Table 4.1 ) are presented in Figure 4.13 in the same form as the previous series. It can be seen that the results for the two tests are identical.
These results were further compared to the results o f shear tests using the Biaxial Tester ( M. Jamebozorgi, 1997 ) on samples o f the same material with the same sample preparation method, same stress path and at the same stress level. Figure 4.14 shows
that data obtained in the Biaxial Tester corresponds well to that o f Tests 1.2.1 and l.l.a . Failure results in the Biaxial Tester showed higher ( c i - C ] )/40 than those o f the DSC. This could be due to the fact that the very last recording o f stresses and strains in the Biaxial Tester results corresponds to a state o f the sample after the onset o f failure planes where the last recording in the DSC corresponds to the state ju st prior to onset o f failure planes and after failure no stress-strain response could be recorded in DSC testing. Biaxial Tester measurements o f strain were from external dial gauges. It seems that DSC tests 1.2.1 and l.l .a reliably represent behaviour o f this powder under shear.
Further confirmation o f the apparatus can be found when comparing other workers results in which measurements o f the intermediate principal stress was taken.
The intermediate principal stress measurements from Test 1.2.1 are now considered and are shown in Figure 4.15.
Prior to biaxial loading the sample had already been loaded in the G2 direction to a
stress o f (jJ2=20 kpa during sample preparation. Starting with such a sample a biaxial consolidation (AB) led to a state in which = ^2 = 0 2 at 10 kpa. It can be seen that
during the remaining part o f the biaxial consolidation stage, the principal stress orthogonal to the plane o f strain ( termed G2 ) was the smallest principal stress.
Stresses are also shown in Figure 4.16 where stresses were plotted against resulting strains.
Figure 4.16 shows that up to 10 Kpa the sample was behaving as an over consolidated material. After increasing the biaxial stresses above 10 Kpa strains develop and plastic deformation results. This is seen more clearly in Figure 4.17.b, the volumetric strain
during biaxial loading and unloading (Figure 4.17.a) are plotted which is used to show the virgin consolidation line.
At point B the shearing process started in which a j and Cg were increased and decreased respectively, keeping ( Gi + G3 )/2 = 40kpa constant. As shearing continued Figure 4.15.b shows that (jj remained constant. A switch o f the intermediate and minor principal stresses took place at approximately a2~c>3=25 kpa which was at an early stage o f the shearing process and occurred when the minor principal strain began to develop. At this stage strain is close to one dimensional consolidation. The switch is also marked by an arrow in Figure 4.13 where strain during shear prior to the switch is very small. This switch in Uj directions can not be prevented and will always occur during shear.
These results are comparable to the results o f other researchers eg J. Harder and J. Schwedes (1985) and L.P.Maltby (1993) who also used Biaxial Testers to shear cubic Calcite powder samples.
Harder and Schwedes used a rigid boundary biaxial tester where stress orthogonal to the plane o f strain (02) was measured using a special load cell. Samples were prepared in layers and were further consolidated along the deposition direction. This sample preparation method is very similar to the method employed by the author in this study. Starting with such a sample, biaxial consolidation in the plane o f strain o f rigid boundary Biaxial Tester led to a sample state o f stresses where to CTi= C73=cr2- Harder and Schwedes termed this the hydrostatic state. This hydrostatic state in the authors results corresponds to a state where a i= a3=(j2=10 kpa. A complete state o f stress on the samples during shearing was not given and further comparison could not be made.
A more through comparison could be made from results obtained by L.P.Maltby, 1993.
Maltby used a flexible boundary Biaxial Tester and measured the stress orthogonal to the plane o f strain (0-2) using a load cell placed in the top plane strain platen o f the apparatus. Samples were prepared by depositing loose powder into the cubic sample space. These samples were not prestressed in G2 direction during sample preparation. Data obtained from a test in which calcite powder sample was subjected to the stress path ( Figure 4.2.d ) to kpa is shown in Figure 4.18. The curves show the principal stresses magnitudes in X, Y and Z directions as Gy and These symbols could be matched to those used by author Gj, G3 and G2 respectively.
Biaxial loading o f the Calcite powder sample showed that Gz was always the smallest principal stress. Here no over consolidation stresses existed. From point D shearing started, G% was increased and Gy reduced keeping ( Gx + Gy )/2=10 kpa constant. It can be seen that Gg remained constant during shearing ( D to DI ). A switch o f the intermediate and the minor principal stresses took place where Gy = Gz = 6 .6 kpa. He stated that when the sample was deforming at almost constant volume the intermediate principal stress was related to the major and minor principal stresses by
Gz= 0.35 ( Gx + Gy )
His figure shows that this relationship applies throughout the shearing process. The author data from Test 1.2.1 ( Figure 4.16) gives
G2= 0.3 1 ( Gj + G3 )
This small difference could be due to the difference in material tested. Maltby used calcite powder whereas material used here is "dry kaolinite powder". The difference could be due to the measurement o f intermediate principal stress in the Z direction . A
load cell was used at the top plane strain face o f the Biaxial Tester whereas a constant volume water filled plane strain bag was used in the DSC.
Testes 1.2.2 and 1.2.3:
These tests are described in Table 4.1. The stress-strain response o f the powder samples during shearing are presented in Figure 4.19.
It can be seen that the results corresponding to Test 1.2.2, where shear sheets were placed around the sample but not used to apply shear stresses to the faces o f the sample, showed softer response in the stress - strain curve but were quite dilatant. These results are also compared to Tests l.l .a and 1.2.1 ( Figure 4.19 ). It can be seen that results o f the Tests l.l.a , 1.2.1 and 1.2.2 where no shear sheets were placed are identical and were taken to be representative of the behaviour of dry Kaolinite powder when tested in the DSC. This comparison also indicated that slight differences between stress paths o f Figure 4.9.a and Figure 4.9.b had no influence in the results obtained from the shear tests. This was because Biaxial consolidation components o f both o f these stresses paths caused the samples to undergo the same volumetric compression prior to shear (Figure 4.17.b).
The only difference which was appreciable when comparing results o f Tests l.l.a , 1.2.1, 1.2.2 and 1.2.3 (Figure 4.19), was the effect o f having the shear sheets. This test ( 1.2.3 ) was repeated several times and results were identical to the one presented in Figure 4.19.
Close observation o f radiographs o f samples surrounded by the shear sheets at large strain showed the general deformed shape illustrated in figure 4.20. It seems that in tests with shear sheets the major principal stress bags were inclined to punch into the
sample. A similar process o f outward punching took place in the minor principal stress direction. These punching effects of normal pressure bags resulted non uniform strain and a softer response to be recorded.
This non uniform deformed shape was not observed in radiographs o f the samples in the tests where no shear sheets was used.
As this punching effect could not be cured it was decided that in constant direction monotonie shear tests where v|/= 0° or 90° only, placing o f the shear sheets should be avoided as these tests could better be performed without the shear sheets. It is shown in chapter 5 that with tests which included a change in major principal stress direction during shearing that, provided the first loading was not with \|/= 0° or 90° , then the shear sheets had little influence when subsequent loadings were in these directions.
4.6.2- Results of Series 2 :
Stress-strain data during shear for Series 2 tests ( \|/= 15°, 30° and 45° ) are shown in Figure 4.21. The curve obtained from results o f the Tests 1.2.1, l .l .a and 1.2.2 {\\f =
0°) which was taken to be the representative behaviour o f this powder under shear are also included. There should have been no differences to the principal stress-strain data obtained from these tests ( v|/= 0°, 15°, 30° and 45° ) as all samples followed the same sample preparation procedure and the same stress path. The results should have produced a unique stress-strain curve .
It can be seen (Figure 4.21 ) while data relating to \|/=15° corresponds very well with that for \|/=0° the data for vj/= 30° and 45° do not correspond with that for v|/= 0° and 15°. It was thought that in tests with \\!= 30° and 45° the intended applied boundary shear
result from interlocking o f the two adjacent pairs o f shear sheets at the points o f emergence ( Figure 4.22.a ).
Up to this stage the major components o f the DSC remained unchanged to that described in Chapter 2. The apparatus was suitable for sand testing, but alteration to the shear sheets proved to be necessary for successful testing o f powder in the DSC. It is important to question why this problem was not detected during tests on sand samples. It is thought that the lock up is a result o f the combined effect o f volumetric compression and increasing shear load supplied by the pistons to the shear sheets. Almost no volumetric compression accrued during shearing o f sand samples whereas powder samples showed further volumetric compression during shear. The effect o f this volumetric compression coupled with increasing shear load supplied by the pistons especially for tests requiring high shear stresses ( i|/= 45° and 30° ), on performance o f the apparatus could result in lock up at the point o f emergence o f the shear sheets ( Figure 4.22.a ).
The greatest change in sample shape is when \\f is 45° and the sample changes from square to diamond (Figure 4.22). The volumetric compression coupled with the greatest increasing shear load supplied by the pistons caused interlocking o f the shear sheets where they emerge. In addition it was noted that the tails o f the shear sheets (Figure 4.22.a ) at the back end o f each shear sheet became bent. It was thought that they could reduce normal and shear stress at the back ends o f the sample.
Therefore alterations were made to the shear sheets. Cuts o f very short length were made through the shear sleeves at the points o f emergence ( Figure 4.22.b ) and also the tails at the back end o f each non-stretching pulling strips were shortened. Figure 4.23 shows the shear sheets before and after modification.
After alteration, tests of Series 2 w ere conducted again. Figure 4.24 shows the new data relating to different constant direction monotonie shear tests. All results were very close and a single curve drawn through the data could quite well describe the stress-strain behaviour. Close inspection at failure showed higher (cTi-a3)/40 for v|/=0° which reduced with increasing v}/ to a minimum for \|/=45°.
This difference could be explained with the aid o f Mohr circles at failure and comparing a test with i|/=0° where a j and p, were zero (Figure 3.16) with the test at \|/=45° where and pi were maximum and equal. Angles a j and pj were measured from radiographs o f samples for \|/=45° just before failure and showed ai=Pi=3°. Figure 4.25 illustrates the Mohr circle at failure for the test with v|/=0° as circle (a) . For the test with n/=45° when a i and p, were not taken into account is shown by circle (b). Circle (c) was drawn after correction to the applied stress for circle (b). It can be seen that after the adjustment the Mohr circle for test with \j/=45° becomes tangential to (|)cv=41° similar to the test with v)/=0°.
The uniqueness o f the curve shown in Figure 4.24 could be further confirmed from strain distribution within the samples and by comparing the assumed principal stress directions with the measured strain increments directions. Coincidence o f the two confirms that correct combination o f stresses are applied. This procedure was described in Chapter 3.
The uniformity o f strain within a sample sheared in the DSC could be evaluated from strain data calculated from deformation o f the tungsten grid (Figure 3.13) (Arthur and Wong, 1985). A uniform distribution o f strain suggests that samples have been
given in Figure 4.26 and illustrates the magnitudes and distribution o f major principal strains measured in a sample sheared to (cT|-a3)/40= 10 with i|/=45^. Each printed strain in the Figure 4.26 represents the strain between four adjacent markers (Figure 3.13). The average for the zones defined in Figure 3.14 are also presented in Figures 4.26. It can be seen that the different zones within the samples strained to an approximately mean strain o f 4.6%. This is an indication o f uniformity o f the deformation within the sample.
As was described in Chapter 3 a better evaluation o f the strain distribution and assessment o f scatter in data could be made using the coefficient o f variation, a parameter which is defined as the standard deviation divided by mean strain.
Figures 4.27 illustrates the coefficient o f variation o f the major principal strain for tests where samples were sheared monotonically under different principal stress directions ( \|/= 0°, 15°, 30° and 45° ) using strain data from Area 2 (central 25%) o f the sample . In this figure the coefficient o f variation, corresponding to the system accuracy, is shown by the hatched zone. The system accuracy is based on a standard deviation o f 0.10% (Chapter 3).
The coefficient o f variation is dominated by the measurement uncertainty at major principal strain magnitudes less than 0.5% and only when the measured strain is greater it is controlled by the sample behaviour as the curve obtained from data follows the system accuracy curve shape up to a strain o f 5%. As the strain proceeds the coefficient o f variation starts to rise again with larger deformation . This is an indication o f increasing non-uniformity in the sample as it approaches failure. It can be noticed all the tests exhibited almost the same degree o f uniformity although the principal stress direction was different for each test.
The principal stress directions in each test were calculated from the assumed applied boundary stresses. The directions o f principal strain increments were found from the displacement o f the markers on successive radiographs (Chapter 3). There should be coincidence o f axes o f stress and strain increment direction for all samples as they were prepared at right angles to the direction o f the plane o f strain and were tested with a fixed direction o f major principal stress. Figure 4.28 shows the calculated applied principal stress direction and the measured principal strain increment direction. The deviation between them is defined as A.
Coincidence o f the axes A=0° implies that the sample was subjected to the correct combination o f normal and shear stresses. Figure 4.28 shows the data obtained from tests \\f= 0°, 15°, 30° and 45° . It can be seen that the deviation o f the measured strain increment direction and applied principal stress direction are within ±2°. As this deviation is small, it could be concluded that coincidence o f the axes were achieved during tests.
The monotonie tests data presented in this section (Figure 4.24), illustrates the stress and strain curves o f dry powder samples sheared with different fixed orientations o f major principal stress direction. Computation o f strain o f small elements within the samples provided means o f assessing the uniformity o f strain and coincidence o f axes o f stress and strain increment to be examined for all tests.
These results (Figures 4.24, 4.27 and 4.28) showed that the developed test procedure and sample preparation m ethod resulted a unique stress- strain curve to be achieved.
L o a d in g d ir e c tio n during sa m p le preparation 'l a n e of S tr a i n Y P la n e of S tr a i n P la n e of S tr a i n 03 .^ 2 (b)
Figure 4.1: (a) Plane of strain in the samples (b) Direction of principal stresses
e w CO CD (I) 03 CD s z (D a 1 a'3 a N o r m a l S t r e s s a (a) CO CO 2 œ CD JZ CO Ga = G, = G =a^
Biaxial C on solid ation s tr e s s path
(b) N o r m a l S t r e s s g -1 s tr e s s path A a T Vertical s tr e s s path A a (c) (d)
Figure 4.2: (a) M ajor and m inor principal stresses on an element
(b) Biaxial consolidation stress path (c) Stress path -1 slope (d) V ertical stress path
(a)
(b)
Figure 4.3: (a) Hollow acrylic sample preparation box (b) Pouring powder into the box
(a)
(b)
(a)
(b)
Figure 4.6: (a) Gently squeezed norm al pressure bags (b) Sample after biaxial consolidation and prio r to positioning the shear sheets around it
(a)
(b)
Figure 4.7: (a) Loosely plaeed shear sheets and placing the reinforced edges (b) DSC p rio r to placing the top plane strain platen
40 62 a
(a)
S a fjn p le
Markers at middle a n d lower pla n es in th e s a m p l e for te s t 1.1 .a.
(b)
Figure 4.8: (a) Stress paths for Series 1.1 tests.
(b) Positions of the t>vo grids of m arkers for test with a<.=40 ( kpa)
T e s t No S t r e s s Path S h e a r s h e e t s T o p platen No No Y e s 0 , - 4 0 K p a Y e s No No 1.2.2 No Y e s Y e s .IS Kpa 1.2.3 No No Y e s CT. - 40 Kpa
X A a = 40 Kpa (a) X A 3 5 K pa a = 40 Kpa c (b)
F ig u re 4.9: Schem atic p resen tatio n of the stress paths fo r tests (a) 1.2.1 and 1.1.a (b) 1.2.2 an d 1.2.3