Biaxial Fracture Stresses for Filled-epoxy Tubes

Paper reports on the strength behavior of a brittle material at room temperature. Tests were conducted in all biaxial-stress quadrants

by Richard E. Ely

ABSTRACT--Strength results for a filled epoxy are reported for biaxial stresses. Three test methods and several speci- men configurations were employed. Strength magnitudes were quite dependent upon specimen geometry under com- pression-compression stresses. For other stress states, this dependence disappeared and the modified Coulomb-Mohr theory f i t the data well. Similar data for graphite are cited.

Introduction The f rac ture-s t rength behavior of man-made br i t - tle materials has received considerable at tention un- der tension-tension (TT) and tension-compression (TC) stresses. Ear ly studies 1 were conducted with "bri t t le" metal l ic materials and concrete. Many cur- rent studies 2 are concerned with the s t rength be- havior of ceramic or other heat - res is tant materials such as used in aerospace applications. In only a few of these studies were data reported for compression- compression (CC) stresses. This shortcoming great ly hinders the selection of appropr ia te fai lure cri teria and it can be at t r ibuted largely to the exper imenta l difficulties encountered in testing bri t t le mater ia ls under CC as well as uniaxial compression stresses.S

In the present study, the pr imary objectives were to provide CC strength results and to gain some knowl- edge about the effect of test method on results. A fi l led-epoxy mater ia l was chosen as the test ma- terial because of its availability, ease of fabrication and low cost. A 'pressure-vessel ' test method was developed for conducting tests in all biaxial-s tress quadrants; this method permits the use of tubular specimens of ei ther constant or different lengths. Two previously employed test methods also were employed to obtain s t rength data in two stress quadrants.

Test Procedure Apparatus

Three test methods were employed. The 'pres-

Richard E. Ely is Research Aerospace Engineer, U.S. Army Missile Command, Redstone Arsenal, AL 35809. Paper was presented at 1973 SESA Fall Meeting held in Indianapolis, IN on October 16-19.

sure-vessel ' method, which is described below, can be used to conduct tests in all biaxial-stress quad- rants. The ' long-tube ' method 4 can be used for tests in both the TT and TC stress quadrants. The 'short- tube' method 2,5 is restr icted to tests in the TC stress quadrant. For both the long- and shor t - tube test

Fig. I---Tubular specimen-piston assembly

492 i December 1974

methods, tubular specimens are subjected to various combinations of axial loads and ~nternal fluid pres- sure. Specimen configurations for the three test meth- ods are significantly different (see below).

The pressure-vessel apparatus consists pr imar i ly of a specimen-pis ton assembly, Fig. 1, which is in- serted into a cylindrical pressure vessel, Fig. 2. Dur- ing testing, this apparatus is located on the weigh- ing table of a universal tester where the tester is used for measur ing loads generated by the ap- paratus. The specimen had a luminum plug-type, end-adapters which were cemented to the specimen ID surface wi th epoxy resin, Fig. 3. The p lug- in- sertion length was 1 in. (25.4 ram). For tests in the TT and TC stress quadrants, the end-adapters also include an a luminum ring (not shown) which is ce- mented to the OD surface of the specimen and where the r ing length matched the plug length. Each end- adapter has a central hole which mates wi th a piston shaft, Fig. 3, in order to insure al ignment; al ign- ment also was mainta ined using these holes and suitable fixtures dur ing assembly of the specimen and end-adapters . The piston diameters were 3 in. (76.2 mm) .

The operat ing principle of the pressure-vessel ap-

/ / / / / / / / / / / / / / / / / / / / / / / I I

0

~ P ~iiiiiiiiiii!i~i~i~ii!iiiiiiiii!!)~i~iii!i!i!i~iiii iiiiiiiiii!i!i!i!!!!iiiiiilililili!i!i i i i ii!i!iiii

BOTTOM PISTON

Po

I

/ / / / / / / / / / / / / / / / / / / / / / /

Fig. 2--Schematic of pressure-vessel apparatus

paratus is similar to that used in other studies.6. 7 Specimen loading, Fig. 2, was accomplished by various combinations of internal fluid pressure, p~, external fluid pressure, Po, and pressure, Pz, which is applied to the lower surface of the bot tom piston. For the present study, only one or two of the vol- umes were pressurized for a test. Discrete pressure ratios were obtained using booster assemblies (not shown), so that the biaxial loading was applied in a proport ional manner dur ing a test for 9 of the 10 loading paths employed. Being able to measure axial loads ex terna l ly is a unique feature of the ap- paratus and this capabili ty provided a means for evaluat ing the frictional forces on the top piston when rubber O-r ing seals were used (po > 0); cor- rections for the frictional forces never exceeded 3.9 percent, s The pr imary pressure source was bot- t led ni t rogen gas. This source was used in the pres- sure vessel to load the specimen and to actuate the booster when a booster was employed. When a booster was used, water at room tempera ture was used as the second fluid in the system. Per t inent pressures were recorded as a function of t ime and m ax im um pressures were measured with bourdon- tube- type dial gages.

Test Specimens

The test mater ia l was a cast epoxy* which was filled with a black product and which can be readily machined. It was received in the form of tubular stock with a 1-in. (25.4-mm) ID and a 1-1/2-in. (38.1-mm) OD. Test specimen ID and OD surfaces were machined and then polished in a uni form manner. Specimens were machined f rom two ship- ments of stock, designated as Lots A and B.

Test specimens for the pressure-vessel method had a 1.190-in. (30.2-ram) ID and a uniform wall th ick- ness of 0.075 in. (1.90 mm) . Several gage lengths were employed; the gage length was 2 in. (50.8 ram) less than the total tube length because of the plug- type, end-adapters. The short - tube specimens also were r ight circular cylinders but without end- adapters. In this case, the axial compression loads were t ransmit ted to the specimen ends through tub- ular epoxy "seats". The dimensions of the shor t - tube specimens were 1.186-in. (30.I-mm) ID, 0.065-in. (1.65-mm) uniform wall thickness and a 1-1/2-in. (38.I-ram) length. The long- tube specimens had either threaded or unthreaded bosses at each end. These specimens had a 1-in. (25.4-mm) ID, a 0.60-in. (1.52-mm) gage-wal l thickness, a 3-1/2-in. (88.4- mm) gage length with 1-in. (25.4-mm) external radii at each end for transit ion to the th icker -wal l sections. 4

Q HysoI, Type II, CP2-4289, The Dexter Corp.

Fig. 3--Pistons and specimen-adapter assemblies

Experimental Mechanics I 493

S t r e s s C a l c u l a t i o n s

All s t r e s se s w e r e c o m p u t e d u s ing t he in i t i a l d i - m e n s i o n s of the test-specimen gage sections. Hoop

stresses were computed for the ID surface using the

thick-walled formula. For hoop tension stress, the

expression was

Pi (D2 + d2) ~t -- (1)

(D2 -- d 2)

a n d for hoop c o m p r e s s i o n s t ress ,

-- 2poD2 ~t - (2)

(D2 -- d2)

w h e r e

p~ _-- i n t e r n a l p r e s s u r e Po = e x t e r n a l p r e s s u r e D = s p e c i m e n OD d = s p e c i m e n ID

T h e ax i a l s t r e s s was c a l c u l a t e d b y v a r i o u s e x p r e s - s ions d e p e n d i n g u p o n t he s t r e s s s t a t e a n d t e s t m e t h o d .

W h e n p r e s s u r e s Pi a n d Po w e r e u s e d in t he p r e s s u r e - vesse i m e t h o d (Fig. 2) , t h e ax i a l s t ress was c o m p u t e d u s ing

~z : (3) D 2 _ d 2

Fo r p r e s s u r e s Pi a n d Pz,

(') p~d2_ ~- (L)

~z = (4) D2 _ d 2

a n d for p r e s s u r e s Po a n d Pz

TABLE I--STRENGTH RESULTS FOR PRESSURE-VESSEL TEST METHOD, l- IN. GAGE LENGTH, LOT B MATERIAL

Hoop Stress Axial Stress Stress State psi (MPa) psi (MPa)

CT --4130 (--28.5) 8080 (55.7) CT --4140 (--28.5) 8110 (55.9) T - - - - 8400 (57.9) TT 8920 (61.5) 3930 (27.1) TT 8930 (61.6) 3940 (27.2) TC 7510 (51.8) --8510 (--58.7) TC 7060 (48.7) --8140 (--56.1) TC 6050 (41.7) -- 14200 (--97.9) TC 5730 (39.5) --13400 (--92,4) C - - - - - - 2 1 0 0 0 (--144,8) C - - - - - - 20800 ( - - 143.4) CC --5030* (--34.7) --23100* (--159.3) CC --4620* (--31.9) --22600* (--155.8) CC - - 1 1 5 0 0 (--79.3) - - 2 4 9 0 0 (--171.7) CC - - 1 1 4 0 0 (--78.6) - - 2 4 3 0 0 (--167,5) CC - - 1 9 1 0 0 (--131.7) - - 2 1 2 0 0 (--146,2) CC - - 1 8 3 0 0 (--126.2) - - 2 0 7 0 0 (--142.72)

* N o n p r o p o r t i o n a l l o a d i n g .

( ' ) po ( A 2 - D e ) - - ~ ( L + f )

~z = (5) D 2 _ d 2

w h e r e

L ---- m e a s u r e d ax ia l load _-- f r i e t i o n a l l o a d ( e x p e r i m e n t a l )

A ---- ID of p r e s s u r e vesse l

a n d t h e o t h e r s y m b o l s a re as be fore . In eqs (3) a n d (5) t h e f r i c t i on load was t a k e n as zero w h e n po -~ 0 s ince no r u b b e r O - r i n g seals w e r e used on t h e t op p is ton , Fig. 2, ( m e a s u r e d f r i c t i o n a l loads w e r e n e g l i - g ib le for th i s p r o c e d u r e ) . Fo r t h e s h o r t - a n d l o n g - t u b e t e s t m e t h o d s , eq (4) is used ; t he s ign in t h e

TABLE 2--STRENGTH RESULTS FOR PRESSURE-VESSEL TEST METHOD, 2-iN. GAGE LENGTH, LOT B MATERIAL

I II III

Hoop Stress Axial Stress Stress State psi (MPa) psi (IVIPa)

CT --3930 (--27.1) 7680 (53.0) CT --3790 (--26.1) 7390 (51.0) TT 8310 (57.3) 3660 (25.2) TT 8050 (55.5) 3550 (24.5) TC 8160 (56.3) --680 (4.7) TC 74t0 (51.1) --8610 (--59.4) TC 5540 (38.2) --13000 (--89.6) TC 6360 (43.9) - - 14900 ( - - 102.7) C - - - - - - 20300 ( - - 140.0) C - - - - --21100 (--145.5) CC --5580* (--38.5) --22700* (--156.5) CC --10300 (--71.0) --22100 (--152.4) CC -- 10200 ( - - 70.3) - - 20800 ( - - 143.4) CC --15200 (--104.8) --17300 (--119.3) CC --15800 (--108.9) --17900 (--123.4)

* Nonproportional loading,

TABLE 3--STRENGTH RESULTS FOR PRESSURE-VESSEL TEST METHOD. 3-IN. GAGE LENGTH, LOT B MATERIAL

I II

H O o p S t r e s s A x i a l S t r e s s

Stress State psi (MPa) psi (MPa)

CT --4470 (--76.2) 8760 (60.4) TT 8290 (57.2) 3650 (25.2) TT 9340 (64.4) 4120 (24,4) TC 6350 (43.8) --6830 (--47.1) TC 7320 (50.5) --8230 (--56.7) TC 5840 (40.3) -- 13500 (--93.1) TC 5990 (41.3) --13700 (--94.5) C - - - - --22500 (--155.1) C - - - - --22300 (--153.8) CC --3880* (--26.8) --21590* (--148.2) CC --3820* (--26.3) --23100* (--159.3) CC --8330 (--57.4) --16800 (--115.8) CC --8840 (--60.9) --18000 (--124.1) CC --11600 (--80.0) --13000 (--89.6) CC --11500 (--79.3) --12900 (--88.9)

* N o n p r o p o r t i o n a l l o a d i n g .

494 I December 1974

numerator is negative for the TC stress quadrant 2 .~ and positive for the TT stress quadrant.

Results Fracture-strength results for a filled-epoxy ma-

terial under biaxial stresses were obtained at room temperature using three test methods. Data for the pressure vessel method are given in Tables I, 2 and 3 for specimen gage lengths of I, 2 and 3 in. (25.4, 50.8 and 76.2 ram), respectively. Fracture times for these tests ranged from about 30 to 130 s where the shortest and longest times are for tests conducted in the TT and CC stress quadrants, respectively.

In Fig. 4, data for the pressure-vessel test method are presented for three specimen gage lengths; these data were normalized using a uniaxial tensile strength, ~o, of 8300 psi (57.2 MPa). The first sig- nificant observation is that the strength magnitudes under CC stresses are very dependent upon the gage length, but such a dependence disappears in the other stress quadrants. A second observation is that the curves for the different gage lengths 'cross over' one another in the CC stress quadrant. Also, under uniaxial compression the greatest strength was obtained with the longest gage length. Strength data s for gage lengths of 1-i/2 and 2-1/2 in. (38.1 and 63.5 ram) also were obtained in the CC stress quadrant but are not reported here; curves for these data would fit precisely the behavior shown and described above. Also, 42 graphite tubes of dif- ferent gage lengths were fractured in the CC stress quadrant using the pressure-vessel test method. 8 The strength behavior obtained was identical to that re- ported here for the filled-epoxy material.

As noted above for the CC stress quadrant, the strength curves crossed over one another in a particu- lar biaxial loading 'zone', Fig. 4. This behavior is significant since it means the strength magnitudes for the tubular specimens are independent of their gage lengths. The reason for this behavior is pre- dicted by elastic theory, s namely, specimen and load conditions are such that the specimen diameter does not change or, in other words, specimen longitudinal bending is at a minimum. It is suggested that strength data in the cross-over zone are reliable. Clockwise

TABLE 4--AVERAGE STRENGTH RESULTS FOR LONG-TUBE METHOD, LOT A MATERIAL

Num- ber of Stress

Test State Hoop Stress Axial Stress

psi (MPa)' psi (MPa)

3 T - - - -

3 TT 3187 (3.4)* (22.0) 3 TT 6660 (5.6) (45.9) 3 TT 6963 (5,1) (48.0) 5 T 7362 (3.1) (50.8) 3 TC 7077 (11.4) (48.8) 4 TC 7038 (5.0) (48.5) 3 TC 5590 (1.1) (38.5) 3 TC 3883 (2.8) (26.8) 4 C - - - -

* Coefficient of variation, percent.

7176 (5.1)* (49.5) 7220 (3.2) (49.8) 6643 (5.5) (45.8) 3103 (5.2) (21.4)

--2927 (11.2) (--20.2) --6475 (5.3) (--44.6)

--11533 (1.3) (--79.5) --14033 (2.7) (--96.8) --18575 (1.4) (--128.1)

CT

i i i -3 --2 --1 0

CC

--1

3 IN. GAGE LENGTH %

2 IN. \ ~ -2

,,N % i o \ z~ - - .

0 o = 8300 psi 157.2 MPa)

--4

Fig. 4--Filled-epoxy strength data for pressure-vessel test method

" t

"C

I

2 otl%

from this zone, the longitudinal bending becomes quite severe and the test method is inadequate. Average strength results for the epoxy material are

given in Tables 4 and 5 for the long- and short-tube test methods. For the long-tube method, fracture times ranged from about 50 to 200 s and about 40 to I00 s for the short-tube method. In Fig. 5, average strength results are presented

for 105 tests, the 3 test methods and 2 lots of the epoxy material. For the pressure-vessel method (Tables I, 2 and 3), the data points are average values for the results for all three gage lengths, ex- cept for uniaxial compression where only data for the longest gage length was used. These data, Fig. 5, were normalized for each test method using different values for the tensile strength. The highest tensile strength, 8300 psi (57.2 MPa), was for Lot B material and the pressure-vessel method while the lowest strength, 7200 psi (49.6 MPa) was for Lot A material and the long-tube method. In the CC stress quadrant, only the data in the crossover zone were employed.

The average-strength data, Fig. 5, are compared with the predictions of the Leon ,q and the modified

TABLE 5--AVERAGE STRENGTH RESULTS FOR SHORT-TUBE TEST METHOD, LOT B MATERIAL, TC QUADRANT

Number of

Tests

5 5 5 4

5

Hoop Stress Axial Stress psi (MPa) psi (MPa)

7814 (1.8)* (53.9) --1984 (3.3)* (--13.7) 6968 11.0) (48.0) --6440 (1 .9 ) (--44.4) 4610 (1.4) 131.8) --13960 (1.2) (--96.3) 2080 (2.0) (14.3) --17550 13.1) (--121.0)

- - - - - - 18540 (9.4) ( - - 127.8)

* Coefficient of variation, percent.

Experimental Mechanics I 495

MODIFIED COULOMB - MOHR - - ~

\

t f - i i --3 -2 -1 0

-1

2 -Oz/O o

-2 z~

�9

D

x

�9 TABLES I, II & III, o o = 8300 psi (57.2 MPa) o TABLE IV, (~o = 7200 psi (49.6 MPa) �9 TABLE III, COMPRESSIVE STRENGTH A TABLE V, o o = 7600 psi (52.4 MPa)

Fig. 5--Fil led epoxy, average-strength data

i 2

crt/o o

Coulomb-Mohr lo theories and the fit is best for the latter. Both of these theories are based on a genera l - ized s shear-stress, in ternal - f r ic t ion concept. Their s trength predictions in the TT and CC stress quad- rants are identical. S t rength predictions for the modified s t ra in-energy theory~ would fit the data reasonably well; predictions of the modified distor- tion energy 11 would not.

The fracture modes for the epoxy mater ia l indi- cated it to be a bri t t le material . For the pressure- vessel test method, for example, the test specimens were l i teral ly pulverized under CC stresses. Under TC stresses, the specimen remains consisted of some powder and many small pieces; under TT stresses the pieces were considerably larger but still numer - ous. In no case was there any indication of plastic flow.

Discussion The results of this study indicated that the devel -

opment of test methods for determining the strength behavior of bri t t le materials under CC stresses is a much more difficult task than for ei ther TT or TC stresses. The strength behavior reported for the epoxy mater ia l in the TT and TC stress quadrants, Fig. 5, is considered representa t ive since it was obtained using ei ther four or five tes t -specimen configurations. In the CC stress quadrant, it was suggested that only the strength data in the cross-over zone are valid.

One controversial question which arises from this study is whe ther or not there is a s t rength increase under CC stresses re la t ive to the simple compressive s t rength for a t ru ly bri t t le material , that is, a ma- terial which fractures with very l i t t le or no evidence of plastic flow under all biaxial-s tress states. Most theories, which have been proposed for bri t t le ma-

terials, do not predict a s t rength increase for equal biaxial compressive stresses, s Exper imental ly , CC strength studies are quite l imited in both number and scope. Coffin's s t rength data for gray cast iron 12 has received wide attention. He reported data for 3 uniaxial compression and 4 CC tests where the lat ter were at or near the cross-over zone. A mod- erate s trength increase was obtained. For these tests, short uni form cylindrical specimens were em- ployed and, in all cases, the fai lure mode was either a buckling or instabil i ty type, that is, significant plastic flow occurred. Thus, under CC stress, this part icular cast iron probably should not be con- sidered as a bri t t le material . S t rength data for an A T J - S graphite I3 support the proposition of no s t rength increase under CC stresses in the cross- over zone. Outside this zone, both no increases and increases were obtained; effects of specimen geom- e t ry were not investigated. Data reported for alu- mina, 3,14 on the other hand, tend to support the fact that s t rength increases are possible under CC stresses. For these studies, however, the tests were conducted for the single-stress state of 2:1 hoop-to- axial CC stresses, a region remote f rom the cross- over zone. For graphite tested at this stress state wi th the pressure-vessel method, s both s t rength in- creases and decreases were obtained, depending upon the specimen gage length. It can only be concluded that additional exper imenta l work is needed in the CC stress quadrant.

A c k n o w l e d g m e n t

The author is indebted to Will iam O. Hayes, Physical Sciences Directorate, U.S. Army Missile Command, for conducting most of the exper imenta l program.

R e ] e r e n c e s

1. Coronet, I., and Grassi, R. C., "'A Study of Theories of Fracture Under Combined Stresses," Jnl. of Basic Engineering, 83, 33-44 (March 1961).

2. Ely, 1l. E., "Strength of Titania and Aluminum Silicate Under Combined Stresses," 1. Amer. Ceramic Soc., 55(7), 347-350 (July 1972).

3. Sine, G., An Experimental Study on the Compressive Biaxial Strength of Ceramics, Univ. of Calif., Report UCLA-ENG-7172, (Oct. 1971).

4. Ely, 1l. E., "Biaxial Stress Testing of Acrylic Tube Specimens," Polymer Engineering and Science, 7(1), 40-44 (Jan. i967).

5. Ely, R. E., "'Strength of Magnesium Silicate and Graphite Under Biaxial Stresses," Amer. Ceramic Soc. Bull., 47(5), 489-492 (7 May 1968).

6. Weng, 7'., "Biaxial Fracture Strength and Mechanical Proper- ties of Glaphite-Base Refractory Composites," A IAA ]hi., 7(5), 851-858 (May 1969).

7. Broutman, L. ]., Krishnakumar, S. M., and MaIlick, P. K., "'Effects o1' Combined Stresses on Fracture of Alumina and Graphite," Jnl. Amer. Ceramic Soc., 53(12), 649-654 (Dec. 1970).

8. Ely, 1t. E., "Strength Results for Two Brittle Materials Under Biaxial Stresses," U. S. Army Missile Command, Redstone Arsenal, Alabama, Report No. RR-72-11 (Sept. 1972).

9. Romano, M., "On Leon's Criterion," Meccanica (AIMETA), 4(1), 48-66 (March 1969).

10. Paul, B., "A Modification of the Coulomb-Mohr Theory of Fracture," ]. of Appl. Mech., 28, Trans. ASME, 83, 259-268 (June 1961).

11. Fisher, ]. C., "'A Criterion for the Failure of Cast Iron," ASTM Bull., (181), 74-75 (Apri! 2952).

12. Co~n, L. F., "The Flow and Fracture of a Brittle Material," ]nl. of Appl. Mech., 17, Trans. ASME, 72, 233-248 (1950).

13. Perkins, R. D., et al., Multiaxial Loading Behavior of Four Materials Including AT]S Graphite and RAD-6300 Carbon Phenolic, General Motors Corp., Materials and Structures Laboratory, Technical Report SAMSO-TR-69-393, 1, Contract F04701-68-C-0161 (Aug. 1970).

14. Broutman, L. L, and Cornish, R. H., "'Effect of PoIyaxial Stress States on Failure Strength o~ Alumina Ceramics," ]nl. Amer. Ceramic Soc., 48(10), 519-524 (Oct. 1965).

496 [ December 1974