28 Stress Rate Dependency of the Tensile Strength of Fiber-Reinforced Plastics TAICHI FUJII D e p a r t m e n t of M e c h a n i c a l E n g i n e e r i n g , O s a k a City U n i v e r s i t y , Sugimoto-cho, S u m i y o s h i - k u , O s a k a 558, J a p a n
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MITSUNORI MIKI D e p a r t m e n t of M e c h a n i c a l E n g i n e e r i n g , O s a k a M u n i c i p a l R e s e a r c h Institute, O g i m a c h i , K i t a - k u , O s a k a 530, J a p a n
Several r e s e a r c h e r s have r e p o r t e d t h a t t h e tensile s t r e n g t h of glassfiber reinforced plastics (GRP) increases with increasing strain rate (l,2,3,4,5). Though many materials become more brit tle u n d e r i m p a c t conditions, t h e i m p a c t strength and fracture strain o f GRP c o m p o s i t e s increase. Under i m p a c t conditions GRP c o m p o s i t e show increased ductile fracture behavior. In addition, t h e i m p a c t e n e r g y is considerably l a r g e r t h a n t h e static fracture energy. T h i s a d v a n t a g e o f GRP c o m p o s i t e s will become increasing ly i m p o r t a n t in future applications. E v e n t h o u g h this phenomena is well r e c o g n i z e d , it r e m a i n s t o be a n a l y z e d . With increasing stress rate, the constituent materials react mechanically in a different way t h a n t h e y do u n d e r static condi tions. The resin-fiber interface may also react differently. Accordingly, we p r o p o s e two m o d e l s : a t i m e d e p e n d e n t probabilistic failure model and a m e c h a n i c a l model w h i c h recognizes t h e non linear b e h a v i o r at t h e interface. These two m o d e l s c a n be u s e d t o a n a l y z e t h e strength d e p e n d e n c y of GRP on t h e stress rate. Time Dependent P r o b a b i l i s t i c F a i l u r e M o d e l Damage F u n c t i o n . Damage f u n c t i o n i s d e f i n e d a s t h e i n t e g r a l of the p r o b a b i l i t y o f f r a c t u r e per u n i t time o r the t r a n s i t i o n p r o b a b i l i t y o f f r a c t u r e w i t h r e s p e c t t o t i m e . L e t Ρ be t h e p r o b a b i l i t y o f f r a c t u r e o f a m a t e r i a l and D be t h e damage f u n c t i o n . Then P=l-exp(-D) The f r a c t u r e p r o c e s s i s r e g a r d e d a s a 2 - s t a t e , 1 - s t e p s t o c h a s t i c p r o c e s s ( s e e R e f . §_ f o r s t o c h a s t i c f r a c t u r e p r o c e s s ) . The t r a n s i t i o n p r o b a b i l i t y m has been e x p r e s s e d so f a r a s f o l -
0-8412-0567-l/80/47-132-379$05.00/0 © 1980 A m e r i c a n C h e m i c a l S o c i e t y
May; Resins for Aerospace ACS Symposium Series; American Chemical Society: Washington, DC, 1980.
380
RESINS F O R A E R O S P A C E
lows(7), m=yS
(γ,δ;
c o n s t a n t s , S; s t r e s s )
In t h i s case, t h e f r a c t u r e s t r e s s
(1+6)S 1+6
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i s given by
2+δ 1+6
Ύ
(2)
(3)
where S i s s t r e s s r a t e a n d Γ d e n o t e s gamma f u n c t i o n . I t s h o u l d be n o t e d t h a t t h e f r a c t u r e s t r e s s c h a n g e s b y 4 7 % when t h e s t r e s s r a t e c h a n g e s b y 10 t i m e s i f δ=5.0 ( 8 ) . However, t h e i n c r e a s e i n t h e s t r e n g t h o f GRP c o m p o s i t e i s a b o u t 1 0 % when t h e s t r e s s r a t e c h a n ges b y 10 t i m e s . To remove t h i s d i s c r e p a n c y , t h e f o l l o w i n g f o r m o f t h e t r a n s i t i o n p r o b a b i l i t y i s proposed. u
m(t;S)= S f(t) Y
(4)
where f i s a n a r b i t r a r y f u n c t i o n w h i c h i s d e f i n e d f o r t = 0 a n d c a n be i n t e g r a t e d w i t h r e s p e c t t o t i m e . When t h e c o n s t a n t s t r e s s S i s a p p l i e d t o a s p e c i m e n f o r t i m e t , t h e damage f u n c t i o n D i s d e f i n e d as t . D ( t ; S ) = f m(x;S)dT=YS°F(t) (5) 0 where F(t)=/
f(t)dx
(6)
0 Reduced Time Method When t h e a p p l i e d s t r e s s c h a n g e s , t h e e v a l u a t i o n o f t h e damage f u n c t i o n i s p e r f o r m e d b y " t h e r e d u c e d t i m e method". F i g u r e 1 shows t h e s i m p l e s t s t r e s s h i s t o r y . L e t IÏIQ b e t h e t r a n s i t i o n p r o b a b i l i t y f o r S = S Q a n d m^ f o r S = S j .
• «YSo 0
f i t 3
(7a)
(7b) These are shown i n F i g . 2. The value o f the damage f u n c t i o n at t = t j , D j , i s equal t o the area OABC. D
r^ o V s
F(
(8)
The damage f o r So, D^, should be s h i f t e d t o an equivalent damage i n order t o evaluate the damage occurred from t = t j t o t=t2» T h i s
May; Resins for Aerospace ACS Symposium Series; American Chemical Society: Washington, DC, 1980.
Tensile
Strength of
381
FRP
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F u j n AND M i K i
Figure 2.
Transition probability
May; Resins for Aerospace ACS Symposium Series; American Chemical Society: Washington, DC, 1980.
RESINS F O R
382
AEROSPACE
c a n be done b y p u t t i n g YS^F(tJ)=D where
*
(9)
1
i s t h e reduced time o f t j .
* From E q s . ( 8 ) and ( 9 ) , t j c a n
be o b t a i n e d . ^0
F(t*) =
F(tp
(10)
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When t = t ^ , t h e damage f u n c t i o n D i s e x p r e s s e d a s D=YS^F(tJ t -t ) +
2
(11)
1
F r a c t u r e S t r e s s under a C o n s t a n t - s t r e s s - r a t e Loading C o n d i t i o n I n t h i s p a p e r , f ( t ) i n a t r a n s i t i o n p r o b a b i l i t y i s assumed a s the f o l l o w i n g form f(t)=t
X
(λ: c o n s t a n t )
t , F ( t ) = f f (x)dT=Y~Y t 0 A c o n s t a n t - s t r e s s - r a t e l o a d i n g c o n d i t i o n i s expressed as
(12)
Then
1
S=at
+
(13)
λ
(a; constant)
(14)
S i n c e t h e damage f u n c t i o n i s e v a l u a t e d b y " t h e r e d u c e d t i m e me thod", t h e i n c r e a s i n g s t r e s s i s approximated t o t h e combination o f c o n s t a n t s t r e s s e s a s shown i n F i g . 3. L e t A t b e t h e t i m e i n t e r v a l where t h e s t r e s s i s c o n s t a n t , then t =rAt (Γ=1,2,···,η) (15) r
and S =at = a r A t r r
(16)
K
J
U s i n g E q s . ( 1 3 ) a n d ( 1 6 ) , E q . ( 1 0 ) c a n be w r i t t e n a s
t*= r
fr-1
(t*^+ât)
(17)
where ξ=6/(1+λ) Therefore,
May; Resins for Aerospace ACS Symposium Series; American Chemical Society: Washington, DC, 1980.
(18)
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F u j n AND M i K i
Tensile
Strength
of
FRP
Time Figure 3.
Approximation of stress history
May; Resins for Aerospace ACS Symposium Series; American Chemical Society: Washington, DC, 1980.
RESINS FOR AEROSPACE
384
t*= η
n-1 η n-1 η
n-1
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n-1
At+
n-2
n-2 n-1
CtJJ_ +At)
n-3
+ *·· +
2
η
At
n-1 = Σ At r=0
(19)
From E q . ( 1 5 ) , A t i s g i v e n b y (20)
A t = t /n η Substitution
o f Eq. (20) i n t o Eq. (19) y i e l d s
t*=t η η
n-1 Σ ~ η r=0
(21)
A
When η a p p r o a c h e s i n f i n i t y ,
t*=lim t η _ η Substitution
t
E q . ( 2 1 ) becomes
n-1 - r \ ^ Σ — — η η
1
=
Λ
t
n
ζ
J
χ 0
d
x
=
1 û f *η
(22)
o f E q . (18) i n t o E q . ( 2 2 ) y i e l d s
*&Ji2L η 1+λ+δ
t
(23)
η
When t = t , t h e damage f u n c t i o n and ( 2 3 ) ? η 1+λ D=1+λ 1+λ+δ
i s e v a l u a t e d b y u s i n g E q s . ( 5 ) , (13),
1+λ
1
(24)
η
S u b s t i t u t i n g E q . ( 1 4 ) i n t o E q . ( 2 4 ) a n d l e t t i n g D=D (D : c r i t i c a l v a l u e o f damage f u n c t i o n . D =0.693 when t h e p r o b a b i l i t y o f f a i l u r e i s 5 0 % ) , the f r a c t u r e s t r e s s under constant-stress-rate l o a d i n g c o n d i t i o n c a n now be o b t a i n e d a s c
May; Resins for Aerospace ACS Symposium Series; American Chemical Society: Washington, DC, 1980.
28.
F u j n AND M i K i
Tensile
Strength
of
385
FRP
1+λ 1+X 1+λ+δ 1+λ+δ f ( l + X + ô ) D c l S =a c l γ(ΐ+λ)'
(25)
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I f λ=0, t h a t i s , t h e t r a n s i t i o n p r o b a b i l i t y does n o t i n f l u e n c e d by t i m e , E q . (25) i s r e d u c e d t o 1 1+δ 1+δ (26) S =a c Ύ w h i c h i s same a s E q . ( 3 ) e s s e n t i a l l y . M e c h a n i c a l Model w i t h N o n - l i n e a r I n t e r f a c i a l
Behavior
When t h e r e l a t i o n b e t w e e n s h e a r d e f o r m a t i o n a n d f o r c e i n t h e f i b e r m a t r i x i n t e r f a c e i n f i b r o u s composite m a t e r i a l s i s l i n e a r or p e r f e c t l y p l a s t i c , the a n a l y t i c a l treatment i s completed(9). But t h e s h e a r d e f o r m a t i o n i n t h e i n t e r f a c e becomes n o n - l i n e a r a c c o r d i n g t o t h e i n c r e a s e o f l o a d , l o a d i n g r a t e a n d t h e change o f environmental c o n d i t i o n a t aerospace. I t c a n b e t h o u g h t t h a t many p a t t e r n s e x i s t a s t h e n o n - l i n e a r s h e a r i n g d e f o r m a t i o n a n d f o r c e re lation. F o r t h e s i m p l i c i t y o f t h e a n a l y t i c a l c o n s i d e r a t i o n , we a d o p t e d t h e r e l a t i o n o f t h e m-th power o f i n t e r f a c i a l s h e a r f o r c e i s p r o p o r t i o n a l t o s h e a r d e f o r m a t i o n . Then t h e r e l a t i o n becomes s o f t o r h a r d a c c o r d i n g t o m>l o r m
