Strength of Materials, Vol. 27, Nos. 11.12, 1995

S O M E F E A T U R E S O F T H E M E C H A N I C A L P R O P E R T I E S O F

R O C K S AND B U I L D I N G M A T E R I A L S B A S E D O N S I L I C A

G. M. Okhrimenko and A. A. Sidorko UDC 539.411:679.85

Experimental estimates have been made on the mechanical properties of granite and marble under conditions

of uniaxial compression and pure bending. Estimates have been made of how some design, technological,

and working factors affect the strength characteristics and elasticity. The data are compared with the strength characteristics in compression and bending for materials based on silica. Failure features are considered.

INTRODUCTION

Granite and marble are used largely in facing buildings and structures and for making memorials and sculptures. Accumulat-

ed experience shows that these materials can be used also as constructional ones, e.g., for making the frames of precision

lathes, and also rods, plates, and other items that operate under compression and bending. This makes it necessary to

determine the constructional strength of this class of material. Also, there is a need for such data in devising optimal processes for making blanks in quarries and preparing pieces of given shape and required dimensions from them.

There is only a restricted amount of published data on the physicomechanical characteristics of granites and marble

[1-5], so the objects of this paper were the experimental evaluation of how the strength and elastic characteristics vary for

two grades of granite mined in the Ukraine (the Korostishiv and Tokov deposits), and also for marble from the Koelign deposit in the Ural, as these are affected by certain design, technological, and working factors that occur in making and using items from these materials. The characteristics of the materials taken from literature sources are given in Table 1.

METHODS

The compression tests were done by a method previously developed, whose correctness has been demonstrated by polarization optical methods and by means of moir6 fringes [6-8]. We used cylindrical specimens with diameters of 10 and 20 ram, which were one-third of the lengths, and also disks of diameter 35 and 42 mm and thicknesses 11 and 14 mm respec- tively. We also used parallelepipeds 9 x 22 • 22 mm in size under compression conditions.

The disks were loaded by compressive forces along the axis. The boundary contact stresses [9] were found by dividing the boundary force P by the product of the disk diameter d and the thickness h:

P (1)

Hertz's theory indicates that a planar state of stress arises at the center of such a plate, in which the principal stresses tr I and tr 3 are related as 1 / ( -3 ) and are given [9] by

2 6 (2)

Strength Problems Institute, Ukrainian National Academy of Sciences, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 11-12, pp. 89-99, November-December, 1995. Original article submitted January 23, 1995.

702 0039-2316/95/1112-0702512.50 �9 Plenum Publishing Corporation

TABLE 1. Physicomechanical Characteristics of Granites and Marble

Characteristic

Structure

Korostishiv gray granite [5]

H'olocrys~alline '

Tokov red granite [5] Crystalline

Porosity, % 0.90 0.86 Density, g/cm j 2,68 2,63

Mineralogical composition, % Quartz - 28 Quartz-49 Plagioclase- 31 Microcline - 36 Feldspar-47 Biodte- 5 Mica - 4

Grain.size, mm 0,75 5.0 0.20

g. 10- s, MPa 0,62 0,75 0,40 Poisson's ratio 0,25 0,23 0,26 Schreiner hardness Pm , MPa

Koelginwhite marble [4]

' ' Tr;anular 0,50 2.70

Calcite - 98 Impurities- 2

2700 2600 790 Compressive strength aco, MPa 170 180 63,5

The tests under bending conditions were done by Gogotsi 's method [10-12], which was developed for ceramics and

refractories. We used specimens as rectangular rods of length 70 mm and transverse dimensions height h = 2.5 mm and

width b = 4.0 m m (red granite) or 3.5 x 4.5 mm (gray granite).

The length of the segment where a constant bending moment acted was 40 ram, while the distance between the end

supports was 60 m m (the length of one segment where a variable bending moment acted was a = 10 mm). An inductive

sensor was used in the zone of constant bending moment with a base length l t = 20 m m to record the bending, while a

special dynamometer recorded P. The displacement speed of the test-machine beam was 0.0015-0.0020 mm/sec .

This automatically generated diagrams for the force P against the deflection 6, which were nonlinear for all the

materials. Figure I shows examples of the tensile strain as a function of the tensile stresses in the zone of pure bending for

the red and gray granites. As the strain diagrams are nonlinear, Navier ' s formula [13] gives only the nominal boundary

stresses for this class of material:

~ = 3 a p t, h2 " ( 3 )

The real boundary stresses in bending erb are [14] related to the nonlinearity of the strain diagrams (Fig. I) and are

given [10-12, 15] by

o . = , , . + - (4)

in which t t is the tensile strain in the pure-bending zone,

�9 t = 4tdlt2t.

In addition to arb, the plots of P against 6 were used to derive the static elastic modulus Est from [10-12, 15]

(5)

3al~ dP �9 ~ for P ~ O. (6) Est = 4bh 3 d6

The limit to the tensile strain ett was given by (5), where 6 = 6max.

To calculate a brittleness measure, we used the Fig. 1 diagrams [10-12, 15] with

703

%

t8

t2

6

MPa

j o 5

1,e z,7 t~ */o

d.d

Fig. 1. Experimental dependence of tensile strain on tensile stress for pure

bending in specimens of red granite (1 and 2) and gray granite (3-5).

~ i n X=

lt' 2.~, f crd~

o

(7)

which satisfies the conditions

0 <z -< 1. (8)

The maximum value x = 1 occurs in materials having linear strain diagrams, e.g., technical glass or glass ceramic [16, 17].

We also measured the dynamic modulus E d in dynamic tests made with a UKIOP upgraded apparatus, on the basis

of measurements of transit time at 0.15 MHz [15].

RESULTS

Tables 2 and 3 give the results from static compression and bending of gray and red granites and white marble, in which v is the sample coefficient of variation, n o the number of specimens tested, and Vy the speed of ultrasound. All the

specimens in the form of cylinders and disks were worked with a diamond tool to give a microroughness height of 2.5-10

tzm, while those in the form of rectangular rods were worked to 0.63-1.25 ~m.

One can compare Table 2 with the characteristics in compression for silicate materials [16, 17], which shows that the

limiting stresses under compression for gray granite and white marble are from 10 to 15% of the analogous parameters for sheet glass and MKR-1 glass [16]. An important point for these materials and for glass and ceramics [16, 17] is that the

limiting stress in compression is dependent on the boundary conditions: the ratio aeo/a s did not exceed 0.8, and a s was largely dependent, as with optical materials, on the height of the microrougimess on the side surfaces [18], while for rectan-

gular specimens 9 • 22 • 22 ram, a s may exceed by a factor two the analogous characteristic for a cylindrical rod (Table 2). Tests on gray granite in liquids showed the same tendency as for silicate and ceramic materials: aqueous media

reduced the boundary stresses by about 20%, while oil media raised them by up to 10%.

704

T A B L E 2. Limiting Stresses under Conditions o f Compress ion for Gray Granite and White

Marble

Material size, mm

Gray d = 42, granite h ffi 14

d = ~ , h-- ll

d = 10, hffi30

dffi 20, hffi60

22• 22x 9

White 22x 22x 9 marbie

Specimen Character- istics, MPa

a c = 20,0

a t = + 12,7

v.%

0 3 = 3 8 . 2

no, Specimen

0.9

G' c = 9.5 9,1 6

o I = + 6.0 6,0 6

a 3 = 18,1 18 ,1 6

items batch

Oco = 123,0 5,0 7 2

Us ffi 152.0 7,0 4 2

as= 121,0" 11,3 4 2

as = 170,0"" 5,6 4 2

asffi 166,0 4,0 4 2

asffi 310,0 19,0 12 3

a s = 121,0 14 25,8

*Measured in f resh water; **in machine oil.

TABLE 3. Limiting Stresses in Bending and Elastic Characteristics of Gray and Red Gran-

ites

Characteristic v, % no, items Red granite

Orb - 18,7 MPa 18.5" 2

Oin ',, 16,3 MPa 13,4 2

�9 lt-10-2 ffi 3 ,48~ 28.4 2

Est" IOS ffi 0,702 MPa 1,44 2

Ed' IO s ~ 0,761 MPa _ 1

vu ffi 5380 m/sec - - I

= 0,60 21.7 2

G.ray granite

O'rb ffi 17.3 MPa 9,9 6

O'in ffi 13,9 MPa 21,6 6

~it" LO- 2 : 4,10 %

Es t" l0 s ffi 0,646 MPa

20,9

10,1 6

E d- l o s == 0 , 7 1 7 M P a - - 1

ru - 5248 m/sec - 1

Z = 0,5~1 36.0 6

D I S C U S S I O N

The ratio of the real boundary of stress in bending arb f rom (4) to a s for gray granite did not exceed 0.06 and is of

the same order as that characterist ic o f silicate materials.

These proper t ies for gray and redgranites and for white marble and silicate materials can serve as basis when these

materials are used in simulating technological and design factor effects when specimens o f materials based on silica are used

under conditions o f compress ion .

Figure 2 and 3 give the relationship o f am/arb and the static Young ' s modulus Est to the brittleness parameter X.

These relationships are expressed to a first approximation by straight lines:

705

ain/Crrb Est" 10 -5, MPa q~

o Z �9 o o ! 0 e 2

. , I . - - . / �9 * / . I 1 1 - o

o o,~ o,.~ O7.0 O,~ 0,50 t~" x ago R-~ o~'o ~ x

Fig. 2 Fig. 3

Fig. 2. Experimental dependence of Crin/arb on the brittleness parameter X for red granite (1)

and gray granite (2).

Fig. 3. Experimental dependence of the static Young's modulus Est on the brittleness param-

eter X for red granite (1) and gray granite (2).

cry,, lamb - 0,7 + 0,2;C; (9)

e ~ , - ( o , s s + o,17x), lO s Mrs, (10)

where the coefficients to X and the free terms have been determined by least-squares fitting [19] on the basis of the experi-

mental data. The limiting value of the right-hand side in (9) differs by less than 10% from ~rrn/Crrb as assumed for silicate and ceramic materials. At the same time, the limiting value on the right-hand side in (10) for X = 1 attains 78,000 MPa and is of

the same order as Young's modulus for sheet glass and for technical grades of the glass ceramics A-1 and A-3, and for glasses 13v, MKR-t, and ZS-9 [16].

Table 2 shows that the strength of a component made of gray granite is dependent on the dimensions: when the volume of a cylindrical specimen increases by a factor four or the thickness of a disk along the loading axis is doubled, a s

and % increase by factors 1.1 and 2.0 respectively.

A similar regularity has been observed [20, 2I] in compression of hard limestones and other rocks with smooth metal disks. On the other hand, the lkerature gives other data on the effects of specimen size on strength characteristics for brittle

materials. These can be divided conditionally into two groups. The first group includes the technical and optical glass

ceramics A-l, SOll5M, and STL-10, as well as refractory periclase-chromite PKhPPP and periclase-spinel-chromite PShPCh) material, for which a s decreases as the specimen volume increases [22-24]. The second group consists of any brittle

material whose Crin and a s in bending (A-1 and STL-10 glass ceramics [25]) and in compression (electrotechnical porcelain [26]) increase at first, but after reaching a maximum value for specimens of any size, the strength characteristics fall as the volumes increase further.

Our results for gray granite and all the published data [20-26] correspond to theoretical concepts on the effects of specimen size on the limiting stress, e.g., in uniaxial tension of wires [27] and compression and bending of brittle materials [28].

Carmarsh (1958) and Weibull [27, 28] proposed formulas for the effects of the diameter d of a wire or the volume V of a brittle-material specimen on the strength:

ar - a + B / d ; (11)

G i n , a s - D / V l / ~ w (12)

706

TABLE 4. Results on Uniaxial Compression of Cylindrical Electrotecimical Porcelain Speci-

mens

Specimensize , m m

d = 7,5; h = 22,61

d = 8,95; h = 27,10

d = 10,85; h = 33,10

Characteristic

Y = 0,999 - 0,007 cm 3

a s = 1160 MPa

Y = 1,70 cm 3

v,%

1 , 7 3

2,40

Number of specimens

12

12

- - 10

a s = 1213 MPa 3 , 8 4 10

Y = 3,06 c m 3 - - 10

a s = 1151 MPa 5,10 10

I n (~s 74

t 2 o $,8

-2.5 0 2,5 5,0 lnv

In els, lar tr~

&7

Z"

2.2 0,85 ~,~0 t95 Zn V

Fig. 4 Fig. 5

Fig. 4. Effects of volume V of a cylindrical specimen of electrotechnical porcelain on the

limiting stress a s in uniaxial compression: 1) [26] data; 2) experiment.

Fig. 5. Dependence of limiting stress a s in uniaxial compression of cylindrical specimens (2)

and of the stren~-r& a c in compression for disks along the axis (1) for gray granite in relation

to specimen volume V.

in which A, B, and D are constants for a given material that are to be determined by experiment [19], while ro w is the

Weibull modulus. Then (11) and (12) imply that the strength may increase for V 1/mB _< 1 in gray granite and also in hard limestones

and other rocks [20, 21]. The same applies to the strength of metal wires for d _< 1.

I have found that the compression of electrotechnical porcelain [26] gives rise to two characteristic segments in the

dependence of the strength on pressure in accordance with (12) as the specimen volume varies, and in addition experiments

were performed under conditions of uniaxial compression of specimens of electrotechnical porcelain having three sets of

dimensions, which were made from the material used in [26]. Table 4 gives the specimen characteristics and limiting stresses

in compression. The cylindrical surfaces of the specimens having the first two sizes were worked with a diamond tool to a

microroughness height R a _< 2 .5 -1 .25 /zm. The cylindrical surface of the specimen with diameter 10.85 ram was not

worked mechanically after heat treatment at up to t350~ As in the case of the granite specimens (Table 2), the ratio of height to diameter was close to three. It is evident

from [26] that the limiting stresses for electrotechnical porcelain specimens of volume 1 cm 3 are 1187 MPa, so the volume of

the specimen of the first size was taken as virtually 1 cm 3. The theoretical and published values (Table 4) indicate that the

differences are not more than 3 %. The values can be taken as a first approximation as represented by piecewise-linear curves,

which after mathematical transformation become

a s - 1,'0'~ (MPa) for V < 18,85 cm3; (13)

a s - v - O , O 7 . e T , 4 (MPa) for V _> 1 8 , 8 5 c m 3. (14)

707

TABLE 5. Limiting Strength in Uniaxial Tension for Gray Granite Given by the Pisarenko-

Lebedev Theory

I Characteristic, MPa Limiting strength, MPa i Calculation error, % O" c O" s I

9 . 5 152 6,0(0.35) [ 11.35 20.0 166 12.6(0.73) [ ..... 7.2

We have found in this study that these granite as constructional materials have features closely similar to those of

silicate and ceramic materials, so it can be assumed that as = f(V) and a c = f(V), for granite can be approximated (Table 2)

in logarithmic coordinates by analogy with the formulas for electrotecknical porcelain (Fig. 4) and other brittle materials [23]

as piecewise-linear curves. The experimental data for gray granite are shown in Fig. 5 as a linear function of specimen

volume, and where (1) and (2) may be approximated respectively by the following functions:

o" K - vl '2 .e-0 '6 (MPa); (15)

ar (MPa). (16)

A conclusion from the experiments is that there is a limiting value for the specimen volumes for gray granite used in

uniaxial compression and in compression of disks along the axis for which (15) and (16) apply, and also there is a different dependence of a c = f(V) and a s = f(V), in which the limiting values of the strength decrease as the specimen volume increases. From (13) and (16), there is a fairly important dependence for electrotechnical porcelain and gray granite: the

Weibull moduli differ by not more than 15 %. The design of strong and reliable structures made of gray granite and white marble requires one to use strength

theory in the strenbnh calculations. The limiting transverse tensile strain in uniaxial compression of gray granite specimens 20 mm in diameter is greater by a factor 1.5 than the limiting tensile strain in bending for that material, and the second classical

theory of strength (the theory of maximal tensile strain) is not suitable for it. As an example, we use a c (Table 2) to calculate the limiting stress in uniaxial tension crf for gray granite from the

formula [9]

of =AI ae, (17)

in which A 1 is a coefficient that is dependent on the strength theory. It has been shown [9, 16, 17] that for silicate, ceramic, and similar materials (including gray granite and marble), the

experimental data for conditions of planar or bulk stress under conditions of limiting loading are in good agreement with the

theoretical values given by the Pisarenko-Lebedev strength theory [29]. From that theory, (17) takes the form

2Csac = - ( 1 8 )

Calculations from (18) indicate that the limiting stress in uniaxial tension for gray granite is also dependent on the scale factor (Table 5). In parentheses in Table 5 we give the ratio of af to the real strength in bending arb (Table 3). By analogy with silicate and ceramic materials [15-17], those ratios are less than one (in the range 0.4-0.7). This shows that

values calculated from (18) are in good agreement with the actual characteristics of gray granite under conditions of uniaxial tension.

The maxima likely in the calculated quantities (11.3 and 7.2% for a s and a c in Table 5) were determined by calculat-

ing the variance S, f from a formula derived from (18) in accordance with the rules of error theory [30]:

2 (19)

708

in which Sac2 and 5% 2 are the variances that characterize the test method for a c and a s .

A difference from the latter formula is that Sa f for (18) can also be expressed as

O ' s ~ a c -1- 2 Saf = (~z2o.2s -- 48o2c) 3 (20)

This calculation error can be compared with the sample coefficient of variation v (Table 2), the two being of the same order.

The failure features of these materials are also important. In uniaxial compression of gray or red granite, the specimen splits into two large parts in the working zone with the simultaneous formation of several small pieces (largest

dimension up to 3 ram). About 20% of the specimens with diameters 10 and 20 nun had clear-cut failure surfaces in the form

of ellipses, in which the major axis formed an angle close to 20 ~ with the axis of the specimen. That angle is indicated by the

[14] data as less than 45 ~ The failure surface took the form of roughness composed of multiface pyramids of height up to 1.5 mm.

A very different mode of failure occurred on uniaxial compression of rectangular white marble specimens. The

working part was transformed into a large number of pieces (maximum dimension up to 0.2 mm), which formed a spatial patten on the metal pistons, which resembled a four-face covering. The failure occurred over surfaces whose form has been

given in [14]. There were differences in failure mode for granite and marble possible on account of features of the chemical composition (Table 1).

The disks on loading along the axis split into two parts in the plane of action of the forces. At the points of force transmission to the specimen, there were high contact stresses, which crushed the material to a depth of up to 2 ram.

We are indebted to Candidate V. P. Zavadi for assistance in the experiments.

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