n t e d in Great B r i t a i n
AN ELASTIC-PLASTIC P L A N E STRESS SOLUTION USING T H E INCREMENTAL T H E O R Y A . N . PALAZOTTO D e p a r t m e n t of Mechanical Engineering, University of Bridgeport, Bridgeport, Connecticut, U.S.A.
and N. F. MORRIS D e p a r t m e n t of Civil Engineering, New York University, Bronx, New York, U.S.A.
(Received 22 June 1970, and in revised form 8 October 1970) S u m m a r y - - A numerical procedure is presented to determine the elastic-plastic plane stress distribution for a rectangular plate acting under any t y p e of partial edge loading. The incremental theory of P r a n d t l - R e u s s is used in conjunction with a Ramberg-Osgood stress-strain relati~nship. A linear partial differential equation is arrived a t through the required load-history dependent constitutive equations. The necessity of incompressible material throughout b o t h the initial and subsequent stress range is eliminated. Finally, because of the assumed Ramberg-Osgood equation, a solution in plasticity can be carried out without a n y fixed yield point. A finite difference scheme is incorporated to solve the partial differential equation and application is made considering a highly concentrated b o u n d a r y loading. Comparison between the plastic and elastic solution shows t h a t for increasing load situations, the plastic effective stress is larger than an elastic equivalent effective stress in the so-called plastic region. NOTATION rectangular Cartesian co-ordinates t o t a l incremental strain tensor (Cartesian) incremental elastic strain tensor incremental plastic strain tensor Y Poisson's ratio E modulus of elasticity (7 shear modulus da~t incremental stress tensor Kronecker delta deviatoric stress tensor equivalent stress H" slope of the equivalent stress-plastic strain curve B material constant present in Ramberg-Osgood stress-strain equation exponent present in Ramberg-Osgood equation n Ca.] m a t r i x of known stress components equation (9) ¢ incremental function Pit_m2 stress constants present in equation (14) dT~, dTy x a n d y components of the external incremental b o u n d a r y tractions stress molecule coefficients present in Fig. 1 aspect ratio, width-height of plate known stress function ~ at b o u n d a r y nodes 97 X, y, Z d~j de~ de 5
98
A.N. P~-LAZOTTOand N. F. MORRIS INTRODUCTION
THIS p a p e r presents a plane stress solution for a p l a t e m a d e of an e l a s t i c plastic m a t e r i a l which o b e y s t h e v o n Mises yield criterion a n d the P r a n d t l Reuss i n c r e m e n t a l s t r e s s - s t r a i n relationship. I t is also a s s u m e d t h a t t h e uniaxial s t r e s s - s t r a i n c u r v e can be m a t h e m a t i c a l l y s t a t e d w i t h t h e help of a R a m b e r g - 0 s g o o d equation. T h e b o u n d a r y condition chosen gives an indeterm i n a t e stress d i s t r i b u t i o n within this plate. T h e a p p r o a c h used for t h e solution is similar to elasticity w h e r e b y a stress function satisfies t h e equilibrium a n d c o m p a t i b i l i t y equations. This leads to a p a r t i a l differential e q u a t i o n which is e v e n t u a l l y solved b y m e a n s of t h e finite difference n u m e r i c a l m e t h o d . I n a p p l y i n g t h e i n c r e m e n t a l t h e o r y of P r a n d t l - R e u s s , t h e e q u a t i o n s are f o u n d to be linear o v e r an i n c r e m e n t a l load. Therefore w i t h t h e use of a n u m e r i c a l a p p r o a c h , t h e p a r t i a l differential e q u a t i o n a r r i v e d a t is linear. This result has only r e c e n t l y been noticed 1, 2 a n d because of it, t h e a p p r o a c h t a k e s on t h e properties of a n elasticity solution. T h e difference is in t h e stress function, which is r e l a t e d to small i n c r e m e n t s o f load. T h e i n c r e m e n t a l s t r e s s - s t r a i n law has been p r e v i o u s l y applied to w o r k - h a r d e n i n g m a t e r i a l using a n u m e r i c a l a p p r o a c h for plates u n d e r t r a n s v e r s e loading, a T h e m a t e r i a l was a s s u m e d to be rigid plastic and, therefore, t h e p r o b l e m i n t r o d u c e d b y elastic surfaces n e v e r e n t e r e d these solutions. This p r o b l e m has, in general, been a v o i d e d b y assuming t h e m a t e r i a l to be incompressible t h r o u g h o u t . T h e n u m e r i c a l a p p r o a c h , i n c o r p o r a t i n g the i n c r e m e n t a l t h e o r y a n d t h e R a m b e r g - O s g o o d m a t e r i a l equation, does n o t h a v e to restrict Poisson's ratio. T h e r e is a l w a y s c o m p a t i bility b e t w e e n the so-called elastic region a n d plastic region as discussed in ref. 4. Moreover, the n u m e r i c a l m e t h o d follows the physical characteristics of t h e i n c r e m e n t a l flow t h e o r y in a c o n v e n i e n t m a t h e m a t i c a l form. THEORY The constitutive relations based upon the Prandtl-Reuss theory and the von Mises yield criterion have been presented in a number of references. 5, 6 The basic part of this development is presented here for completeness. The strain increments for increasing load and isotropic material may be written in tensor notation for convenience as e d e i i : detj -b de/s,
( 1)
where dens is the elastic strain increment which is equal to de~.~-
d~i~ v 2G ~ 8~ daii,
(2)
in which ais = the stress tensor, E = Young's modulus, 8i~ = 1 (i = j) and 0 (i ¢ j) and v = Poisson's ratio. The repeated indices imply summation. The term de~ is the plastic strain increment and is equal to dens = 3sis d~
2~H' '
(3)
where sis = the deviatoric stress tensor [si~- = a i s - ~is(ai~'/3)], 8 = the equivalent stress and is equal to = (~s~ss,)~,
(4)
A n elastic-plastic plane stress solution using t h e incremental t h e o r y
99
H ' = t h e slope of t h e e q u i v a l e n t stress-plastic strain c u r v e which is t h e s a m e as t h e simple tension t e s t curve. The R a m b e r g - O s g o o d e q u a t i o n can be s t a t e d as follows: s = N+
~
,
14a)
Therefore H" is
where B is a m a t e r i a l c o n s t a n t a n d n a function of strain hardening. n u m e r i c a l l y equal to B" H' - - -
(5)
n(~)._x.
E q u a t i o n (1) m a y n o w be w r i t t e n for continuous loading where d ~ > 0 as dais. ( 1 - - 2 v ) ~
deij = " ~ "5-T
daii
3sijd~
0tt y
"}"~2~H •
(6)
F o r unloading where d~ < 0 t h e elastic strain is t h e only t e r m used and dso- . (1 -- 2v) dee = ~ - G - + ~ i J
dais' 3 "
(7)
The t e r m G = shear m o d u l u s of elasticity. E q u a t i o n (6) in engineering n o t a t i o n gives t h e following e q u a t i o n s :
dsy
--- [a~j]
• dy~y
day
,
(8)
davy
where -
E
v
2
~+K(2a~--ay), - - ~ + K ( ay-a~), 6Ka~y
1 1
[ai'] =
- E + KK(2a~-aY)' l + KK(2aY-ax)' 6KKaxv I
(O)
O+v) 1 LL( 2a~-- ay), LL( 2ay - a~), LL( 6a~y) + 2 ~ J and K
ax-ay/2 2~ n H "
'
KK
=
ay-a~/2 2~ ~ H '
'
LL-
3axy
2~ ~ H ' "
(10)
I t should be n o t e d t h a t e q u a t i o n s (8)-(10) contain the properties of plastic incompressibility and t h e strain h a r d e n i n g characteristics as well as elastic compressibility. F u r t h e r m o r e these e q u a t i o n s can be used in the elastic p o r t i o n of the b o d y considered since the H" t e r m tends to infinity in this range. I t becomes a p p a r e n t also t h a t e q u a t i o n (8) gives a physical m e a n i n g to t h e i n c r e m e n t a l flow a p p r o a c h in t h a t it is based u p o n the previous history of loading. The differential t e r m m a y be f u r t h e r looked at r e l a t i v e to t h e R a m b e r g Osgood e q u a t i o n and it can be observed t h a t in order to produce a differential i n c r e m e n t of strain, a small v a l u e of load i n c r e m e n t is r e q u i r e d as w o u l d be e x p e c t e d f r o m the concept of the differential. I n light of this discussion, it is n o t e d t h a t the t o t a l stress t e r m s indicated in e q u a t i o n (9) are k n o w n values e v a l u a t e d at t h e i m m e d i a t e l y prior load level allowing the strain i n c r e m e n t to be c o m p u t e d w i t h o u t a n y i t e r a t i o n procedure once the small stress i n c r e m e n t is found. The stress increments must, of course, be in equilibrium. This equilibrium is n o t only based on the i n c r e m e n t a l stresses b u t also the t o t a l stresses as p o i n t e d o u t in ref. 7. This i n c r e m e n t a l e q u i l i b r i u m e q u a t i o n m a y be s t a t e d in a f o r m similar to the plane stress e q u a t i o n for t h e t o t a l stresses.
dax,~Tday~,y = 0;
da~y.~+day,y = 0.
(11)
(The notationf, x denotes partial differentiation w i t h respect to x, etc. and is used t h r o u g h out.) A stress function ¢(x, y) is chosen to satisfy e q u a t i o n (11) in t e r m s of the i n c r e m e n t a l 8
A. N. PA~ZOTTO a n d N. F. MORRIS
100 stresses such t h a t
¢,~z = da~;
~b,v~ = da~;
-~b,x~ = d a ~ .
(12)
The e q u a t i o n for t h e solution of ¢ is found b y considering the c o m p a t i b i l i t y equations for plane stress in incremental f o r m : de~,v~+de~,~ = dy~,zy.
(13)
E q u a t i o n s (12) and (8) are n e x t s u b s t i t u t e d into e q u a t i o n (13). The t e r m s in t h e m a t r i x [a~¢] are d e p e n d e n t u p o n t h e k n o w n t o t a l stress values which v a r y from p o i n t to p o i n t in the plate, therefore coefficients a~j are d e p e n d e n t u p o n position. The resulting e q u a t i o n in t e r m s of ~b and t h e k n o w n stress coefficient m a t r i x [a~] can be s t a t e d as : P1 ~b,,,,, + P 2 ~ , ~
+ P3 ~b. . . . . + P 4 ~ . ~ + P 5
~ . . . . ~ + P 6 ~ . . . . + P 7 ~ . ~ z , + P 8 ~b,~x
+P9~.,,,,,,+PIO~.~+Pll~b,~,~+P12~b,,~,~
= 0.
(14)
The P coefficients become l e n g t h y b u t for illustrative purposes the expression P 7 will be shown P 7 = 2aiz,~,- 2a~a,®-aa~,~ +aaa,~, (15) where a ~ , ass, etc. are elements of the m a t r i x [a~]. I n a d d i t i o n to satisfying e q u a t i o n (14), t h e stress function m u s t yield t h e r e q u i r e d b o u n d a r y stress values at a point. This condition can be satisfied b y d T ~ = ~s ~ (4.,);
d T , = - ~ s (~b.,),
(16)
where dT~ and dT~ are the x and y components, respectively, of t h e e x t e r n a l incremental forces acting a t a p o i n t on t h e b o u n d a r y . E v e n t h o u g h the stress-strain relations are non-linear, it m a y be observed from the previous equations t h a t w i t h the use of t h e linear i n c r e m e n t a l assumption, t h e solution to t h e stress function is v e r y similar to t h e elastic, anisotropic problem. I t m u s t be realized t h a t a l t h o u g h t h e a p p e a r a n c e of the plane stress equations are the same, t h e coefficients are b y t h e i r nature, completely different. The coefficients present in t h e plasticity e q u a t i o n are established directly f r o m t h e incremental constitutive relationships. These relationships are based u p o n the a s s u m p t i o n of isotropie material. The coefficients in t h e elastic, anisotropie e q u a t i o n for plane stress are characterized b y directional variations in t h e s t r e n g t h constants. F u r t h e r m o r e , t h e ~ t e r m in e q u a t i o n (14) is an i n c r e m e n t a l stress f u n c t i o n c o m p a r e d to t h e elastic stress f u n c t i o n which is a t o t a l function. NUMERICAL
METHODS
The solution to e q u a t i o n (14) has been carried o u t w i t h t h e aid of t h e finite difference numerical a p p r o x i m a t i o n using a 12 × 12 m e s h a r r a n g e m e n t . The n e t w o r k of elements forming t h e a p p r o x i m a t i o n is shown in Fig. 1, and to allow for an appreciation of t h e contents of these elements, several of t h e m are presented. Using )t = a/b : T1 = 6a n )t4 + 4(az2 + a~z + aa3 ) 2t2 +
6a~
-- 2[(a12,~)
+ a22,~ ~ -- a32,xy ]
_ 2[an,v ~ _t_a21,zz_ aal,xu ] ~2,
(17)
T8 - (a12 + a21 + aaa ) )t~ + 0"5[ - (axa + a31)] ~t3 + 0.5[ - (a,8 + aa~)] -
0.5(2a12,, - 2a2a,, - aa2,x + aas,,) ~ + 0"5(2a21,x - 2ala,,
-
aal,~ + aaa,z ) ~ - 0.25( - a13,v~ - a2a,~z + a3a,~) ~t.
(18)
These t w e n t y - o n e elements m a k i n g up the n e t w o r k are n o t s y m m e t r i c a l a b o u t t h e m a i n node p o i n t as t h e y t u r n o u t to be in t h e elastic case which can be described b y t h i r t e e n finite difference elements. A n additional difference b e t w e e n t h e plastic incremental stress function e q u a t i o n a n d the elastic e q u a t i o n is present in the P coefficients. These expressions can be observed to contain partial d e r i v a t i v e s of t h e stress constants a~¢, therefore in their finite difference e v a l u a t i o n stress values at each b o u n d a r y node p o i n t m u s t be found. This, of course, is n o t t r u e in t h e elastic solution. A f u r t h e r conclusion
An elastic-plastic plane stress solution using the incremental theory
101
from the finite difference approach to the solution of equation (14) can be stated. An incremental stress function must be chosen along the boundary edge which has a derivative in at least one co-ordinate direction at each corner point because of the particular requirements of the H" t e r m in the Ramberg-Osgood equation. This limits the number of incremental functions, 4, which can be chosen and this requirement of choice becomes even more pronounced in the solution to partially distributed edge loadings. Consideration of this peculiarity is not necessary in the elastic solution since the stress function is independent of material constants.
FIG. 1. Plane stress molecule. I n addition to the above discussion, the finite difference method cannot compute boundary stresses accurately when the incremental stress function changes equations along the boundary. This inaccuracy occurs at or adjacent to the point in which the change comes about. The stress function used in this analysis is continuous along the boundaries. Certain incremental stress functions m a y be established which are not continuous and yet satisfy the boundary loading condition in closed form. I f these functions are used, it m a y be found t h a t not only are some stress increments incorrect in value b u t also sign. This can lead to additional inconsistencies in evaluating the stresses. .applying the finite difference network at each node point within the plate leads to a set of linear algebraic equations whose number is dependent upon the sum of the nodes. This series of equations can be stated as [A] {~} = {V},
(19)
where the A terms are known coefficients produced by the network at each node point and V corresponds to the known values of the stress function ¢ at the given boundary node. The solution to equation (19) has been carried out by the Gauss elimination scheme. The known ¢ values in turn can be used in finding the stress increments and to~al stress values at each node point. F u r t h e r use of the total stress values occur in establishing the stress coefficients atj and P present in equation (14). These coefficients will be used in the n e x t load increment. The process can be repeated until a preassigned boundary load is reached. I n order to establish a general program for evaluating any type of edge loading, it became necessary to find what stress inaccuracy m a y be established in the use of a constant mesh dimension in a given direction. To do this, two separate programs were developed, one which considered a quarter plate subdivided by a 12 x 12 mesh and the other, more
A. N. P~AZOTrO and N. F. MORRIS
102
general arrangement, considered the entire plate subdivided by a 12 x 12 mesh. A partial edge load was then positioned symmetrically with the horizontal axis shown in Fig. 2. The total load equaUed 24 ksi and all material constants stated in equation (4a) were used.
I
b0
°TL X
-
- I - P
o' Ib' ,2
FIG. 2. Partial edge load. I t was found t h a t the largest discrepancy occurred at the comers but stress values were rather small for these positions and any percentage difference would be magnified out of proportion. Points near the load compared to within 10 per cent while all remaining interior points had stress comparisons less than 5 per cent. I t was therefore decided to use equal mesh distances in a given direction without changing the mesh size for load concentrations.
"~
p(K.S.D
a • 48"
t (Sym)
% I
V---~X
~L_
.p=II-725 __~.~_~ ~p-7,6 _ L -
~elast
P • I 1.725
~,5-8 12p
FIG. 3. Plastic area for given boundary loads.
RESULTS The partial edge loading considered for solution is shown in Fig. 3. The loading is highly concentrated along the lower boundary, because of this certain inherent errors
A n elastic-plastic plane stress solution using t h e i n c r e m e n t a l t h e o r y
103
occur in t h e v i c i n i t y of t h e loading due to t h e n u m e r i c a l solution. Y e t significant findings are observed which, as y e t , h a v e n o t been published b y anyone, a t least to b o t h t h e a u t h o r s ' knowledge. T h e solution is d e v e l o p e d b y a s s u m i n g the following m a t e r i a l constants in t h e R a m b e r g - O s g o o d s t r e s s - s t r a i n c u r v e shown in Fig. 4 where e = N+
N
,
(4a)
where E = 10,500 ksi, B = 72.3 ksi and n = 12. Poisson's ratio is t a k e n equal to 0.333.
60
50
40
30
20 n-12
I0
I O00e
I 0004 Sfrain,
I 0006
I 0008
I 0010
in./in.
:FIG. 4. S t r e s s - s t r a i n curve. The i n c r e m e n t a l a p p r o a c h is carried o u t w i t h o u t establishing the elastic-plastic region w i t h i n t h e plate. Fig. 3 m a k e s an a t t e m p t to define such a b o u n d a r y for g i v e n loadings d i s t r i b u t e d along t h e u p p e r edge of t h e plate. T h e v a l u e of 35 ksi was established as t h e lower limit of plasticity since t h e R a m b e r g - O s g o o d curve shows significant d e v i a t i o n f r o m a s t r a i g h t line at this point. Therefore, Fig. 2 is a plot of the 35 ksi effective stress c o n t o u r for g i v e n b o u n d a r y loads. The contour described w i t h i n t h e plate for a 11.725 ksi loading indicates a large plastic region building u p along each side of t h e plate. I t is a r a t h e r n a r r o w region which will h a v e significance in t h e stress c o m p o n e n t considerations. Figs. 5-8 depict t h e stress c o m p o n e n t s a~ a n d a~ along stress lines 1 a n d 2. Lines 1 and 2 are located 2 and 4 in., respectively, f r o m t h e lower b o u n d a r y . The stress distribution shown in these figures is for one-half of t h e s y m m e t r i c a l l y loaded plate. Also included is t h e stress distribution found f r o m a separate elastic solution u n d e r certain m a g n i t u d e s of t h e edge loading. T h e r e is a significant difference in t h e a~ distribution b e t w e e n t h e i n c r e m e n t a l and elastic solution for an 11.725 ksi loading. This difference is n o t present to the same degree w h e n a~ is considered. P a r t i c u l a r a t t e n t i o n should be focused u p o n line 2 for t h e a~ distribution. The i n c r e m e n t a l solution presents horizontal compressive stress in t h e v i c i n i t y of the n a r r o w a~ region. This created u n l o a d i n g of some a d j a c e n t points along t h e considered line. I t m a y be s t a t e d t h a t t h e effective stress distribution for 11-725 ksi is larger using t h e i n c r e m e n t a l t h e o r y as c o m p a r e d to t h e elasticity equation.
104
A . N . P~AZOTrO and N. F. MORRIS
I t iS p r i m a r i l y due to t h e increase in a= values o v e r the a f o r e m e n t i o n e d region which m a y h a v e occurred because of the horizontal fiber's a t t e m p t to hold b a c k the spread of t h e plastic zone.
(Sym) ~ p , 1 1 , 7 2 5
v~
/
~
p" boundary
p • 111"725
~'~
~ . , ? . ~
"~p.5.8 11 ' J
z~U
I
\'~V~"
~
~ , ¢ ~
":"r'Tl I
\
o.~
o.~
~
,.ooo
2x @
F I e . 5. ax along line 1-1 for given b o u n d a r y loads. (Sym)
ol.
I
0,355" 0`667~"
°
I.OOO
\~
~ p=7.6
v
p • boundary load
~ -20
:=
u)
48"--'-'-"
I I
--p=s.s --p=(11-725 elalf)
I I
t
p• 11.725
FIG. 6. a , along line 1-1 for given b o u n d a r y loads. Stresses in the plate were c o m p u t e d , b o t h w i t h a p r o g r a m including elastic unloading and one which did not. I t was found, as m e n t i o n e d above, t h a t certain points u n l o a d e d [see e q u a t i o n (7)]. This unloading began at top edge pressures of 7.55 ksi. The case in which elastic unloading was n o t considered led to unrealistic stress values at a larger edge pressure of I1.5 ksi. The stress function a r r i v e d at b e c a m e u n s y m m e t r i c a l w i t h respect to the y axis. This u n b a l a n c e occurred despite the fact t h a t e q u a t i o n (19) was still n u m e r i c a l l y satisfied. Since lack of s y m m e t r y only occurred a t high stress values, one possible reason for its occurrence m a y be due to t h e sensitivity of those stiffnesses found in the relatively fiat portion of the stress-strain curve p i c t u r e d in Fig. 4. Results, w i t h
An elastic-plastic plane stress solution using the incremental theory
105
(Sym)
I0
v
0
\ \ ~
|
-
-
0.333 2_.~x
p= boundary load
a
p=7.6 p" 11'725 p=(I 1-725 elost)
\
-.,,,,.._~l o(x)=5'8 '=-'1"---1~=7.6
-I0
48"----1 >=I 1.725
L FIG. 7. (~= along line 2-2 for given boundary loads.
(Sym)
|1 t - . ~ p = ( I 1.725)
elast)
p=boundary m load -30 I-"----48" ~
~ p = 7 " 6 p=11"725
-
Fro. 8. e~ along line 2-2 for given boundary loads. elastic unloading included, did not have this defect although, of course, more computer time became necessary. A n investigation of the in-plane compressive stresses shown in Figs. 5-8 indicates a probability that, unless the plate was very thick, buckling would occur. This is a n obvious limitation on results obtained, although in a problem of this nature, the plane stress
106
A.N. PALAZO~rOand N. F. MORRIS
solution presented by the authors, or some similar solution, would be required before a buckling load could be computed. CONCLUSIONS A n u m e r i c a l m e t h o d has been p r e s e n t e d for t h e plane stress solution using t h e i n c r e m e n t a l t h e o r y of plasticity. T h o u g h t h e s t r e s s - s t r a i n relationship is highly nonlinear, t h e a p p r o a c h used is to solve a linearized set o f e q u a t i o n s in t e r m s of a given i n c r e m e n t a l stress function. T h e e q u a t i o n for solution t a k e s on t h e a p p e a r a n c e of t h e anisotropic elasticity p l a n e stress expression. A finite difference solution was carried o u t for a p l a t e acting u n d e r highly c o n c e n t r a t e d edge loadings. A c o m p a r i s o n was m a d e b e t w e e n t h e i n c r e m e n t a l flow t h e o r y a n d t h e elasticity solution. T h e conclusion was d r a w n t h a t , for increased b o u n d a r y stress, t h e effective stress using p l a s t i c i t y t h e o r y b e c a m e m u c h higher in t h e so-called plastic region t h a n the elastic effective stress in the s a m e region. This a p p a r e n t l y is due to t h e r e l a t i v e l y large horizontal stress p r o d u c e d using the flow t h e o r y . Mention should be m a d e of the f a c t t h a t t h e c o m p l e t e plastic solution was d e t e r m i n e d w i t h o u t considering a m a t e r i a l yield point. F u r t h e r m o r e , a Poisson's r a t i o different f r o m 0.5 can be used in t h e elastic p o r t i o n of the c o n s t i t u t i v e relationships. Acknowledgements--The work presented herein formed part of the thesis presented by the
initial author to New York University School of Engineering and Science in partial fulfilment of the requirement for the degree of Doctor of Philosophy. Financial support was supplied by the N.S.F. under Grant No. 67168. Further work was supported partially by N.S.F. Grant No. GK-5084.
1. 2. 3. 4. 5. 6. 7. 8.
REFERENCES P. V. M ~ c ~ and I. P. KING, lnt. J. mech. Sci. 9, 143 (1967). P. V. M~a~c~, Int. J. mech. Sci. 7, 229 (1965). C. HWA~G, J. appl. Mech., Trans. A m . Soc. mech. Engrs 81, 594 (1959). D. S. BROOKS, Int. J. mech. Sci. 11, 75 (1969). 1%. HILL, The Mathematical Theory of Plasticity, p. 39. Clarendon Press, Oxford (1950). A. MENDELSON, Plasticity: Theory and Application, p. 100. Macmillan, New York (1968). 1%. I-IxLL, The Mathematical Theory of Plasticity, p. 54. Clarendon Press, Oxford (1950). I. S. SOKOL~IKOFF,Mathematical Theory of Elasticity, p. 260. McGraw-Hill, New York (1956).
AN ELASTIC-PLASTIC P L A N E STRESS SOLUTION USING T H E INCREMENTAL T H E O R Y A . N . PALAZOTTO D e p a r t m e n t of Mechanical Engineering, University of Bridgeport, Bridgeport, Connecticut, U.S.A.
and N. F. MORRIS D e p a r t m e n t of Civil Engineering, New York University, Bronx, New York, U.S.A.
(Received 22 June 1970, and in revised form 8 October 1970) S u m m a r y - - A numerical procedure is presented to determine the elastic-plastic plane stress distribution for a rectangular plate acting under any t y p e of partial edge loading. The incremental theory of P r a n d t l - R e u s s is used in conjunction with a Ramberg-Osgood stress-strain relati~nship. A linear partial differential equation is arrived a t through the required load-history dependent constitutive equations. The necessity of incompressible material throughout b o t h the initial and subsequent stress range is eliminated. Finally, because of the assumed Ramberg-Osgood equation, a solution in plasticity can be carried out without a n y fixed yield point. A finite difference scheme is incorporated to solve the partial differential equation and application is made considering a highly concentrated b o u n d a r y loading. Comparison between the plastic and elastic solution shows t h a t for increasing load situations, the plastic effective stress is larger than an elastic equivalent effective stress in the so-called plastic region. NOTATION rectangular Cartesian co-ordinates t o t a l incremental strain tensor (Cartesian) incremental elastic strain tensor incremental plastic strain tensor Y Poisson's ratio E modulus of elasticity (7 shear modulus da~t incremental stress tensor Kronecker delta deviatoric stress tensor equivalent stress H" slope of the equivalent stress-plastic strain curve B material constant present in Ramberg-Osgood stress-strain equation exponent present in Ramberg-Osgood equation n Ca.] m a t r i x of known stress components equation (9) ¢ incremental function Pit_m2 stress constants present in equation (14) dT~, dTy x a n d y components of the external incremental b o u n d a r y tractions stress molecule coefficients present in Fig. 1 aspect ratio, width-height of plate known stress function ~ at b o u n d a r y nodes 97 X, y, Z d~j de~ de 5
98
A.N. P~-LAZOTTOand N. F. MORRIS INTRODUCTION
THIS p a p e r presents a plane stress solution for a p l a t e m a d e of an e l a s t i c plastic m a t e r i a l which o b e y s t h e v o n Mises yield criterion a n d the P r a n d t l Reuss i n c r e m e n t a l s t r e s s - s t r a i n relationship. I t is also a s s u m e d t h a t t h e uniaxial s t r e s s - s t r a i n c u r v e can be m a t h e m a t i c a l l y s t a t e d w i t h t h e help of a R a m b e r g - 0 s g o o d equation. T h e b o u n d a r y condition chosen gives an indeterm i n a t e stress d i s t r i b u t i o n within this plate. T h e a p p r o a c h used for t h e solution is similar to elasticity w h e r e b y a stress function satisfies t h e equilibrium a n d c o m p a t i b i l i t y equations. This leads to a p a r t i a l differential e q u a t i o n which is e v e n t u a l l y solved b y m e a n s of t h e finite difference n u m e r i c a l m e t h o d . I n a p p l y i n g t h e i n c r e m e n t a l t h e o r y of P r a n d t l - R e u s s , t h e e q u a t i o n s are f o u n d to be linear o v e r an i n c r e m e n t a l load. Therefore w i t h t h e use of a n u m e r i c a l a p p r o a c h , t h e p a r t i a l differential e q u a t i o n a r r i v e d a t is linear. This result has only r e c e n t l y been noticed 1, 2 a n d because of it, t h e a p p r o a c h t a k e s on t h e properties of a n elasticity solution. T h e difference is in t h e stress function, which is r e l a t e d to small i n c r e m e n t s o f load. T h e i n c r e m e n t a l s t r e s s - s t r a i n law has been p r e v i o u s l y applied to w o r k - h a r d e n i n g m a t e r i a l using a n u m e r i c a l a p p r o a c h for plates u n d e r t r a n s v e r s e loading, a T h e m a t e r i a l was a s s u m e d to be rigid plastic and, therefore, t h e p r o b l e m i n t r o d u c e d b y elastic surfaces n e v e r e n t e r e d these solutions. This p r o b l e m has, in general, been a v o i d e d b y assuming t h e m a t e r i a l to be incompressible t h r o u g h o u t . T h e n u m e r i c a l a p p r o a c h , i n c o r p o r a t i n g the i n c r e m e n t a l t h e o r y a n d t h e R a m b e r g - O s g o o d m a t e r i a l equation, does n o t h a v e to restrict Poisson's ratio. T h e r e is a l w a y s c o m p a t i bility b e t w e e n the so-called elastic region a n d plastic region as discussed in ref. 4. Moreover, the n u m e r i c a l m e t h o d follows the physical characteristics of t h e i n c r e m e n t a l flow t h e o r y in a c o n v e n i e n t m a t h e m a t i c a l form. THEORY The constitutive relations based upon the Prandtl-Reuss theory and the von Mises yield criterion have been presented in a number of references. 5, 6 The basic part of this development is presented here for completeness. The strain increments for increasing load and isotropic material may be written in tensor notation for convenience as e d e i i : detj -b de/s,
( 1)
where dens is the elastic strain increment which is equal to de~.~-
d~i~ v 2G ~ 8~ daii,
(2)
in which ais = the stress tensor, E = Young's modulus, 8i~ = 1 (i = j) and 0 (i ¢ j) and v = Poisson's ratio. The repeated indices imply summation. The term de~ is the plastic strain increment and is equal to dens = 3sis d~
2~H' '
(3)
where sis = the deviatoric stress tensor [si~- = a i s - ~is(ai~'/3)], 8 = the equivalent stress and is equal to = (~s~ss,)~,
(4)
A n elastic-plastic plane stress solution using t h e incremental t h e o r y
99
H ' = t h e slope of t h e e q u i v a l e n t stress-plastic strain c u r v e which is t h e s a m e as t h e simple tension t e s t curve. The R a m b e r g - O s g o o d e q u a t i o n can be s t a t e d as follows: s = N+
~
,
14a)
Therefore H" is
where B is a m a t e r i a l c o n s t a n t a n d n a function of strain hardening. n u m e r i c a l l y equal to B" H' - - -
(5)
n(~)._x.
E q u a t i o n (1) m a y n o w be w r i t t e n for continuous loading where d ~ > 0 as dais. ( 1 - - 2 v ) ~
deij = " ~ "5-T
daii
3sijd~
0tt y
"}"~2~H •
(6)
F o r unloading where d~ < 0 t h e elastic strain is t h e only t e r m used and dso- . (1 -- 2v) dee = ~ - G - + ~ i J
dais' 3 "
(7)
The t e r m G = shear m o d u l u s of elasticity. E q u a t i o n (6) in engineering n o t a t i o n gives t h e following e q u a t i o n s :
dsy
--- [a~j]
• dy~y
day
,
(8)
davy
where -
E
v
2
~+K(2a~--ay), - - ~ + K ( ay-a~), 6Ka~y
1 1
[ai'] =
- E + KK(2a~-aY)' l + KK(2aY-ax)' 6KKaxv I
(O)
O+v) 1 LL( 2a~-- ay), LL( 2ay - a~), LL( 6a~y) + 2 ~ J and K
ax-ay/2 2~ n H "
'
KK
=
ay-a~/2 2~ ~ H '
'
LL-
3axy
2~ ~ H ' "
(10)
I t should be n o t e d t h a t e q u a t i o n s (8)-(10) contain the properties of plastic incompressibility and t h e strain h a r d e n i n g characteristics as well as elastic compressibility. F u r t h e r m o r e these e q u a t i o n s can be used in the elastic p o r t i o n of the b o d y considered since the H" t e r m tends to infinity in this range. I t becomes a p p a r e n t also t h a t e q u a t i o n (8) gives a physical m e a n i n g to t h e i n c r e m e n t a l flow a p p r o a c h in t h a t it is based u p o n the previous history of loading. The differential t e r m m a y be f u r t h e r looked at r e l a t i v e to t h e R a m b e r g Osgood e q u a t i o n and it can be observed t h a t in order to produce a differential i n c r e m e n t of strain, a small v a l u e of load i n c r e m e n t is r e q u i r e d as w o u l d be e x p e c t e d f r o m the concept of the differential. I n light of this discussion, it is n o t e d t h a t the t o t a l stress t e r m s indicated in e q u a t i o n (9) are k n o w n values e v a l u a t e d at t h e i m m e d i a t e l y prior load level allowing the strain i n c r e m e n t to be c o m p u t e d w i t h o u t a n y i t e r a t i o n procedure once the small stress i n c r e m e n t is found. The stress increments must, of course, be in equilibrium. This equilibrium is n o t only based on the i n c r e m e n t a l stresses b u t also the t o t a l stresses as p o i n t e d o u t in ref. 7. This i n c r e m e n t a l e q u i l i b r i u m e q u a t i o n m a y be s t a t e d in a f o r m similar to the plane stress e q u a t i o n for t h e t o t a l stresses.
dax,~Tday~,y = 0;
da~y.~+day,y = 0.
(11)
(The notationf, x denotes partial differentiation w i t h respect to x, etc. and is used t h r o u g h out.) A stress function ¢(x, y) is chosen to satisfy e q u a t i o n (11) in t e r m s of the i n c r e m e n t a l 8
A. N. PA~ZOTTO a n d N. F. MORRIS
100 stresses such t h a t
¢,~z = da~;
~b,v~ = da~;
-~b,x~ = d a ~ .
(12)
The e q u a t i o n for t h e solution of ¢ is found b y considering the c o m p a t i b i l i t y equations for plane stress in incremental f o r m : de~,v~+de~,~ = dy~,zy.
(13)
E q u a t i o n s (12) and (8) are n e x t s u b s t i t u t e d into e q u a t i o n (13). The t e r m s in t h e m a t r i x [a~¢] are d e p e n d e n t u p o n t h e k n o w n t o t a l stress values which v a r y from p o i n t to p o i n t in the plate, therefore coefficients a~j are d e p e n d e n t u p o n position. The resulting e q u a t i o n in t e r m s of ~b and t h e k n o w n stress coefficient m a t r i x [a~] can be s t a t e d as : P1 ~b,,,,, + P 2 ~ , ~
+ P3 ~b. . . . . + P 4 ~ . ~ + P 5
~ . . . . ~ + P 6 ~ . . . . + P 7 ~ . ~ z , + P 8 ~b,~x
+P9~.,,,,,,+PIO~.~+Pll~b,~,~+P12~b,,~,~
= 0.
(14)
The P coefficients become l e n g t h y b u t for illustrative purposes the expression P 7 will be shown P 7 = 2aiz,~,- 2a~a,®-aa~,~ +aaa,~, (15) where a ~ , ass, etc. are elements of the m a t r i x [a~]. I n a d d i t i o n to satisfying e q u a t i o n (14), t h e stress function m u s t yield t h e r e q u i r e d b o u n d a r y stress values at a point. This condition can be satisfied b y d T ~ = ~s ~ (4.,);
d T , = - ~ s (~b.,),
(16)
where dT~ and dT~ are the x and y components, respectively, of t h e e x t e r n a l incremental forces acting a t a p o i n t on t h e b o u n d a r y . E v e n t h o u g h the stress-strain relations are non-linear, it m a y be observed from the previous equations t h a t w i t h the use of t h e linear i n c r e m e n t a l assumption, t h e solution to t h e stress function is v e r y similar to t h e elastic, anisotropic problem. I t m u s t be realized t h a t a l t h o u g h t h e a p p e a r a n c e of the plane stress equations are the same, t h e coefficients are b y t h e i r nature, completely different. The coefficients present in t h e plasticity e q u a t i o n are established directly f r o m t h e incremental constitutive relationships. These relationships are based u p o n the a s s u m p t i o n of isotropie material. The coefficients in t h e elastic, anisotropie e q u a t i o n for plane stress are characterized b y directional variations in t h e s t r e n g t h constants. F u r t h e r m o r e , t h e ~ t e r m in e q u a t i o n (14) is an i n c r e m e n t a l stress f u n c t i o n c o m p a r e d to t h e elastic stress f u n c t i o n which is a t o t a l function. NUMERICAL
METHODS
The solution to e q u a t i o n (14) has been carried o u t w i t h t h e aid of t h e finite difference numerical a p p r o x i m a t i o n using a 12 × 12 m e s h a r r a n g e m e n t . The n e t w o r k of elements forming t h e a p p r o x i m a t i o n is shown in Fig. 1, and to allow for an appreciation of t h e contents of these elements, several of t h e m are presented. Using )t = a/b : T1 = 6a n )t4 + 4(az2 + a~z + aa3 ) 2t2 +
6a~
-- 2[(a12,~)
+ a22,~ ~ -- a32,xy ]
_ 2[an,v ~ _t_a21,zz_ aal,xu ] ~2,
(17)
T8 - (a12 + a21 + aaa ) )t~ + 0"5[ - (axa + a31)] ~t3 + 0.5[ - (a,8 + aa~)] -
0.5(2a12,, - 2a2a,, - aa2,x + aas,,) ~ + 0"5(2a21,x - 2ala,,
-
aal,~ + aaa,z ) ~ - 0.25( - a13,v~ - a2a,~z + a3a,~) ~t.
(18)
These t w e n t y - o n e elements m a k i n g up the n e t w o r k are n o t s y m m e t r i c a l a b o u t t h e m a i n node p o i n t as t h e y t u r n o u t to be in t h e elastic case which can be described b y t h i r t e e n finite difference elements. A n additional difference b e t w e e n t h e plastic incremental stress function e q u a t i o n a n d the elastic e q u a t i o n is present in the P coefficients. These expressions can be observed to contain partial d e r i v a t i v e s of t h e stress constants a~¢, therefore in their finite difference e v a l u a t i o n stress values at each b o u n d a r y node p o i n t m u s t be found. This, of course, is n o t t r u e in t h e elastic solution. A f u r t h e r conclusion
An elastic-plastic plane stress solution using the incremental theory
101
from the finite difference approach to the solution of equation (14) can be stated. An incremental stress function must be chosen along the boundary edge which has a derivative in at least one co-ordinate direction at each corner point because of the particular requirements of the H" t e r m in the Ramberg-Osgood equation. This limits the number of incremental functions, 4, which can be chosen and this requirement of choice becomes even more pronounced in the solution to partially distributed edge loadings. Consideration of this peculiarity is not necessary in the elastic solution since the stress function is independent of material constants.
FIG. 1. Plane stress molecule. I n addition to the above discussion, the finite difference method cannot compute boundary stresses accurately when the incremental stress function changes equations along the boundary. This inaccuracy occurs at or adjacent to the point in which the change comes about. The stress function used in this analysis is continuous along the boundaries. Certain incremental stress functions m a y be established which are not continuous and yet satisfy the boundary loading condition in closed form. I f these functions are used, it m a y be found t h a t not only are some stress increments incorrect in value b u t also sign. This can lead to additional inconsistencies in evaluating the stresses. .applying the finite difference network at each node point within the plate leads to a set of linear algebraic equations whose number is dependent upon the sum of the nodes. This series of equations can be stated as [A] {~} = {V},
(19)
where the A terms are known coefficients produced by the network at each node point and V corresponds to the known values of the stress function ¢ at the given boundary node. The solution to equation (19) has been carried out by the Gauss elimination scheme. The known ¢ values in turn can be used in finding the stress increments and to~al stress values at each node point. F u r t h e r use of the total stress values occur in establishing the stress coefficients atj and P present in equation (14). These coefficients will be used in the n e x t load increment. The process can be repeated until a preassigned boundary load is reached. I n order to establish a general program for evaluating any type of edge loading, it became necessary to find what stress inaccuracy m a y be established in the use of a constant mesh dimension in a given direction. To do this, two separate programs were developed, one which considered a quarter plate subdivided by a 12 x 12 mesh and the other, more
A. N. P~AZOTrO and N. F. MORRIS
102
general arrangement, considered the entire plate subdivided by a 12 x 12 mesh. A partial edge load was then positioned symmetrically with the horizontal axis shown in Fig. 2. The total load equaUed 24 ksi and all material constants stated in equation (4a) were used.
I
b0
°TL X
-
- I - P
o' Ib' ,2
FIG. 2. Partial edge load. I t was found t h a t the largest discrepancy occurred at the comers but stress values were rather small for these positions and any percentage difference would be magnified out of proportion. Points near the load compared to within 10 per cent while all remaining interior points had stress comparisons less than 5 per cent. I t was therefore decided to use equal mesh distances in a given direction without changing the mesh size for load concentrations.
"~
p(K.S.D
a • 48"
t (Sym)
% I
V---~X
~L_
.p=II-725 __~.~_~ ~p-7,6 _ L -
~elast
P • I 1.725
~,5-8 12p
FIG. 3. Plastic area for given boundary loads.
RESULTS The partial edge loading considered for solution is shown in Fig. 3. The loading is highly concentrated along the lower boundary, because of this certain inherent errors
A n elastic-plastic plane stress solution using t h e i n c r e m e n t a l t h e o r y
103
occur in t h e v i c i n i t y of t h e loading due to t h e n u m e r i c a l solution. Y e t significant findings are observed which, as y e t , h a v e n o t been published b y anyone, a t least to b o t h t h e a u t h o r s ' knowledge. T h e solution is d e v e l o p e d b y a s s u m i n g the following m a t e r i a l constants in t h e R a m b e r g - O s g o o d s t r e s s - s t r a i n c u r v e shown in Fig. 4 where e = N+
N
,
(4a)
where E = 10,500 ksi, B = 72.3 ksi and n = 12. Poisson's ratio is t a k e n equal to 0.333.
60
50
40
30
20 n-12
I0
I O00e
I 0004 Sfrain,
I 0006
I 0008
I 0010
in./in.
:FIG. 4. S t r e s s - s t r a i n curve. The i n c r e m e n t a l a p p r o a c h is carried o u t w i t h o u t establishing the elastic-plastic region w i t h i n t h e plate. Fig. 3 m a k e s an a t t e m p t to define such a b o u n d a r y for g i v e n loadings d i s t r i b u t e d along t h e u p p e r edge of t h e plate. T h e v a l u e of 35 ksi was established as t h e lower limit of plasticity since t h e R a m b e r g - O s g o o d curve shows significant d e v i a t i o n f r o m a s t r a i g h t line at this point. Therefore, Fig. 2 is a plot of the 35 ksi effective stress c o n t o u r for g i v e n b o u n d a r y loads. The contour described w i t h i n t h e plate for a 11.725 ksi loading indicates a large plastic region building u p along each side of t h e plate. I t is a r a t h e r n a r r o w region which will h a v e significance in t h e stress c o m p o n e n t considerations. Figs. 5-8 depict t h e stress c o m p o n e n t s a~ a n d a~ along stress lines 1 a n d 2. Lines 1 and 2 are located 2 and 4 in., respectively, f r o m t h e lower b o u n d a r y . The stress distribution shown in these figures is for one-half of t h e s y m m e t r i c a l l y loaded plate. Also included is t h e stress distribution found f r o m a separate elastic solution u n d e r certain m a g n i t u d e s of t h e edge loading. T h e r e is a significant difference in t h e a~ distribution b e t w e e n t h e i n c r e m e n t a l and elastic solution for an 11.725 ksi loading. This difference is n o t present to the same degree w h e n a~ is considered. P a r t i c u l a r a t t e n t i o n should be focused u p o n line 2 for t h e a~ distribution. The i n c r e m e n t a l solution presents horizontal compressive stress in t h e v i c i n i t y of the n a r r o w a~ region. This created u n l o a d i n g of some a d j a c e n t points along t h e considered line. I t m a y be s t a t e d t h a t t h e effective stress distribution for 11-725 ksi is larger using t h e i n c r e m e n t a l t h e o r y as c o m p a r e d to t h e elasticity equation.
104
A . N . P~AZOTrO and N. F. MORRIS
I t iS p r i m a r i l y due to t h e increase in a= values o v e r the a f o r e m e n t i o n e d region which m a y h a v e occurred because of the horizontal fiber's a t t e m p t to hold b a c k the spread of t h e plastic zone.
(Sym) ~ p , 1 1 , 7 2 5
v~
/
~
p" boundary
p • 111"725
~'~
~ . , ? . ~
"~p.5.8 11 ' J
z~U
I
\'~V~"
~
~ , ¢ ~
":"r'Tl I
\
o.~
o.~
~
,.ooo
2x @
F I e . 5. ax along line 1-1 for given b o u n d a r y loads. (Sym)
ol.
I
0,355" 0`667~"
°
I.OOO
\~
~ p=7.6
v
p • boundary load
~ -20
:=
u)
48"--'-'-"
I I
--p=s.s --p=(11-725 elalf)
I I
t
p• 11.725
FIG. 6. a , along line 1-1 for given b o u n d a r y loads. Stresses in the plate were c o m p u t e d , b o t h w i t h a p r o g r a m including elastic unloading and one which did not. I t was found, as m e n t i o n e d above, t h a t certain points u n l o a d e d [see e q u a t i o n (7)]. This unloading began at top edge pressures of 7.55 ksi. The case in which elastic unloading was n o t considered led to unrealistic stress values at a larger edge pressure of I1.5 ksi. The stress function a r r i v e d at b e c a m e u n s y m m e t r i c a l w i t h respect to the y axis. This u n b a l a n c e occurred despite the fact t h a t e q u a t i o n (19) was still n u m e r i c a l l y satisfied. Since lack of s y m m e t r y only occurred a t high stress values, one possible reason for its occurrence m a y be due to t h e sensitivity of those stiffnesses found in the relatively fiat portion of the stress-strain curve p i c t u r e d in Fig. 4. Results, w i t h
An elastic-plastic plane stress solution using the incremental theory
105
(Sym)
I0
v
0
\ \ ~
|
-
-
0.333 2_.~x
p= boundary load
a
p=7.6 p" 11'725 p=(I 1-725 elost)
\
-.,,,,.._~l o(x)=5'8 '=-'1"---1~=7.6
-I0
48"----1 >=I 1.725
L FIG. 7. (~= along line 2-2 for given boundary loads.
(Sym)
|1 t - . ~ p = ( I 1.725)
elast)
p=boundary m load -30 I-"----48" ~
~ p = 7 " 6 p=11"725
-
Fro. 8. e~ along line 2-2 for given boundary loads. elastic unloading included, did not have this defect although, of course, more computer time became necessary. A n investigation of the in-plane compressive stresses shown in Figs. 5-8 indicates a probability that, unless the plate was very thick, buckling would occur. This is a n obvious limitation on results obtained, although in a problem of this nature, the plane stress
106
A.N. PALAZO~rOand N. F. MORRIS
solution presented by the authors, or some similar solution, would be required before a buckling load could be computed. CONCLUSIONS A n u m e r i c a l m e t h o d has been p r e s e n t e d for t h e plane stress solution using t h e i n c r e m e n t a l t h e o r y of plasticity. T h o u g h t h e s t r e s s - s t r a i n relationship is highly nonlinear, t h e a p p r o a c h used is to solve a linearized set o f e q u a t i o n s in t e r m s of a given i n c r e m e n t a l stress function. T h e e q u a t i o n for solution t a k e s on t h e a p p e a r a n c e of t h e anisotropic elasticity p l a n e stress expression. A finite difference solution was carried o u t for a p l a t e acting u n d e r highly c o n c e n t r a t e d edge loadings. A c o m p a r i s o n was m a d e b e t w e e n t h e i n c r e m e n t a l flow t h e o r y a n d t h e elasticity solution. T h e conclusion was d r a w n t h a t , for increased b o u n d a r y stress, t h e effective stress using p l a s t i c i t y t h e o r y b e c a m e m u c h higher in t h e so-called plastic region t h a n the elastic effective stress in the s a m e region. This a p p a r e n t l y is due to t h e r e l a t i v e l y large horizontal stress p r o d u c e d using the flow t h e o r y . Mention should be m a d e of the f a c t t h a t t h e c o m p l e t e plastic solution was d e t e r m i n e d w i t h o u t considering a m a t e r i a l yield point. F u r t h e r m o r e , a Poisson's r a t i o different f r o m 0.5 can be used in t h e elastic p o r t i o n of the c o n s t i t u t i v e relationships. Acknowledgements--The work presented herein formed part of the thesis presented by the
initial author to New York University School of Engineering and Science in partial fulfilment of the requirement for the degree of Doctor of Philosophy. Financial support was supplied by the N.S.F. under Grant No. 67168. Further work was supported partially by N.S.F. Grant No. GK-5084.
1. 2. 3. 4. 5. 6. 7. 8.
REFERENCES P. V. M ~ c ~ and I. P. KING, lnt. J. mech. Sci. 9, 143 (1967). P. V. M~a~c~, Int. J. mech. Sci. 7, 229 (1965). C. HWA~G, J. appl. Mech., Trans. A m . Soc. mech. Engrs 81, 594 (1959). D. S. BROOKS, Int. J. mech. Sci. 11, 75 (1969). 1%. HILL, The Mathematical Theory of Plasticity, p. 39. Clarendon Press, Oxford (1950). A. MENDELSON, Plasticity: Theory and Application, p. 100. Macmillan, New York (1968). 1%. I-IxLL, The Mathematical Theory of Plasticity, p. 54. Clarendon Press, Oxford (1950). I. S. SOKOL~IKOFF,Mathematical Theory of Elasticity, p. 260. McGraw-Hill, New York (1956).