Experimental Investigation on Fracture of Viscoelastic Materials under Biaxial-stress Fields

Paper describes the investigation on the fracture behavior of two viscoelastic materials for several biaxial-stress fields corresponding to tension-tension quadrant of principal-stress plane

by M. G. Sharma and C. K. Lim

ABSTRACT---Fracture behavior of viscoelastic materials under various biaxial-stress fields was studied experi- mentally in a specially developed apparatus. The bi- axial stresses were applied at various time rates of stress to study the effects of rate of loading on fracture behavior. Examination of experimental data indicated that a simple relationship could be established between octa- hedral shear stress and octahedral shear strain at fracture corresponding to various biaxial stresses. Finally, a criterion of failure based on the total strain energy at fracture was suggested. The strain energy at fracture predicted from the linear viscoelastic theory agrees reasonably well with that determined experimentally.

Introduction In recent years, great interest is being shown in the understanding of fracture of viscoelastic materials. M a n y materials used for aerospace and related ap- plications are viscoelastic in nature, in tha t they display t ime-dependent mechanical response to ex- ternal loading. Although considerable progress has been made in the understanding of fracture of viscoelastic materials under uniaxial loading, 1 very little work can be found on fracture of these mater- ials under biaxial loading. 2, 3

Fracture behavior of materials has been studied f rom different viewpoints. The first approach seeks to relate failure to various microstructural mechanisms. 4 -8 Although this method has led to a better understanding of the influence of various s tructural variables, the results are still qualitative in nature and have limited significance for engineer- ing-design purposes. The second procedure at- tempts to determine the stress field in the vicinity of an existing crack and, f rom this information, finds the condition necessary for the crack to prop- agate and give rise to catastrophic failure. This method has been used extensively for perfect elastic materials.g, 10 At tempts are being made to extend the method to include viscoelastic materials. 11 The third approach a t tempts to s tudy the circumstances

M. G. Sharma and C. K . Lim are associated with The Pennsylvania State University, University Park, Pa. Paper was presented at 1967 S E S A Spring Meeting held in Ottawa, Ont., Can. on May 16-19.

under which failure occurs. This is characterized by the existence of certain critical functions of stress or strain which when exceeded in a body under complex stress distribution, give rise to failure. The above procedure has been extensively applied to metallic materials.~2 At tempts have been made to extend the same to polymeric materials.2, ~ The critical function of stress or strain tha t determines failure must be determined from multiaxial-failure experiments. Stress states chosen for the multi- axial-failure experiments must correspond to stress components of all eight octants of the principal stress or strain space. Effects of rate of loading and temperature introduce additional complica- tions in the evaluation of critical functions. This paper describes the investigations on the fracture behavior of two viscoelastic materials for several biaxial-stress fields corresponding to tension-ten- sion quadrant of principal-stress plane. The ex- perimental results have been carefully scrutinized to detect any general characteristics of fracture of viscoelastic materials. Finally, a t tempts have been made to develop criteria of fracture of visco- elastic materials based on the strain energy at fracture.

0.375 -\ , ~ i ,

1.000

1 J

~oooo t

Fig. 1--A typical flat specimen (Janaf)

I - ; . 5 , , 1 1.5

S~CTION A-A

Fig. 2--A typical tubular specimen

202 I May 1968

6 C

5,G

~ 3D

2O

I.C

2, ,'o ,~ ,~ ,ooo TIME (HOURS)

Fig. 3--Variation of tension-creep compliance with time for the composite material

Experimental Program

Materials Used in the Experiments

Two mater ia ls were used in this invest igat ion. The first one is a composi te e las tomer t ha t is a copolymer of Butadiene and acrylic acid, cross- l inked with Epon 828. Finely divided a luminum of 10-micron particle size was used in the prepara- t ion of the composi te material . The propor t ion of ingredients in the mater ial is as follows:

Percent H y c a r 2000 X 131 (B. F. Goodrich

Chemical Co.) 24 .4 E p o n 828 (Shell Deve lopment ) 5 .7 H-10 Aluminum (Valley Metal lurgical

Processing) 69.9

T h e second mater ia l used in the s t u d y was Paracr i l RF-1 liquid polymer t ha t is also a copolymer of Butadiene and acrylic acid. The liquid po lymer was combined with Epon 828 to provide a solid mater ia l of appropr ia te stiffness. The propor t ion of ingredients in this test mater ia l is as follows:

Percent Paracri l RF-1 (U.S. R ubbe r Co.) 86 Epon 828 (Shell Deve lopment ) 14

Two types of specimens were used in the t es t p rogram. The fiat specimen (see Fig. 1) was used for s tudying uniaxial tension s t ress - s t ra in and creep behavior and the tubula r specimen (see Fig. 2) was used for uniaxial and biaxial-fracture studies. The respect ive specimens were cas t in molds and were prepared according to a recipe developed in our labora tory .

Stress-Strain Behavior of Test Materials

THE COMPOSITE MATERIAL ( H Y C A R ) - - T h e be- havior of the mater ia l in creep was s tudied by sub- ject ing flat specimens (Fig. 1) to cons tan t values of loads in a s t andard creep machine and observing the elongation in axial direction of specimens. T h e creep da ta are presented in the form of var ia t ion of

5o

30

?o

IO

t B~ - - - - T H E O R E T I C A L CURVE

o 0 & O 0

0 �9 []

0 I I 0

STEADY PRESSURE 460 PSI

I0 o TIME (HOURS)

Fig. 4--Variation of bulk-creep compliance with time for the composite material

creep compliance funct ion D(t)* with log t (where t = t ime) in Fig. 3. I t can be seen f rom the figure

t ha t the creep-compliance funct ion varies wi th s t ress g0, implying the mater ia l is sl ightly nonlinear viscoelastic. However , the same figure shows t h a t the creep-compliance funct ion does not v a r y wi th stress in a consis tent fashion. Therefore, a mean compliance funct ion was compu ted and found to obey the following relation.

[ ( :)1 D(t) -- Do + D 1 - exp -- 4- -- (1)

where

Do = initial compliance, (3.6 X 10-3ps i -~) D = re ta rded elastic compliance (4.5 X 10 -4

psi -1) r = re ta rda t ion t ime in tensile d e f o r m a t i o n

(1.09 hr) yv = flow viscosi ty (6.67 • 103 psi-hr)

Equa t ion (1) represents a four-e lement Kelv in model (see Fig. 3).

The behavior of the mate r ia l under hydros ta t i c stress (triaxial compression) was found to be visco- elastic. In Fig. 4, is shown plo t ted the volumetr ic creep compliance B(t) observed a t var ious t imes in volumetr ic-creep experiments . 1~ I t was found t h a t the creep behavior corresponded to a th ree-e lement model (see Fig. 4) leading to the following rela- tionship.

B(t) = B0 + B [ 1 - e x p ( - ; ) 1 (2)

where

B0 = initial volumetr ic compliance (218 X 10-7 psi-1)

B = volumetr ic re tarded elastic compliance (5.28 X 10-7 psi-~)

k = re ta rda t ion t ime in volumetr ic deformat ion (2.5 hr)

* The creep-c~mplianee function is the rot& o f observed t ime-dependent strain ~(0 to constant value o f stress ao in a creep test.

Experimental Mechanics I 203

PARACRIL RF-1---Unlike the previous material, Paracril RF-1 showed t ime-independent stress- strain behavior for the range of rate of loading studied. However, the fracture properties were found to depend on rate of loading. Uniaxial tension data for Paracril RF-1 is shown plotted in Fig. 5, where a, representing the ratio of nominal tangential stress to nominal axial stress, is zero for uniaxial tension loading. The nominal stresses are the stresses based upon the undeformed cross- sectional area. Figure 5 shows tha t the data for various stress ratios and rates fall on a smooth curve, al though the scatter of da ta points is larger beyond an extension ratio of 1.5. The tension data (see Fig. 5) indicates tha t Paracil RF-1 dis- plays large deformation. For a material capable of large deformation, the relationship between stresses and the associated displacements has been well established using the finite deformation theory. 14 This theory requires the knowledge of a strain-

I ~ 0

1 2 0

5 I 0 0

~ so

<

~ 4o

2 0

- - 7 9 § q~ . K = NOMIN ,~L STRESS RATE e /

_ I = I D E N T I C A L T E S T S

- ~ '7

L O I . I 1.2 1.3 I.'# 1.5 1.6 I .T 1.8 A X I A L E X T E N S I O N R A T I O , "~'1

Fig. 5--Variation of true axial stress with axial extension ratio for Paracril RF-1

1.9

2 ~

22

ZO

..~ l e

h 4' 5

oP 6

I NOMINAL. STRESS RATIO ~%0 i IJ ~ K I ao~5 I 0o9,~1o,~5e I z,~'3 I

I K 2 NOMINAL STRESS RATE IDENTICAL TESTS '~

I I I I I t I I I 0 5 0 055 060 065 070 0.75 0,90 085 0 9 0 095 tOO

I / A t

Fig. ~--Variation of au/[2(Xt ~ -- 1/Xl)] with (1/Xx) in uniaxial- tension experiment for Paracril RF-1

energy function for any isotropic material under consideration, in terms of the three strain invari- ants. Using the finite deformation theory, the re- lationship between stress and the resulting deforma- t ion in uniaxial tension for an incompressible material can be shown to be

= 2 + = : . - (3)

where

0"11 = true stress in the direction of tensile force

)̀ 1 = axial extension ratio, the ratio of deformed length to the original length

W = strain-energy function, representing the strain energy stored per unit volume of the undeformed material

I1 and 12 = strain invariants. 14

The ratio ~11/[2()`~ 2 - 1/)`1)] is shown plotted in Fig. 6 against 1/~l. Although there exists con- siderable scatter, the data in Fig. 6 fit to a hori- zontal straight line. The deviations from the horizontal straight line corresponding to an ordi- nate value of 20.5 in the figure are greater at low extension ratios. This is due to inaccuracies in the measurements of small deformations during the experiments. Figure 6 indicates tha t the strain- energy function of the material can be expressed as

W = C(I~ - 3) (4) where

C = 20.5, a constant I1 = )̀ x 2 + )`52 + ),32 )̀ ~, )`2,

)`3 = principal extension ratios.

D e s c r i p t i o n o f B i a x i a l - s t r e s s A p p a r a t u s

The biaxial-stress apparatus developed for this s tudy is based on the principle of construction of a similar apparatus '5 used for metallic materials. The impor tant feature of the apparatus is tha t the effect of rate of loading on fracture behavior of low- modulus viscoelastic materials can be simul- taneously studied for different biaxial-stress fields. The complete biaxial apparatus with other acces- sories is shown in Fig. 7. The biaxial-stress ap- paratus as such consists of a lower head which is common for all biaxial-stress-field tests and a top head which is variable depending on biaxial-stress field under consideration (see Fig. 8). The appara- tus is equipped with several top heads to cover the various stress fields corresponding to biaxial-stress ratio a equal to 0 to co (a = ~22/~11, the nominal principal-stress ratio) in the tension-tension quad- rant of principal-stress coordinate plane. Figure 8 is a picture of a complete assembly of the top head and the bo t tom head with the specimen. The manner of subjecting specimens to biaxial-tension- stress fields can be explained as follows (see Fig. 9).

204 I M a y 1968

Fig. 7--Experimental arrangement for multiaxial-fracture studies Fig, 8--Closeup view of specimen-head assembly

Nit rogen gas from storage tanks fed th rough por t A pressurizes the specimen along the inner wall. At the same time, the gas flows through por t B into chamber C. The gas pressure act ing on face D produces an axial load on the specimen propor t iona l to the area of face D. A load cell E measures the t rue axial load on the specimen, devoid of any fric- t ional effects a t the pis ton-cyl inder contact .

A complete schemat ic of the biaxial-s tress ap- pa ra tus with various accessories is shown in Fig. 10. The specimen in a par t icu lar b iaxia l - f rac ture tes t can be subjected to prede te rmined pressure his- tories by a closed loop feed-back control sys tem consisting of a D a t a T rak funct ion generator, a Servac controller, a servovalve, a pressure t rans- ducer and a high-pressure ni t rogen-gas supply (see Fig. 10). The sys tem is capable of imposing l inear pressure ra tes up to 200 psi /sec.

Strain Measurement

The deformat ion of tubu la r specimens during bi- axial-stress exper iments was eva lua ted by measur- ing axial e longat ion and the var ia t ions in in ternal and external d iameters of specimens during the test. The measurements were made by clip gages. These clip gages are special ly sui ted for the mea- surement of soft, rubber l ike mater ia ls . The de- ta i led drawing of all the three gages used for mea- surement of axial extension, in ternal d iameter and external d iameter is given in Fig. 11. The clip gages used for measurement of in terna l and external d iameters are provided with shoes at their ends to conform to the inner and outer curved surfaces of

specimens. The clip gages, together with 906C Honeywel l recorder, can read d isp lacement with an accuracy of 0.002 in.

EYE BOLT FOR CYLINDER HEAD COUNTER WEIGHT

~ . \ \ \ I l;J l~.\\X.~l ~ P/ST~cYL, NDERAND III A"RANOE ENT

~H ~ESCAPE TO A TMOSPHERE

OSE CLAMP GRIP

~ ~ i i ECIMEN

AXIAL LOAD TRANSDUCER IL GAUGES

~ ~ ~ % ~ ~ TRANSDUCER / I / / / / / / I / T " / 1 1 1 / / / / LOAD WIRE FOR~/-j A PRESSURE / / LOAD TRANSDUCER INLET

Fig. 9--Biaxial-stress-test apparatus

Experimental Mechanics I 205

Exper imenta l Procedure and Results

Tubular specimens were subjected to fixed ratios of internal pressure and axial load. These fixed ratios of loads were obtained by using top heads of different sizes. The time rate of pressure was held constant for the durat ion of each test. The tests were continued up to fracture and a continuous record of pressure, axial load, changes in diameter and length of the specimens was made during the tests. F rom the observed values at fracture, true or nominal axial and tangential stresses were determined.

For the composite material (Hycar), fracture be- havior was investigated for biaxial-stress ratios equal to 0, 0.32, 0.82, 1.29, 1.68 and 2.29. The be- havior under these biaxial-stress fields was observed for two stress rates, namely k = 0.01 and 10 psi/sec (where k represents nominal stress rate in the maxi- m u m principal stress direction). The fracture be- havior of Paracril RF-1 polymer was studied for five biaxial-stress ratios, namely a = 0, 0.32, 0.82, 2.35 and oo. The s tudy was extended to various stress rates ranging from 0.013 to 4.5 psi/sec. Figure 12 shows the limiting curves of fracture for Hycar and Paracril R F 1.

Each above-mentioned experiment was repeated at least three times to check repeatabili ty of the data. All experiments were conducted under room temperature conditions of (75 -4- 1) ~ and (50 • 2) percent relative humidity.

Using the fracture data for various biaxial-stress fields, true octahedral shear stress and strain values were calculated for both materials. These values, when plotted against each other, seem to fall along a smooth curve obeying the following relationship:

roc~ = A sinh 7o0t (5) where

ro0~ = I/3[(~,1 - ~=)~ + (~= - ~3)2 + (~33 -~lt)~] 1/2 = octahedral shear stress (6)

~o0~ = ~/~[(~1~ - ~ ) ~ + ( ~ = - ~ ) ~ + (E33 -- e~t)2] 1/2 = octahedral shear strain (7)

~tt, ~22, and a33 are true principal-stress compo- nents.

el~, e22, and e33 are large principal-strain compo- nents.

The principal strains can be expressed in terms of experimentally observed extension ratios as follows:

Xl 2 -- I

~il = 2 (8)

similarly for e22 and e~3. Figure 13 shows tha t the fracture data for the

test materials fall on straight lines when ro~t is plot ted against sinh 7o~t.

Theoretical A n a l y s i s

Rivlin and Thomas ~6 have suggested a criterion of fracture for rubberlike materials which is an ex- tension of the Griffith fracture hypothesis for brittle

I NITROGEN SUPPLY I

~ o r 19o6c ( I HONEYWELL Pressure { V/S/CORDER ~

(q~,q , i ) [ At

_ . Q so::~i ere

' i ER

Pressure';"" ": 'O ('Q .~: ..~. o~': ~ "" ':"

I FUNCTION [ GENERATOR I

(DATA -TRACK I ~=~OGRAM ) I

I SERVAC CONTROLLER

, 1

=1PRESSURE "]TRANSDUCER

Fig. lO--Schemat ic of mul t iax ia l - loading system

materials. According to this criterion, catastro- phic tearing occurs in a rubberlike material when the critical tearing energy T is related to the elastic strain energy as follows.

5W

where

to

W

C

= thickness of specimen

= elastic strain energy

= crack length

( S W / 5 C ) ~ = change in elastic strain energy with the change in the length of the crack corresponding to constant over-all length in a tension test.

For a material with linear response, the change in elastic energy due to the formation of a small sharp crack can be shown to be 16

A W = - K ' C % W o (10)

where

W0 - - t h e elastically stored energy per unit volume of material in simple tension

g ' = a constant.

Using eq (10), eq (9) becomes

T = 2 K ' C W o (11)

Equat ion (11) represents the critical tear e n e r g y for uniaxial tension. This energy can be deter- mined if K ' and C are known and W0 is calculated. I n principle K ' can be obtained experimentally.

206 ] May 1968

.01 THICK SPRING S T E E L

9 ,52

F i l l I m

O U T S I D E G A U G E

. : r N7 _.] o ,o , ,

t'gH -4F _,_ / 1 . 1 ] 3

S H O E S /1 a n d B

Fig. l l - -C l ip gages

I N S I D E G A U G E

+

A X I A L E X T E N S I O N G A U G E

o | 1_

T. W I ~ N G D I A G R A M

Equation (11) could be extended to a linear visco- elastic material provided K' is considered to be a function of time rather than a constant. In the case of a viscoelastic material, the total work done by external forces during deformation is partly stored as recoverable strain energy and partly dis- sipated in internal friction. In order to apply eq (11) to a viscoelastic material, the recoverable strain energy just before fracture must be evalu- ated and substituted for 1410 in eq (11). In prac- rice, it is very difficult to evaluate the recoverable strain energy both experimentally and analytically. For a material with little energy dissipation, the total work done by external forces during visco- elastic deformation can be approximated to total recoverable strain energy. In the following, the total strain energy at fracture for the composite material is calculated using the creep compliance functions in tension and hydrostatic stress and the Bol tzmann superposition principle.

Total Energy at Fracture

The three-dimensional stress-strain relations for a linear viscoelastic material in creep ~ can be shown to be

.,,= ( - , , + - 4

2 0 0

150

I 0 0

D

"~ 5 0

~ 0

-- RATE OF LOADING K ( P S I / S E C )

- - / 0 0 0 0 1

B/AXIAL STRESS RATIO

A 2.29 ' A V ~ 1 6 8

0 0 8 2 X 0 52 + 0

MATERIAL A V [3 HYCAR (FILLED)

.xX ,,,,~ ~ d'-'" ~ x ^ \

I .++_/!J I I~+ I 50 IOO ~5o aoo z s o

NOMINAL A X I A L STRESS c~ I (PSI)

AVERAGE RATE OF" 5 0 LOAD K ( P S I / S E C )

i3 0.015 -.J A 0.094,

0 0 . 4 5 6 4o F v z. 173

I MATERIAL: PARACRIL RF- I

2 0 ~ V

10

,,I I 0 2 0 3 0 4 0 5 0 GO 70 8 0

~l'l {PSi)

Fig. 12--Fracture curves for the test materials

Experimental Mechanics I 207

Using eqs (12) and the Bol tzmann superposition principle, the strain components for any given stress history can be evaluated as follows:

d~n . , t - t ')

d aaa)dt,l B(t 6 t') ) -~, (cr2~ +

d~*~ s (D( t - t') D(t - t') ~ dr' -

B(t - t ' ) ) d ] 6 ~ (~a + an)dt'

daaa s (D( t - t') D(t - - t') dr-- ~ dt' - - 2

B ( t i t ' ) ) d ~ , (o'~ + ~ ) d t ' ]

(13) t = present time; t ' = past time.

For biaxial loading corresponding to

0"22 stress ratio e -

0"11

stress rate Cn = kt' eq (13) becomes

E( o~ = 1 -- k D(t - t')dt' +

~ ? ~- B ( t - t ' ) d t '

~= = [ ( ~ ) k f D(t-t')dt' +

In the above equation, aaa is assumed as zero and true-stress components are identical to nominal- stress components as a result of deformations being small. Using eq (14), the total energy at fracture can be determined for various biaxial-stress fields and stress rates as follows:

r % ) s r(~-~Os Ws = ~ n d ~ + ~.~d~2 (15)

J 0 J 0

where (en)l, (e2~)i are strain components at fracture calculated from eq (14) for a given time to fracture tr in a biaxial-stress experiment. Figure 14 shows the total strain energy at fracture for the composite material evaluated from eq (15) for various biaxial- stress ratios. The same figure shows also the ex- perimentally determined values. The comparison seems to be reasonably good beyond the stress ratio of 0.5.

For Paracril RF-1 which showed large deforma- tion and no time effects in the stress-strain be-

90

80

70

60

50

40

O~ 3O

43

6

FRACTURE DATA MATERIAL ,' PARACRIL RF- I

,o

~ o 0,4 0 .8 L2 1 6 2 O 241 2,8 3 2

O~ H Y P E R B O L I C O C T H E D R A L S H E A R S T R A I N

-A 1 4 0 [ ~ ~2-

I

O.Ol 6 c - 9 -o -o-

~ L MATERIAL : HYCAR I ~ 1 K = N O M I N A L STRESS RATE (MAX. PRINCIPAL STRESS)

_̂~ | ct'= ct~2/a,~ o-a~=-p ~ / ~. o . Yoc~'SASED ON KIRC.,O~F V ~ . ~ 1

o~ - MEASURE . # ~ ' FRACTURE DATA ~"

% c t = A ~ N H Yoc, ,a=145.0 ~ ,. V f "

I I I I I I L 0.1 0 2 0 3 0 4 0 ,5 0 6 0 .7

H Y P E R B O L I C O C T H E D R A L S H E A R S T R A I N

BIA)qAL STRESS RATIO Ct ~ K ~ Z.35 0.82 O.32

o.ol5 6 ~ r~ s 0 . 0 9 4 O- ~7- (3- o456 ~ 7 9 2.173 -0 ~7 -{3

K = NOMINAL STRESS RATE RADIAL STRESS, ~ 3 = - P STRAIN BASED ON KIRCHHOFF MEASURE ~oct = A SINH Yoct , A=. -~85

Fig, 13--Variation of time octahedral-shear stress with hyperbolic octahedral-shear strain for the test materials

I 0.8

havior, the tearing-energy expression essentially remains the same as eq (11), except K ' must de- pend on the axial extension ratio X1. In the ab- sence of the knowledge of how K ' varies with X~, the strain energy at fracture which contributes a major extent to the critical tear energy is evaluated. Figure 14 also shows the total strain energy at fracture for various biaxial-stress fields.

Discussion of Results The results from biaxial-stress experiments in-

dicate tha t the composite material (Hycar) is vis- coelastic not only in stress-strain behavior but also in fracture behavior (see Figs. 3, 4 and 12). The same is not true for Paracril RF-1 which does not display any t ime-dependent stress-strain behavior

0 6 9

3.6

I 0.9

208 I May 1968

9C

8C

7C

C "

~ ~o ~J "~ ~o LU zo

L)

~ l~ I k.

. J

0 0.01 V I0 .0

M A T E R I A L .

- - - - B Y VISCOELASTIC THEORY \

�9 , I I I I 0 .5 t O / 5 2 0

B I A X I A L S T R E S S R A T I O cl

RATE OF" LOADING K ( P S I / S E C )

H Y C A R (FILLED) - - EXPERIMENTAL

2 5

VERAGE RATE OF LOADING (PSI /SECJ

0 0 0 1 5 0 0 9 4 0%1)'456

V 2 IF3 MATERIAL:

PARACRIL R F - I

, , , I I o , I o 0 2 0 4 o 6 0 8 i o 0 8 0 6 0 4 0 2 o I N V E R S E O F B / A X I A L S T R E S S R A T I O (I/c~) B / A X I A L S FRESS RATIO ct

Fig. 14--Variation of total strain energy at facture with the biaxial-stress ratio for the test materials

as seen from Fig. 5. However, the fracture proper- ties of Paracril RF-1 depend appreciably on the rates of loading (see Fig. 12). This peculiar be- havior may be due to the onset of crystallization very near to fracture resulting in a steep rise in the stress strain curve, as usually observed for rubber- like materials. ~s It is quite possible that this part of the stress-strain curve is appreciably influenced by the rate of loading. Fracture data for the com- posite material and Paracri] RF-1 plotted in nominal principal-stress plane indicate that the fracture stresses for pure tension in axial (a = 0) and circumferential (a = oo) directions of the cylindrical specimen are not the same. This may be due to directional effects resulting from the geometry of the specimen. Figure 14 shows that the total strain energy at fracture predicted from the linear viscoelastic theory compares reasonably well with experimentally determined values for stress ratios a greater than 0.5, at higher rate of loading. However, the comparison between the- oretical and experimental results is poor for stress ratios less than 0.5�9 This is due to the occurrence of large deformations for stress ratios less than 0�9

limiting the applicability of the linear viscoelastic theory.

Figure 13 indicates that a linear relationship exists between octahedral shear stress and hyper- bolic sine value of octahedral shear strain. The data for the composite material lie closer to a straight line than for Paracril RF-1. The constant A representing the slope of the lines in Fig. 13 is independent of rates of loading and states of stress. The relation between ro~ and ~'o~t provides a simple general criterion of failure for the two viscoelastic materials studied.

Conclusions

(1) The new biaxial-stress apparatus described in this paper is very suitable for studying the frac- ture behavior of viscoelastic materials under multi- axial loading.

(2) The total strain energy at fracture for the composite material calculated from the linear vis- coelastic theory agrees reasonably well with exper- imental results.

(3) Fracture data obtained from biaxial-stress experiments for the two test materials indicate that there exists a linear relationship between octahe- dra] shear stress and hyperbolic sine value of octa- hedra] shear strain.

A cknowledgments

The content of this paper is a part of the research project sponsored by the Jet Propulsion Laboratory, under Contract No. 950875.

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forced E P R and S BR Vulcanizates," Rubber Chemistry and T~chnology, 3 7 (1964).

7. Taylor, G. R�9 and Darin, S. R�9 "7['he Tensile Strength of Elasto- mers," Journal of Polymer Science, 17 (1955).

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