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A study of variables in the mechanical testing of beryllia
JOURNAL
OF NUCLEAR
MATERIALS
14 (1964) 39ti-463@ NORTH-HOLLAND
A STUDY OF VARIABLES
IN THE MECHANICAL
K. VEEVERS, Materials
Division,
J. F. WHATHAM
AA EC
Research
PUBLISHING
TESTING
CO., AMSTERDAM
OF BERYLLIA
and W. J. WRIGHT
Establishment,
Lucas
Heights,
Australia
Hot-pressed beryllia of 2.9 to 3.0 g/cm* density and 16-20 microns grain size was studied to determine the effects of specimen size, shape, surface finish, strain rate; prior mechanical treatment, and loading geometry on the modulus of rapture measured in four point bending. The modulus of rupture was found to be insensitive to variations in volume of the test piece, prestraining, and loading geometry over the range studied; when considered as a normally distributed sample, the variations in modulus of rupture at the extremes of volume were not significant at the 96 per cent confidence level. The standard deviation of results for a normal population .was about 1900 psi. The results were analysed according to the Weibull
“weakest link” theory. Values of the Weibull “constant” varied from 10 to 36 with a mean of 16; the variations in modulus of rupture with volume under stress, predicted from this model were therefore small but probably significant at the 96 per cent confidence level for a normal distribution. Specimens prepared with various surface finishes (20 micron to 160 micron abrasives) showed no significant variation in modulus of rupture. An increase in strain rate from 0.06 per cent per minute (extreme fibre strain) to 1.2 per cent per minute showed no significant increase in strength, but further increase in strain rate to 48 per cent per minute resulted in 26 per cent increase in the modulus of rupture.
1. Introduction
of polycrystalline ceramics at much lower average stresses than apply in “perfect” crystals. WeibulP ) extended the principles of the Griffith theory to explain the size-effect on stress to cause failure. He proposed the existence of a range of “flaws” in ceramics, which could cause failure at a low average stress; these could be features such as inclusions, pores, micro-cracks, and pile-ups of dislocations at impurity atoms, grain boundaries, or intersecting dislocation systems. Thus the material was assumed to contain a variety of flaw types, randomly distributed in the volume under stress, and the presence of one flaw, stressed to the “characteristic” failure stress of that flaw, could cause failure through the whole section. These flaws were recognized to be more effective in causing failure under tensile stresses than under compressive stresses. The Weibull theory suggested that the probability of failure at a given stress increases with larger volumes under tensile stress since the probability of the existence of a “critical”
Theories of brittle fracture invariably suggest that the measured strength of ceramics depends on a number of variables in the testing method. Perhaps the most important condition of interest in reactor core design is the dependence of the measured strength upon the volume of specimen under stress but other conditions such as surface finish and strain rate in the test may also be important. This paper reports the first phase of an investigation of the dependence of measured modulus of rupture upon various testing conditions. 2. Statistical Theories of Failure in Ceramics The original theory of brittle fracture, proposed by Griffith I) suggested the existence of “flaws” in brittle solids which could raise the local stress level and initiate cracks to produce failure. This theory largely explains the failure
VI.
FABRICATION
AND MECHANICAL
PROPERTIES
K.
396
VEEVERS,
J.
F.
WHATHAM
flaw in the section will be higher. The low (apparent) strength shown by ceramics in tension compared with bending is thus a consequence of the fact that the volume under tensile stress is larger in the tensile test and the probability of failure at a given stress is therefore higher. The theory also predicted that the strength measured in bending depends on the geometry (for example 3 or 4 point bending, gauge length) and the section under stress. The derivations of probability equations from the Weibull theory have been given previously (for example Bortz “). The probability of failure in bending may be written: Pg=l-exp
[
-
V, 2(m+l)
W.
J.
00
v,1 -_=Cl --me u2
v2
The constant, m, arises from the stress function which Weibull used to character&e the occurrence of various flaws with failure stress; m is thus a measure of the scatter in strength values arising from a flaw distribution characteristic of the material.
The variations predicted by the Weibull theory for measured modulus of rupture as a function of volume under tensile stress in bending are shown in fig. 1; for high values of m, that is, highly reproducible material, the effects of volume are small, but poorly characterised material, for example concrete, may show large effects of volume. A further conclusion from the Weibull analysis is that the strength of such solids need not be normally distributed about the mean. If an extremely large section of material is under test, there is a high probability of occurrence of a flaw which will cause failure at a low average value of strength. If, however, the same section is cut into a number of smaller pieces for testing, some of these will not contain the “critical” flaw and will therefore require a higher stress to cause failure (that is, another type of flaw). Thus the distribution curve will tend to skew to low values with increasing volume and the scatter of results will become less. The assumption of non-normal distribution, and a varying distribution, greatly complicates the analysis of results; it is desirable to retain the advantages of calculation from a normal distribution and to examine the departures from normality suggested by the Weibull analysis. This report examines the effects, on measured modulus of rupture, of surface finish, strain rate, and volume under stress, for beryllium oxide tested at room temperature. 3. Experimental
VP
v, Fig.
WRIGHT
01 - (1) C7m
(see section 7, Notation) and for the same probability of failure, the effects of volume on stress may be written:
01
AND
1. The effects of volume on measured strength predicted by the Weibull theory.
as
Methods
All measurements of modulus of rupture were made with specimens in four point bending. Thirteen 4~ 4~ 2 inch blocks of hotpressed Brush UOX grade beryllia of 2.9 to 3.0 g/cm3 density and 16-20 microns grain size were used in these experiments. The uniformity of strength between blocks was examined on +x ) inch (6.3 mm x 6.3 mm) test pieces; no significant differences between blocks were evident and results from all blocks were therefore grouped. The significance of grouping these
MECHANICAL
TESTING
results is discussed in section 6. Specimens were machined from these blocks; the tolerances on dimensions were &O.OOl inch (0.06 mm) on the cross-section, and the faces of the specimens were parallel to f0.0005 inch (0.02mm). Specimens were finished by grinding with wheels impregnated with diamond of 100-120 microns grit size at a peripheral wheel speed of 5000 ft min-l. Comers along the length of the specimens were rounded to 0.015 inch radius. Before testing, all specimens were examined by X-radiography using both radio-opaque and fluorescent-penetrant techniques; specimens containing any obvious defects were excluded
Fig.
2. Typical
modulus
of rupture
397
at the grain boundaries and in association with the needles. Most of the testing was done on a Hounsfield Tensometer at a strain rate of 0.0626 inch (1.50 mm) per minute. The effects of strain rate were investigated on a 60 000 pound Universal Tensile Machine at rates from 0.0025 (0.5 mm) to 2 inches (5 cm) per minute cross-head movement. For these tests a high speed recorder was necessary to measure the fracture loads because the machine recording system did not respond fast enough at the high strain rates. Specimens were tested with the following loading geometries:
specimens
from the testing programme. Typical specimens are shown in fig. 2; the grey colour is thought to be due to carbon picked up from the hotpressing die and is mainly associated with specimens cut from the outside areas of the pressings. The strength was apparently unaffected by these variations. The microstructure of the material (fig. 3) showed equiaxed grains of 15-20 microns diameter with a random distribution throughout of needles of beryllia up to 160 or 200 microns long. The needles were present in the as-received powder to the extent of approximately 1 vol Oh). The porosity was mainly found
OF BERYLLIA
showing
surface
and type of failure.
Method A. Span 2.5 inch (6.25 cm). Gauge Length 1.25 inch (3.1 cm). Method B. Span 1.5 inch (3 cm). Gauge Length 0.75 inch (1.8 cm). At least ten specimens were tested under each condition; the modulus of rupture was calculated using the elastic relationship: bg=-. M.Y I The accuracy of the stress measurements was about f2.5 per cent on the Hounsfield Tensometer or about &2 per cent on the Universal Tensile Machine. VI.
FABRICATION
AND MECHANICAL
PROPERTIES
398
K.
VEEVERS, J. F. WHATHAM AND W. J. WRIGHT
4. Results 4.1. EFFECTS OF SURFACEFINISH
Specimens ix 4 inch (6.3 mm x 6.3 mm) were finish ground with wheels of various diamond grit sizes as shown in table 1. The grinding schedule was such as to remove a total depth of 0.040 inch (1 mm) from the surface, initially with cuts of 0.005 inch (0.15 mm) and, for the last 0.010 inch (0.27 mm) with cuts of 0.002 inch (0.07 mm). One series was given a further polish in a slurry of 20 micron alumina on a vibratory polishing machine for three weeks. The coarseground specimen had deep
Each series of ten specimens was tested in four-point bending by Method A at a strain rate of 0.0625 inch (1.50 mm) per minute cross-head movement and the results are shown in table 1. No consistent effect of surface finish was found, and, in particular, the series with the relatively fine finish (20 microns) showed no improvement over those with a coarse finish (150 microns). With the relatively fine finish obtained by polishing, there still remained a large number of surface scratches up to 20 microns deep. In brittle ceramics a flaw 2 microns in depth is known “) to reduce the strength markedly and a surface preparation more extensive than given here is almost certainly required to show a significant effect of surface finish on modulus of rupture. The range of finishes studied was, therefore, probably insufficient to affect the modulus of rupture but was sufficient to suggest that no practical mechanical method of surface preparation was available whereby significant improvement in strength could be achieved. 4.2. EFFECTS OF STRAIN RATE
Fig. 3. Photomicrograph 10 per cent.
of hot pressed beryllia, etched in NH*F/HF solution. x200.
parallel scratches along its length while the polished specimen appeared to be free from any oriented scratches. TABLE 1 The effect of surface
finish on modulus of rupture
(Specimens ax & inch tested by Method A. Span 2.6 inch gauge length 1.26 inch) Surface Finish Microns
Modulus of Rupture a psi
Standard Deviation S psi
Specimens ix * inch (6.3 mmx6.3 mm) were tested by Method A on the Universal Tensile Machine at the strain rates shown in table 2. The effect of increasing the strain rate over the range 0.06-48 per cent per minute (fig. 4) was to increase the modulus of rupture by about 6000 psi or 25 per cent. From 0.06 to 1.2 per cent per minute, a small increase in modulus of rupture was noted which was not significant at the 95 per cent confidence limit; as the strain rate was increased from 1.2 to 24 per cent per minute, the modulus of rupture increased by about 15 per cent. A further increase of 5 per cent was noted as the strain rate was increased to 48 per cent per minute. The increase in modulus of rupture with increase in strain rate could be due to a stress corrosion mechanism as described by Charles and Shaw “) and by Carniglia ‘), 4.3. EFFECTS OF PRE-STRAINING Specimens of various sizes were tested in
MECHANICAL TABLE The
effect
(Specimens
2
of strain rate on modulus
0.06 0.06 0.06 0.241 1.206 1.206 24.10 48.20
of rupture
f x 2 inch tested by Method A. Span 2.6 inch gauge length 1.25 inch)
-
Specimen Strain Rate % min-i
TESTING
Modulus of Rupture u psi
Standard Deviation S psi
m
24 600 23 660 23 300 24 300 24 800 24 600 21600 29 100
18.3 11.3 20 18.3 18.3 19.1 16.2 21.1
1670 2400 1600 1600 1760 1800 2600 1460
bending by Method A and the broken halves of these specimens were re-tested by Method B. A similar number of specimens of each size, not previously broken, were tested by Method B. A comparison of results was made of the modu-
20000 Id’
e
significant for the number of tests involved. Two possible effects of pre-straining, (i) strain hardening, and (ii) generation of flaws under stress, were considered. The extent of local plastic deformation of beryllia before failure is less than 0.1 per cent and thus the extent of strain hardening was probably negligible. There was no evidence from these mechanical tests, or from the microstructures, of further weakening due to flaw generation. 4.4. EFFECTS
OF VOLUME
effects of volume were investigated initially by varying the loading geometry (span and gauge length) and the cross-section shape. Results (table 4) showed no significant effect of these parameters on modulus of rupture, and hence no effect of volume within the range considered. Further tests were made on specimens with a wider range of cross-section The
1
I
I
Id’
I
IO
SPECIMEN
Fig. 4. Effect
399
OF BERYLLIA
STRAIN
RATE
of strain rate on modulus
lus of rupture determined on the same testing geometry with and without pre-straining; the results are shown in table 3. Five of the seven pairs tested showed a small decrease in modulus of rupture on the prestrained samples but these variations were not significant when considered as a normal distribution at the 95 per cent confidence level. Similarly the standard deviation of the unstrained samples was lower than the standard deviation of the strained sample (1700 psi compared with 2100 psi) but this trend was not
per
cw11
nine’
of rupture.
areas to cover a volume ratio between specimens of 13 to 1. In a number of cases broken halves of specimens tested by Method A were retested by Method B. The results are shown in figs. 5 and 6; the open points are original specimens and the closed points are pre-strained specimens. Within the range of volumes studied, no trend of decreasing strength with increasing volume under stress was noted. The lines representing the grand mean of all samples and the 95 per cent confidence limit on the means of VI.
FABRICATIONANDMECHANICALPROPERTIES
K. VEEVERS,
400
J.
F. WHATHAM
AND
W.
J. WRIGHT
TARLE 3 The effect of pre-straining Specimens
Cross-
Volume
section
G.L.
in s
in s
in
tested by Method
7
on modulus
Unstrained
Modulus of Rupture ur psi
of rupture
B. Span 1.6 inch, gauge length 0.76 inch
- Strained
-.
Standard Deviation S, psi
-
Modulus of Rupture a$ psi
Standard Deviation S, psi
m2
%-Us psi
-
0.0116 0.0116
23 800 24 160
1960 1840
18.9 19
26 300 23 600
1000 2400
26.2 10
- 1600 + 660
0.0212
20 600
1900
16
21 100
2600
12.26
- 600
0.0468
24 600
1600
20
22 600
2600
9.1
+ 1900
0.0470
22 800
1660
21
22 700
1900
16
+ 100
0.084 0.094
22 800 22 200
1360 1300
19.7 22
22 700 21000
2600 2440
18.6 11.6
+100 + 1200
I1
ten individual tests are shown in fig. 5. The Weibull constant m, calculated from eq. (1) on the basis of ten individual samples, ranged from 10 to 35 with a mean value of 16; the effect of volume predicted from the theory for a value of m = 16 is also shown in fig. 5. This prediction is drawn through the position of mean strength and mean volume of all samples and extrapolated to the extremes of volume using eq. (2). From these data the variations in strength predicted from the Weibull theory are within the 95 per cent confidence limits of the means for a normal distribution of tests.
-
-
5. Discussion The standard deviation within batches of ten samples tested under identical conditions in this work was typically 1900 psi (fig. 6); thus the standard error in the means of ten samples should be 1900/2/10or 600 psi. This is the best estimate available of the reproducibility of the test method and, admitting no other variations, the 95 per cent confidence limit on the line of the grand mean in fig. 5 should be &1200 psi. On this basis a volume change of about 3Q :1 would be the minimum which would result in a significant change in the
TABEL 4 The effect of varying Span & Gauge Length in Effect of Varying Span
span and cross-section
Cross-section in s
shape on modulus
Volume in G.L. in a
Modulus of Rupture psi
of rupture Standard Deviation psi
Weibull Constant m
1.60x0.76
0.26 x 0.26
0.0469
26 700
33.4
2.60x 0.76
0.26 x 0.26
0.0469
26 600
8.7
Effect of Varying Gauge Length
2.6 x 0.76
0.26 x 0.26
0.0469
23 200
7.1
2.6 x 1.26
0.26 x 0.26
0.0781
23 600
2 300
7.1
Effect of Varying Shape
1.60 x 0.76 1.60 x 0.76 1.60x0.76
1.188 DIA 0.26 x 0.26 0.20 x 0.60
0.0208 0.0469 0.0760
21 600 21 100 20 600
2 200 1300 2 900
11.7 11.9 7.9
MECHANICAL
TESTING
the residual error in the tests is so large that the relatively small effects of volume as predicted by the theory are assessed to be insignificant in the statistical tests.
measured strength as judged by standard statistical techniques for material characterised by a Weibull constant of 16. When all results are admitted as in fig. 5, the 30
000
-
ZBOOO
1
I
I
401
OF BERYLLIA
’
A
Mdod
0
Method
B
l
Method
B
KEY’
A
(Previously
-
Method 0
2.5
” Span,
~95% _________________-__~~~~~~~~~~Contidence limit of means
_______--__
of
broken
0.75”Gaug.
IO
In
A) Iwyth
9pecimenr
l ; 0
l 26000
Estlmattd --_.-A-.-.-.-
.-A-.
Z-I-
error
in
testin9
A
.-.-.- A
ii Grand
2
mea”
A
I!? v.-. 22 000
%
+.--.-;
-
ln
a
3
---__
0 Q__‘____‘_‘___Q_______________________
A 20
000
95%
Confidence
Ilmlt
0
0 of
maanl
rprcimenr
I
I
I
1
I
I
I
4
6
B
IO
I2
I4
I6
OF
SPECIMEN
GAUGE
LENGTH
between
IN
measured
95 per cent confidence limits on the grand mean are broadened to f2900 psi; if no correla-
in3
I5
x 10-2
of rupture and volume.
The homogeneity of blocks was assessed before testing as noted earlier but it may now I
I
l
modulus
I
-
I
A
Wthod ‘F
0
Method
B
0
Method
B
’ (Rniowly Method
.’
0
2.5”
devlotlon
rlthln
Span.
0.75”
brdwn
in
A) Gaul
length
A
i 3000
A
A
E l 5
0 2000
0
’
-__8--foL:
1000
-
0
AZ Typical
0 ! z
IO
,
Fig. 6. Relationships
x
01
2 VOLUME
4000
01 m=16-
-
IB 000
5000
Curve
A
.
8
At@
z
2
4
Fig. 6. Relationships
batches
A
Q
A
AA
6
VOLUME
A
.tand%d
IN
a GAUGE
0 LENGTH
I2
between measured standard deviation
tion exists with volume as shown, this increased error must arise from variations within or between blocks of material. Thus in fig. 5
I4
I6
IB
f”3 x lO-2
and volume.
be apparent that the extent of these preliminary tests was insufficient to detect the inhomogeneities which may be present both within and VI.
FABRICATION
AND MECHANICAL
PROPERTIES
402
K.
VEEVERS,
J.
F.
WHATHAM
between blocks. Surveys of the properties within a block between the extremes of edge and centre have shown no significant variations, although, again, more extensive tests may be required. The estimate of the standard error of the mean of 600 psi quoted above may in fact be high owing to material variations within each sample; the general scatter of standard deviations shown in fig. 6 suggests that this value could lie between the limits of 350 and 850 psi. The values of the Weibull constant may similarly be in error. The estimates from eq. (1) include all residual errors in the test method and errors arising from material variations; the calculated value of m thus tends to be low compared with the true value and the effects of volume are over-estimated. The relatively high value of the Weibull constant for beryllium oxide derived from eq. (1) for small groups of ten specimens, suggests that the “flaws” in this material are all of similar character and failure occurs at a similar stress for all flaws present. Further, these characteristic flaws must be widely distributed within the material or, conversely, flaws which could cause failure at much lower stresses are relatively infrequent. In the broadest sense, at least two defects are widely distributed in the material and could act as flaws to cause failure at a well defined stress level. The beryllia powder used in this work contained about 1 volume per cent of acicular particles and these were still present after hot pressing, as shown in fig. 3. The misfit of these particles relative to the matrix is considered to be a possible source of crack nucleation. Although no significant effect of surface preparation was observed in these tests the presence of small surface flaws arising from preparation of the bend test pieces almost certainly contributed to failure of the specimens as discussed previously. Further experiments are required to determine the conditions for crack nucleation and propagation in these materials and to define the character and role of Weibull - type flaws in the failure of ceramics.
AND
W.
J.
WRIGHT
In terms of reactor design, the importance of the effects of volume under stress is dependent upon the true reproducibility of the material involved. If the scatter in strength between various batches is wide, the corrections to allow for this variation in design calculations will be more important than the effects of volume; with the best reproducibility of material, the effects of volume on strength will probably become significant for volume changes of 3 to 4. In either case, with values of the Weibull constant greater than 10 to 12, the errors in assuming a normal distribution of values appear to be small and acceptable in design calculations. 6. Conclusions 1. The modulus of rupture of beryllium oxide
was found to be insensitive to changes in volume of the test piece, pre-straining loading geometry, or surface finish, over the range studied. 2. The variations in strength of beryllia with volume under stress were examined and compared with those predicted from the Weibull theory. Significant variations in strength (as measured at the 95 per cent confidence level in a normal distribution) were predicted for volume changes of 3 to 4 based on the testing of reproducible material; variations in material within and between batches are shown to mask any effects of volume. 3. The presence of acicular beryllia grains or surface imperfections arising from specimen preparation may contribute to the failure of beryllium oxide. 4. The modulus of rupture was found to increase with strain rate in excess of 2.4 per cent per minute. 7. Notation I’, = volume of specimen in the gauge Iength. = extreme fibre stress at the section in 0, pure bending. a characteristic strength for the material. 00 =
MECHANICAL
TESTING
P, = the probability of failure occurring in volume V, at a stress uB. m = the Weibull material constant describing the scatter of modulus of repture results. M = the maximum bending moment. y = the distance of extreme fibres from the neutral axis. I = themoment of inertia of the section about the neutral axis. S = standard deviation. References 1) A. A. Griffith,
Phenomenon of Rupture and Flow in Solids. Transactions Royal Society. London A221 (1921) 163-198
‘) 8) ‘1 9 9
‘)
403
OF BERYLLIA
W. Weibull, The Phenomenon of Rupture in Solids. Proceedings, Royal Swedish Inst. Engineering Research No. 163 (1939) S. A. Bortz, The Effect of Structural Size, The Zero Strength. Task 1. Technical Document Report No. ASD-TR-61-628 (1962) M. J. Bannister, Sinterability Studies on Various Be0 Powders. This Conference, p. 303 J. Washburn, Mechanical Behaviour of Materials at Elevated Temperatures. Edited by J. E. Dorn (McGrawHill Co. Inc., 1961) R. J. Charles and R. R. Shaw, Delayed Failure of Polycrystalline and Single Crystal Alumina. General Electric Research Laboratory, Report No. 62-RL-3081M (1962) S. C. Camiglia, Private Communication. Atomics International, California (1962)
VI.
FABRICATION
AND MECHANICAL
PROPERTIES
OF NUCLEAR
MATERIALS
14 (1964) 39ti-463@ NORTH-HOLLAND
A STUDY OF VARIABLES
IN THE MECHANICAL
K. VEEVERS, Materials
Division,
J. F. WHATHAM
AA EC
Research
PUBLISHING
TESTING
CO., AMSTERDAM
OF BERYLLIA
and W. J. WRIGHT
Establishment,
Lucas
Heights,
Australia
Hot-pressed beryllia of 2.9 to 3.0 g/cm* density and 16-20 microns grain size was studied to determine the effects of specimen size, shape, surface finish, strain rate; prior mechanical treatment, and loading geometry on the modulus of rapture measured in four point bending. The modulus of rupture was found to be insensitive to variations in volume of the test piece, prestraining, and loading geometry over the range studied; when considered as a normally distributed sample, the variations in modulus of rupture at the extremes of volume were not significant at the 96 per cent confidence level. The standard deviation of results for a normal population .was about 1900 psi. The results were analysed according to the Weibull
“weakest link” theory. Values of the Weibull “constant” varied from 10 to 36 with a mean of 16; the variations in modulus of rupture with volume under stress, predicted from this model were therefore small but probably significant at the 96 per cent confidence level for a normal distribution. Specimens prepared with various surface finishes (20 micron to 160 micron abrasives) showed no significant variation in modulus of rupture. An increase in strain rate from 0.06 per cent per minute (extreme fibre strain) to 1.2 per cent per minute showed no significant increase in strength, but further increase in strain rate to 48 per cent per minute resulted in 26 per cent increase in the modulus of rupture.
1. Introduction
of polycrystalline ceramics at much lower average stresses than apply in “perfect” crystals. WeibulP ) extended the principles of the Griffith theory to explain the size-effect on stress to cause failure. He proposed the existence of a range of “flaws” in ceramics, which could cause failure at a low average stress; these could be features such as inclusions, pores, micro-cracks, and pile-ups of dislocations at impurity atoms, grain boundaries, or intersecting dislocation systems. Thus the material was assumed to contain a variety of flaw types, randomly distributed in the volume under stress, and the presence of one flaw, stressed to the “characteristic” failure stress of that flaw, could cause failure through the whole section. These flaws were recognized to be more effective in causing failure under tensile stresses than under compressive stresses. The Weibull theory suggested that the probability of failure at a given stress increases with larger volumes under tensile stress since the probability of the existence of a “critical”
Theories of brittle fracture invariably suggest that the measured strength of ceramics depends on a number of variables in the testing method. Perhaps the most important condition of interest in reactor core design is the dependence of the measured strength upon the volume of specimen under stress but other conditions such as surface finish and strain rate in the test may also be important. This paper reports the first phase of an investigation of the dependence of measured modulus of rupture upon various testing conditions. 2. Statistical Theories of Failure in Ceramics The original theory of brittle fracture, proposed by Griffith I) suggested the existence of “flaws” in brittle solids which could raise the local stress level and initiate cracks to produce failure. This theory largely explains the failure
VI.
FABRICATION
AND MECHANICAL
PROPERTIES
K.
396
VEEVERS,
J.
F.
WHATHAM
flaw in the section will be higher. The low (apparent) strength shown by ceramics in tension compared with bending is thus a consequence of the fact that the volume under tensile stress is larger in the tensile test and the probability of failure at a given stress is therefore higher. The theory also predicted that the strength measured in bending depends on the geometry (for example 3 or 4 point bending, gauge length) and the section under stress. The derivations of probability equations from the Weibull theory have been given previously (for example Bortz “). The probability of failure in bending may be written: Pg=l-exp
[
-
V, 2(m+l)
W.
J.
00
v,1 -_=Cl --me u2
v2
The constant, m, arises from the stress function which Weibull used to character&e the occurrence of various flaws with failure stress; m is thus a measure of the scatter in strength values arising from a flaw distribution characteristic of the material.
The variations predicted by the Weibull theory for measured modulus of rupture as a function of volume under tensile stress in bending are shown in fig. 1; for high values of m, that is, highly reproducible material, the effects of volume are small, but poorly characterised material, for example concrete, may show large effects of volume. A further conclusion from the Weibull analysis is that the strength of such solids need not be normally distributed about the mean. If an extremely large section of material is under test, there is a high probability of occurrence of a flaw which will cause failure at a low average value of strength. If, however, the same section is cut into a number of smaller pieces for testing, some of these will not contain the “critical” flaw and will therefore require a higher stress to cause failure (that is, another type of flaw). Thus the distribution curve will tend to skew to low values with increasing volume and the scatter of results will become less. The assumption of non-normal distribution, and a varying distribution, greatly complicates the analysis of results; it is desirable to retain the advantages of calculation from a normal distribution and to examine the departures from normality suggested by the Weibull analysis. This report examines the effects, on measured modulus of rupture, of surface finish, strain rate, and volume under stress, for beryllium oxide tested at room temperature. 3. Experimental
VP
v, Fig.
WRIGHT
01 - (1) C7m
(see section 7, Notation) and for the same probability of failure, the effects of volume on stress may be written:
01
AND
1. The effects of volume on measured strength predicted by the Weibull theory.
as
Methods
All measurements of modulus of rupture were made with specimens in four point bending. Thirteen 4~ 4~ 2 inch blocks of hotpressed Brush UOX grade beryllia of 2.9 to 3.0 g/cm3 density and 16-20 microns grain size were used in these experiments. The uniformity of strength between blocks was examined on +x ) inch (6.3 mm x 6.3 mm) test pieces; no significant differences between blocks were evident and results from all blocks were therefore grouped. The significance of grouping these
MECHANICAL
TESTING
results is discussed in section 6. Specimens were machined from these blocks; the tolerances on dimensions were &O.OOl inch (0.06 mm) on the cross-section, and the faces of the specimens were parallel to f0.0005 inch (0.02mm). Specimens were finished by grinding with wheels impregnated with diamond of 100-120 microns grit size at a peripheral wheel speed of 5000 ft min-l. Comers along the length of the specimens were rounded to 0.015 inch radius. Before testing, all specimens were examined by X-radiography using both radio-opaque and fluorescent-penetrant techniques; specimens containing any obvious defects were excluded
Fig.
2. Typical
modulus
of rupture
397
at the grain boundaries and in association with the needles. Most of the testing was done on a Hounsfield Tensometer at a strain rate of 0.0626 inch (1.50 mm) per minute. The effects of strain rate were investigated on a 60 000 pound Universal Tensile Machine at rates from 0.0025 (0.5 mm) to 2 inches (5 cm) per minute cross-head movement. For these tests a high speed recorder was necessary to measure the fracture loads because the machine recording system did not respond fast enough at the high strain rates. Specimens were tested with the following loading geometries:
specimens
from the testing programme. Typical specimens are shown in fig. 2; the grey colour is thought to be due to carbon picked up from the hotpressing die and is mainly associated with specimens cut from the outside areas of the pressings. The strength was apparently unaffected by these variations. The microstructure of the material (fig. 3) showed equiaxed grains of 15-20 microns diameter with a random distribution throughout of needles of beryllia up to 160 or 200 microns long. The needles were present in the as-received powder to the extent of approximately 1 vol Oh). The porosity was mainly found
OF BERYLLIA
showing
surface
and type of failure.
Method A. Span 2.5 inch (6.25 cm). Gauge Length 1.25 inch (3.1 cm). Method B. Span 1.5 inch (3 cm). Gauge Length 0.75 inch (1.8 cm). At least ten specimens were tested under each condition; the modulus of rupture was calculated using the elastic relationship: bg=-. M.Y I The accuracy of the stress measurements was about f2.5 per cent on the Hounsfield Tensometer or about &2 per cent on the Universal Tensile Machine. VI.
FABRICATION
AND MECHANICAL
PROPERTIES
398
K.
VEEVERS, J. F. WHATHAM AND W. J. WRIGHT
4. Results 4.1. EFFECTS OF SURFACEFINISH
Specimens ix 4 inch (6.3 mm x 6.3 mm) were finish ground with wheels of various diamond grit sizes as shown in table 1. The grinding schedule was such as to remove a total depth of 0.040 inch (1 mm) from the surface, initially with cuts of 0.005 inch (0.15 mm) and, for the last 0.010 inch (0.27 mm) with cuts of 0.002 inch (0.07 mm). One series was given a further polish in a slurry of 20 micron alumina on a vibratory polishing machine for three weeks. The coarseground specimen had deep
Each series of ten specimens was tested in four-point bending by Method A at a strain rate of 0.0625 inch (1.50 mm) per minute cross-head movement and the results are shown in table 1. No consistent effect of surface finish was found, and, in particular, the series with the relatively fine finish (20 microns) showed no improvement over those with a coarse finish (150 microns). With the relatively fine finish obtained by polishing, there still remained a large number of surface scratches up to 20 microns deep. In brittle ceramics a flaw 2 microns in depth is known “) to reduce the strength markedly and a surface preparation more extensive than given here is almost certainly required to show a significant effect of surface finish on modulus of rupture. The range of finishes studied was, therefore, probably insufficient to affect the modulus of rupture but was sufficient to suggest that no practical mechanical method of surface preparation was available whereby significant improvement in strength could be achieved. 4.2. EFFECTS OF STRAIN RATE
Fig. 3. Photomicrograph 10 per cent.
of hot pressed beryllia, etched in NH*F/HF solution. x200.
parallel scratches along its length while the polished specimen appeared to be free from any oriented scratches. TABLE 1 The effect of surface
finish on modulus of rupture
(Specimens ax & inch tested by Method A. Span 2.6 inch gauge length 1.26 inch) Surface Finish Microns
Modulus of Rupture a psi
Standard Deviation S psi
Specimens ix * inch (6.3 mmx6.3 mm) were tested by Method A on the Universal Tensile Machine at the strain rates shown in table 2. The effect of increasing the strain rate over the range 0.06-48 per cent per minute (fig. 4) was to increase the modulus of rupture by about 6000 psi or 25 per cent. From 0.06 to 1.2 per cent per minute, a small increase in modulus of rupture was noted which was not significant at the 95 per cent confidence limit; as the strain rate was increased from 1.2 to 24 per cent per minute, the modulus of rupture increased by about 15 per cent. A further increase of 5 per cent was noted as the strain rate was increased to 48 per cent per minute. The increase in modulus of rupture with increase in strain rate could be due to a stress corrosion mechanism as described by Charles and Shaw “) and by Carniglia ‘), 4.3. EFFECTS OF PRE-STRAINING Specimens of various sizes were tested in
MECHANICAL TABLE The
effect
(Specimens
2
of strain rate on modulus
0.06 0.06 0.06 0.241 1.206 1.206 24.10 48.20
of rupture
f x 2 inch tested by Method A. Span 2.6 inch gauge length 1.25 inch)
-
Specimen Strain Rate % min-i
TESTING
Modulus of Rupture u psi
Standard Deviation S psi
m
24 600 23 660 23 300 24 300 24 800 24 600 21600 29 100
18.3 11.3 20 18.3 18.3 19.1 16.2 21.1
1670 2400 1600 1600 1760 1800 2600 1460
bending by Method A and the broken halves of these specimens were re-tested by Method B. A similar number of specimens of each size, not previously broken, were tested by Method B. A comparison of results was made of the modu-
20000 Id’
e
significant for the number of tests involved. Two possible effects of pre-straining, (i) strain hardening, and (ii) generation of flaws under stress, were considered. The extent of local plastic deformation of beryllia before failure is less than 0.1 per cent and thus the extent of strain hardening was probably negligible. There was no evidence from these mechanical tests, or from the microstructures, of further weakening due to flaw generation. 4.4. EFFECTS
OF VOLUME
effects of volume were investigated initially by varying the loading geometry (span and gauge length) and the cross-section shape. Results (table 4) showed no significant effect of these parameters on modulus of rupture, and hence no effect of volume within the range considered. Further tests were made on specimens with a wider range of cross-section The
1
I
I
Id’
I
IO
SPECIMEN
Fig. 4. Effect
399
OF BERYLLIA
STRAIN
RATE
of strain rate on modulus
lus of rupture determined on the same testing geometry with and without pre-straining; the results are shown in table 3. Five of the seven pairs tested showed a small decrease in modulus of rupture on the prestrained samples but these variations were not significant when considered as a normal distribution at the 95 per cent confidence level. Similarly the standard deviation of the unstrained samples was lower than the standard deviation of the strained sample (1700 psi compared with 2100 psi) but this trend was not
per
cw11
nine’
of rupture.
areas to cover a volume ratio between specimens of 13 to 1. In a number of cases broken halves of specimens tested by Method A were retested by Method B. The results are shown in figs. 5 and 6; the open points are original specimens and the closed points are pre-strained specimens. Within the range of volumes studied, no trend of decreasing strength with increasing volume under stress was noted. The lines representing the grand mean of all samples and the 95 per cent confidence limit on the means of VI.
FABRICATIONANDMECHANICALPROPERTIES
K. VEEVERS,
400
J.
F. WHATHAM
AND
W.
J. WRIGHT
TARLE 3 The effect of pre-straining Specimens
Cross-
Volume
section
G.L.
in s
in s
in
tested by Method
7
on modulus
Unstrained
Modulus of Rupture ur psi
of rupture
B. Span 1.6 inch, gauge length 0.76 inch
- Strained
-.
Standard Deviation S, psi
-
Modulus of Rupture a$ psi
Standard Deviation S, psi
m2
%-Us psi
-
0.0116 0.0116
23 800 24 160
1960 1840
18.9 19
26 300 23 600
1000 2400
26.2 10
- 1600 + 660
0.0212
20 600
1900
16
21 100
2600
12.26
- 600
0.0468
24 600
1600
20
22 600
2600
9.1
+ 1900
0.0470
22 800
1660
21
22 700
1900
16
+ 100
0.084 0.094
22 800 22 200
1360 1300
19.7 22
22 700 21000
2600 2440
18.6 11.6
+100 + 1200
I1
ten individual tests are shown in fig. 5. The Weibull constant m, calculated from eq. (1) on the basis of ten individual samples, ranged from 10 to 35 with a mean value of 16; the effect of volume predicted from the theory for a value of m = 16 is also shown in fig. 5. This prediction is drawn through the position of mean strength and mean volume of all samples and extrapolated to the extremes of volume using eq. (2). From these data the variations in strength predicted from the Weibull theory are within the 95 per cent confidence limits of the means for a normal distribution of tests.
-
-
5. Discussion The standard deviation within batches of ten samples tested under identical conditions in this work was typically 1900 psi (fig. 6); thus the standard error in the means of ten samples should be 1900/2/10or 600 psi. This is the best estimate available of the reproducibility of the test method and, admitting no other variations, the 95 per cent confidence limit on the line of the grand mean in fig. 5 should be &1200 psi. On this basis a volume change of about 3Q :1 would be the minimum which would result in a significant change in the
TABEL 4 The effect of varying Span & Gauge Length in Effect of Varying Span
span and cross-section
Cross-section in s
shape on modulus
Volume in G.L. in a
Modulus of Rupture psi
of rupture Standard Deviation psi
Weibull Constant m
1.60x0.76
0.26 x 0.26
0.0469
26 700
33.4
2.60x 0.76
0.26 x 0.26
0.0469
26 600
8.7
Effect of Varying Gauge Length
2.6 x 0.76
0.26 x 0.26
0.0469
23 200
7.1
2.6 x 1.26
0.26 x 0.26
0.0781
23 600
2 300
7.1
Effect of Varying Shape
1.60 x 0.76 1.60 x 0.76 1.60x0.76
1.188 DIA 0.26 x 0.26 0.20 x 0.60
0.0208 0.0469 0.0760
21 600 21 100 20 600
2 200 1300 2 900
11.7 11.9 7.9
MECHANICAL
TESTING
the residual error in the tests is so large that the relatively small effects of volume as predicted by the theory are assessed to be insignificant in the statistical tests.
measured strength as judged by standard statistical techniques for material characterised by a Weibull constant of 16. When all results are admitted as in fig. 5, the 30
000
-
ZBOOO
1
I
I
401
OF BERYLLIA
’
A
Mdod
0
Method
B
l
Method
B
KEY’
A
(Previously
-
Method 0
2.5
” Span,
~95% _________________-__~~~~~~~~~~Contidence limit of means
_______--__
of
broken
0.75”Gaug.
IO
In
A) Iwyth
9pecimenr
l ; 0
l 26000
Estlmattd --_.-A-.-.-.-
.-A-.
Z-I-
error
in
testin9
A
.-.-.- A
ii Grand
2
mea”
A
I!? v.-. 22 000
%
+.--.-;
-
ln
a
3
---__
0 Q__‘____‘_‘___Q_______________________
A 20
000
95%
Confidence
Ilmlt
0
0 of
maanl
rprcimenr
I
I
I
1
I
I
I
4
6
B
IO
I2
I4
I6
OF
SPECIMEN
GAUGE
LENGTH
between
IN
measured
95 per cent confidence limits on the grand mean are broadened to f2900 psi; if no correla-
in3
I5
x 10-2
of rupture and volume.
The homogeneity of blocks was assessed before testing as noted earlier but it may now I
I
l
modulus
I
-
I
A
Wthod ‘F
0
Method
B
0
Method
B
’ (Rniowly Method
.’
0
2.5”
devlotlon
rlthln
Span.
0.75”
brdwn
in
A) Gaul
length
A
i 3000
A
A
E l 5
0 2000
0
’
-__8--foL:
1000
-
0
AZ Typical
0 ! z
IO
,
Fig. 6. Relationships
x
01
2 VOLUME
4000
01 m=16-
-
IB 000
5000
Curve
A
.
8
At@
z
2
4
Fig. 6. Relationships
batches
A
Q
A
AA
6
VOLUME
A
.tand%d
IN
a GAUGE
0 LENGTH
I2
between measured standard deviation
tion exists with volume as shown, this increased error must arise from variations within or between blocks of material. Thus in fig. 5
I4
I6
IB
f”3 x lO-2
and volume.
be apparent that the extent of these preliminary tests was insufficient to detect the inhomogeneities which may be present both within and VI.
FABRICATION
AND MECHANICAL
PROPERTIES
402
K.
VEEVERS,
J.
F.
WHATHAM
between blocks. Surveys of the properties within a block between the extremes of edge and centre have shown no significant variations, although, again, more extensive tests may be required. The estimate of the standard error of the mean of 600 psi quoted above may in fact be high owing to material variations within each sample; the general scatter of standard deviations shown in fig. 6 suggests that this value could lie between the limits of 350 and 850 psi. The values of the Weibull constant may similarly be in error. The estimates from eq. (1) include all residual errors in the test method and errors arising from material variations; the calculated value of m thus tends to be low compared with the true value and the effects of volume are over-estimated. The relatively high value of the Weibull constant for beryllium oxide derived from eq. (1) for small groups of ten specimens, suggests that the “flaws” in this material are all of similar character and failure occurs at a similar stress for all flaws present. Further, these characteristic flaws must be widely distributed within the material or, conversely, flaws which could cause failure at much lower stresses are relatively infrequent. In the broadest sense, at least two defects are widely distributed in the material and could act as flaws to cause failure at a well defined stress level. The beryllia powder used in this work contained about 1 volume per cent of acicular particles and these were still present after hot pressing, as shown in fig. 3. The misfit of these particles relative to the matrix is considered to be a possible source of crack nucleation. Although no significant effect of surface preparation was observed in these tests the presence of small surface flaws arising from preparation of the bend test pieces almost certainly contributed to failure of the specimens as discussed previously. Further experiments are required to determine the conditions for crack nucleation and propagation in these materials and to define the character and role of Weibull - type flaws in the failure of ceramics.
AND
W.
J.
WRIGHT
In terms of reactor design, the importance of the effects of volume under stress is dependent upon the true reproducibility of the material involved. If the scatter in strength between various batches is wide, the corrections to allow for this variation in design calculations will be more important than the effects of volume; with the best reproducibility of material, the effects of volume on strength will probably become significant for volume changes of 3 to 4. In either case, with values of the Weibull constant greater than 10 to 12, the errors in assuming a normal distribution of values appear to be small and acceptable in design calculations. 6. Conclusions 1. The modulus of rupture of beryllium oxide
was found to be insensitive to changes in volume of the test piece, pre-straining loading geometry, or surface finish, over the range studied. 2. The variations in strength of beryllia with volume under stress were examined and compared with those predicted from the Weibull theory. Significant variations in strength (as measured at the 95 per cent confidence level in a normal distribution) were predicted for volume changes of 3 to 4 based on the testing of reproducible material; variations in material within and between batches are shown to mask any effects of volume. 3. The presence of acicular beryllia grains or surface imperfections arising from specimen preparation may contribute to the failure of beryllium oxide. 4. The modulus of rupture was found to increase with strain rate in excess of 2.4 per cent per minute. 7. Notation I’, = volume of specimen in the gauge Iength. = extreme fibre stress at the section in 0, pure bending. a characteristic strength for the material. 00 =
MECHANICAL
TESTING
P, = the probability of failure occurring in volume V, at a stress uB. m = the Weibull material constant describing the scatter of modulus of repture results. M = the maximum bending moment. y = the distance of extreme fibres from the neutral axis. I = themoment of inertia of the section about the neutral axis. S = standard deviation. References 1) A. A. Griffith,
Phenomenon of Rupture and Flow in Solids. Transactions Royal Society. London A221 (1921) 163-198
‘) 8) ‘1 9 9
‘)
403
OF BERYLLIA
W. Weibull, The Phenomenon of Rupture in Solids. Proceedings, Royal Swedish Inst. Engineering Research No. 163 (1939) S. A. Bortz, The Effect of Structural Size, The Zero Strength. Task 1. Technical Document Report No. ASD-TR-61-628 (1962) M. J. Bannister, Sinterability Studies on Various Be0 Powders. This Conference, p. 303 J. Washburn, Mechanical Behaviour of Materials at Elevated Temperatures. Edited by J. E. Dorn (McGrawHill Co. Inc., 1961) R. J. Charles and R. R. Shaw, Delayed Failure of Polycrystalline and Single Crystal Alumina. General Electric Research Laboratory, Report No. 62-RL-3081M (1962) S. C. Camiglia, Private Communication. Atomics International, California (1962)
VI.
FABRICATION
AND MECHANICAL
PROPERTIES