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On the elastic-plastic behaviour of a reinforced layer
Int. J. mech. Sei. Pergamon Press. 1970. Vol. 12, pp. 893-903. Printed in Great Britain
ON T H E E L A S T I C - P L A S T I C B E H A V I O U R REINFORCED LAYER
OF A
Yu. V. •EMIROVSKY Institute of Hydrodynamics, Siberian Branch of the U.S.S.R. Academy of Sciences, Novosibirsk, U.S.S.R. (Received 10 February 1969, and in revised form 31 July 1969)
Summary--On the basis of the structural analysis of a model reflecting all the main features of a reinforced layer, equations are obtained for the interdependence of stresses and strains beyond the elastic limit. The type of these equations as well as the conditions under which residual strains m a y occur are shown to depend essentially on the character of reinforcement and on the properties of the component elements. It is characteristic that even with the ideally plastic properties of component elements, the interdependence between averaged stresses and strains in the reinforced layer in the elastic-plastic deforming region remains mutually single-valued. Examples are given showing that the experimentally determined characteristics of the reinforced material (Young's modulus, shear modulus, Poisson's ratio and yield or breaking stress) as well as strain diagrams can be obtained by calculation. NOTATION
tk(1, 2, 3) averaged stresses in the layer ~k
~ ( i , j = 1, 2) a.(n = 1, 2. . . . . N) e(n) COn [iS
Eo, Ec,,
E, E n
strain components stresses in the binding material stresses in the reinforcement elements strain of angular reinforcement elements specific intensities of the elements of reinforcement direction cosines of reinforcement elements secant moduli of the binding materials and the reinforcing elements at tension ( + ) and compression ( - ) yield stresses of binding materials and reinforcing elements at tension ( + ) and compression ( - ) Young's moduli INTRODUCTION
NEW COMBINATIONS o f m a t e r i a l s , s u c h as r e i n f o r c e d plastics a n d concrete, m e t a l s s t r e n g t h e n e d b y fibres a n d others, are finding wide a p p l i c a t i o n in m o d e r n engineering. T h e i r m a i n a d v a n t a g e lies in t h e possibility o f r e g u l a t i n g t h e i r m e c h a n i c a l c h a r a c t e r i s t i c s in a n y direction. This possibility is so w i d e t h a t t o s p e a k a b o u t a n y definite class o f r e i n f o r c e d m a t e r i a l h a s little m e a n i n g . The mechanical properties of reinforced material depend on those of the binding and the reinforcing materials, their percentage composition and the c h a r a c t e r o f r e i n f o r c e m e n t . H e n c e p r a c t i c a l l y e v e r y class o f r e i n f o r c e d m a t e r i a l r e q u i r e s a special a n a l y s i s t h e m a i n p u r p o s e o f w h i c h is t o o b t a i n r e l a t i o n s d e s c r i b i n g i n t e r d e p e n d e n c i e s b e t w e e n stresses a n d s t r a i n s in t h e r e i n f o r c e d material. 893
894
Yu. V. N]:MIROVSKY
There are two approaches to o b t a i n i n g these relations : (a) phenomenological, where the reinforced m e d i u m is considered to be uniform, monolithic and anisotropic 1, ~ and (b) an a p p r o a c h based on the s t r u c t u r a l analysis of t h e reinforced m a t e r i a l in accordance with the c h a r a c t e r of its s t r u c t u r e a n d the mechanical properties of its components, a, 4 In the elastic range if some specific effects (stress c o n c e n t r a t i o n in the neighbourhood of reinforcing elements, n o n - u n i f o r m i t y of d e f o r m a t i o n a m o n g t h e m , etc.) are neglected, both approaches yield in essence the same equations of interdependence b e t w e e n the a v e r a g e d stresses and strains a n d in this sense are equivalent. The phenomenological a p p r o a c h to o b t a i n i n g equations d e t e r m i n i n g t h e elasticplastic b e h a v i o u r of anisotropic m e d i a has been applied in a n u m b e r of investigations. ~-7 H o w e v e r , in the equations o b t a i n e d it is necessary to refer to "physically anisotropie m e d i a " whose a n i s o t r o p y is due to the crystalline structure. E a c h t y p e of constructed anisotropy, including reinforced materials, m u s t he analysed separately since b e y o n d the limit of elasticity t h e specific features of each t y p e affect the f o r m of the plasticity conditions ;a. a and, as will be seen later, also affects the c h a r a c t e r of t h e equations describing the interdependence b e t w e e n stresses and strains. Hence, t h e relations describing this i n t e r d e p e n d e n c e b e y o n d the limit of elasticity can be o b t a i n e d only b y m e a n s of the s t r u c t u r a l analysis of the reinforced m a t e r i a l on the basis of a m o d e l reflecting its specific features. The a d v a n t a g e of the s t r u c t u r a l analysis also lies in the fact t h a t it p e r m i t s an e s t i m a t e to be m a d e of the efficiency of each element of the composition and t h u s opens a w a y to t h e purposeful regulation of reinforcement c h a r a c t e r for increasing the strength properties of reinforced materials. The present p a p e r analyses the elastic-plastic b e h a v i o u r of a reinforced layer u n d e r the action of forces in its plane on the basis of a proposed m o d e l a n d some additional simplifying suggestions. 1. G E N E R A L
CASE
The elemental layer under consideration is a c o m p a r a t i v e l y t h i n isotropic plate with a reinforcing layer incorporated in it (see Fig. 1). The l a t t e r represents a n e t of thin onedimensional threads running at an angle an (n = 1, 2, ..., N) w i t h direction 1.
C) C) Q
(D (D C)
FIG. l.
I h
It is assumed t h a t
(i) the m a t e r i a l of each element of t h e composition is elastic-plastic and in the general case is different for each e l e m e n t ; (ii) the n u m b e r of reinforcing elements is sufficiently great to consider the composite m a t e r i a l as quasi-homogeneous; (iii) t h e space b e t w e e n the reinforcing elements is sufficiently great c o m p a r e d with t h e i r dimensions, and at t h e same t i m e sufficiently small c o m p a r e d w i t h the plate dimensions, t h a t local effects in the n e i g h b o u r h o o d of t h e t h r e a d s a n d irregularity of strain b e t w e e n the threads can be neglected; (iv) the b o n d b e t w e e n t h e composition elements is ideal, i.e. there is no slip between the reinforcing elements and t h e binding m a t e r i a l ; (v) each thread, if incorporated into the binding material, can w i t h s t a n d b o t h tensile and compressive stresses. H o w e v e r , due to a form of instability which m a y occur in compression, t h e yield stresses or b r e a k i n g stresses (as well as t h e m o d u l u s of s t r e n g t h e n i n g for s t r e n g t h e n e d materials) in tension a n d compression are considered to be different, Y o u n g ' s moduli being assumed the same for tension and compression; and
On t h e e l a s t i c - p l a s t i c b e h a v i o u r of a r e i n f o r c e d l a y e r
895
(vi) t h e isotropic b i n d i n g m a t e r i a l satisfies t h e t h e o r y of p l a s t i c d e f o r m a t i o n w i t h t h e s a m e c h a r a c t e r i s t i c s u n d e r t e n s i o n a n d c o m p r e s s i o n a n d for s i m p l i c i t y is a s s u m e d t o u n d e r g o n o c h a n g e of v o l u m e b o t h i n t h e elastic a n d in t h e p l a s t i c regions. W h e r e n e c e s s a r y it is possible t o r e j e c t t h e r e q u i r e m e n t o f c o n s t a n t v o l u m e a n d t o use a t y p e of i n c r e m e n t t h e o r y i n s t e a d of t h a t of s t r a i n . L e t ~on (n = 1, 2 . . . . . N ) b e t h e specific i n t e n s i t i e s o f t h r e a d s in t h e r e i n f o r c i n g l a y e r r u n n i n g a t a n a n g l e an (n = 1, 2, ..., 5/) w i t h d i r e c t i o n 1, a n d let h b e t h e t h i c k n e s s of t h e r e i n f o r c e d layer, ¢o~ is t h e specific i n t e n s i t y of t h e r e i n f o r c e d l a y e r in t h e p l a t e ' s t h i c k n e s s , Fig. 2. T h e n i n t h e o r t h o g o n a l s y s t e m 1-2 c o m p o n e n t s o f t h e i n t e r n a l forces, see Fig. 3, in t h e c o m p o s i t e l a y e r w o u l d b e e x p r e s s e d as follows:
t k = aa~j+ ~ o J . anli. l~.
TIj
tj = ~ - ,
TI.
t3 = ~ - ,
(i,j= 1,2;k = 1,2,3),
lxn= cosa,,
l~,=sinan,
O<~a,~<.lr, a = 1--~ z, ¢o,~=nnFn/AFh, C
°I_ I-
(1)
¢o~=3/h.
B
d~,'111111 ~A -I
FIG. 2.
T22 FIG. 3. H e r e Tis a r e forces, a ~ are stresses in t h e filler, an a r e stresses in t h e r e i n f o r c e m e n t t h r e a d s , F n a r e cross-sectional a r e a s of t h e r e i n f o r c e m e n t e l e m e n t s a n d n n are t h e n u m b e r s o f t h r e a d s of t h e r e i n f o r c e m e n t e l e m e n t s in t h e s e g m e n t s AF, Fig. 2. A t s m a l l s t r a i n s o n t h e b a s i s of a s s u m p t i o n (iv) t h e following r e l a t i o n s b e t w e e n t h e s t r a i n s e(,~ of t h e r e i n f o r c e m e n t e l e m e n t s a n d t h o s e of t h e filling l a y e r are o b t a i n e d :
e~.) = el l~. + e2 l~. + ea lln 12..
(2)
H e r e ca, e2 a r e t h e s t r a i n c o m p o n e n t s of t h e filling l a y e r in d i r e c t i o n s 1 a n d 2, r e s p e c t i v e l y ; ea is t h e s h e a r s t r a i n .
896
Yu. V. NEMIROVSKY
I n c o n f o r m i t y w i t h the a b o v e assumptions the internal stresses of the composite layer elements are related to the strains b y the following equalities: a°l = ~Eo(el + ½~2), a~2 --- ~Ec(e2 + ½el), ~ 0 = 1Ece3,
(3)
an = E ~ n e . .
H e r e Ec, E$n are the secant moduli of the filler materials and of the reinforcing elements in tension ( + ) and compression ( - ). I f all t h e elements of the composition u n d e r the g i v e n forces tk (k = 1, 2, 3) r e m a i n elastic, t h e n all the secant moduli are equal to the corresponding Y o u n g ' s moduli Ec=E
,
(4)
E~n=E,,.
Then, s u b s t i t u t i n g expressions (3) into (1) the following relations b e t w e e n the forces tk and the strains e~ are o b t a i n e d : t=
I[akmlle,
t
[[tl, t2, t3[I -x,
e = !]bkm[It, ~=
][bkm][ = [Iakm][-l'
Ii~l, e2, e3][ -1
1
(5)
1,2,3).
(k,m=
The coefficients of the m a t r i x [I akin [I are equal to: N
a. = kaE+ ~.(o.E.l~., n=l N
ale = a21 = ~ a E + ~. (on En l~n l~n, n=l
aia = a3i =
N ~ (onEnl~nlsn,
(6)
nffil N
a33 = ½ a E + Z ( o , , E n l ~ , , l ~
( i , j = 1,2; i ¢ j ) .
n=l
S u b s t i t u t i n g the strain c o m p o n e n t s f r o m e q u a t i o n (5) into (3) and t a k i n g into a c c o u n t e q u a t i o n (4) the internal stresses in all t h e elements composing t h e reinforced layer are found. W i t h t h e i r help it is possible to d e t e r m i n e u n d e r w h a t loads different elements of t h e composite layer pass into t h e plastic state. So t h e reinforcing elements r e m a i n elastic if the following inequalities are satisfied : -a~
ltk(blkl~+b2~l~+b3kll~12~)
]
(n = 1 , 2 . . . . . N )
(7)
and t h e layer of t h e binding m a t e r i a l remains elastic if the inequality o2 all o ( 7 o 2 + a ozu 2 + 3q12 02 < °'02 a11--
(8)
is satisfied. The expression (3) m u s t be s u b s t i t u t e d instead of o~j t a k i n g into aecotmt equations (4) and (5). I n equations (7) a n d (8) a0 and a .~ denote, respectively, t h e yield stresses of the binding m a t e r i a l a n d the m a t e r i a l of the reinforcing elements in tension or compression. The violation of a n y of the inequalities (7) and (8) leads to the occurrence of plastic strain in t h e corresponding reinforcing elements or the filler. F o r example, u n d e r some c o m b i n a t i o n of forces tk, such t h a t f . ( t l , t 2, t3) = 0 (9) let a n y of t l ~ inequalities (7) be violated. T h e n e q u a t i o n (9) in stress space t 1, t 2, t 3 determines the yield surface for the reinforced m a t e r i a l w i t h the c o m p o n e n t elements possessing the elastic-plastic properties. Indeed, for all stresses w i t h i n this surface there are no residual strains after t h e unloading of the layer. F o r the stresses outside this surface some residual strain remains in t h e reinforced layer after unloading. Analogous surfaces can also be o b t a i n e d in stress space w h e n all r e m a i n i n g inequalities (7) or (8) are v i o l a t e d
On t h e elastic-plastic b e h a v i o u r of a reinforced layer
897
(individually or in combinations). T h u s t h e c o m p l e t e yield surface for t h e reinforced m a t e r i a l consists of a n u m b e r of " p i e c e s " of various analytic surfaces a n d its f o r m depends on t h e c h a r a c t e r of t h e r e i n f o r c e m e n t a n d on t h e properties of t h e c o m p o n e n t elements. Moreover, as will be seen, t h e f o r m of t h e stress-strain interdependence equations of t h e reinforced layer also depends on t h e c h a r a c t e r of the reinforcement. F o r example, let t h e p l a s t i c i t y condition h a v e the f o r m of e q u a t i o n (9) a n d the s t r e s s - s t r a i n d i a g r a m be elastic linear strain hardening. T h e n for t h e stresses lying in the v i c i n i t y of b u t n o t w i t h i n surface (9) : E c = E, E ~ = E ~ , ( p = 1,2 . . . . . N , p ¢ n ) (10) and t h e law of strain in t h e reinforcement elements of angle an has t h e f o r m a~ = Ep. 8(.) +_ a~(1 - E # J E ~ ) .
(11)
Signs ( + ) or ( - ) are d e p e n d e n t on t h e violation of the right- or left-hand p a r t of i n e q u a l i t y (8) a t stresses satisfying e q u a t i o n (9). E ~ are the t a n g e n t m o d u l i for tension ( + ) or compression ( - ). F u r t h e r , assume for definiteness the dependence e q u a t i o n (11) a t signs ( + ) . Then, t a k i n g into a c c o u n t equations (2) and (10) and s u b s t i t u t i n g equations (3) and (11) into e q u a t i o n (1) the following relations b e t w e e n the forces a n d strains are o b t a i n e d : t"
=
t'=
H akm ]l ~, '
e
IIt'l,t'2, t'a]] -1,
t; = t , - f l ,
l~,,
[Ibkm][ t,
=
'
k,m=
f[ bkm H '
=
H "
a~m][
-1
1,2,3,
t's = t a - f l ~ l l , 1 2 ~ ,
' /
)
(12)
f~, = w ~ a + [ 1 - E + ~ / E , ] .
I n e q u a t i o n (12) the coefficients of the m a t r i x I] a~m H h a v e the same form as those of t h e m a t r i x II akin ]1 f r o m e q u a t i o n (6), if in t h e l a t t e r E~ is replaced b y E$~. R e l a t i o n s (12) describe t h e elastic-plastic b e h a v i o u r of the reinforced layer in t h e case w h e n t h e plasticity condition for it has t h e f o r m of e q u a t i o n (9). These relations will hold until one of t h e inequalities (7) or (8) is v i o l a t e d ff in the l a t t e r b~ and tk are replaced b y b~ and t~ (i = 1, 2; k = 1, 2, 3), respectively. I f inequalities of t h e t y p e (7) are violated, t h e n the subsequent t r a n s f o r m a t i o n s of t h e equations can be m a d e in t h e same w a y as in o b t a i n i n g equations (12) and as a result linear e q u a t i o n s similar to equations (12) will be obtained. I f t h e m a t e r i a l of t h e reinforcing elements is ideally elastic-plastic, t h e n in equations (12) and in those similar to t h e m t h e corresponding t a n g e n t m o d u l i m u s t be e q u a t e d to zero. I n contrast to the isotropie or " p h y s i c a l l y a n i s o t r o p i c " ideally plastic layer the reinforced layer w i t h ideally elastic-plastic reinforcing elements displays a one-to-one relation b e t w e e n stresses and strains for non-elastic strains. If, a t first, i n e q u a l i t y (8) alone is v i o l a t e d while inequalities (7) r e m a i n valid, t h e plastic strain starts in t h e layer of the filler and is a c c o m p a n i e d b y elastic strains in t h e reinforcing elements. H e n c e it should be assumed t h a t E$~ = E . ,
E c = ¢(8),
I
J
(13)
H e r e 8 is the i n t e n s i t y of strain in the layer of t h e filler and ~(8) is t h e function d e t e r m i n e d b y t h e plastic p a r t of t h e tension diagram. S u b s t i t u t i n g expressions (3) and (2) into equations (1) for the values of equations (13) for the secant m o d u l i we o b t a i n tl = a l l 81 + a12 82 -t- a l a 83 -I- ~ [ ~ ( 8 ) - - E ] (81 -I- ~82), t~ = a12 e I -~- a22 82 + a2a 8a + ~ [ ¢ ( 8 ) -- E ] (8~ + -~81) , t s = a13 e 1 -{-
(14)
ass 8s + a33 8 s -}- ½ [ ¢ ( 8 ) - - E ] 8 s.
I n c o n t r a s t to e q u a t i o n (12) these equations are essentially non-lineax e v e n for a linear s t r e n g t h e n i n g of t h e filler material. I f the m a t e r i a l of t h e filler is ideally plastic w i t h yield stress a 0 t h e n in equations (14) it m u s t be assumed t h a t ~(8) = a0/8. I n this case e q u a t i o n s (14) d e t e r m i n e the one-to-one relation b e t w e e n t k and 8~ if only a~m¢0
(k,m=
1,2,3).
898
Ytr. V. NEMIROVSKY
A s s u m e t h a t a t first i n e q u a l i t y (7) is v i o l a t e d i m p l y i n g p l a s t i c s t r a i n in t h e r e i n f o r c i n g elements which make the angle a, with direction 1 and causing strains in the reinforced m a t e r i a l t o o c c u r in a c c o r d a n c e w i t h t h e c o n d i t i o n s (12), a n d f u r t h e r t h a t i n e q u a l i t y (8) is v i o l a t e d a t some v a l u e of t k (k = 1, 2, 3). T h e n t h e law of s t r a i n will h a v e t h e f o r m of e q u a t i o n (14) if t k a n d ak~ are s u b s t i t u t e d for t~ a n d a ~ , respectively. R e l a t i o n s of t h e t y p e (12)-(14) allows n o t o n l y t h e c h a r a c t e r of t h e defo1~ning e l a s t i c p l a s t i c l a y e r t o b e d e s c r i b e d b u t also, t o g e t h e r w i t h e q u a t i o n s (2) a n d (13), t h e d e t e r m i n a t i o n of t h e efficiency of all e l e m e n t s of t h e composite. D e p e n d i n g o n w h a t e l e m e n t s of t h e c o m p o s i t e are d e f o r m e d plastically, t h e s e r e l a t i o n s h a v e a l i n e a r or a n o n - l i n e a r c h a r a c t e r , so t h a t in a n u m b e r of cases some i n d i r e c t e v i d e n c e o n t h e efficiency of t h e g i v e n t y p e of r e i n f o r c e m e n t c a n b e o b t a i n e d i m m e d i a t e l y f r o m t h e e x p e r i m e n t a l d i a g r a m s of s t r a i n in m a t e r i a l s or c o n s t r u c t i o n s . 2. E X A M P L E :
THE TENSION REINFORCED
OF A UNI-DIRECTIONAL LAYER
C o n s i d e r t h e p r o b l e m of e x t e n d i n g , u n d e r t h e force tl, a l a y e r r e i n f o r c e d b y m o n o d i r e c t i o n a l t h r e a d s m a k i n g a n a n g l e al = a w i t h t h e d i r e c t i o n of loading. F o r s i m p l i c i t y a s s u m e t h e m a t e r i a l o f all c o m p o n e n t s to b e ideally e l a s t i c - p l a s t i c . T h e n in t h i s case it must be assumed that t2 = t3 =
O,
N
=
l,
E~
=
O,
5(~)
=
ao/e.
As a r e s u l t in t h e elastic region we o b t a i n tl - - 8 , aEel E~
-
~
O"1 E1 a E e l - 2 - E a a 81'
1-
'
---- a l l A -- als A 1 -- a13 A s
ass
a23,
-
A1
e~
" A'
3~2
A/ ,
AI=
aaa
E3 _
As
e1
A'
E- -~ I = -
(15)
(16)
~'
81 = 2A cos 2 a - - A s sin 2a -- 2A 1 sin ~ a '
aEA
A=lass
~
eS =
/
A
a12
a23,
alz
a33
A 2 = [ a2s ass
a12 , a13
(17)
J
O n Fig. 4 a r e g i v e n p l o t s of t h e v a l u e s of t l / a E e 1 (full lines) a n d e2/¢1 ( d a s h e d lines) c a l c u l a t e d f r o m t h e f o r m u l a e (15) a t eo1 E 1 / a E = 1, 5, 10 . . . . . T h e p l a s t i c i t y c o n d i t i o n for t h e r e i n f o r c i n g e l e m e n t s in t h i s case h a s t h e f o r m _tl_ = + 28 E a~ aao
(18)
- 81 E 1 (~o'
w h e r e t h e u p p e r signs c o r r e s p o n d to ell > 0 [the s t r a i n of t h e r e i n f o r c e m e n t e l e m e n t s ~11 is c a l c u l a t e d f r o m e q u a t i o n (2) w i t h e q u a t i o n (15) t a k e n i n t o a c c o u n t ] a n d t h e lower ones, t o exl < 0. A s s u m e f u r t h e r for definiteness t h a t a + = a~-. T h e n it c a n easily b e seen t h a t i n e q u a t i o n (18) o n l y t h e u p p e r signs m u s t b e t a k e n . I n Fig. 5 t h e d a s h e d lines r e p r e s e n t t h e r e l a t i o n s of t l / a a o o n a c a l c u l a t e d a t ¢o1 a + / a a o = 0.5, 1.0, 2.0 a n d w 1 E 1 / a E = 5. A t t h e v a l u e s of t h e loads l o c a t e d b e t w e e n t h e d o t t e d a n d d a s h e d lines in Fig. 5 t h e stresses i n t h e r e i n f o r c i n g e l e m e n t s r e m a i n c o n s t a n t a n d e q u a l t o a +. T h e r e l a t i o n s b e t w e e n t h e stress a n d t h e s t r a i n h a v e t h e f o r m aEel
=
tl
-
wl a+(cos 2 a - ½ sin s a),
a E e 2 = - ½ t I - t o 1 a+(sin 2 a -
½ cos 2 a),
(19)
a E e 3 = ~oJ 1 a + sin 2a.
T h u s t h e s h e a r s t r a i n does n o t d e p e n d o n t h e load and, o n t h e c o n t r a r y , " P o i s s o n ' s r a t i o " d e p e n d s o n it. R e l a t i o n s (19) a r e v a l i d u p to t h e v a l u e s of t h e l o a d a t w h i c h p l a s t i c s t r a i n s o c c u r in t h e binding material.
On the elastic-plastic b e h a v i o u r of a reinforced layer _.E_~2 aEE, 0.8 ,5
I0
4-5
3.7 5
%
-S
4.0
%
-0.6
r,2: 3-5
-0.5
~k
'3.0
-o.a ~\
.2.5
-0-3
\\ g
2.0
-0.2
1
llo -0.1
1.5
I-0
0
I0
20
30
40
50
60
70
80
90*
! 70
I 80
1 90
~o
FIG. 4.
. t , i'
°%1
3-5
3.0,
Z D,
2.5
4 2 I,
'2 • 0
° o 1.5
'~110 1.0
o~ ~"
0.5 0
I I0
I 20
f 30
T I 40 50
T 60
FIG. 5.
a ~1'~
.
899
900
Y r . V. NEmROVSKY
T h e c o r r e s p o n d i n g l o a d v a l u e s are d e t e r m i n e d f r o m t h e e q u a t i o n (tl - c o l a+ cos2 a) 2 + (t, - c o l a+ cos2 a) wl al+ sin s a + (wl a+) 2 sin 4 ~ + ¼(wl a+) 2 sin 2 2a = (av0) 2.
(20)
T h e d e p e n d e n c e o f t l / a a o on ~ c a l c u l a t e d w i t h t h e help o f t h i s e q u a t i o n a t col a+/aao = 0.5, 1.0, 2.0
is p l o t t e d o n Fig. 5 b y d o t t e d lines. G r e a t e r loads c a n n o t b e w i t h s t o o d b y t h e reinforced m a t e r i a l w i t h ideally p l a s t i c e l e m e n t s . F o r m u l a e (15) a n d (19) d e t e r m i n e t h e m a t e r i a l t e n s i l e d i a g r a m s in t h e cases w h e n p l a s t i c s t r a i n s first o c c u r i n t h e r e i n f o r c i n g e l e m e n t s . T h e c o r r e s p o n d i n g d i a g r a m s a t ¢ol E 1 / a E - - - - 5 , w I a+/aao = 0.5 (full lines) a n d 1.0 ( d a s h e d lines) a n d a = 0 °, 20 ° a r e r e p r e s e n t e d i n Fig. 6. T h e h o r i z o n t a l s e c t i o n s c o r r e s p o n d t o t h e l i m i t i n g loads for t h e layer. A ease is also possible w h e n p l a s t i c s t r a i n s o c c u r first i n t h e b i n d i n g m a t e r i a l while t h e r e i n f o r c i n g e l e m e n t s r e m a i n elastic. T h e n in e q u a t i o n (10) t h e i n e q u a l i t y sign m u s t b e r e p l a c e d b y t h a t o f e q u a l i t y a n d a f t e r s u b s t i t u t i n g i n t o it expressions (16) t h e following d e p e n d e n c e is o b t a i n e d : 2 t l / a a o = ~/(3) 8A(A 2 -- AA~ + A~)-½.
(21)
T h e g r a p h o f t h i s d e p e n d e n c e a t col E 1 / a E = 5 is r e p r e s e n t e d in Fig. 5 b y t h e full line. U n d e r loads e x c e e d i n g t h e v a l u e s of e q u a t i o n (21 ), u s i n g for s i m p l i c i t y t h e a s s u m p t i o n o f t h e ideally p l a s t i c m a t e r i a l , t h e following d e p e n d e n c e s b e t w e e n t h e stress t 1 a n d t h e s t r a i n s el, £2, ~a occur: 2a(2el + e2) a0 + 3col E1 eel cos 2 ~ = 3tl e, 2a(2e~ + e l ) a 0 +
3colE1 ee I sin 2 a
= 0,
)
(22)
2ass a0 + 3col E1 ~Sl sin 2a ---- 0. S i m p l e t r a n s f o r m a t i o n s of t h e s e e q u a t i o n s yield t h e following d e p e n d e n c e s : 6 t g a(1 - B t l / a a o ) + (tg 2 a - 2) e3/el = 0,
\aa0 A
/
el
1 e3 t g a ( t g 2 a + 4) - 2(tg 2 a - 2) 4(l+tg2a )
B = ~
12+
(tg2a+4)
.
T h e stresses in t h e r e i n f o r c i n g e l e m e n t s are e x p r e s s e d as follows: co1 a l / a a o = A / B .
F o r m u l a e (23) d e t e r m i n e s t h e s i n g l e - v a l u e d r e l a t i o n b e t w e e n t h e force el, e2, ea a n d t h i s in s p i t e of t h e ideal p l a s t i c i t y of t h e b i n d i n g m a t e r i a l . n o t e d t h a t i n c o n t r a s t t o e q u a t i o n (19) t h i s d e p e n d e n c e is n o n - l i n e a r . t h e l o a d - e x t e n s i o n d i a g r a m s c a l c u l a t e d a c c o r d i n g t o f o r m u l a e (15) a n d E / a o = 40,
(24) t1 a n d t h e strains I t s h o u l d also b e Fig. 7 r e p r e s e n t s (23) a t
col E 1 / a E = 1 , 5 .
R e l a t i o n s (23) w o u l d r e m a i n v a l i d u p t o t h e v a l u e s of t , / a a o a t w h i c h t h e stresses in t h e r e i n f o r c i n g e l e m e n t s will r e a c h t h e yield p o i n t . T h i s load v a l u e w o u l d b e l i m i t i n g for t h e g i v e n m a t e r i a l . A t t h i s l o a d a h o r i z o n t a l s e c t i o n occurs in t h e t e n s i o n d i a g r a m . I n Fig. 7 t h e s t r a i g h t lines c o r r e s p o n d t o s o l u t i o n (15) a n d t h e c u r v e d p o r t i o n s t o s o l u t i o n (23). T h e t r a n s i t i o n p o i n t s a r e d e t e r m i n e d a c c o r d i n g t o f o r m u l a (21). I t s h o u l d b e n o t e d t h a t f o r m u l a e (15) a n d (19) c a n b e a p p l i e d also to d e s c r i b i n g t h e e x t e n s i o n d i a g r a m of a m a t e r i a l r e i n f o r c e d b y ideally b r i t t l e t h r e a d s . I n t h i s case f o r m u l a
On the elastic-plastic b e h a v i o u r of a reinforced layer
901
(18) determines t h e loads u n d e r which the threads are broken. I f a t these load values t h e yield condition for the filler is n o t violated, t h e n t h e further b e h a v i o u r of t h e m a t e r i a l is c h a r a c t e r i z e d b y a m o m e n t a r y skip to t h e d i a g r a m described b y e q u a t i o n s (19) a t a + = O.
tl O0"O
2.C
/ s// •
1.6
/"
/o .................
20 °
s"
¢ • ~
••
o.
20*
1.2 0.8
0.4
1 04
I
1
I
l
0.8
1.2
1"6
2"0
: E~= a'o
FIG. 6.
Ii o0"o 1.730
1.720
ca=E= O.E
=5
1"440 --
1-430
oE
1.420
I
, 0-2
o
I 0.4
I 0.6
I 0.8
~
el
FIG. 7.
3. S H E A R
OF UNI-DIRECTIONAL
REINFORCED
MATERIAL
Consider t h e shear of uni-directional reinforced m a t e r i a l as a n o t h e r example. this case
t~=t z=0,
N=
1,
~=~,
Eh=0,
¢{e)=ao/e,
In
902
Y u . V. NEMIROVSKY
in the elastic region we have ta
- 8,
al = E181 t
aEea
~
a~j~ =
a,
3ESAa
2ts(2A1 + As) - 3~AE~a
a°~
ta
E
3SEa"
(25)
2(2A 2 + A1) t a
E
a~l E =
'
Here 8 = Aaaa-- ala A1 + as3 As ,
aEA
A =
all a21
a]2 , a22
A~ =
81 =
als a23
(A sin 2a - 2A 1 cos ~ a -- 2A, sin s a) 2A
a12 ,
A2 : I a l l I a21
a22
als a2a
i
On Fig. 8 are plotted graphs of the relation of the shear modulus and the angle a calculated a f t e r f o r m u l a e (25) a t eo1 E 1 / a E = 1, 5, 10.
J~
aEe 3 1.2
1.0
0"8
0-6
0.4
0.2
I
t
I0 20
I
I
50 40
I
I
50 60
I
70
I
80
I
90a~
F r o . 8. T h e p l a s t i c i t y c o n d i t i o n f o r t h e r e i n f o r c i n g e l e m e n t s in t h i s c a s e h a s t h e f o r m (26)
Q/aao -- E S a + / E 1 8 1 (7o.
T h e g r a p h s o f t h i s d e p e n d e n c e a t co1 E 1 / a E = 5 a n d ¢o1 a+/aao = 0.5, 1.0 a r e r e p r e s e n t e d i n F i g . 9 b y d a s h e d lines. I f t h e l o a d s e x c e e d t h e v a l u e s o f e q u a t i o n (26), t h e n t h e e l a s t i c p l a s t i c b e h a v i o u r is d e t e r m i n e d b y t h e r e l a t i o n s a E e l = - oJ1 a+(cos s a -
½ sin 2 ~),
a E ~ 8 -- 3[ta-½oJ 1 a + sin 2a], aa°s = --wx a + s i n S a ,
aEe2 = - co1 a + ( s i n 2 a - ½ cos s a),
aa~l = - w 1 a + cos s a,
(27)
aa°s = ta--½oJ 1 a + s i n 2a.
R e l a t i o n s (27) r e m a i n v a l i d u n t i l t h e s t r e s s e s all,° a,~,° a°~ v i o l a t e i n e q u a l i t i e s (18). T h e c o r r e s p o n d i n g l i m i t i n g l o a d is d e t e r m i n e d b y t h e i n e q u a l i t y t3
aa o
_
_l coI a+ s i n 2a -42 aa o
1 -- ( ~°1 ° i t (cos4 ~ -- s i n ' ~ c o s ' ~ + sin 4 ~)
.
\ a(7o !
T h e g r a p h s o f t h e r e l a t i o n o f t h i s l o a d o n a a t wl a+/aao = 0-5, 1.0 a r e r e p r e s e n t e d in F i g . 9 b y d o t t e d lines. T h e solid lines i n t h e s a m e F i g . 9 r e p r e s e n t t h e r e l a t i o n s o f l o a d s o n t h e a n g l e a a t eo1 E 1 / a E = 5 in w h i c h c a s e t h e p l a s t i c s t r a i n s f i r s t o c c u r i n t h e b i n d i n g m a t e r i a l . T h e c o r r e s p o n d i n g e q u a t i o n for t h e m is o b t a i n e d b y r e p l a c i n g in e q u a t i o n (8) t h e s i g n o f i n e q u a l i t y b y t h a t o f e q u a l i t y a n d s u b s t i t u t i n g i n t o it e x p r e s s i o n s (25) for a~l, a~s, a~,.
On the elastic-plastic behaviour of a reinforced layer
903
The equations for the elastic-plastic behaviour of the reinforced layer for plastic strain of the binding material and for elastic strain of the reinforcing elements are obtained from equation (14) at t z = t~ = 0 and ¢(e) = ao/e. Similar answers can be obtained for any other type of loading or reinforcing of the layer.
oo-o I-C 0.9 1
/ ." ,-r-.. \
0.8 0.7
0,6
/',
/\
0.~
\
/ \
0.4
I 10
I 20
I o-s,,
I " ',I-4- "1" 30 40 50
I 60
I 1 70 80
IL_ 90a,,
F I G . 9.
I n conclusion it should be noted t h a t the model of ideally plastic material accepted here for calculation is used for definiteness and for emphasizing some peculiarities of relations between stresses and strains in the elastic-plastic straining of reinforced materials. When necessary the corresponding calculations can be made readily for any particular strength law. REFERENCES 1. V. L. BAZHANOV, I. I. GOLDENBLAT, V. A. KOP>TOV, A. D. POSPELOV and A. M. SI~YUKov, Soprotivleniye Stekloplastikov ("Strength of Fibre Glass Plastic"). Izd-vo "Mashinostroyeniye", Moskva (1968). 2. A. K. MAL~:EISTER, V. P. TAMUZH and G. A. TETERS, Soprotivleniye Zhostkikh Poll. mernykh Materialov. ("Strength of Rigid Polymer Materials"). Izd-vo "Zinatne", 1%iga (1967). 3. V. V. BOLOTIN, Mekhanika polimerov 2, 27 (1965). 4. G. A. VA~ F. FY, Mekhanika polimerov 4, 539 (1966). 5. 1%. ]:JILL, The Mathematical Theory of Plasticity. Oxford (1950). [Russian translation Gost. izdat (1956)]. 6. I. V. GOLDEI~BLAT,Izvestiya Akad. Nauk SSSR, OTN, 2, 60 (1955). 7. V. A. LOMAKZN, Izvestiya Akad. Nauk S S S R , Mekhanika i mashinostroyeniye 4, 47 (1960). 8. Y c . V. NEMIROVSKYand Yu. N. 1%ABOTNOV,Non-classical Shell Problems, pp. 786-807, North Holland, Amsterdam (1964). 9. Yu. V. NE~IIROVSKY, Inzhenerniy Zhurnal Mekhanika tvyordogo tyela 6, 80--89 (1969).
60
ON T H E E L A S T I C - P L A S T I C B E H A V I O U R REINFORCED LAYER
OF A
Yu. V. •EMIROVSKY Institute of Hydrodynamics, Siberian Branch of the U.S.S.R. Academy of Sciences, Novosibirsk, U.S.S.R. (Received 10 February 1969, and in revised form 31 July 1969)
Summary--On the basis of the structural analysis of a model reflecting all the main features of a reinforced layer, equations are obtained for the interdependence of stresses and strains beyond the elastic limit. The type of these equations as well as the conditions under which residual strains m a y occur are shown to depend essentially on the character of reinforcement and on the properties of the component elements. It is characteristic that even with the ideally plastic properties of component elements, the interdependence between averaged stresses and strains in the reinforced layer in the elastic-plastic deforming region remains mutually single-valued. Examples are given showing that the experimentally determined characteristics of the reinforced material (Young's modulus, shear modulus, Poisson's ratio and yield or breaking stress) as well as strain diagrams can be obtained by calculation. NOTATION
tk(1, 2, 3) averaged stresses in the layer ~k
~ ( i , j = 1, 2) a.(n = 1, 2. . . . . N) e(n) COn [iS
Eo, Ec,,
E, E n
strain components stresses in the binding material stresses in the reinforcement elements strain of angular reinforcement elements specific intensities of the elements of reinforcement direction cosines of reinforcement elements secant moduli of the binding materials and the reinforcing elements at tension ( + ) and compression ( - ) yield stresses of binding materials and reinforcing elements at tension ( + ) and compression ( - ) Young's moduli INTRODUCTION
NEW COMBINATIONS o f m a t e r i a l s , s u c h as r e i n f o r c e d plastics a n d concrete, m e t a l s s t r e n g t h e n e d b y fibres a n d others, are finding wide a p p l i c a t i o n in m o d e r n engineering. T h e i r m a i n a d v a n t a g e lies in t h e possibility o f r e g u l a t i n g t h e i r m e c h a n i c a l c h a r a c t e r i s t i c s in a n y direction. This possibility is so w i d e t h a t t o s p e a k a b o u t a n y definite class o f r e i n f o r c e d m a t e r i a l h a s little m e a n i n g . The mechanical properties of reinforced material depend on those of the binding and the reinforcing materials, their percentage composition and the c h a r a c t e r o f r e i n f o r c e m e n t . H e n c e p r a c t i c a l l y e v e r y class o f r e i n f o r c e d m a t e r i a l r e q u i r e s a special a n a l y s i s t h e m a i n p u r p o s e o f w h i c h is t o o b t a i n r e l a t i o n s d e s c r i b i n g i n t e r d e p e n d e n c i e s b e t w e e n stresses a n d s t r a i n s in t h e r e i n f o r c e d material. 893
894
Yu. V. N]:MIROVSKY
There are two approaches to o b t a i n i n g these relations : (a) phenomenological, where the reinforced m e d i u m is considered to be uniform, monolithic and anisotropic 1, ~ and (b) an a p p r o a c h based on the s t r u c t u r a l analysis of t h e reinforced m a t e r i a l in accordance with the c h a r a c t e r of its s t r u c t u r e a n d the mechanical properties of its components, a, 4 In the elastic range if some specific effects (stress c o n c e n t r a t i o n in the neighbourhood of reinforcing elements, n o n - u n i f o r m i t y of d e f o r m a t i o n a m o n g t h e m , etc.) are neglected, both approaches yield in essence the same equations of interdependence b e t w e e n the a v e r a g e d stresses and strains a n d in this sense are equivalent. The phenomenological a p p r o a c h to o b t a i n i n g equations d e t e r m i n i n g t h e elasticplastic b e h a v i o u r of anisotropic m e d i a has been applied in a n u m b e r of investigations. ~-7 H o w e v e r , in the equations o b t a i n e d it is necessary to refer to "physically anisotropie m e d i a " whose a n i s o t r o p y is due to the crystalline structure. E a c h t y p e of constructed anisotropy, including reinforced materials, m u s t he analysed separately since b e y o n d the limit of elasticity t h e specific features of each t y p e affect the f o r m of the plasticity conditions ;a. a and, as will be seen later, also affects the c h a r a c t e r of t h e equations describing the interdependence b e t w e e n stresses and strains. Hence, t h e relations describing this i n t e r d e p e n d e n c e b e y o n d the limit of elasticity can be o b t a i n e d only b y m e a n s of the s t r u c t u r a l analysis of the reinforced m a t e r i a l on the basis of a m o d e l reflecting its specific features. The a d v a n t a g e of the s t r u c t u r a l analysis also lies in the fact t h a t it p e r m i t s an e s t i m a t e to be m a d e of the efficiency of each element of the composition and t h u s opens a w a y to t h e purposeful regulation of reinforcement c h a r a c t e r for increasing the strength properties of reinforced materials. The present p a p e r analyses the elastic-plastic b e h a v i o u r of a reinforced layer u n d e r the action of forces in its plane on the basis of a proposed m o d e l a n d some additional simplifying suggestions. 1. G E N E R A L
CASE
The elemental layer under consideration is a c o m p a r a t i v e l y t h i n isotropic plate with a reinforcing layer incorporated in it (see Fig. 1). The l a t t e r represents a n e t of thin onedimensional threads running at an angle an (n = 1, 2, ..., N) w i t h direction 1.
C) C) Q
(D (D C)
FIG. l.
I h
It is assumed t h a t
(i) the m a t e r i a l of each element of t h e composition is elastic-plastic and in the general case is different for each e l e m e n t ; (ii) the n u m b e r of reinforcing elements is sufficiently great to consider the composite m a t e r i a l as quasi-homogeneous; (iii) t h e space b e t w e e n the reinforcing elements is sufficiently great c o m p a r e d with t h e i r dimensions, and at t h e same t i m e sufficiently small c o m p a r e d w i t h the plate dimensions, t h a t local effects in the n e i g h b o u r h o o d of t h e t h r e a d s a n d irregularity of strain b e t w e e n the threads can be neglected; (iv) the b o n d b e t w e e n t h e composition elements is ideal, i.e. there is no slip between the reinforcing elements and t h e binding m a t e r i a l ; (v) each thread, if incorporated into the binding material, can w i t h s t a n d b o t h tensile and compressive stresses. H o w e v e r , due to a form of instability which m a y occur in compression, t h e yield stresses or b r e a k i n g stresses (as well as t h e m o d u l u s of s t r e n g t h e n i n g for s t r e n g t h e n e d materials) in tension a n d compression are considered to be different, Y o u n g ' s moduli being assumed the same for tension and compression; and
On t h e e l a s t i c - p l a s t i c b e h a v i o u r of a r e i n f o r c e d l a y e r
895
(vi) t h e isotropic b i n d i n g m a t e r i a l satisfies t h e t h e o r y of p l a s t i c d e f o r m a t i o n w i t h t h e s a m e c h a r a c t e r i s t i c s u n d e r t e n s i o n a n d c o m p r e s s i o n a n d for s i m p l i c i t y is a s s u m e d t o u n d e r g o n o c h a n g e of v o l u m e b o t h i n t h e elastic a n d in t h e p l a s t i c regions. W h e r e n e c e s s a r y it is possible t o r e j e c t t h e r e q u i r e m e n t o f c o n s t a n t v o l u m e a n d t o use a t y p e of i n c r e m e n t t h e o r y i n s t e a d of t h a t of s t r a i n . L e t ~on (n = 1, 2 . . . . . N ) b e t h e specific i n t e n s i t i e s o f t h r e a d s in t h e r e i n f o r c i n g l a y e r r u n n i n g a t a n a n g l e an (n = 1, 2, ..., 5/) w i t h d i r e c t i o n 1, a n d let h b e t h e t h i c k n e s s of t h e r e i n f o r c e d layer, ¢o~ is t h e specific i n t e n s i t y of t h e r e i n f o r c e d l a y e r in t h e p l a t e ' s t h i c k n e s s , Fig. 2. T h e n i n t h e o r t h o g o n a l s y s t e m 1-2 c o m p o n e n t s o f t h e i n t e r n a l forces, see Fig. 3, in t h e c o m p o s i t e l a y e r w o u l d b e e x p r e s s e d as follows:
t k = aa~j+ ~ o J . anli. l~.
TIj
tj = ~ - ,
TI.
t3 = ~ - ,
(i,j= 1,2;k = 1,2,3),
lxn= cosa,,
l~,=sinan,
O<~a,~<.lr, a = 1--~ z, ¢o,~=nnFn/AFh, C
°I_ I-
(1)
¢o~=3/h.
B
d~,'111111 ~A -I
FIG. 2.
T22 FIG. 3. H e r e Tis a r e forces, a ~ are stresses in t h e filler, an a r e stresses in t h e r e i n f o r c e m e n t t h r e a d s , F n a r e cross-sectional a r e a s of t h e r e i n f o r c e m e n t e l e m e n t s a n d n n are t h e n u m b e r s o f t h r e a d s of t h e r e i n f o r c e m e n t e l e m e n t s in t h e s e g m e n t s AF, Fig. 2. A t s m a l l s t r a i n s o n t h e b a s i s of a s s u m p t i o n (iv) t h e following r e l a t i o n s b e t w e e n t h e s t r a i n s e(,~ of t h e r e i n f o r c e m e n t e l e m e n t s a n d t h o s e of t h e filling l a y e r are o b t a i n e d :
e~.) = el l~. + e2 l~. + ea lln 12..
(2)
H e r e ca, e2 a r e t h e s t r a i n c o m p o n e n t s of t h e filling l a y e r in d i r e c t i o n s 1 a n d 2, r e s p e c t i v e l y ; ea is t h e s h e a r s t r a i n .
896
Yu. V. NEMIROVSKY
I n c o n f o r m i t y w i t h the a b o v e assumptions the internal stresses of the composite layer elements are related to the strains b y the following equalities: a°l = ~Eo(el + ½~2), a~2 --- ~Ec(e2 + ½el), ~ 0 = 1Ece3,
(3)
an = E ~ n e . .
H e r e Ec, E$n are the secant moduli of the filler materials and of the reinforcing elements in tension ( + ) and compression ( - ). I f all t h e elements of the composition u n d e r the g i v e n forces tk (k = 1, 2, 3) r e m a i n elastic, t h e n all the secant moduli are equal to the corresponding Y o u n g ' s moduli Ec=E
,
(4)
E~n=E,,.
Then, s u b s t i t u t i n g expressions (3) into (1) the following relations b e t w e e n the forces tk and the strains e~ are o b t a i n e d : t=
I[akmlle,
t
[[tl, t2, t3[I -x,
e = !]bkm[It, ~=
][bkm][ = [Iakm][-l'
Ii~l, e2, e3][ -1
1
(5)
1,2,3).
(k,m=
The coefficients of the m a t r i x [I akin [I are equal to: N
a. = kaE+ ~.(o.E.l~., n=l N
ale = a21 = ~ a E + ~. (on En l~n l~n, n=l
aia = a3i =
N ~ (onEnl~nlsn,
(6)
nffil N
a33 = ½ a E + Z ( o , , E n l ~ , , l ~
( i , j = 1,2; i ¢ j ) .
n=l
S u b s t i t u t i n g the strain c o m p o n e n t s f r o m e q u a t i o n (5) into (3) and t a k i n g into a c c o u n t e q u a t i o n (4) the internal stresses in all t h e elements composing t h e reinforced layer are found. W i t h t h e i r help it is possible to d e t e r m i n e u n d e r w h a t loads different elements of t h e composite layer pass into t h e plastic state. So t h e reinforcing elements r e m a i n elastic if the following inequalities are satisfied : -a~
ltk(blkl~+b2~l~+b3kll~12~)
]
(n = 1 , 2 . . . . . N )
(7)
and t h e layer of t h e binding m a t e r i a l remains elastic if the inequality o2 all o ( 7 o 2 + a ozu 2 + 3q12 02 < °'02 a11--
(8)
is satisfied. The expression (3) m u s t be s u b s t i t u t e d instead of o~j t a k i n g into aecotmt equations (4) and (5). I n equations (7) a n d (8) a0 and a .~ denote, respectively, t h e yield stresses of the binding m a t e r i a l a n d the m a t e r i a l of the reinforcing elements in tension or compression. The violation of a n y of the inequalities (7) and (8) leads to the occurrence of plastic strain in t h e corresponding reinforcing elements or the filler. F o r example, u n d e r some c o m b i n a t i o n of forces tk, such t h a t f . ( t l , t 2, t3) = 0 (9) let a n y of t l ~ inequalities (7) be violated. T h e n e q u a t i o n (9) in stress space t 1, t 2, t 3 determines the yield surface for the reinforced m a t e r i a l w i t h the c o m p o n e n t elements possessing the elastic-plastic properties. Indeed, for all stresses w i t h i n this surface there are no residual strains after t h e unloading of the layer. F o r the stresses outside this surface some residual strain remains in t h e reinforced layer after unloading. Analogous surfaces can also be o b t a i n e d in stress space w h e n all r e m a i n i n g inequalities (7) or (8) are v i o l a t e d
On t h e elastic-plastic b e h a v i o u r of a reinforced layer
897
(individually or in combinations). T h u s t h e c o m p l e t e yield surface for t h e reinforced m a t e r i a l consists of a n u m b e r of " p i e c e s " of various analytic surfaces a n d its f o r m depends on t h e c h a r a c t e r of t h e r e i n f o r c e m e n t a n d on t h e properties of t h e c o m p o n e n t elements. Moreover, as will be seen, t h e f o r m of t h e stress-strain interdependence equations of t h e reinforced layer also depends on t h e c h a r a c t e r of the reinforcement. F o r example, let t h e p l a s t i c i t y condition h a v e the f o r m of e q u a t i o n (9) a n d the s t r e s s - s t r a i n d i a g r a m be elastic linear strain hardening. T h e n for t h e stresses lying in the v i c i n i t y of b u t n o t w i t h i n surface (9) : E c = E, E ~ = E ~ , ( p = 1,2 . . . . . N , p ¢ n ) (10) and t h e law of strain in t h e reinforcement elements of angle an has t h e f o r m a~ = Ep. 8(.) +_ a~(1 - E # J E ~ ) .
(11)
Signs ( + ) or ( - ) are d e p e n d e n t on t h e violation of the right- or left-hand p a r t of i n e q u a l i t y (8) a t stresses satisfying e q u a t i o n (9). E ~ are the t a n g e n t m o d u l i for tension ( + ) or compression ( - ). F u r t h e r , assume for definiteness the dependence e q u a t i o n (11) a t signs ( + ) . Then, t a k i n g into a c c o u n t equations (2) and (10) and s u b s t i t u t i n g equations (3) and (11) into e q u a t i o n (1) the following relations b e t w e e n the forces a n d strains are o b t a i n e d : t"
=
t'=
H akm ]l ~, '
e
IIt'l,t'2, t'a]] -1,
t; = t , - f l ,
l~,,
[Ibkm][ t,
=
'
k,m=
f[ bkm H '
=
H "
a~m][
-1
1,2,3,
t's = t a - f l ~ l l , 1 2 ~ ,
' /
)
(12)
f~, = w ~ a + [ 1 - E + ~ / E , ] .
I n e q u a t i o n (12) the coefficients of the m a t r i x I] a~m H h a v e the same form as those of t h e m a t r i x II akin ]1 f r o m e q u a t i o n (6), if in t h e l a t t e r E~ is replaced b y E$~. R e l a t i o n s (12) describe t h e elastic-plastic b e h a v i o u r of the reinforced layer in t h e case w h e n t h e plasticity condition for it has t h e f o r m of e q u a t i o n (9). These relations will hold until one of t h e inequalities (7) or (8) is v i o l a t e d ff in the l a t t e r b~ and tk are replaced b y b~ and t~ (i = 1, 2; k = 1, 2, 3), respectively. I f inequalities of t h e t y p e (7) are violated, t h e n the subsequent t r a n s f o r m a t i o n s of t h e equations can be m a d e in t h e same w a y as in o b t a i n i n g equations (12) and as a result linear e q u a t i o n s similar to equations (12) will be obtained. I f t h e m a t e r i a l of t h e reinforcing elements is ideally elastic-plastic, t h e n in equations (12) and in those similar to t h e m t h e corresponding t a n g e n t m o d u l i m u s t be e q u a t e d to zero. I n contrast to the isotropie or " p h y s i c a l l y a n i s o t r o p i c " ideally plastic layer the reinforced layer w i t h ideally elastic-plastic reinforcing elements displays a one-to-one relation b e t w e e n stresses and strains for non-elastic strains. If, a t first, i n e q u a l i t y (8) alone is v i o l a t e d while inequalities (7) r e m a i n valid, t h e plastic strain starts in t h e layer of the filler and is a c c o m p a n i e d b y elastic strains in t h e reinforcing elements. H e n c e it should be assumed t h a t E$~ = E . ,
E c = ¢(8),
I
J
(13)
H e r e 8 is the i n t e n s i t y of strain in the layer of t h e filler and ~(8) is t h e function d e t e r m i n e d b y t h e plastic p a r t of t h e tension diagram. S u b s t i t u t i n g expressions (3) and (2) into equations (1) for the values of equations (13) for the secant m o d u l i we o b t a i n tl = a l l 81 + a12 82 -t- a l a 83 -I- ~ [ ~ ( 8 ) - - E ] (81 -I- ~82), t~ = a12 e I -~- a22 82 + a2a 8a + ~ [ ¢ ( 8 ) -- E ] (8~ + -~81) , t s = a13 e 1 -{-
(14)
ass 8s + a33 8 s -}- ½ [ ¢ ( 8 ) - - E ] 8 s.
I n c o n t r a s t to e q u a t i o n (12) these equations are essentially non-lineax e v e n for a linear s t r e n g t h e n i n g of t h e filler material. I f the m a t e r i a l of t h e filler is ideally plastic w i t h yield stress a 0 t h e n in equations (14) it m u s t be assumed t h a t ~(8) = a0/8. I n this case e q u a t i o n s (14) d e t e r m i n e the one-to-one relation b e t w e e n t k and 8~ if only a~m¢0
(k,m=
1,2,3).
898
Ytr. V. NEMIROVSKY
A s s u m e t h a t a t first i n e q u a l i t y (7) is v i o l a t e d i m p l y i n g p l a s t i c s t r a i n in t h e r e i n f o r c i n g elements which make the angle a, with direction 1 and causing strains in the reinforced m a t e r i a l t o o c c u r in a c c o r d a n c e w i t h t h e c o n d i t i o n s (12), a n d f u r t h e r t h a t i n e q u a l i t y (8) is v i o l a t e d a t some v a l u e of t k (k = 1, 2, 3). T h e n t h e law of s t r a i n will h a v e t h e f o r m of e q u a t i o n (14) if t k a n d ak~ are s u b s t i t u t e d for t~ a n d a ~ , respectively. R e l a t i o n s of t h e t y p e (12)-(14) allows n o t o n l y t h e c h a r a c t e r of t h e defo1~ning e l a s t i c p l a s t i c l a y e r t o b e d e s c r i b e d b u t also, t o g e t h e r w i t h e q u a t i o n s (2) a n d (13), t h e d e t e r m i n a t i o n of t h e efficiency of all e l e m e n t s of t h e composite. D e p e n d i n g o n w h a t e l e m e n t s of t h e c o m p o s i t e are d e f o r m e d plastically, t h e s e r e l a t i o n s h a v e a l i n e a r or a n o n - l i n e a r c h a r a c t e r , so t h a t in a n u m b e r of cases some i n d i r e c t e v i d e n c e o n t h e efficiency of t h e g i v e n t y p e of r e i n f o r c e m e n t c a n b e o b t a i n e d i m m e d i a t e l y f r o m t h e e x p e r i m e n t a l d i a g r a m s of s t r a i n in m a t e r i a l s or c o n s t r u c t i o n s . 2. E X A M P L E :
THE TENSION REINFORCED
OF A UNI-DIRECTIONAL LAYER
C o n s i d e r t h e p r o b l e m of e x t e n d i n g , u n d e r t h e force tl, a l a y e r r e i n f o r c e d b y m o n o d i r e c t i o n a l t h r e a d s m a k i n g a n a n g l e al = a w i t h t h e d i r e c t i o n of loading. F o r s i m p l i c i t y a s s u m e t h e m a t e r i a l o f all c o m p o n e n t s to b e ideally e l a s t i c - p l a s t i c . T h e n in t h i s case it must be assumed that t2 = t3 =
O,
N
=
l,
E~
=
O,
5(~)
=
ao/e.
As a r e s u l t in t h e elastic region we o b t a i n tl - - 8 , aEel E~
-
~
O"1 E1 a E e l - 2 - E a a 81'
1-
'
---- a l l A -- als A 1 -- a13 A s
ass
a23,
-
A1
e~
" A'
3~2
A/ ,
AI=
aaa
E3 _
As
e1
A'
E- -~ I = -
(15)
(16)
~'
81 = 2A cos 2 a - - A s sin 2a -- 2A 1 sin ~ a '
aEA
A=lass
~
eS =
/
A
a12
a23,
alz
a33
A 2 = [ a2s ass
a12 , a13
(17)
J
O n Fig. 4 a r e g i v e n p l o t s of t h e v a l u e s of t l / a E e 1 (full lines) a n d e2/¢1 ( d a s h e d lines) c a l c u l a t e d f r o m t h e f o r m u l a e (15) a t eo1 E 1 / a E = 1, 5, 10 . . . . . T h e p l a s t i c i t y c o n d i t i o n for t h e r e i n f o r c i n g e l e m e n t s in t h i s case h a s t h e f o r m _tl_ = + 28 E a~ aao
(18)
- 81 E 1 (~o'
w h e r e t h e u p p e r signs c o r r e s p o n d to ell > 0 [the s t r a i n of t h e r e i n f o r c e m e n t e l e m e n t s ~11 is c a l c u l a t e d f r o m e q u a t i o n (2) w i t h e q u a t i o n (15) t a k e n i n t o a c c o u n t ] a n d t h e lower ones, t o exl < 0. A s s u m e f u r t h e r for definiteness t h a t a + = a~-. T h e n it c a n easily b e seen t h a t i n e q u a t i o n (18) o n l y t h e u p p e r signs m u s t b e t a k e n . I n Fig. 5 t h e d a s h e d lines r e p r e s e n t t h e r e l a t i o n s of t l / a a o o n a c a l c u l a t e d a t ¢o1 a + / a a o = 0.5, 1.0, 2.0 a n d w 1 E 1 / a E = 5. A t t h e v a l u e s of t h e loads l o c a t e d b e t w e e n t h e d o t t e d a n d d a s h e d lines in Fig. 5 t h e stresses i n t h e r e i n f o r c i n g e l e m e n t s r e m a i n c o n s t a n t a n d e q u a l t o a +. T h e r e l a t i o n s b e t w e e n t h e stress a n d t h e s t r a i n h a v e t h e f o r m aEel
=
tl
-
wl a+(cos 2 a - ½ sin s a),
a E e 2 = - ½ t I - t o 1 a+(sin 2 a -
½ cos 2 a),
(19)
a E e 3 = ~oJ 1 a + sin 2a.
T h u s t h e s h e a r s t r a i n does n o t d e p e n d o n t h e load and, o n t h e c o n t r a r y , " P o i s s o n ' s r a t i o " d e p e n d s o n it. R e l a t i o n s (19) a r e v a l i d u p to t h e v a l u e s of t h e l o a d a t w h i c h p l a s t i c s t r a i n s o c c u r in t h e binding material.
On the elastic-plastic b e h a v i o u r of a reinforced layer _.E_~2 aEE, 0.8 ,5
I0
4-5
3.7 5
%
-S
4.0
%
-0.6
r,2: 3-5
-0.5
~k
'3.0
-o.a ~\
.2.5
-0-3
\\ g
2.0
-0.2
1
llo -0.1
1.5
I-0
0
I0
20
30
40
50
60
70
80
90*
! 70
I 80
1 90
~o
FIG. 4.
. t , i'
°%1
3-5
3.0,
Z D,
2.5
4 2 I,
'2 • 0
° o 1.5
'~110 1.0
o~ ~"
0.5 0
I I0
I 20
f 30
T I 40 50
T 60
FIG. 5.
a ~1'~
.
899
900
Y r . V. NEmROVSKY
T h e c o r r e s p o n d i n g l o a d v a l u e s are d e t e r m i n e d f r o m t h e e q u a t i o n (tl - c o l a+ cos2 a) 2 + (t, - c o l a+ cos2 a) wl al+ sin s a + (wl a+) 2 sin 4 ~ + ¼(wl a+) 2 sin 2 2a = (av0) 2.
(20)
T h e d e p e n d e n c e o f t l / a a o on ~ c a l c u l a t e d w i t h t h e help o f t h i s e q u a t i o n a t col a+/aao = 0.5, 1.0, 2.0
is p l o t t e d o n Fig. 5 b y d o t t e d lines. G r e a t e r loads c a n n o t b e w i t h s t o o d b y t h e reinforced m a t e r i a l w i t h ideally p l a s t i c e l e m e n t s . F o r m u l a e (15) a n d (19) d e t e r m i n e t h e m a t e r i a l t e n s i l e d i a g r a m s in t h e cases w h e n p l a s t i c s t r a i n s first o c c u r i n t h e r e i n f o r c i n g e l e m e n t s . T h e c o r r e s p o n d i n g d i a g r a m s a t ¢ol E 1 / a E - - - - 5 , w I a+/aao = 0.5 (full lines) a n d 1.0 ( d a s h e d lines) a n d a = 0 °, 20 ° a r e r e p r e s e n t e d i n Fig. 6. T h e h o r i z o n t a l s e c t i o n s c o r r e s p o n d t o t h e l i m i t i n g loads for t h e layer. A ease is also possible w h e n p l a s t i c s t r a i n s o c c u r first i n t h e b i n d i n g m a t e r i a l while t h e r e i n f o r c i n g e l e m e n t s r e m a i n elastic. T h e n in e q u a t i o n (10) t h e i n e q u a l i t y sign m u s t b e r e p l a c e d b y t h a t o f e q u a l i t y a n d a f t e r s u b s t i t u t i n g i n t o it expressions (16) t h e following d e p e n d e n c e is o b t a i n e d : 2 t l / a a o = ~/(3) 8A(A 2 -- AA~ + A~)-½.
(21)
T h e g r a p h o f t h i s d e p e n d e n c e a t col E 1 / a E = 5 is r e p r e s e n t e d in Fig. 5 b y t h e full line. U n d e r loads e x c e e d i n g t h e v a l u e s of e q u a t i o n (21 ), u s i n g for s i m p l i c i t y t h e a s s u m p t i o n o f t h e ideally p l a s t i c m a t e r i a l , t h e following d e p e n d e n c e s b e t w e e n t h e stress t 1 a n d t h e s t r a i n s el, £2, ~a occur: 2a(2el + e2) a0 + 3col E1 eel cos 2 ~ = 3tl e, 2a(2e~ + e l ) a 0 +
3colE1 ee I sin 2 a
= 0,
)
(22)
2ass a0 + 3col E1 ~Sl sin 2a ---- 0. S i m p l e t r a n s f o r m a t i o n s of t h e s e e q u a t i o n s yield t h e following d e p e n d e n c e s : 6 t g a(1 - B t l / a a o ) + (tg 2 a - 2) e3/el = 0,
\aa0 A
/
el
1 e3 t g a ( t g 2 a + 4) - 2(tg 2 a - 2) 4(l+tg2a )
B = ~
12+
(tg2a+4)
.
T h e stresses in t h e r e i n f o r c i n g e l e m e n t s are e x p r e s s e d as follows: co1 a l / a a o = A / B .
F o r m u l a e (23) d e t e r m i n e s t h e s i n g l e - v a l u e d r e l a t i o n b e t w e e n t h e force el, e2, ea a n d t h i s in s p i t e of t h e ideal p l a s t i c i t y of t h e b i n d i n g m a t e r i a l . n o t e d t h a t i n c o n t r a s t t o e q u a t i o n (19) t h i s d e p e n d e n c e is n o n - l i n e a r . t h e l o a d - e x t e n s i o n d i a g r a m s c a l c u l a t e d a c c o r d i n g t o f o r m u l a e (15) a n d E / a o = 40,
(24) t1 a n d t h e strains I t s h o u l d also b e Fig. 7 r e p r e s e n t s (23) a t
col E 1 / a E = 1 , 5 .
R e l a t i o n s (23) w o u l d r e m a i n v a l i d u p t o t h e v a l u e s of t , / a a o a t w h i c h t h e stresses in t h e r e i n f o r c i n g e l e m e n t s will r e a c h t h e yield p o i n t . T h i s load v a l u e w o u l d b e l i m i t i n g for t h e g i v e n m a t e r i a l . A t t h i s l o a d a h o r i z o n t a l s e c t i o n occurs in t h e t e n s i o n d i a g r a m . I n Fig. 7 t h e s t r a i g h t lines c o r r e s p o n d t o s o l u t i o n (15) a n d t h e c u r v e d p o r t i o n s t o s o l u t i o n (23). T h e t r a n s i t i o n p o i n t s a r e d e t e r m i n e d a c c o r d i n g t o f o r m u l a (21). I t s h o u l d b e n o t e d t h a t f o r m u l a e (15) a n d (19) c a n b e a p p l i e d also to d e s c r i b i n g t h e e x t e n s i o n d i a g r a m of a m a t e r i a l r e i n f o r c e d b y ideally b r i t t l e t h r e a d s . I n t h i s case f o r m u l a
On the elastic-plastic b e h a v i o u r of a reinforced layer
901
(18) determines t h e loads u n d e r which the threads are broken. I f a t these load values t h e yield condition for the filler is n o t violated, t h e n t h e further b e h a v i o u r of t h e m a t e r i a l is c h a r a c t e r i z e d b y a m o m e n t a r y skip to t h e d i a g r a m described b y e q u a t i o n s (19) a t a + = O.
tl O0"O
2.C
/ s// •
1.6
/"
/o .................
20 °
s"
¢ • ~
••
o.
20*
1.2 0.8
0.4
1 04
I
1
I
l
0.8
1.2
1"6
2"0
: E~= a'o
FIG. 6.
Ii o0"o 1.730
1.720
ca=E= O.E
=5
1"440 --
1-430
oE
1.420
I
, 0-2
o
I 0.4
I 0.6
I 0.8
~
el
FIG. 7.
3. S H E A R
OF UNI-DIRECTIONAL
REINFORCED
MATERIAL
Consider t h e shear of uni-directional reinforced m a t e r i a l as a n o t h e r example. this case
t~=t z=0,
N=
1,
~=~,
Eh=0,
¢{e)=ao/e,
In
902
Y u . V. NEMIROVSKY
in the elastic region we have ta
- 8,
al = E181 t
aEea
~
a~j~ =
a,
3ESAa
2ts(2A1 + As) - 3~AE~a
a°~
ta
E
3SEa"
(25)
2(2A 2 + A1) t a
E
a~l E =
'
Here 8 = Aaaa-- ala A1 + as3 As ,
aEA
A =
all a21
a]2 , a22
A~ =
81 =
als a23
(A sin 2a - 2A 1 cos ~ a -- 2A, sin s a) 2A
a12 ,
A2 : I a l l I a21
a22
als a2a
i
On Fig. 8 are plotted graphs of the relation of the shear modulus and the angle a calculated a f t e r f o r m u l a e (25) a t eo1 E 1 / a E = 1, 5, 10.
J~
aEe 3 1.2
1.0
0"8
0-6
0.4
0.2
I
t
I0 20
I
I
50 40
I
I
50 60
I
70
I
80
I
90a~
F r o . 8. T h e p l a s t i c i t y c o n d i t i o n f o r t h e r e i n f o r c i n g e l e m e n t s in t h i s c a s e h a s t h e f o r m (26)
Q/aao -- E S a + / E 1 8 1 (7o.
T h e g r a p h s o f t h i s d e p e n d e n c e a t co1 E 1 / a E = 5 a n d ¢o1 a+/aao = 0.5, 1.0 a r e r e p r e s e n t e d i n F i g . 9 b y d a s h e d lines. I f t h e l o a d s e x c e e d t h e v a l u e s o f e q u a t i o n (26), t h e n t h e e l a s t i c p l a s t i c b e h a v i o u r is d e t e r m i n e d b y t h e r e l a t i o n s a E e l = - oJ1 a+(cos s a -
½ sin 2 ~),
a E ~ 8 -- 3[ta-½oJ 1 a + sin 2a], aa°s = --wx a + s i n S a ,
aEe2 = - co1 a + ( s i n 2 a - ½ cos s a),
aa~l = - w 1 a + cos s a,
(27)
aa°s = ta--½oJ 1 a + s i n 2a.
R e l a t i o n s (27) r e m a i n v a l i d u n t i l t h e s t r e s s e s all,° a,~,° a°~ v i o l a t e i n e q u a l i t i e s (18). T h e c o r r e s p o n d i n g l i m i t i n g l o a d is d e t e r m i n e d b y t h e i n e q u a l i t y t3
aa o
_
_l coI a+ s i n 2a -42 aa o
1 -- ( ~°1 ° i t (cos4 ~ -- s i n ' ~ c o s ' ~ + sin 4 ~)
.
\ a(7o !
T h e g r a p h s o f t h e r e l a t i o n o f t h i s l o a d o n a a t wl a+/aao = 0-5, 1.0 a r e r e p r e s e n t e d in F i g . 9 b y d o t t e d lines. T h e solid lines i n t h e s a m e F i g . 9 r e p r e s e n t t h e r e l a t i o n s o f l o a d s o n t h e a n g l e a a t eo1 E 1 / a E = 5 in w h i c h c a s e t h e p l a s t i c s t r a i n s f i r s t o c c u r i n t h e b i n d i n g m a t e r i a l . T h e c o r r e s p o n d i n g e q u a t i o n for t h e m is o b t a i n e d b y r e p l a c i n g in e q u a t i o n (8) t h e s i g n o f i n e q u a l i t y b y t h a t o f e q u a l i t y a n d s u b s t i t u t i n g i n t o it e x p r e s s i o n s (25) for a~l, a~s, a~,.
On the elastic-plastic behaviour of a reinforced layer
903
The equations for the elastic-plastic behaviour of the reinforced layer for plastic strain of the binding material and for elastic strain of the reinforcing elements are obtained from equation (14) at t z = t~ = 0 and ¢(e) = ao/e. Similar answers can be obtained for any other type of loading or reinforcing of the layer.
oo-o I-C 0.9 1
/ ." ,-r-.. \
0.8 0.7
0,6
/',
/\
0.~
\
/ \
0.4
I 10
I 20
I o-s,,
I " ',I-4- "1" 30 40 50
I 60
I 1 70 80
IL_ 90a,,
F I G . 9.
I n conclusion it should be noted t h a t the model of ideally plastic material accepted here for calculation is used for definiteness and for emphasizing some peculiarities of relations between stresses and strains in the elastic-plastic straining of reinforced materials. When necessary the corresponding calculations can be made readily for any particular strength law. REFERENCES 1. V. L. BAZHANOV, I. I. GOLDENBLAT, V. A. KOP>TOV, A. D. POSPELOV and A. M. SI~YUKov, Soprotivleniye Stekloplastikov ("Strength of Fibre Glass Plastic"). Izd-vo "Mashinostroyeniye", Moskva (1968). 2. A. K. MAL~:EISTER, V. P. TAMUZH and G. A. TETERS, Soprotivleniye Zhostkikh Poll. mernykh Materialov. ("Strength of Rigid Polymer Materials"). Izd-vo "Zinatne", 1%iga (1967). 3. V. V. BOLOTIN, Mekhanika polimerov 2, 27 (1965). 4. G. A. VA~ F. FY, Mekhanika polimerov 4, 539 (1966). 5. 1%. ]:JILL, The Mathematical Theory of Plasticity. Oxford (1950). [Russian translation Gost. izdat (1956)]. 6. I. V. GOLDEI~BLAT,Izvestiya Akad. Nauk SSSR, OTN, 2, 60 (1955). 7. V. A. LOMAKZN, Izvestiya Akad. Nauk S S S R , Mekhanika i mashinostroyeniye 4, 47 (1960). 8. Y c . V. NEMIROVSKYand Yu. N. 1%ABOTNOV,Non-classical Shell Problems, pp. 786-807, North Holland, Amsterdam (1964). 9. Yu. V. NE~IIROVSKY, Inzhenerniy Zhurnal Mekhanika tvyordogo tyela 6, 80--89 (1969).
60