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Strain fluctuations in heterogeneous rocks
Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol. 12, pp. 181-189.
Pergamon
Press 1975. Printed
in Great
Britain
Strain Fluctuations in Heterogeneous Rocks JERZY GUSTKIEWICZ* Attention has been drawn to the dispersion of the results of the strain measurements in heterogeneous rocks. The representative strain as the mean value of a sequence of strains has been defined at a point of a heterogeneous rock specimen. The difference between the strain of the sequence and the representative strain is called the strain fluctuation. In a homogeneous specimen this diflerence practically vanishes. The strain fluctuation is random in this meaning that it assumes random values from point to point of the specimen. Thejeld of the representative strains within the cross-section of uniaxially compressed granite specimen as well as the corresponding strain fluctuations on its side surface were determined. For this purpose the longitudinal strains were measured on the side surface of the specimen. The mean strain fluctuation in the form of the standard deviation has been expressed as the function of the strain gauge measurement length and of the representative strain. The representative length of the strain gauge has been introduced as the length ascribed to the given bounded above value of the mean strain fluctuation. From the other side, the representative length of the strain gauge bounds below the size of the heterogeneous specimen.
INTRODUCI’ION In rock mechanics attention is drawn to dispersion in the results of observations of mechanical quantities, including strains. This dispersion is the reason for the fact that the numerical data cannot be interpreted univocally. This observed dispersion is thus transferred to any projections made from the results. The reasons for dispersion are: insufficient accuracy of the measuring apparatus, possible errors in the observation method, the effects of environment on the functioning of the apparatus, and random heterogeneity of the rock material. The present paper deals with the influence of the last mentioned factor on the results of strain measurements while neglecting the other factors as being relatively small and of a higher order. The aim of this study is to demonstrate the way in which the result of strain measurement depends on the length of the section of a material line, called the strain measurement base, in a heterogeneous medium. The difference in strains measured at one and the same place and in the same direction on bases of different lengths has been called the strain fluctuation. Attention has been drawn to this problem in the study [l]. The studies [2,3], on the other hand, deal with the fluctuations in concrete and in loose sand, respectively. * Polish Academy of Sciences, Strata Mechanics lishment, ul. Reymonta 27, Krakbw, Poland.
Research
REPRESENTATIVE STRAIN, LOCAL STRAIN FLUCHJATION A rock is a heterogeneous medium with a grain structure. Size, shape, type, the distribution of grains and their properties in identical volume elements of the rock of one and same kind have a random variation. In particular, the strain field which is formed under the influence of definite loading is a random function of a material point with the following meaning. Assume a sequence of specimens of identical dimensions and shapes collected by the same method from a rock of one and the same type. In each of these let us introduce in an identical way a separate system of material coordinates. Consider a sequence of points taken one from each specimen, having the corresponding coordinates equal to each other. Subjecting all the specimens to the influence of an identical boundary condition we shall induce a strain in each point of the sequence. A sequence of strains formed in this way, as a result of random heterogeneity of the samples, contains random values. The strains of the sequence are oscillating around a mean value which we shall call the representative strain of the sequence. The symbol E is used to denote an arbitrary strain of the sequence of strains under consideration, and q the corresponding representative strain. The difference
Estab-
S=E-_YI ,181
(11
1X2
Jcrzy Gustkicwicz
will be called the local fluctuation of strain at a point of an arbitrary specimen. To the sequence of points under consideration each of which belongs to a different specimen there is ascribed a sequence of strains E. the representative strain r) and the sequence of fluctuations 6. Under the influence of the imposed boundary condition a field of strains is formed in the specimen. This condition along with all the specimens of the sequence induce a field of representative strains and a corresponding field of strain fluctuations in the specimen. The field of strains E and the field of strain fluctuations S are random, meaning that they assume random values from point to point. A strain is an invariant depending on a point and on direction. To determine it by measurement we need a section of a material line of a definite direction and a finite length, called the strain measurement base. The result of the measurement gives a value of strain, equal to a certain mean value, differing in general from the strain value at the considered point, localized on the base. The strain measured on the smallest possible base must necessarily be assumed as local at a point. If, however, the strain of the field of representative strains along the direction of the measurement is constant, then lengthening of the base will produce values of the measured strains that will be increasingly close to the value of the representative strain. The measurement of strain along the direction of a constant reprcsentative strain on the smallest and sufficiently long base gives an approximate value of fluctuation (1). Use of the representative strains is frequently made in measuring practice. This follows from the fact that a material such as rock is replaced by a homogeneous model. Then we have two available procedures regarding the strains. The first consists in assuming a definite model of a medium and, with the boundary conditions given. in a theoretical anticipation of the strains forming in it. Their control by measurement refers to the representative strains. The second procedure is related to the problem from which the model of the medium corresponds to the behaviour of the rock under the defined conditions. This problem can be solved by a laboratory test which consists of establishing the relation between a definite loading of the sample and its strain. The strain to be measured represents the total behaviour of the sample under a uniformly loaded condition i.e. the representative strain.
sured result can be compared with the representative strain, which is constant along the base, if we are able to determine the field of such strains in the material under investigation. The requirement of uniform loading of a sequence of samples, that could be repeated, is difficult to realise in practice. A decisive role is played here by the bending moment of the sample due to the testing machine, varying at random. Its existence results from the lack of parallelism between the loaded surfaces of the sample, insufficient accuracy as regards its central position in the loading machine and the structural imperfections of the testing machine itself. Accepting some assumptions concerning the reprcsentative strains we can investigate the fluctuations on a single specimen instead of a sequence. When the platens of the loading machine are made of a rigid material they induct in the cross-section of a continuous, homogeneous and elastic sample a field of strains, which being a function of the section point may be expressed in the form of the equation of the plane 141. In particular. this field may be uniform if the bending moment of the sample is equal to zero. It is assumed that in such a cross-section of a heterogeneous, elastic sample the field of representative strains is also a flat, continuous function of the section point. If the sample is statistically homogeneous, then each cross-section at a sufficient distance from the loaded surfaces will have an identical field of representative strains. This means that along the gcneratrix of the sample, in the area of these cross-sections. the represcntativc strain will be constant. A plane of distribution of longitudinal strains can be drawn over a flat surface of ,the cross-section of a cylindrical sample. The distance between a point on the strain distribution plane and the surface of the section is a measure of the strain. The trace of the strain distribution plane on the side surface of the sample gives an ellipse. When developed on a plane this trace gives a sinusoidal curve. Let K denote the radius of the sample. Then the area of the cross-section of the sample is determined in the polar coordinates 11,cp by the inequalities 0 d /, d I<.
0 < cp 6 2rc.
Using the symbol yeto denote the axis of the representative strains which run vertically to the surface (2) of its centre, then the equation of the distribution plane of the representative strains will assume the form given by equation (3) in the cylindrical coordinates p. 41. Y rJ = B,, + B,P cos cp + H,p sin cp.
POSSIBILITY OF THE DETERMINATION OF THE REPRESENTATIVE STRAINS WITHIN THE HETEROGENEOUS SPECIMEN
From the remarks given above there arises the problem of the relation between the length of the base and the value of the strain measured on it. The mea-
(2)
(3)
where B,, B,, B, are constant, and .P, cp satisfy (2). E, is the representative strain in the centre of the section. EXPERIMENTAL
RESULTS
The following experiment was carried out. In the mid-way position along its length a granite specimen
Strain Fluctuations
Fig. 1. Arrangement
of samples
in Heterogeneous
for test.
of a radius R = 2.5 cm and the slenderness ratio 2 strain gauges were mounted to measure the longitudinal strains. The number of gauges used was 40, each with a base length of 6mm. The sample is shown in Fig. 1. The average length of the grains of granite was of the order of the strain gauge base length. The grains of the specimen are illustrated in Fig. 2. The sample was compressed uniaxially in a loading machine within the limits of elasticity by load steps of 1 ton. With each step of loading the strains were registered by means of an automatic apparatus for strain measurements, of the type TSA-63 made by Mikrotechna in Prague. The measuring accuracy was )l x 10-5. For each series of measurement results the function of representative strain (3) has been approximated by the least squares method. It has been assumed that all the cross-sections of the sample within the bases of the strain gauges have identical distributions of representative strains.
Fig. 2. Cross-section
of a granite
Rocks
test sample.
Fig. 3. Measured
and representative strains on the developed of a cylindrical granite test sample.
surface
The sample was repeatedly subject to a cyclic loading. Each cycle gave repeatable results. In Fig. 3 the results are shown in the form of strains on the side surface of the sample in its flat development, for a few steps of loading. The smooth curves are diagrams of the strain q, expressed by the formula (3) for p = R. The angle cp or the gauge number are the argument. The brokenlinesillustrate the measured stainse. The horizontal lines determine the mean strain co from all 40 gauges. It appears that in the centre of the cross-section of the sample that e. = B,. This can be seen from Table 1 in which the respective values B, and Bz are contained according to the formula (3). Table 2 contains an example of the numerical data corresponding to the diagrams for (TV= 240 bars in Fig. 3. * For the sake of comparison in Fig. 4 analogous results are shown which were obtained from a sample of homogeneous limestone, of identical specimen size [4]. The measurement was performed by means of 16 gauges with a 12 mm base. The results obtained are smooth within the accuracy of the measuring equipment. In both cases the same equipment and strain gauge types have been used. The results obtained for granite display relatively large differences between the locally measured strains E and the local representative strains Y),i.e. large local fluctuations (1). The measured strains are in a visible way random functions of a point of the sample. The corresponding fluctuations are also random functions of a point. It appears, however, that at a given point we may consider the measured strain and the corresponding fluctuation with a sufficiently great accuracy as function of the loading level determined by the given boundary
fX
IO'
118 109 122 135 116 145 140 126 130 120
CT,,= 240 bars.
i
137.1 132.8 128.1 123.1 118.1 113.0 108.0 103.3 99.0 95.1
?/x IO5
Bi
11 12 13 14 15 16 17 18 19 20
- 19.1 - 23.8 -6.1 11.9 -2.1 32.0 32.0 22.7 31.0 24.9 99 77 81 71 74 71 68 44 97 99
< x lo5
i
I
6 x lo5
cm3.851 4.808 5,720 6,456 7.453
68.45 82.33 94.98 106.50 117.95
48 96 144 192 240
RI x IO5
68.40 82.30 95.00 106.50 1 17.90
co x lo5
2
BZ
91.8 89.1 87.1 85.9 85.5 85.8 87.0 88.9 91.6 94.8
‘I x 105
- 6,622 - 7.559 - 8.544 -9.314 - 10.149
cm-’ 48.7 59.3 68.5 71.6 85.5
10’
105
15.4 17.7 19.7 21.7 23.2
6mm
21 22 23 24 25 26 27 28 29 30
i
TABLE 2.
88.1 105.3 121.5 136.0 150.3
lo5
)Imar
7.2 ~ 12.1 -6.1 - 14.9 -11.5 -14.8 - 19.0 - 44.9 5.4 4.2
6x
%mn
TABLE 1.
IO5
110 104 114 116 88 110 131 156 158 186
EX
12.2 13.9 15.6 17.2 18.5
12mm
98.7 103.0 107.7 112.7 117.7 122.8 127.8 132.5 136.8 140.7
17 x IO5
7.2 8.2 9.2 10.1 10.7
11.3 I.0 6.3 3.3 - 29.7 - 12.8 3.2 23.5 21.2 45.3
6 x IO’
2.5 2.6 3.0 3.2 3.3
48 mm
s x IO5
24 mm
i 31 32 33 34 35 36 37 38 39 40
1.9 2.2 2-6 2.9 3.1
60 mm
143 168 183 174 109 110 129 134 100 153
E x IO5
1.5 1.8 2.2 2.3 2.5
120mm
144.0 146.7 148.7 149.9 150.3 150.0 148.8 146.9 144.2 141.0
‘I x 105
-1.0 21.3 34.3 24.1 -41.3 - 40.0 - 19.8 - 12.9 - 44.2 12.0
5 x IO5
2 z K 5' -z 5. N
kT ;J Y
Strain _Fluctuations in Heterogeneous
I 3
I 5
I 7
I 9
I II
45
, I
I
1
1
I
,
I
I
90
135
I80
225
270
315
3M)
I+P
Fig. 4. Comparable
I 15
strains
185
number
i gouge
1
I 13
Rocks
Fig. 6. Local
angle
observed on a homogeneous sample.
limestone
condition. This is clear so far as the sample was being loaded within the limits of elasticity. This finding was to be expected judging from the shape of the broken lines in Fig. 2 for different loadings. Besides, it can be seen also from the diagrams of the local fluctuations 6 as functions of e0 in Fig. 5. The strain e0 on account of the elasticity limits of the sample may be assumed as a measure of loading of the sample. The numbers on the diagram correspond to the gauge numbers. The fluctuations 6 as functions of the corresponding representative strains q have similar shapes as can be seen from Fig. 6. The initial, non-linear parts of the diagrams in Figs. 5 and 6 are evidently related to the existence of microcracks in the sample and the process of their closing when loaded. The existence of microcracks is confirmed by the behaviour of the sample shown in the diagram Fig. 7. This is a stress-strain diagram, where (T,,is the mean stress. Additionally, (T,,has been represented as a function of the minimum and maximum representative strain, respectively, in the sample cross-section. The strains Yl,in and Y],,, are localized at two opposite points on the sample’s circumference [4]. These values are contained in Table 1. The horizontal span of the fork formed by the diagrams for the given stress rrO
fluctuations
as functions
of representative
constitutes the variation interval for the representative strains in the sample cross-section. The mean elasticity modulus being given, it is possible to obtain the variation interval of stresses in the sample cross-section and to estimate the bending moment of the sample in the loading machine. TOTAL
MEAN STRAIN
FLUCTUATIONS
When determining the relations between the stress and strain during uniaxial testing of materials we have available the mean stress as the quotient of the force and the cross-sectional area of the sample, and the mean strain. The mean strain is calculated from the strains measured along the bases on the sample which are situated symmetrically with respect to its axis. For this purpose at least two bases are required. The mean stress and strain are devoid of the corresponding corn; ponents, resulting from the bending moment of the sample. Considering the axial symmetry of the sample, such symmetry of the measurement of strains and the flat distribution of the representative strains (3) the mean
I-
600
500
7
mm
6.
-
x t?
?
300
-
200
-
I
50 400
oi
~
50
150
100
2ooxlo-5
co
Fig. 5. Local
fluctuations
as functions
of measured
strain.
strain.
Fig. 7. Observed
stress-strain
relationships
~rna,
Jerzy Gustkiewicz
1X6
strain should equal B,. However, on account of the fluctuations the mean value may be different from B,. From the population of 40 strain gauges mounted on the sample it was possible to obtain 6 sets of gauges localized symmetrically with respect to the sample axis. If, in a given set, the mean strain was calculated from II gauges localized at every angle cp, = 2rc/n, then it was possible to obtain k = 40/n of such mean strains. The corresponding values of n, k, cp,, are contained in Table 3. Let us denote the mean strain from II gauges by $1, i = 1, . . . , k. In particular, for IZ= 40 and k = 1 we have E(~O)= E(). By the fluctuation of the ith mean from II gauges we shall define the difference $0I = c!n) _ B 09 1
i=
k.
I...,...,
(4)
For II = 40 we have E’;C”= 0 because e\““’ = co = Bo. On the other hand, to the set contained in Table 3 we shall join II = 1 and k = 40, respectively, i.e. the results obtained from particular gauges. Then, with regard to (4) and in agreement with (1) we shall write g!‘) -- & = Ei I
'li,
i =
1,.
,
,40,
(4a)
where 6i denotes the strain measured by ith gauge, and vi-the corresponding representative strain. As a measure of the total mean fluctuation in the investigated area of the sample we shall assume the mean deviation having the form
The mean strain from II gauges, each of which having the base length Lo, corresponds to the strain measured by a gauge of the base n&. However, in the case under consideration the gauges from which the mean strain is being calculated, do not lie on a single generatrix of the sample with the representative strain constant. The differences between the representative strains result from the bending of the sample and they disappear in each of the mean strains. On the other hand, the local fluctuations are linear functions of the strains 4 and E(),aside from the beginning of loading, i.e. their increments are proportional to those values. The case of the increments of strains will be dealt with later on. It appears thus, that it would be possible to lengthen the base in the given way by taking the mean values of strains the more so as this is done testing the samples. Taking the mean values according to the given rule does not take into consideration the possible correla-
50
L.
mm
Fig. 8. The mean strain fluctuation as a function of (a) measured strain, (b) the base.
tions between the strains measured at neighbouring points on the same sample generatrix, and so the mean fluctuation (5) may vary from the fluctuation obtained along such a generatrix having a constant representative strain. The problem of the correlations of strain fluctuations is dealt with later. Table 1 contains the calculation results of the mean fluctuation according to the scheme of choice of the mean strains given in Table 3. Using Table 1, the diagrams of fluctuation s as a function of the base L, with the strain co = B, as a parameter, and s as a function of lo with the parameter L, have been plotted. The diagrams are shown in Fig. 8. The first of the diagrams outside the area corresponding to the closing of microcracks are linear.The linear range corresponds to the linear part of the stressstrain diagram in Fig. 7. It can be seen from the other family of diagrams in Fig. 8 that for the investigated granite the mean fluctuation relatively quickly decreases at first, as the base increases. Starting with base L = 60mm the decrease of the fluctuation goes on very slowly. The line on the upper diagrams referring to this base indicates the practically proportional relation between the mean fluctuation and the mean strain co. If we assume a system of rectangular coordinates L, Q, s in space. then we can imagine a surface s = (p(&,,) such that the diagrams in Fig. 8 are lines on this surface, for L = const. and E(, = const., respectively. This is illustrated in Fig. 9. The results obtained in Fig. 8 are qualitatively in good agreement with those obtained in the studies [2,3] for concrete and loose sand, respectively. FZUCT UATIONS OF STRAIN INCREMENTS USC is often made of strains determined with respect to the state of a material which is already deformed. In such a case the results of experiments on the above
Strain Fluctuations
in Heterogeneous
-30
Fig.
Fig. 9. The mean
strain
fluctuation as a combined and measured strain.
function
of base
granite sample subjected to initial mean strain e0 = 51 x lo- * illustrate the fluctuations of strains. When this strain occurs the process of intensive closing of the sample’s microcracks is finished, which can be seen from Fig. 7. Assuming the sample in this condition to be an undeformed one the particular strains have been registered which occur under successive loadings. The diagrams in Fig. 10 show the registered values of strains, the representative and the mean strains from all gauges on the side surface of the sample. It can be seen from the figure that the fluctuations are correspondingly smaller than for the case shown in Fig. 3. This can be also seen from Fig. 11 illustrating the local fluctuations as functions of representative strains. These fluctuations are practically proportional to the representative strains. The diagrams are accordingly
187
Rocks
-
11. Local
fluctuations of strain as functions strain. (Numbers indicate gauges.)
of representative
_
parallel to the straight parts of the diagrams in Fig. 6. The mean global fluctuations, as functions of the mean from all the gauges, behave in a similar manner, as can be seen from Fig. 12. The lines obtained here are parallel to the corresponding lines in Fig. 8. The other family of diagrams, however, have been straightened in comparison with those shown in Fig. 8. This can be seen from the confrontation of the corresponding lines marked l-5. Table 4 contains the numerical data from the experiment. Let us introduce for the given base the quotient
e=i. We shall call it the relative global fluctuation. This is represented in Fig. 12 as a slope of straight lines, and in Table 5 its values are given for different bases. The diagram of dependence of the relative global fluctuation on the base is shown in Fig. 13. This diagram replaces both families of diagrams shown in Fig. 12. CORRELATION OF THE STRAIN FLUCTUATIONS Let us consider the case where the strain fluctuations are proportional to the respective representative strains. This case is illustrated in Fig. 10. We can define pi =
5
(i = 1, . . . .
yli
(7)
as a relative local strain fluctuation. It is illustrated by a slope of the ith line in Fig. 11. Table 6 contains the values &, 9i and the respective values pi multiplied by 100. The mean value z;:i i=Y=
/l.
-0.001173
i.e. it practically vanishes. Then we may write
Fig.
IO. Observed
representative and mean already deformed.
strains
on
samples
(81
Strain Fluctuations I
I
in Heterogeneous
Rocks
189
gauge the number of pairs mounted on the surface of the same grain is much lower than that of the first one.
s
c
CONCLUSIONS
(b)
L,
mm
Fig. 12. The mean global fluctuations as functions of (a) the measured strain, (b) the base.
coefficient the set of pi has been taken into consideration. The correlation coefficient of strain fluctuations obtained from the neighbouring strain gauges is defined by the formula
where m = 40 is the number of the strain gauges. For every second strain gauge we have
as the correlation coefficient. If we put the set of the values of pi from Table 6 and equation (9) into (10) and (11) we have K, = 04464
K, = 0.2154
respectively. The relative importance of the value of K1 may be connected with the fact that there exist pairs of neighbouring strain gauges such that each of them has been mounted on the surface of the same sample grain. Hence, they are influenced by the strain of the same grain. This influence decreases rapidly with the increase of the distance between the gauges, which can be seen from the value of K2. As we remember, the dimensions of the grain surface and of the gauge surface are of the same order. Thus, in the case of every second strain
author would like to thank Mr B. Brodzitiski, Mr M. Mugosz and Mrs Z. Kusmierczyk for their assistance in carrying out the experiments.
Acknowledgements--The
Received 4 December 1974.
REFERENCES Dantu P. Etude des contraintes dans les milieux hhte’rog&es Application au biton, Publication No 567 de Laboratoire Central
9 03t02
The above considerations are evidently related to the problem of the dispersion of the observations results in rock mechanics. The influence of the measurement base on the magnitude of dispersion of the strain measurements is particularly visible. The value of the global strain fluctuation tells us that the measured result on a definite base represents the average response of the material. Thus we may say that the given base is representative within the accuracy of the corresponding mean fluctuation. In particular, the base determines the size of the tested sample. For instance, considering those parts of the sample disturbed by the platens, for a sample with the slenderness ratio 2 the base of a longitudinal gauge should not exceed the diameter of the sample. In the case of granite described above we have noticed that for a base above L = 60 mm the fluctuation decreases imperceptibly. If we want to determine the elasticity modulus for the linear section of the diagram in Fig. 6, on this base, then the relative fluctuation 9 = 0.03, as we are interested in the increment of strains. This means that the base is representative within an accuracy of kO.03. Having decided to use two gauges on a sample we shall take each of them to be 30mm long. Then the maximum height of the sample with the slenderness ratio 2 will be 60mm. However, if we want to control the bending moment acting on the sample we should determine the plane of the distribution of strains (3). Then we shall need at least 3 gauges each with a 60 mm base. This determines the minimum height of the sample, 120mm. The plane will be defined within the average accuracy of the relative fluctuation of strain 20.03, whereas the elasticity modulus will be defined with much greater accuracy.
-
1 50
.
!
I
I
I
100
L.
mm
Fig. 13. Dependence of the relative global fluctuation on the measured base length.
des Ponts et Chausstes, Paris (1957). Mtiller R. K. Der Einguss der Messlange auf die Ergebnisse bei Dehnungsmessungen an Beton, Messtechnische Briefe fur Elektrisches Messen Mechanischer Grossen Hottinger Baldwin Messtechnik, Nr 3, Darmstadt (1966). Klein G. The Length of Measurement Basis and The Fluctuation Value of Strains in a Loose Medium, Zeszyty Problemowe, Gornictwa, Krakow, 13, No. (1975). Gustkiewicz J. Uniaxial compression testing of brittle rock specimens with special consideration of the effects of bending moment. Int. J. Rock Mech. Min. Sci. 12, 13-25 (1975).
Pergamon
Press 1975. Printed
in Great
Britain
Strain Fluctuations in Heterogeneous Rocks JERZY GUSTKIEWICZ* Attention has been drawn to the dispersion of the results of the strain measurements in heterogeneous rocks. The representative strain as the mean value of a sequence of strains has been defined at a point of a heterogeneous rock specimen. The difference between the strain of the sequence and the representative strain is called the strain fluctuation. In a homogeneous specimen this diflerence practically vanishes. The strain fluctuation is random in this meaning that it assumes random values from point to point of the specimen. Thejeld of the representative strains within the cross-section of uniaxially compressed granite specimen as well as the corresponding strain fluctuations on its side surface were determined. For this purpose the longitudinal strains were measured on the side surface of the specimen. The mean strain fluctuation in the form of the standard deviation has been expressed as the function of the strain gauge measurement length and of the representative strain. The representative length of the strain gauge has been introduced as the length ascribed to the given bounded above value of the mean strain fluctuation. From the other side, the representative length of the strain gauge bounds below the size of the heterogeneous specimen.
INTRODUCI’ION In rock mechanics attention is drawn to dispersion in the results of observations of mechanical quantities, including strains. This dispersion is the reason for the fact that the numerical data cannot be interpreted univocally. This observed dispersion is thus transferred to any projections made from the results. The reasons for dispersion are: insufficient accuracy of the measuring apparatus, possible errors in the observation method, the effects of environment on the functioning of the apparatus, and random heterogeneity of the rock material. The present paper deals with the influence of the last mentioned factor on the results of strain measurements while neglecting the other factors as being relatively small and of a higher order. The aim of this study is to demonstrate the way in which the result of strain measurement depends on the length of the section of a material line, called the strain measurement base, in a heterogeneous medium. The difference in strains measured at one and the same place and in the same direction on bases of different lengths has been called the strain fluctuation. Attention has been drawn to this problem in the study [l]. The studies [2,3], on the other hand, deal with the fluctuations in concrete and in loose sand, respectively. * Polish Academy of Sciences, Strata Mechanics lishment, ul. Reymonta 27, Krakbw, Poland.
Research
REPRESENTATIVE STRAIN, LOCAL STRAIN FLUCHJATION A rock is a heterogeneous medium with a grain structure. Size, shape, type, the distribution of grains and their properties in identical volume elements of the rock of one and same kind have a random variation. In particular, the strain field which is formed under the influence of definite loading is a random function of a material point with the following meaning. Assume a sequence of specimens of identical dimensions and shapes collected by the same method from a rock of one and the same type. In each of these let us introduce in an identical way a separate system of material coordinates. Consider a sequence of points taken one from each specimen, having the corresponding coordinates equal to each other. Subjecting all the specimens to the influence of an identical boundary condition we shall induce a strain in each point of the sequence. A sequence of strains formed in this way, as a result of random heterogeneity of the samples, contains random values. The strains of the sequence are oscillating around a mean value which we shall call the representative strain of the sequence. The symbol E is used to denote an arbitrary strain of the sequence of strains under consideration, and q the corresponding representative strain. The difference
Estab-
S=E-_YI ,181
(11
1X2
Jcrzy Gustkicwicz
will be called the local fluctuation of strain at a point of an arbitrary specimen. To the sequence of points under consideration each of which belongs to a different specimen there is ascribed a sequence of strains E. the representative strain r) and the sequence of fluctuations 6. Under the influence of the imposed boundary condition a field of strains is formed in the specimen. This condition along with all the specimens of the sequence induce a field of representative strains and a corresponding field of strain fluctuations in the specimen. The field of strains E and the field of strain fluctuations S are random, meaning that they assume random values from point to point. A strain is an invariant depending on a point and on direction. To determine it by measurement we need a section of a material line of a definite direction and a finite length, called the strain measurement base. The result of the measurement gives a value of strain, equal to a certain mean value, differing in general from the strain value at the considered point, localized on the base. The strain measured on the smallest possible base must necessarily be assumed as local at a point. If, however, the strain of the field of representative strains along the direction of the measurement is constant, then lengthening of the base will produce values of the measured strains that will be increasingly close to the value of the representative strain. The measurement of strain along the direction of a constant reprcsentative strain on the smallest and sufficiently long base gives an approximate value of fluctuation (1). Use of the representative strains is frequently made in measuring practice. This follows from the fact that a material such as rock is replaced by a homogeneous model. Then we have two available procedures regarding the strains. The first consists in assuming a definite model of a medium and, with the boundary conditions given. in a theoretical anticipation of the strains forming in it. Their control by measurement refers to the representative strains. The second procedure is related to the problem from which the model of the medium corresponds to the behaviour of the rock under the defined conditions. This problem can be solved by a laboratory test which consists of establishing the relation between a definite loading of the sample and its strain. The strain to be measured represents the total behaviour of the sample under a uniformly loaded condition i.e. the representative strain.
sured result can be compared with the representative strain, which is constant along the base, if we are able to determine the field of such strains in the material under investigation. The requirement of uniform loading of a sequence of samples, that could be repeated, is difficult to realise in practice. A decisive role is played here by the bending moment of the sample due to the testing machine, varying at random. Its existence results from the lack of parallelism between the loaded surfaces of the sample, insufficient accuracy as regards its central position in the loading machine and the structural imperfections of the testing machine itself. Accepting some assumptions concerning the reprcsentative strains we can investigate the fluctuations on a single specimen instead of a sequence. When the platens of the loading machine are made of a rigid material they induct in the cross-section of a continuous, homogeneous and elastic sample a field of strains, which being a function of the section point may be expressed in the form of the equation of the plane 141. In particular. this field may be uniform if the bending moment of the sample is equal to zero. It is assumed that in such a cross-section of a heterogeneous, elastic sample the field of representative strains is also a flat, continuous function of the section point. If the sample is statistically homogeneous, then each cross-section at a sufficient distance from the loaded surfaces will have an identical field of representative strains. This means that along the gcneratrix of the sample, in the area of these cross-sections. the represcntativc strain will be constant. A plane of distribution of longitudinal strains can be drawn over a flat surface of ,the cross-section of a cylindrical sample. The distance between a point on the strain distribution plane and the surface of the section is a measure of the strain. The trace of the strain distribution plane on the side surface of the sample gives an ellipse. When developed on a plane this trace gives a sinusoidal curve. Let K denote the radius of the sample. Then the area of the cross-section of the sample is determined in the polar coordinates 11,cp by the inequalities 0 d /, d I<.
0 < cp 6 2rc.
Using the symbol yeto denote the axis of the representative strains which run vertically to the surface (2) of its centre, then the equation of the distribution plane of the representative strains will assume the form given by equation (3) in the cylindrical coordinates p. 41. Y rJ = B,, + B,P cos cp + H,p sin cp.
POSSIBILITY OF THE DETERMINATION OF THE REPRESENTATIVE STRAINS WITHIN THE HETEROGENEOUS SPECIMEN
From the remarks given above there arises the problem of the relation between the length of the base and the value of the strain measured on it. The mea-
(2)
(3)
where B,, B,, B, are constant, and .P, cp satisfy (2). E, is the representative strain in the centre of the section. EXPERIMENTAL
RESULTS
The following experiment was carried out. In the mid-way position along its length a granite specimen
Strain Fluctuations
Fig. 1. Arrangement
of samples
in Heterogeneous
for test.
of a radius R = 2.5 cm and the slenderness ratio 2 strain gauges were mounted to measure the longitudinal strains. The number of gauges used was 40, each with a base length of 6mm. The sample is shown in Fig. 1. The average length of the grains of granite was of the order of the strain gauge base length. The grains of the specimen are illustrated in Fig. 2. The sample was compressed uniaxially in a loading machine within the limits of elasticity by load steps of 1 ton. With each step of loading the strains were registered by means of an automatic apparatus for strain measurements, of the type TSA-63 made by Mikrotechna in Prague. The measuring accuracy was )l x 10-5. For each series of measurement results the function of representative strain (3) has been approximated by the least squares method. It has been assumed that all the cross-sections of the sample within the bases of the strain gauges have identical distributions of representative strains.
Fig. 2. Cross-section
of a granite
Rocks
test sample.
Fig. 3. Measured
and representative strains on the developed of a cylindrical granite test sample.
surface
The sample was repeatedly subject to a cyclic loading. Each cycle gave repeatable results. In Fig. 3 the results are shown in the form of strains on the side surface of the sample in its flat development, for a few steps of loading. The smooth curves are diagrams of the strain q, expressed by the formula (3) for p = R. The angle cp or the gauge number are the argument. The brokenlinesillustrate the measured stainse. The horizontal lines determine the mean strain co from all 40 gauges. It appears that in the centre of the cross-section of the sample that e. = B,. This can be seen from Table 1 in which the respective values B, and Bz are contained according to the formula (3). Table 2 contains an example of the numerical data corresponding to the diagrams for (TV= 240 bars in Fig. 3. * For the sake of comparison in Fig. 4 analogous results are shown which were obtained from a sample of homogeneous limestone, of identical specimen size [4]. The measurement was performed by means of 16 gauges with a 12 mm base. The results obtained are smooth within the accuracy of the measuring equipment. In both cases the same equipment and strain gauge types have been used. The results obtained for granite display relatively large differences between the locally measured strains E and the local representative strains Y),i.e. large local fluctuations (1). The measured strains are in a visible way random functions of a point of the sample. The corresponding fluctuations are also random functions of a point. It appears, however, that at a given point we may consider the measured strain and the corresponding fluctuation with a sufficiently great accuracy as function of the loading level determined by the given boundary
fX
IO'
118 109 122 135 116 145 140 126 130 120
CT,,= 240 bars.
i
137.1 132.8 128.1 123.1 118.1 113.0 108.0 103.3 99.0 95.1
?/x IO5
Bi
11 12 13 14 15 16 17 18 19 20
- 19.1 - 23.8 -6.1 11.9 -2.1 32.0 32.0 22.7 31.0 24.9 99 77 81 71 74 71 68 44 97 99
< x lo5
i
I
6 x lo5
cm3.851 4.808 5,720 6,456 7.453
68.45 82.33 94.98 106.50 117.95
48 96 144 192 240
RI x IO5
68.40 82.30 95.00 106.50 1 17.90
co x lo5
2
BZ
91.8 89.1 87.1 85.9 85.5 85.8 87.0 88.9 91.6 94.8
‘I x 105
- 6,622 - 7.559 - 8.544 -9.314 - 10.149
cm-’ 48.7 59.3 68.5 71.6 85.5
10’
105
15.4 17.7 19.7 21.7 23.2
6mm
21 22 23 24 25 26 27 28 29 30
i
TABLE 2.
88.1 105.3 121.5 136.0 150.3
lo5
)Imar
7.2 ~ 12.1 -6.1 - 14.9 -11.5 -14.8 - 19.0 - 44.9 5.4 4.2
6x
%mn
TABLE 1.
IO5
110 104 114 116 88 110 131 156 158 186
EX
12.2 13.9 15.6 17.2 18.5
12mm
98.7 103.0 107.7 112.7 117.7 122.8 127.8 132.5 136.8 140.7
17 x IO5
7.2 8.2 9.2 10.1 10.7
11.3 I.0 6.3 3.3 - 29.7 - 12.8 3.2 23.5 21.2 45.3
6 x IO’
2.5 2.6 3.0 3.2 3.3
48 mm
s x IO5
24 mm
i 31 32 33 34 35 36 37 38 39 40
1.9 2.2 2-6 2.9 3.1
60 mm
143 168 183 174 109 110 129 134 100 153
E x IO5
1.5 1.8 2.2 2.3 2.5
120mm
144.0 146.7 148.7 149.9 150.3 150.0 148.8 146.9 144.2 141.0
‘I x 105
-1.0 21.3 34.3 24.1 -41.3 - 40.0 - 19.8 - 12.9 - 44.2 12.0
5 x IO5
2 z K 5' -z 5. N
kT ;J Y
Strain _Fluctuations in Heterogeneous
I 3
I 5
I 7
I 9
I II
45
, I
I
1
1
I
,
I
I
90
135
I80
225
270
315
3M)
I+P
Fig. 4. Comparable
I 15
strains
185
number
i gouge
1
I 13
Rocks
Fig. 6. Local
angle
observed on a homogeneous sample.
limestone
condition. This is clear so far as the sample was being loaded within the limits of elasticity. This finding was to be expected judging from the shape of the broken lines in Fig. 2 for different loadings. Besides, it can be seen also from the diagrams of the local fluctuations 6 as functions of e0 in Fig. 5. The strain e0 on account of the elasticity limits of the sample may be assumed as a measure of loading of the sample. The numbers on the diagram correspond to the gauge numbers. The fluctuations 6 as functions of the corresponding representative strains q have similar shapes as can be seen from Fig. 6. The initial, non-linear parts of the diagrams in Figs. 5 and 6 are evidently related to the existence of microcracks in the sample and the process of their closing when loaded. The existence of microcracks is confirmed by the behaviour of the sample shown in the diagram Fig. 7. This is a stress-strain diagram, where (T,,is the mean stress. Additionally, (T,,has been represented as a function of the minimum and maximum representative strain, respectively, in the sample cross-section. The strains Yl,in and Y],,, are localized at two opposite points on the sample’s circumference [4]. These values are contained in Table 1. The horizontal span of the fork formed by the diagrams for the given stress rrO
fluctuations
as functions
of representative
constitutes the variation interval for the representative strains in the sample cross-section. The mean elasticity modulus being given, it is possible to obtain the variation interval of stresses in the sample cross-section and to estimate the bending moment of the sample in the loading machine. TOTAL
MEAN STRAIN
FLUCTUATIONS
When determining the relations between the stress and strain during uniaxial testing of materials we have available the mean stress as the quotient of the force and the cross-sectional area of the sample, and the mean strain. The mean strain is calculated from the strains measured along the bases on the sample which are situated symmetrically with respect to its axis. For this purpose at least two bases are required. The mean stress and strain are devoid of the corresponding corn; ponents, resulting from the bending moment of the sample. Considering the axial symmetry of the sample, such symmetry of the measurement of strains and the flat distribution of the representative strains (3) the mean
I-
600
500
7
mm
6.
-
x t?
?
300
-
200
-
I
50 400
oi
~
50
150
100
2ooxlo-5
co
Fig. 5. Local
fluctuations
as functions
of measured
strain.
strain.
Fig. 7. Observed
stress-strain
relationships
~rna,
Jerzy Gustkiewicz
1X6
strain should equal B,. However, on account of the fluctuations the mean value may be different from B,. From the population of 40 strain gauges mounted on the sample it was possible to obtain 6 sets of gauges localized symmetrically with respect to the sample axis. If, in a given set, the mean strain was calculated from II gauges localized at every angle cp, = 2rc/n, then it was possible to obtain k = 40/n of such mean strains. The corresponding values of n, k, cp,, are contained in Table 3. Let us denote the mean strain from II gauges by $1, i = 1, . . . , k. In particular, for IZ= 40 and k = 1 we have E(~O)= E(). By the fluctuation of the ith mean from II gauges we shall define the difference $0I = c!n) _ B 09 1
i=
k.
I...,...,
(4)
For II = 40 we have E’;C”= 0 because e\““’ = co = Bo. On the other hand, to the set contained in Table 3 we shall join II = 1 and k = 40, respectively, i.e. the results obtained from particular gauges. Then, with regard to (4) and in agreement with (1) we shall write g!‘) -- & = Ei I
'li,
i =
1,.
,
,40,
(4a)
where 6i denotes the strain measured by ith gauge, and vi-the corresponding representative strain. As a measure of the total mean fluctuation in the investigated area of the sample we shall assume the mean deviation having the form
The mean strain from II gauges, each of which having the base length Lo, corresponds to the strain measured by a gauge of the base n&. However, in the case under consideration the gauges from which the mean strain is being calculated, do not lie on a single generatrix of the sample with the representative strain constant. The differences between the representative strains result from the bending of the sample and they disappear in each of the mean strains. On the other hand, the local fluctuations are linear functions of the strains 4 and E(),aside from the beginning of loading, i.e. their increments are proportional to those values. The case of the increments of strains will be dealt with later on. It appears thus, that it would be possible to lengthen the base in the given way by taking the mean values of strains the more so as this is done testing the samples. Taking the mean values according to the given rule does not take into consideration the possible correla-
50
L.
mm
Fig. 8. The mean strain fluctuation as a function of (a) measured strain, (b) the base.
tions between the strains measured at neighbouring points on the same sample generatrix, and so the mean fluctuation (5) may vary from the fluctuation obtained along such a generatrix having a constant representative strain. The problem of the correlations of strain fluctuations is dealt with later. Table 1 contains the calculation results of the mean fluctuation according to the scheme of choice of the mean strains given in Table 3. Using Table 1, the diagrams of fluctuation s as a function of the base L, with the strain co = B, as a parameter, and s as a function of lo with the parameter L, have been plotted. The diagrams are shown in Fig. 8. The first of the diagrams outside the area corresponding to the closing of microcracks are linear.The linear range corresponds to the linear part of the stressstrain diagram in Fig. 7. It can be seen from the other family of diagrams in Fig. 8 that for the investigated granite the mean fluctuation relatively quickly decreases at first, as the base increases. Starting with base L = 60mm the decrease of the fluctuation goes on very slowly. The line on the upper diagrams referring to this base indicates the practically proportional relation between the mean fluctuation and the mean strain co. If we assume a system of rectangular coordinates L, Q, s in space. then we can imagine a surface s = (p(&,,) such that the diagrams in Fig. 8 are lines on this surface, for L = const. and E(, = const., respectively. This is illustrated in Fig. 9. The results obtained in Fig. 8 are qualitatively in good agreement with those obtained in the studies [2,3] for concrete and loose sand, respectively. FZUCT UATIONS OF STRAIN INCREMENTS USC is often made of strains determined with respect to the state of a material which is already deformed. In such a case the results of experiments on the above
Strain Fluctuations
in Heterogeneous
-30
Fig.
Fig. 9. The mean
strain
fluctuation as a combined and measured strain.
function
of base
granite sample subjected to initial mean strain e0 = 51 x lo- * illustrate the fluctuations of strains. When this strain occurs the process of intensive closing of the sample’s microcracks is finished, which can be seen from Fig. 7. Assuming the sample in this condition to be an undeformed one the particular strains have been registered which occur under successive loadings. The diagrams in Fig. 10 show the registered values of strains, the representative and the mean strains from all gauges on the side surface of the sample. It can be seen from the figure that the fluctuations are correspondingly smaller than for the case shown in Fig. 3. This can be also seen from Fig. 11 illustrating the local fluctuations as functions of representative strains. These fluctuations are practically proportional to the representative strains. The diagrams are accordingly
187
Rocks
-
11. Local
fluctuations of strain as functions strain. (Numbers indicate gauges.)
of representative
_
parallel to the straight parts of the diagrams in Fig. 6. The mean global fluctuations, as functions of the mean from all the gauges, behave in a similar manner, as can be seen from Fig. 12. The lines obtained here are parallel to the corresponding lines in Fig. 8. The other family of diagrams, however, have been straightened in comparison with those shown in Fig. 8. This can be seen from the confrontation of the corresponding lines marked l-5. Table 4 contains the numerical data from the experiment. Let us introduce for the given base the quotient
e=i. We shall call it the relative global fluctuation. This is represented in Fig. 12 as a slope of straight lines, and in Table 5 its values are given for different bases. The diagram of dependence of the relative global fluctuation on the base is shown in Fig. 13. This diagram replaces both families of diagrams shown in Fig. 12. CORRELATION OF THE STRAIN FLUCTUATIONS Let us consider the case where the strain fluctuations are proportional to the respective representative strains. This case is illustrated in Fig. 10. We can define pi =
5
(i = 1, . . . .
yli
(7)
as a relative local strain fluctuation. It is illustrated by a slope of the ith line in Fig. 11. Table 6 contains the values &, 9i and the respective values pi multiplied by 100. The mean value z;:i i=Y=
/l.
-0.001173
i.e. it practically vanishes. Then we may write
Fig.
IO. Observed
representative and mean already deformed.
strains
on
samples
(81
Strain Fluctuations I
I
in Heterogeneous
Rocks
189
gauge the number of pairs mounted on the surface of the same grain is much lower than that of the first one.
s
c
CONCLUSIONS
(b)
L,
mm
Fig. 12. The mean global fluctuations as functions of (a) the measured strain, (b) the base.
coefficient the set of pi has been taken into consideration. The correlation coefficient of strain fluctuations obtained from the neighbouring strain gauges is defined by the formula
where m = 40 is the number of the strain gauges. For every second strain gauge we have
as the correlation coefficient. If we put the set of the values of pi from Table 6 and equation (9) into (10) and (11) we have K, = 04464
K, = 0.2154
respectively. The relative importance of the value of K1 may be connected with the fact that there exist pairs of neighbouring strain gauges such that each of them has been mounted on the surface of the same sample grain. Hence, they are influenced by the strain of the same grain. This influence decreases rapidly with the increase of the distance between the gauges, which can be seen from the value of K2. As we remember, the dimensions of the grain surface and of the gauge surface are of the same order. Thus, in the case of every second strain
author would like to thank Mr B. Brodzitiski, Mr M. Mugosz and Mrs Z. Kusmierczyk for their assistance in carrying out the experiments.
Acknowledgements--The
Received 4 December 1974.
REFERENCES Dantu P. Etude des contraintes dans les milieux hhte’rog&es Application au biton, Publication No 567 de Laboratoire Central
9 03t02
The above considerations are evidently related to the problem of the dispersion of the observations results in rock mechanics. The influence of the measurement base on the magnitude of dispersion of the strain measurements is particularly visible. The value of the global strain fluctuation tells us that the measured result on a definite base represents the average response of the material. Thus we may say that the given base is representative within the accuracy of the corresponding mean fluctuation. In particular, the base determines the size of the tested sample. For instance, considering those parts of the sample disturbed by the platens, for a sample with the slenderness ratio 2 the base of a longitudinal gauge should not exceed the diameter of the sample. In the case of granite described above we have noticed that for a base above L = 60 mm the fluctuation decreases imperceptibly. If we want to determine the elasticity modulus for the linear section of the diagram in Fig. 6, on this base, then the relative fluctuation 9 = 0.03, as we are interested in the increment of strains. This means that the base is representative within an accuracy of kO.03. Having decided to use two gauges on a sample we shall take each of them to be 30mm long. Then the maximum height of the sample with the slenderness ratio 2 will be 60mm. However, if we want to control the bending moment acting on the sample we should determine the plane of the distribution of strains (3). Then we shall need at least 3 gauges each with a 60 mm base. This determines the minimum height of the sample, 120mm. The plane will be defined within the average accuracy of the relative fluctuation of strain 20.03, whereas the elasticity modulus will be defined with much greater accuracy.
-
1 50
.
!
I
I
I
100
L.
mm
Fig. 13. Dependence of the relative global fluctuation on the measured base length.
des Ponts et Chausstes, Paris (1957). Mtiller R. K. Der Einguss der Messlange auf die Ergebnisse bei Dehnungsmessungen an Beton, Messtechnische Briefe fur Elektrisches Messen Mechanischer Grossen Hottinger Baldwin Messtechnik, Nr 3, Darmstadt (1966). Klein G. The Length of Measurement Basis and The Fluctuation Value of Strains in a Loose Medium, Zeszyty Problemowe, Gornictwa, Krakow, 13, No. (1975). Gustkiewicz J. Uniaxial compression testing of brittle rock specimens with special consideration of the effects of bending moment. Int. J. Rock Mech. Min. Sci. 12, 13-25 (1975).