Photoelasto-plastic Studies on Brittle Fracture of High-polymer Solids A simple photoelasto-plastic method for the determination of stress-concentration factors in plane-elasticity problems is suggested on the basis of experimental results

by Katsuhiko Ito

ABSTRACT--The m e c h a n i s m of b r i t t l e f r a c t u r e of highp o l y m e r solids is e x p e r i m e n t a l l y i n v e s t i g a t e d u n d e r oneor t w o - d i m e n s i o n a l stress s t a t e s b y a n e w p h o t o e l a s t o plastic m e t h o d s u g g e s t e d b y t h e a u t h o r . T h e applicat i o n of t h e p h o t o e l a s t o - p l a s t i c m e t h o d o n t h e b r i t t l e f r a c t u r e p r o b l e m is b a s e d o n t h e principle t h a t b r e a k i n g stress c a n b e c o m p u t e d in b r i t t l e f r a c t u r e b y t h e m e a s u r e m e n t of t h e fringe order N B of i s o c h r o m a t i c lines a t f r a c t u r e point. B e n d i n g u n d e r t h r e e - p o i n t a n d f o u r - p o i n t loads, a n d t h e p l a n e p r o b l e m s , some h a v i n g stress c o n c e n t r a t i o n a n d o t h e r s b e i n g u n d e r c o n t a c t i n g load, are e x a m i n e d b y u s i n g rigid p o l y e s t e r c a s t r e s i n c o n t a i n i n g s t y r o l as a m o d e l s p e c i m e n ; a n d , i n conclusion, t h e b r i t t l e f r a c t u r e of h i g h - p o l y m e r solids u n d e r one- or t w o - d i m e n s i o n a l stress s t a t e s is decided b y t h e c o n s t a n t tensile stress, w h o s e m a g n i t u d e d e p e n d s only u p o n t h e m a t e r i a l u s e d as a m o d e l specimen, a n d is larger t h a n its u l t i m a t e tensile strength. M a n y k i n d s of f a c t o r s in f r a c t u r e are defined, a n d s t r e s s - c o n c e n t r a t i o n f a c t o r s i n f r a c t u r e are c o m p a r e d w i t h s t r e s s - c o n c e n t r a t i o n f a c t o r s i n elasticity. A n e w p h o t o e l a s t o - p l a s t i c s i m p l e m e t h o d for t h e d e t e r m i n a t i o n of s t r e s s - c o n c e n t r a t i o n factors i n elasticity is s u g g e s t e d b y u t i l i z a t i o n of t h e e x p e r i m e n t a l r e s u l t s o n t h i s b r i t t l e f r a c t u r e of h i g h - p o l y m e r solids a n d is e x a m i n e d on t h e p e r f o r a t e d p l a n e p r o b l e m h a v i n g finite w i d t h u n d e r t e n s i o n i n c o m p a r i s o n w i t h t h e o r e t i c a l a n a l y s i s ~and t h e experimental results by other measuring methods.

Nomenclature ~ ~t N k

= = = =

NB

= =

t

= ~2) = = = MB = b~BN = @1

-

b~B ~z

=

~T

=

~c

=

tensile stress, K g / m m 2. tensile s t r a i n , %. fringe o r d e r of i s o c h r o m a t i c line, F.O. w a v e l e n g t h O f o m o n o c h r o m a t i c , circularly polarized light, A. photoelastic sensitivity, F.0./Kg/mm. f r a c t u r e fringe order a t f r a c t u r e p o i n t i n t e s t s p e c i m e n , F.O. t h i c k n e s s of p l a t e t e s t specimen, ram. difference of p r i n c i p a l stresses, K g / m m ~. effective stress, K g / m m 2. effective s t r a i n , %. breaking bending moment, Kg/mm. e x t r e m e fiber stress a t f r a c t u r e m e a s u r e d b y p h o t o e l a s t o - p l a s t i c m e t h o d , K g / m m 2. e x t r e m e fiber stress a t f r a c t u r e c o m p u t e d f r o m b r e a k i n g b e n d i n g m o m e n t , K g / m m 2. u l t i m a t e tensile s t r e n g t h of t e s t specimen, K g / m m 2. u l t i m a t e c o m p r e s s i v e s t r e n g t h of t e s t specim e n , K g / m m 2.

Katsuhiko Ito was Research Associate, Division of Engineering, Brown Univ., Providence, R. L, and is now associated with the Institute of Physical & Chemical Research, Bunkyo-ku, Tokyo, Japan. Paper was presented by title at the 1958 S E S A Annual Meeting held in Albany, N . Y., on November 12-14.

c

= d i s t a n c e b e t w e e n t w o s u p p o r t s for a b e a m in t h r e e - p o i n t b e n d i n g , ram. ~ , ~E = stress a t p o i n t s A a n d E , respectively, of p e r f o r a t e d - p l a t e specimen, K g / m m 2. 1 = initial l e n g t h of specimen, r a m . w = i n i t i a l w i d t h of s p e c i m e n , ram. d = i n i t i a l d i a m e t e r of c i r c u l a r hole i n perfor a t e d - p l a t e s p e c i m e n , ram. ~J, Ec' = a p p a r e n t c o m p r e s s i v e stress a n d s t r a i n for p e r f o r a t e d - p l a t e specimen, c o m p u t e d o n t h e basis of t h e cross-sectional a r e a w i t h o u t a hole, K g / m m ~, %. P = a p p l i e d load, Kg. ~ = m e a n c o m p r e s s i v e stress in p l a t e s p e c i m e n in t h e cross section w i t h o u t a hole, K g / m m 2. ctTBm m e a n c o m p r e s s i v e stress i n p l a t e s p e c i m e n i n t h e cross s e c t i o n w i t h o u t a hole a t fract u r e , K g / m m 2. cO-BN f r a c t u r e stress a t f r a c t u r e p o i n t of p l a t e s p e c i m e n u n d e r c o m p r e s s i v e load, m e a s u r e d b y p h o t o e l a s t o - p l a s t i c m e t h o d , K g / m m ~. kA, kB ' - ~ = - a n a l y t i c a l v a l u e of s t r e s s - c o n c e n t r a t i o n f a c t o r i n elasticity a t p o i n t s A a n d E, respectively, for p e r f o r a t e d s p e c i m e n u n d e r u n i f o r m t e n s i o n , c o m p u t e d o n t h e b a s i s of m e a n stress in t h e cross section w i t h a hole. kff, kJ = a n a l y t i c a l v a l u e of s t r e s s - c o n c e n t r a t i o n f a c t o r in elasticity a t p o i n t s A a n d E , respectively, for p e r f o r a t e d s p e c i m e n u n d e r u n i f o r m t e n s i o n , c o m p u t e d o n t h e basis of m e a n stress in t h e cross section w i t h o u t a hole. N~ = m a x i m u m fringe order of i s o c h r o m a t i c lines, F.O. dp = d i a m e t e r of circular, l o a d - c a r r y i n g pin, m m . tffBN f r a c t u r e stress a t f r a c t u r e p o i n t of p l a t e s p e c i m e n u n d e r tensile load, m e a s u r e d b y p h o t o e l a s t o - p l a s t i c m e t h o d , K g / m m ~. t~Bm = m e a n tensile stress in p l a t e s p e c i m e n in t h e cross section w i t h o u t a hole, a t f r a c t u r e , K g / m m 2. PB = b r e a k i n g load, Kg. tPB b r e a k i n g tensile load, Kg. =

=

=

=

Introduction The study of the fracture mechanism for high-polym e r s o l i d s is v e r y i m p o r t a n t b e c a u s e o f t h e i r p r a c t i cal applications in industry. T h e p r o b l e m is c o m plicated and difficult to analyze since fracture mechanisms 1 depend not only on the nature of materials, but also on the various circumstances of applied stress, temperature, humidity, etc. FI~ ~ ~'l'~ The fracture mechanism ~ of metals and their alloys has been studied extensively. Studies of the fracture mechanism of high-polymer solids must be conducted, owing to their wide applications as new

Experimental

Mechanics

[ 159

Type I

Type 2

I0

Type S

IC /

z

,

8

9 o/

/

,

uJ- 6

/

N/

_'

/

~C uY

o

o

,

4

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0

t~t .*-, ,t"~....-""" 2

4

6

,/ o ~ ' /~

o

N

E//~

--o o/I

/

z ~

/

/

/

I/~

o

/~

"

8 I0 (kg/mm2)

/o

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2

4

6

8 10 (kg/mm2)

e Z-~,.

r

2

I

6

4

07

I

8

10

(kg/mr~2)'

Fig. 1--Calibration results of material properties for polyester cast-plate specimens in thickness, t, of 5 mm

industrial "plastics." The complete analysis of this problem is more difficult than that of metals, however, because of the complicated mechanical behavior of high-polymer solids, as described in terms of "rheology" of amorphous materials. As a first step of this study, the brittle fracture of a cast resin of a three-dimensional cross-linked polymer under bending and plane stress states was examined by the new photoelasto-plastic method suggested by the author. The mechanical behavior of high-polymer solids becomes complicated by the addition of various ingredients 3, 4 such as plasticizers, fillers, reinforcements, etc. Consequently, cast resins without such ingredients are used as test specimens in this study. The application of this photoelastoplastic study to the measurement of stress concentration factors in elasticity is suggested. Brittleness and ductility do not depend upon the state of stress 5-7 alone, but also are greatly influenced by temperature. 3, 4, 7 Thus, the term "brittle" used in the title of this study is confined to the mechanical behavior at a room temperature of about 20 ~ C. The Photoelasto-plastic and Test Specimens

Method

The photoelasto-plastic method of fracture analysis is based upon a characteristic of some common brittle high-polymer solids: at a point of impending fracture, the tensile stress, strain and birefringence per unit thickness are material constants and are independent of the stress distribution in the specimen. Thus, fracture of these materials can be predicted upon the basis of photoelasto-plastic fringe order at a critical point in a specimen. This behavior will be illustrated in the following sections for a variety of plane stress problems. Cast plates of unsaturated polyester resin s. 9 were used for specimens in these experiments. Although the unsaturated polyester resin (Types 1 and 2) containing styrol was mainly used, the unsaturated polyester resin (Type 3) containing styrol and triallylcyanulate 9 (TAC) was used for some specimens. This latter formulation produces harder polyester resin because the three-dimensional cross-linked

160 ] May 1961

structure of polyester resin becomes dense by addition of TAC, which has six active radicals in polymerization reaction. Polymerization was carried out at elevated temperature with Luperco ATC (benzoyl-peroxide paste) in the amount of 1%, acting as catalyst. This unsaturated polyester resin can also be cured at room temperature by using both a catalyzer and a prorooter. The resin cured at room temperature, however, does not have a completely developed threedimensional cross-linked structure; therefore, the effect of time-edge-stress in such specimens is expected to be large in the application of the material for the photoelasto-plastic method. The calibration results on the relationship between tensile stress, Ct, tensile strain, e~, and fringe order of isochromatic lines, N~ are shown in Fig. 1 for three different polyester cast-plate specimens having thickness, t, of 5 mm; the calibration was carried to a rather high stress region. Circularly polarized light of wave length X = 5461 fi, was used for the measurements of N. The thickness change associated with Poisson's ratio and the accompanying change of N were considered to be negligibly small. The photoelastic sensitivity, a, is not as large as that for epoxy,9. 10 diallylphthalate,9, 21 phenol12 cast resin, etc. However the low photoelastic sensitivity of these polyester specimens is rather more convenient for the purpose of this study because the fringe order, NB, at the fracture point, is not so large t h a t the measurement of NB is difficult. The following equation is established for the lowelastic-stress region.

N

=

at((ri

--

or2)

(I)

As shown in Fig. 1, the linear relation between ~ and N is not complete in the region of high tensile stress. Considering the relation between zt and et, this suggests a linear relation between birefringence and tensile strain 1~, 1~ in these resins of three-dimensional cross-linked polymers, while the birefringence effect in such ductile linear polymer solids as celluloid 15-~7 is proportional to stress. ~, i6 Although optical behavior in compression might be different l, ~Gfrom that in tension, only tensile values are

required, since brittle fracture of this cast resin is always caused b y tensile stress, as shown in the following sections. These specimen materials, u n s a t u r a t e d polyester resins containing styrol, in which "Fosterite" is included as one example, have an interesting character in t h a t their birefringence in elasticity is different in sign from t h a t in plasticity. Polystyrol,18. 19 Lactoprene ~~and copolymer 9 of styrol and divinyl benzene also have the same character. I t is unnecessary to consider this m a t t e r , however, because the tensile plastic strain of the cast resin used in this study is negligibly small at room temperature. Stress a t this fracture point is computed from the calibration results of Fig. 1, and the brittle fracture of three-dimensional cross-linked polymer solids is thus investigated b y the photoelasto-plastic method. This measuring method has one further merit concerning the problem associated with the effect of time-edge-stress, which in ordinary photoelastic experiments should be kept as small as possible. However, this is unnecessary to watch in this method, because the true critical stress a t fracture point can be measured directly b y the values of NB. T h e effective stress e vs. effective strain ~ diagram for polyester cast resin, T y p e 1, which was the specimen m o s t generally used in this study, is shown in Fig. 2.

Fracture in Bending The cast resin o f three-dimensional cross-linked polymers behaves as a ductile 3 material in compression, and usually exhibits upper and lower yield points. I t s mechanical behavior in tension, how-

15

E

/

/

I

Compression

/o -9

i~ Tension o o

io

20

5o

f_ LO/o/ Fig, 2--"Fliesskurve" (effective stress ~ vs. effective strain ~) for polyester cast resin Type I

ever, is quite different f r o m t h a t in compression. For example, brittle fractm'e in tension occurs at a relatively low tensile strain and is accompanied b y no significant p e r m a n e n t deformation or local contraction. Therefore, fracture in bending is certain to occur in the tension side. T h e experimental d a t a of the ultimate tensile strength, z~, of such cast resins are considerably influenced b y the various kinds of imperfect factors, although these factors have little effect on the elastic properties. These imperfect factors to be considered generally, are as follows. (a) Macroscopic I and microscopic flaws in the

TABLE 1--PURE-BENDING FRACTURE TESTS OF POLYESTER CAST RESINS UNDER 4-POINT LOADING

Fracture Stresses and Their Ratios

Test Model

Polyester cast resin

Type 1 (containing styrol) Type 2 (containing styrol) Type 3 (containing styrol and TAC)

BendingBending. fracture fracture Fiber Stress Fiber Stress Calculated from Measured by Breaking Photoelasto. Bending plastic Method, Moment, bCBN, Kg/rnm~ bCB cat, Kg/mm=

7.6 i 0.3 6.5 -~ 1.0 9.2 i 0.8

7.4 • 1.3 6.2 • 1.2 8.8 • 1.5

Ultimate Tensile Strength, CT, Kg/mm 2

6.3 5.4 7.8

b ffBN/CT

1.21 :J= 0.50 1.20 +- 0.18~ 1.18 •

b O'B e a l / a T

1.17 ::t:=0.21 1.15 • 0.22 1.13=1=0.19

TABLE 2--SIMPLE-BENDING FRACTURE TESTS OF POLYESTER CAST RESINS UNDER 3-POINT LOADING

Fracture Stresses

Model

Polyester cast resin

%

Type 1 (containing styrol) Type 2 (containing styrol) Type 3 (containing styrol and TAC)

BendingBending. fracture fracture Fiber Stress Fiber Stress Calculated from Measured by Breaking Ultimate PhotoelastoBending Tensile plastic Method, Moment, Strength, b a B N , Kg/mm-" bcrB r Kg/mm 2 a T , Kg/mm -~

7.6• 6.7• 9.4•

7.5• 6.8~0.8 9.2•

6.3 5.4 7.8

b O'BN/(YT

b aB cal/CrT

1.21 • 1.24 • 0.13 1.205 -4- 0.04

1.19• 1.26 • 0.15 1.13 • 0.06

F~xperimental Mechanics I 161

8~',,}'"D I o -

E/w=30/~5

~

_

- -

Fig. 3--Perforatedplate specimen

r

I W

~

thickness

0.047 O,

d=2.0mm,

d=4.0mm

d =6.0 ram,

.5

1.0

1.5

2.0

2.5

3.0

d =7.5mm

Fig. 5--Apparent compressive stress-strain curves for perforated-plate specimens having various d/w Patios

I ~ 30 m m , w = 15 m m , t = 5 rnrn

Fig. 4--Perforated-plate specimens having various d/w ratios, before and after fracture in compression

specimen. (b) The unavoidable errors of the specimen in machining, etc. (c) The time-edge-stress in the specimen. (d) The technological errors in the test, etc. Fracture experiments for pure bending (4-point loading) and simple bending (3-point loading) were performed; ten rectangular bar specimens (height, h = 15 mm; thickness, t = 5 mm) were used in order t h a t the deviation of the experimental data could also be obtained in good accuracy. The results obtained are shown in Tables 1 and 2, together with ar and the extreme deviations in the experiments. The extreme fiber stresses ~BN and b~# ~o~at fracture in Tables 1 and 2 were computed by different methods from the experimental data. Namely, bZBN is computed by using the calibration results of Fig. 1 on the basis of experimental values of NB obtained from the photoelasto-plastic method. On the other hand, ~zB ~z is computed by the experimental results of the breaking bending m o m e n t M8 on the basis of theory of elasticity. If the test specimen and the experimental procedure in this bending fracture test were perfect, bZB ~ must theoretically 162 [ M a y 1961

be larger than ~Z~N, because an elastic stress distribution has been assumed in the computation of bO'B cat and, on the other hand, bO'BN has been obtained from the real nonlinear stress distribution by the photoelasto-plastic method. Except for the polyester resin (Type 2) in three-point, simple bending, br c~ is always smaller rather t h a n b~BN, and the deviation of baB c~ is larger rather than t h a t of baBN, as listed in Tables 1 and 2. This is caused b y the fact that MB is greatly influericed by the various imperfect factors, (a)-(d), as pointed out in brittle fracture tension test, in particular, imperfect factor, (c), and so M B is in reality distrustfully small so that the real extreme bending fiber stress at fracture cannot be evaluated from bZB ~o~. On the other hand, bO'BN measured by photoelasto-plastic method is the true value of the fracture stress at the fracture point, measured without the effects of most of the various imperfect factors. Therefore, the extreme deviations of b~B c~ is larger than those of b~N, as listed in Tables 1 and 2. Because the simple-bending fracture test is always determined by the fracture at the central loading section of maximum bending moment, and since there is a smaller probability regarding the effects of various imperfect factors, in particular, (a) and (b), than in the pure bending test, it is quite natural t h a t the extreme deviations of Table 2 are smaller than those of Table 1. Thus, the stress computed from N~ is the correct value of the real fracture stress at the fracture point. The prevention of fracture b y the low-stress 2~ region in the neighborhood of the fracture point m a y be one reason for the fact in gross cross sections that the magnitude of tensile fracture stress is larger than zr. For example, this can reasonably explain the fact t h a t bZB~ and ba~ ~ in simple bending are a little larger than those in pure bending, because the effect of fracture prevention by the region of the smaller bending m o m e n t in the neighborhood of central loading point is further added in the case of simple-bending fracture.

Fracture of Plane-stress Specimens with Stress Concentration

P = 450 k g , c a m = 6,00 k g / m m

2

P = 420 kg, corm = 5.60 k g / m m

2

Fracture of Perforated Plates When tensile load is applied to the perforated-plate specimen shown in Fig. 3, it is quite natural for fracture 2~to occur at the point of maximum tensile stress, point A. I t is very interesting to apply a compressive load to the specimen, however, and observe the fracture phenomenon. The fracture occurs at the largest compressive stress, CA, at point A, or at the smaller tensile stress, ~E, at point E. Since the relationship between ~A and zE is influenced by the dimensions of the specimen, l, w and d, m a n y plate specimens having various configuration were used. 1. EFFECT OF STRESS-CONCENTRATION F A C T O R . - - I f l is n o t large

compared to w, the stress distribution around the hole is affected by the frictional resistance a t both ends of the compression specimen. Therefore, it becomes unreasonable to compare the experimental results in this case with the theoretical analysis obtained for a hypothetical specimen having infinite length. In order that the experimental results can be compared with the theoretical analysis, fracture experiments were performed under the following conditions. (a) Specimen dimensions were held constant: 30 m m length, 15 m m width; for these proportions frictional resistance had negligible influence on the stress distribution around the hole. (b) Various stress concentrations were produced in the plate specimens by change of d/w.

1=30mm, w=lSmm d = 6.0mm, t=Smm

1=30mm, w=15mm d = 6.0 m r n , t = 5.0 m m

Fig. 6--1sochromatic patterns for perforated-plate specimen having d / w = 6/15 (d = 6 ram) in compression P = 400 k g , c~,m = 5.34 k g / r n m ~

I =30mm, w=lSmm d : 7.5 m m , t = 5 m m

P ~ 375 kg, c a m =

5.00 k g / m m

2

I =30mm, w=15mm d = 7.5 m r n , t = 5 mm

Fig. 7--1sochromatic patterns for perforated-plate specimen having d / w =- 7.5/15 (d ----7.5 ram) in compression

Fractured specimens in compression are shown in Fig. 4, together with the no-load specimens. The apparent stress-strain curves in this compression test are shown in Fig. 5. The apparent compressive stress, zc', and strain, ec', are computed on the basis of the dimensions of the cross section without a hole. As shown in Fig. 4, brittle fracture was caused by the smaller tensile stress 2~ at point E in spite of the existence of the larger compressive stress at point A.

The photographs of isochromatic lines for highly stressed specimens are shown in Figs. 6 and 7. Experimental results obtained a r e plotted in Fig. 8, which also shows the relationships between d/w and the analytical values 24-28 of several stress-concentration factors kA, kff, kE, kE'. I t is natural that the change of cZBmwith d/w is caused by the change of stress-concentration factor at point E for the various configurations of test speci-

Experimental Mechanics I 163

(a)l=45mrn 5 (b)l=3Omm

4.5 4

E

.~ 3.5

z

3 )

2.5

1.5

0.5 O(

w=15mm, d=6mm, t=5mrn

d/w

(c)l=20mm

Fig. 8--Experimental results on fracture of perforated-plate specimens having various ratios d/w under compression and analytical values of their related stress-concentration factors in elasticity

,2

]

I

~,o

~o 0 .t"~

-Bo 8

-""" o

.%-

., /~;'/~o ~ 0.6

6

d; l

o~

d/w=6/i~

/

(d)l=lOmm 5

8

Fig. 9--Apparent compressive stress-strain curves of perforated-plate specimens havingvarious d/I ratios

mens. Consequently, the fracture tensile stress kEctr at point E, which is computed from k J and the experimental values of cabin at fracture, remains constant and is in good agreement with czBN2. FRICTIONALEFFECTS AT THE ENDS OF COMPRESSION SPECIMENS.--~t is interesting to observe the transition between brittle tensile fracture at point E and compressive yielding at point A. When l is not large compared to w, the tensile stress at point E is diminished by the action of transverse frictional forces and for small l / d ratios, point A becomes the critical point before the brittle tensile fracture at point E. The apparent compressive stress-strain curves are shown in Fig. 9. This behavior is illustrated in the sequence of specimens shown in Figs. 10 or 11, in which the shortest specimen failed by general compressive yielding, rather than by brittle fracture. This is understood also from the apparent compres-

164 I May 1961

Fig. lOlPerforated-plate specimens having various d/I ratios before and after fracture in compression

sive stress-strain curves, as shown in Fig. 9. In the case of d / l = 6/20, the compressive yielding sometimes occurs at point A, or the tensile fracture sometimes at point E. Figure 12 is a typical one of isochromatic lines in this case, where the cracking has just occurred at point E. Further, the fracture line in this case does not propagate as a straight line, but

('a) l = 20 mm

(b) l = 10 mm

w = 1 5 mm, d =6mm, t = 5 m r n Fig. ll--Dark-backfield photographs of the perforated specimens of d/I = 6/20 and 6/10 before and after fracture in compression

15 E

Apparent Upper Yield Point cOBBN

I _

Fig. 12--1sochromatic lines of perforatedplate specimen having ratio d/I =- 6/20, which shows cracking just occurred at point E

is turned perpendicular to the axis by the frictional resistance of both ends in compression, as shown in Figs. 10 and 11. The relationships between cabin, caBN and d/l are shown in Fig. 13. For the case of d/l = 6/10, the apparent upper yield point is conveniently plotted in Fig. 13, and kEc'~Bm becomes larger than c~B.V when d/l > 3/10. Obviously, the method of photoelastoplastic analysis applies only to configurations in which failure occurs by brittle fracture. Fracture of Perforated-plate Specimen Under Contact Load Appliecl to the Circular Hole Contact loads applied at holes in plate specimens are often encountered in engineering, for example, at pin or rivet joints. Therefore, the stress distribution around a circular hole under contact load has been investigated. The problem has been approached theoretically by Bickley, 27 and Knight, 28 and experimentally by Coker, ~ Frocht and Hill, s0 and Nisida. zl The author 1~has also studied the stress concentration developed by two loaded holes in a plate of finite width and infinite length. I n this problem, the stress-concentration factors are greatly influenced by the clearance 3~between pin

o o

o.I

I o ~ o

~

O/

r _....._.. o

o,2

0.3

~

0.4

0.5

0.6 o.7 d/l Fig. 13--Experimental relations between fracture stresses c~BN and c~Bm and ratio of perforated-plate specimens having various ratios d/I in compression

and hole. I n the case of neatly fitted or press-fitted pins, the maximum fringe order, N . . . . is not a direct measure of the maximum tensile stress. The point Q of the largest tangential tensile stress on a circumference, which is also expected to determine the fracture of the specimen under contact load, is not the point R at which Nm~x is located, but exists on the circumference approaching point A from point R. Unless the contact-pressure distribution on the circumference is assumed, 27 computation of the maxim u m tensile tangential stress from the value of N is impossible. Therefore, it is difficult to predict the brittle fracture of such specimens by the photoelastoplastic method. In such practical problems as pin joints, there usually exists a little clearance between pin and hole. In this case, it is possible for the tangential stress on a circumference to be computed directly from the value of N, except at point E and its vicinity, where contact pressure still remains. The author 12 has confirmed, in his study of the stress distribution in a plate having two circular holes under contact loads, that there is a very large compressive stress at point E and its vicinity, while there is a small tangential Experimental Mechanics ] 165

[oj "

I

Fd4

2.5

E

2.O o / ~

J

i

--

o/ I

~dpI~

I

"~

-W-!

1

1.5

/

/ , x

I thickness

' C~m

.

t

1.0

Fig. 14--(Above left) Perforated-plate specimen loaded in tension through a loose pin Fig. 15--(Above right) Pin-loaded perforated-plate specimens before and after fracture o

Clearance (ram) IO

0

0.I

~O

0.2

m

m

o

o.I

o.2

o.3

d/w Fig. 17--Relationships between the various factors in fracture and d / w of perforated-plate specimens under compression fracture

O I

7

Clearance 0.2

0.1

o

46m 2

~ 0 ~ 0 ~

I I

o,

O

0

i ~ .5

/

5

of"-... ~ Factors

External ~

Load

~

fo ~

Z ~

TABLE 3--VARIOUS FACTORS IN FRACTURE

Type

o m

4

Clearance R6tio (%) Fig. 16--Experimental results for pin-loaded perforated plate

4

I

in

Stress"Apparent . . . . True" concentration Dangerous Dangerous Factors in Factors in Factors in Fracture Fracture Fracture '-

Compressive Load Tensile Load

cO'BN/cO'Bm ac/ccrB.~

,~c/~o'Bz~

tffBN/~0"Bm

0"T/tO'BN

~TT/tO'Bm

O0 tensile stress at point A. Because the relation between the stresses eA and eE at points A and E, respectively, is greatly influenced by the clearance, the brittle fracture of cast resins under contact loads was investigated by the photoelasto-plastic method. The configuration of the test specimen is shown in Fig. 14, and contact load is applied to a circular hole through brass ring. A typical fracture is shown in Fig. 15. Thus, the brittle fracture under contact load is also determined by the constant tensile stress in the neighborhood of point A, in spite of the existence of very large compressive stresses elsewhere. The results obtained are plotted in Fig. 16, with the clearance (d -- dp) and its ratio [(d -- d~)/d] X 100% as the abscissa, and the fracture tensile stress as the ordinate. As clearly illustrated in Fig. 16, the magnitude of fracture tensile stress is constant, being independent of the average stress and the stress-concentration factor. 166 I May 1961

"

j

3 i

Dangerous Factors ~ F r a c t u r e

arious

(ram) 0.3

I

tO~'Bm

I 2

0.5 0.6

0.5

0.3

t%.

61

0.4

Q--O

i

I

2

5

4

Clearance Ratio

5

(%)

Fig. 18--Relationships between the various factors in fracture and pin clearance

The Factors in Fracture and Their Measurement by Photoelasto-plastic Method Definition of Factors in Fracture and Their Measurement in Some Plane Problems by Photoelasto-plastic Method Elastic stress-concentration problems have been explored extensively by m a n y investigators. The brittle fracture, however, of industrial plastics under various plane-stress states is usually determined by a certain stress, which is not equal to aT or ~c. Hence, it is important, for the practical application of such materials, to define dangerous factors in fracture and stress-concentration factors in fracture, as presented in Table 3. I n Table 3, the factors computed on the basis of

3

~1

'

I

I

E ,

i

Results by 01her Mefhods I

2

o

-- o

9

O.I

0,2

Type

Polyester Specimen

Type 3

Polyester Specimen

0,5

0.4

0,5

0.6

0.7

O.B

0.9

1.0

d/. Fig, 19--Stress-concentration factors

for perforated plate in tension

mean stress are named "apparent," and the factors computed on the basis of fracture stress are named "true." I n the cast resin of three-dimensional cross-linked polymers, these factors for fracture in tension are, in general, different from those in compression, owing to the brittle tensile fracture characteristic, whereas the factors in elasticity are equal for tensile and compressive loading. Therefore, the factors in fracture and in elasticity m a y be greatly different. The various factors in fracture for plane-stress problems can be computed from the photoelastoplastic experiments. Their values for the previous plane-stress problems are described in the following sections. ~r and ~c of the polyester cast resin T y p e 1 were 6.3 K g / m m ~and 32 K g / m m 2, respectively. 1. FRACTURE OF A PERFORATED PLATE IN COMPRES-

SION.--From the experimental results of Fig. 8, the relationships between d/w and the various factors in fracture are computed and plotted as shown in Fig. 17. Stress-concentration factors in fracture and "apparent" dangerous factors in fracture increase with increase of d/w, whereas "true" dangerous factors in fracture are constant, since ~zB~ is constant for fracture of these materials. 2 . F R A C T U R E O F P E R F O R A T E D P L A T E U N T E R CONTACT

LOAD.--From the experimental results of Fig. 16, the relationships between the factors in fracture and pin clearance are computed as shown in Fig. 18. These tendencies on the change of the factors are the same as those in Fig. 17.

Physical Meaning of the Factors in Fracture 1.

~ ' A P P A R E N T " AND

" T R U E " D A N G E R O U S FACTORS I N

FRACTURE.--These factors signify the degree of danger in fracture caused by a stress different from zr and Ze- As illustrated in the fracture experiments, a constant tensile stress determines the brittle fracture of cast resin under plane stress, and this stress can be computed from the photoelasto-plastic experiments. Therefore, "true" dangerous factors in fracture depend only upon the material of the test specimen and do not depend upon the configuration of the test specimen. "Apparent" dangerous factors in fracture are dependent upon both the material and the configuration. 2. S T R E S S - C O N C E N T R A T I O N FACTORS I N F R A C T U R E . T h e s e factors are the ratio of stress ~zBr or tzBN at the fracture point to the mean stress ~aB~ or taB~, when the fracture occurs in the specimen. Although these factors are similar to the stress-concentration factors

in elasticity, which have the same values in tension and compression, the magnitude of stress-concentration factors in fracture is not always the same in tension and compression; and furthermore, they are not always equal to the stress-concentration factors in elasticity. The magnitude of stress-concentration factors in fracture is equal to the magnitude of stressconcentration factors in elasticity, providing ~B~u or t~B~ is equal to the maximum stress in the test specimen. In the cast resins of three-dimensional cross-linked polymers, both ~aBN and t~BN are tensile stresses of the same magnitude, owing to brittle fracture in tension. I t is evident t h a t the factor c~B~/~B,~ is not equal to the factor tO-Bhr/tffB m because of the difference between the mean stresses ~ZB~ and tgB~For the fracture experiment in compression described under "Fracture of Planestress Specimens with Stress Concentration," the values of ~aB.~/cz~m are shown in Fig. 17, and these are different from the stress-concentration factors in elasticity. I f tensile load is applied to the specimen of Fig. 3, however, the stress tCB.Vwill be equal to the fracture tensile stress zA at point A, which is the maximum stress of the specimen, and, therefore, it is certain that the dangerous stress-concentration factors in fracture ~ N / ~ will be equal to the stressconcentration factors in elasticity.

A New Simple Photoelasto-plastic Measuring Method of Stress-concentration Factors in Elasticity by Brittle-fracture Experiment of Cast-plate Specimen On the basis of the experimental results on the brittle fracture of these cast resins, approximate values of stress-concentration factors in elasticity can be easily measured by the photoelasto-plastic method suggested by the author. The problem of a perforated plate of finite width is again used as an example. In order to obtain the stress-concentration factors in elasticity, a tensile load must be applied on the perforated-plate specimen. In this case, the fracture of a cast-resin specimen is determined by the tensile stress, zA, at point A, which is the maximum stress in the specimen. I t has already been shown that the fracture tensile stress under plane stress is a constant magnitude which depends only upon the cast resin used. Therefore, the breaking tensile load, tPB, of the perforatedplate specimens having various ratios of d/w were measured with a conventional testing machine. Then the mean tensile stresses, tzBm = tP~/wt, at fracture were computed from these experimental results. Since the value of true-fracture tensile stress, t(rBN, of cast resin at the fracture point has already been determined by the photoelasto-plastic method, it is reasonable that ttTBN/t~TBm iS an approximation to the stress concentration factor in elasticity. These results are shown in Fig. 19, in which the corresponding theoretical analyses 24-26. 29and photoelastic studies3L ~2 by other investigators are shown in a smooth curve for comparison. As shown in Fig. 19, the

Experimental Mechanics I 167

photoelasto-plastic results are in rather good agreement with the k n o w n values. For d/w larger than 0.5, however, the specimen deforms appreciably under the fracture load, which results in a reduction in curvature at point A and a decrease of effective crosssectional area. Therefore, the stress-concentration factors in elasticity, measured by the photoelastoplastic 83 method, are a little smaller than those measured by other methods. In order to improve the accuracy in this new simple measuring method, fracture of the cast-resin model must occur in as brittle a manner as possible. In other words, the smaller and more linear the strain to fracture, the better the accuracy. Therefore, ~t is quite natural that the experimental results of the hardest and most brittle polyester specimen containing TAC, which has six functional radicals in polymerization reaction, and is considered to give a more dense internal molecular structure of the model material, are in better agreement with the theoretical results than those for Type 1 polyester models. The accuracy m a y depend upon stress distribution, too. For models having real stress concentration (i.e., large gradients in the fracture region), materials which exhibit stress-strain behavior similar to that illustrated in Fig. 1 would not permit substantial stress redistribution before fracture; therefore, stress-concentration factors for the perforated-plate experiments were determined with good accuracy. This technique falls into the category of the wellk n o w n brittle-fracture method 34 for determination of stress-concentration factors; only the model material is different. If the values of c(YBN o r tO-BIVof a transparent model of cast resin having a very brittle fracturing character are once measured by the photoelasto-plastic method, the experimental measuring of the stress-concentration factors in elasticity on the basis of this photoelasto-plastic study can be done easily. ~5 It is merely necessary that the breaking load PB of the same cast-model specimens having various configurations be measured with a conventional testing machine. Obviously, the approximation to the true factor improves as the model material approaches an ideally brittle material. Conclusions As one of the mechanical behaviors of high-polymer solids, the mechanism of brittle fracture under plane-stress states is studied experimentally by a new photoelasto-plastic method. The brittle fracture of such high-polymer solids having fully developed three-dimensional cross-linked structure is prescribed by a constant tensile stress and constant isochromatic fringe order, whose magnitudes depend only upon the material used. As a result, factors in fracture are defined; stress-concentration factors in fracture are compared with stress concentration factors in elasticity. A new, simple photoelasto-plastic method for the determination of stress-concentration factors in plane elasticity problems is suggested on the basis of the experimental results.

168

I May

1961

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