Strength of Materials, Vol. 26, No. 10, 1994

E F F E C T S OF S P E C I M E N S I Z E ON L I M I T I N G C O M P R E S S I V E

L O A D I N G F O R S I L I C A T E , C E R A M I C , AND O T H E R M A T E R I A L S

G. M. Okhrimenko UDC 539.4.019:620.25:666.1

Published data are examined on the ultimate strength in uniaxial compression for glass, glass ceramics,

porcelain, crystalline silicon, perictase-spinel-chromite material PSCM, and ferrite in relation to she

specimen dimensions. Two methods are proposed for combined experimental and computational estimation of

the effects from the volume on the limiting load, which are based only on the data obtained from testing

specimens with one or two standard dimensions.

The dimensions of the sections determine the correctness and reliability of results obtained under conditions of

uniaxial compression on composite or monolithic constructional elements made of silicate, ceramic, and other materials. This

is indicated by the dependence of the ultimate strength in uniaxial compression on the specimen volume for technical grades of glass, glass ceramics, porcelain, crystalline silicon, PSCM, and ferrite [1-6]. Such evidence has been obtained at the

Strength Problems Institute, Ukrainian National Academy of Sciences, under the direction of Academician G. S. Pisarenko on the basis of a correct experimental method [1], which is distinguished by the substantial effort involved in making and

preparing several standard sizes of cylindrical, prismatic, or rectangular specimens for test. In that connection, for such cases we have devised a general method that enables one to determine the states of new materials or to develop the optimum

technology for producing high-strength sections from a material with special properties, e.g., with a preset chemical composi-

tion. On this basis, I have examined the published data on the dependence of the ultimate strength in uniaxial compression

% on the volume V of the specimen with the object of defining trends in the effects of the dimensions on the strength and in

order to obtain a rational method involving less consumption of time and material in order to derive such relationships. It has been shown [4, 5] that there is a functional relationship between a c and V for cylindrical specimens of the glass

ceramics A-I, SO115-M, and STL-10, as well as electrotechnical porcelain and single-crystal silicon, which can be represent-

ed to a first approximation in logarithmic coordinates by the equation for a straight line:

in ,r c - aj lnV + bj, (1)

in which aj and bj are the slope and the intercept on the ordinate.

Figure 1 shows analogous results for prismatic and rectangular specimens of A-1 glass ceramic and PSCM [1.5]. A similar dependence of % on V has been found for ferrite [3], so one can say that no matter what the shape of the cross

section in the specimen, the test results for such materials can be described by (1) in logarithmic coordinates. We calculated aj and bj by least-squares fitting [7], and they are given in Table 1, where the symbols are as follows:

j number of standard sizes of specimen, ~cj and Vj ultimate strength in uniaxial compression and volume of specimens with

maximal dimensions, and m w the Weibull modulus, which is related to the sample coefficient of variation v by [8]

= 0 ! V ~ + m2 '

in which the correction 0(1/mw 2) is a quantity of the order 1/mw2.

If we neglect the second term in (2), we get the Weibull modulus for each material (Table 1) from

(2)

Strength Problems Institute, Ukrainian National Academy of Sciences, Kiev. Translated from Problemy Prochnosti,

No. 10, pp. 23-29, October, 1994. Original article submitted May 4, 1993.

0039-2316/94/2610-0729512.50 �9 Plenum Publishing Corporation 729

TABLE 1. Effects of Specimen Volume on the Limiting Loading in Uniaxial Corn-

pression for Nonmetallic Materials

Material Dimensions of minimal specimen, mm

STL-10 glass d - 10, ceramic [1] h - 30

A-1 glass ceramic [1]

[ SOII5M gIass i ceramic [4] .

Crystal- line sili- con [31

d - 1 0 , h - 30

lOx 10x 30

d-- 10, h - 30

d - 10; h - 30 d - l O , h - 30

Ferrite [3] 6x 6 x 18

PSCM [51 lOx 10x 36

Electrotech- d - 6, nical porcel- h - I8 ain [2]

/ [acm=,, o'cm,x V~

J MPa a 4 V,,i .

5 2680 2.17 511

5 2190 1.65 511

5 2680 2, I7 5 t2

5 2558 2,28 51 l

5 2145 2.61 64

3 2145 1,39 1

4 i135 1.21 8,33

5 11 t 2,41 64

5 [249 1,30 4044

94,1"

bj a/ 1 mw 6m,

t %

8,12 -0 ,111 1'7.04 47.1

7.68 - 0.074 t7.30 20.7

7,86 - 0.121 11.90 30.6

8.04 - 0.143 14.91 53.1

7,64 - 0,177 12,08 53,2

6,29 + 0.314 13.66 76.7

6,92 ] - 0.037 13.84 48.8

4.83 I - 0.212 6.79 30.5

7.08 I - 0.021 80.44 40.g

*Ratio between the volume of the maximal specimen and a specimen for which one

obtains cr c max, the maximal limiting load in compression.

t,,%

7.7,

t.O 2,6 4.,2 5,~ [nv a

4.,2

5.7 t.O Z.I 3,Z 4.3 In v

b

Fig, 1. Dependence of the limiting load in

uniaxial compression a c on the specimen

volume V for prismatic specimens of A-I

glass ceramic (a) and periclase- spinel-chromite material (b).

m w - ~ - ~ - + - - + . . . + i,, 2

in which v 1, v 2 . . . . . vj are the sample coefficients of variation for the limiting strength in uniaxial compression for specimens of the standard sizes used.

Exponentiation in (1) for aj < 0 gives

ebj ac - - . ( 4 )

Va~

This expression resembles in form the equation from the statistical theory of brittle failure due to Weibutl [9, I0], which also has the form

730

TABLE 2. Mathematical Expectations of Limiting Loading in Uniaxial Compression

for Porcelain and Specimen Dimensions

Parameter and Mathematical I Coefficient of Number of tests dimensions expectation variation. % of measurements

12 Limiting loading in uni- axial compression, MPa

1160

1188"

2,39

53, 66* * Specimen diam., mm 7.50 0,68 12 Specimen height, mm 22,61 0,38 12 Specimen volume, cm 3 0.999 0,71 12

*Limiting loading in uniaxial compression for electrotechnical porcelain in accor-

dance with (4) and the Table 1 data for V = 1 cm 3.

**Weibull modulus calculated from (3) with j = 1.

e ~ Crc- vl/m w, (5)

in which ebJ is the limiting strength in uniaxial compression for a specimen with unit volume (in our case 1 cm3).

The last column in Table 1 gives the relative difference between m w and that calculated from the absolute value of

the coefficient aj as given by the formula

1 I ] ~m = r a W - a]- 1 raw x100% = 1-(ai'rav) - t • (6)

Table 1 shows that 6 m is 20-60% for materials having aj < 0. This is not obeyed by the test results for three batches

of cylindrical specimens (diameter 10 mm) made of crystalline silicon as prepared from three billets having diameters of

65.0, 80.5, and 106.5 mm. In this case, aj > 0, and c5 m exceeds 75%, and the experimental data can be fitted to

~rr - *'%6i, (7)

in which F is the area in cm 2 of cross section of the crystalline silicon billet. We can thus say that the relative difference between aj-1 as calculated from experimental data obtained on testing

specimens with several standard dimensions (3 < j _< 5) and the Weibull modulus given by (3) is in the range 20-60%.

It should be noted that I am not aware of any papers in which it has been shown by experiment that the numerator in

(4) or (5) is the limiting strength of a material in uniaxial compression for specimens having volume V = 1 cm 3. In that

connection, the following experiment was done. Specimens having volume 1 cm 3 were cut from cylindrical rods of electrotechnical porcelain 11 mm in diameter,

whose composition and production technology did not differ from the analogous characteristics of the rods used in [2] to

make specimens of all the sizes employed. The mechanical working was done with a diamond toot and water cooling in such a way that the microroughness

heights were R a _< 1.25 and R a N 0.63 ~.m respectively on the cylindrical and end surfaces. The preparation parameters

before testing, namely drying and heat treatment, and the loading rates in testing were analogous to those described in [2].

The data in Table 2 and the characteristics of the specimens indicate that the relative differences between the experimental

and theoretical values for the limiting permissible loading in uniaxial compression are less than 5 %, while the differences

between m w = 53.7 from (3) and aj - t (Table l) are slightly more than 10%. Weibull's statistical theory of brittle strength is represented mathematically by (3) and applies for brittle materials in

which the start of failure under force is in accordance with the postulate of weak links. As that postulate applies for the

materials used here [1, 2, 4, 6], and if we use the Table 2 data, one can say that the numerator in (4) is the limiting loading

in uniaxial compression for specimens with volume V = 1 cm 3, while aj and the Weibull modulus given by (3) are related by

4" ! = mw (8)

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TABLE 3. Parameters in the Straight-Line Equation with the Slope

(9)

ar In at/ i InVI InVj Material In

A-1 glass ceram- 7,69 7,19 0,857 7,095 ic [1] (c,ylinders)

A-1 glass ceramic [1] (prisms) ~, 7,67 6,94 1,099 7,336 STL-10 glass ceramic [l] 7,89 7,12 0,857 7,095 SOO115M glass ceramic [4] 7,85 7,02 0,857 7,095

Crystalline 7,65 6,71 0,857 5,016 silicon [3] . PSCM [5] 43 t 3,83 1,099 5,257 Ferrite [3]_ 7,03 6,85 - 0,0973 1,686 Electrotechnic- 7,06 6,87 - 0,675 7.480 al porcelain [2]

S-37 glass [6] 7,46 7,06 0,857 2,936

Calculated from

bj~, ~b, % a/p ha, %

7,76 1,03 - 0.08 7.50

7,79 -8,99 -0.117 -3,42 7,03 - t5.5 -0,109 1,8 7,96 -1.0 -0,133 -7,52 7,86 2,79 - 0,231 23,38

4,94 2,23 - 0,212 0 6.93 0.14 - 0.068 45.58 7,04 5,35 - 0,023 8,69

7,62 - - - 0,192 - -

1 1 . 3 4 *

*Weibull modulus calculated from (3) for j = 2 [6].

Then to reduce the effort and time in estimating the dependence of the limiting loading in uniaxial compression on

specimen volume for silicate, ceramic, and other materials, including crystalline silicon and ferrite, it is necessary to prepare

and test a set of rods having ratio of diameter to height of 3 -1 and volume V = 1 cm 3, with the Weibull modulus estimated

from (3) (with j = I), and (4) used to predict a c for an element with a given volume.

Another approximate approach is to define by experiment the linear dependence in logarithmic coordinates between

the limiting load and the volume. One needs to make and test two standard sizes of specimen having volumes V 1 and V 2 (the

ratio of height to the characteristic transverse dimension equal to three), for which the experimental limiting loads are acl and

ac2. One assumption is made: the experimental a c = f(V) in logarithmic coordinates is a straight line with the slope of (1)

and passes through the two points with those coordinates. It is clear [11] that the equation of this straight line is

In a c - In act InV - inV, (9) In ac2 - In (re, = lnVz - lnV, "

We substitute into the other relation for the quantities and use obvious transformations to get the relationship between

In a c and In V in the (1) form. We use aj and bj in conjunction with (8) and the experimental fact that ebJ = a c for specimens

with volume V = 1 cm 3 to use (5) to predict the effects of rod volumes on the limiting loading in uniaxial compression.

As an example, we calculate the slope ajp and intercept bjp of the (9) line on the ordinate (Table 3). For this purpose,

we use published data for specimens with minimal and maximal volumes. For each of the materials in Table 1, the number

of possible (9) formulas is equal to the number of combinations C of the j standard dimensions taken two at a time:

jl C / = ( j _ 2)!2! " (10)

From Table 3, the discrepancies of the quantities 8 a calculated from (9) and found by experiment (Table 1) between

the slopes aj and ajp is between - 8 and +46%, while the analogous parameter 6 b for bj and bjp is between - 1 6 and +9%.

Comparison of aj and ajp (Tables 1 and 3) shows that the largest discrepancies occur for crystalline silicon and

ferrite. For these materials, one has the least values of the ratio between Vma x and Vmi n.

To estimate the effects of specimen volume on limiting load. it is necessary to make two standard sizes of specimens,

and then plot the data in logarithmic coordinates, and draw the straight line in accordance with (9), whose parameters aj and

bj are used to calculate the limiting loading of an element in uniaxial compression from (5) and (9).

The data on the volume dependence of the limiting load show that one can use the equations from Weibul l ' s statistical

theory of brittle failure for silicate, ceramic, and other materials, including crystalline silicon and ferrite. This has been done

in two combined experimental and calculation methods of evaluating the dependence for brittle materials by the use of

experimental data for one standard size (specimen volume approximately 1 cm 3) or two standard sizes.

732

The conclusions are as follows.

1. In accordance with [4], no matter what the logarithmic coordinate system used (In a c = f(ln V; In Ss; In Se; In d),

in which S s and S e are the side and end surfaces, while d is specimen diameter), the experimental results can be approximat-

ed by straight lines. In these cases, the relative differences between the bj do not exceed a few percent. The relation between

the slopes aj and a s or a e and a d is then

a~s ae ad

which are substituted into the denominators in the other three single-term expressions to derive the exponents, which must be

supplied with a dimensional parameter that influences a c to find the dimensions of the volume.

Then in all these cases, the Weibull modulus can be derived from the following:

1 1,5 1,5 3 m w ~ - - = - - , ~ . ~ ~ - - . aj ~Z s a e a d

2. These data do not enable one to judge the effects of specimen volume on the limiting loading for these materials

under conditions of uniaxial tension and bending, or the resistance to failure in compression by flat metal supports.

3. Table 1 shows that the limiting loading in uniaxial compression for prismatic specimens made of A-1 glass ceramic

is larger than the analogous quantity, derived from specimens of cylindrical form. This occurs because the prismatic rods after

mechanical working at the plant with a carborundum tool had a microroughness height R a __ 0.63/zm, whereas for all the

other rods (Table 1) made with a diamond tool, it was R a _< 2.5 ... 1.25 ~m. Then the [1, 6, 12] data indicate that one way

of increasing the limiting loading for specimens made of technical glass ceramics for use under compression is mechanical

working of the sections with a carborundum tool.

The research on the constructional strength of the materials considered here was financed by the Ukrainian State

Budget. I am indebted to G. S. Pisarenko and Yu. M. Rodichev for assistance in this work.

REFERENCES

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

G. S. Pisarenko, Yu. M. Rodichev, G. M. Okhrimenko, et al., "Researching the scale effect in testing technical

glass ceramics in uniaxial compression," Probl. Prochnosti, No. 10, 47-53 (1977).

Yu. V. Dobrovol'skii, S. I. Likhatskii, G. M. Okhrimenko, et al., "Effects of specimen size on the strength of

electrotechnical porcelain in axial compression," ibid., No. 6, 85-89 (1983).

G. M. Okhrimenko, "Crystalline silicon, piezoceramic, and ferrite underaxial compression," ibid., No. 9, 45-50 (1989).

L. R. Raevskaya, V. L. Stepchenko, Yu. I. Moskvin, and G. M. Okhrimenko, "Estimating the effects of specimen

size for SOl 15M optical glass ceramic on the strength in axial compressing," ibid., No. 9, 55-65 (1990).

G. M. Oktu'imenko, Strength and Elasticity Characteristics of Silicate and Ceramic Materials [in Russian], Preprint,

Inst. Probl. Prochnosti AN Ukrainy, Kiev (1992).

G. S. Pisarenko, Yu. M. Rodichev, G. M. Okhrimenko, et al., "Effects of technological factors on the strengths of

glass ceramics in compression," ProN. Prochnosti, No. 1, 58-63 (1990). A. Worthing and J. Heffner, Methods for Processing Experimental Data [Russian translation], Izd. Inostr. Lit.,

Moscow (1949).

V. V. Bolotin, Forecasting the Working Lives of Machines and Structures [in Russian], Mashinostroenie, Moscow

(1984). G. S. Pisarenko, V. N. Rudenko, G. N. Tret'yachenko, et al., The Strengths of Materials at High Temperatures [in

Russian], Izd. AN Ukr. SSR, Kiev (1966).

G. S. Pisarenko and V. T. Troshchenko, Statistical Strength Theories and their Application to Metal-Ceramic

Materials [in Ukrainian], Vid. AN URSR, Kiev (1961).

I. I. Privalov, Analytical Geometry [in Russian], Fizmatgiz, Moscow (1960).

G. M. Okhrimenko, Effects of Technological Factors on the Limiting States of Constructional Elements Made of

Silicate Materials [in Russian], Preprint, Inst. Probl. Prochnosti AN Ukr., Kiev (1990).

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