The secant stiffness of mixtures can be calculated at the failure point and constant volume state. The shear modulus at each desired point was determined by the following equation:
𝐺𝐺 = 𝜏𝜏
ϒ (4.2) where G is the secant shear modulus, 𝜏𝜏 is the shear stress and ϒ is assigned with the shear strain. Figure4- 13 demonstrates the variation of shear modulus of sand and other treated sands under six different normal stresses.
The results appear to present three types of behaviour, which are closely related to their rubber content. Such results were expected based on the type of additive applied.
The lowest quantity of rubber added increased the rigidity of the composite. Therefore, the mixtures of the ST5R0 and ST10R0 categories had the highest shear modulus, and
0.00E+00
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the most inflexible composite was established by using the combination of 5% of both slag and cement additions with sand. In contrast, the STR20 mixtures were the most flexible composites in which the minimum shear stiffness for treated sand was with 20% rubber only. The effect of normal stress on the shear modulus behaviour is shown in Figure4- 13. Increasing the normal stress leads to progressing the shear modulus of all the samples. This behaviour illustrates the expected response of the composites depending on their combination of mixtures. Thus, the results of the high-pressure loads clearly show the effectiveness rate of the additives. The difference between highest and lowest shear modulus after applying 1000 kPa normal stress was approximately 1.5E+04 greater that those difference with 50 kPa of vertical load.
Figure4- 13. Variation of shear modulus at the failure point under six normal stresses.
Depending on the rubber content (0%, 10%, and 20%) the mixtures showed a sharp increase, a gentle progress and a very slight improvement tendency, respectively.
Hence, the treatment of sand with the high dosage of rubber (20%) created a composite with the lowest impressionability of vertical load variation. With an insignificant effect of normal stress on STR20 mixtures, they tended to illustrate a linear trend.
0.E+00 5.E+03 1.E+04 2.E+04 2.E+04 3.E+04 3.E+04
50 100 250 500 750 1000
Gf,KPa
ϭ,KPa
1-SC0R0 2-SC0R10 3-SC0R20 4-SC5R0
44-SS5R0 5-SC10R0 55-SS10R0 555-SC5S5R0
6-SC5R10 66-SS5R10 7-SC10R10 77-SS10R10
777-SC5S5R10 8-SC5R20 88-SS5R20 9-SC10R20
99-SS10R20 999-SC5S5R20
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Moreover, as discussed earlier, the flexible composite does not have a particularly low shear strength. Therefore, this behaviour may occur owing to the resistance properties of the rubber and its higher dimension to create a more uniform composite with more connections among particles. Figure4- 14 demonstrates the shear resistance of pure sand and stabilised sand under six different normal stresses. The results explain why the composites with higher quantities of rubber tend to have an elastic behaviour.
In compared with the other mixtures, the composites containing the 20% rubber have failed at the longer shear displacement. In addition, the composites failure has occurred roughly at the same three ranges of shear strain. The STR20 composites reached failure after passing approximately three-quarters of their total shear displacement, which it was approximately half and less than half for STR10 and ST mixtures, respectively.
Figure4- 14. Shear strain of composites at their failure point.
The calculation of secant shear stiffness at constant volume area can be hypothesized that the flexible composite can also have great physical strength. A similar trend was obtained after the stability of shear strength and volumetric strain as shown in Figure4- 15. Comparatively, a similar behaviour at constant volume area was
0.00%
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observed that was not a separate tendency at the variation of normal stress. The only division occurred at 750 kPa and 1000 kPa between STR20 and the other composites, which have the higher shear modulus. This could be happened either owing to the dramatic reduction of shear stress of ST mixtures after the failure or a smaller difference between the strain point of failure and the point that needs to start the constant volume state.
Figure4- 15. Shear modulus at constant volume.
To analysis, the above hypothesis, the difference between peak and constant volume shear stress was calculated. The results revealed an enormous reduction of the shear stress of sand treated with stabilisers after reaching the failure point, Figure4- 16.
The composites of the ST group lost their strength ability up to 181 kPa, which means the creation of rigid composites. This remarkable drop was modified by the addition of 10% of rubber to around 20 kPa. In the same manner, the minimum change was achieved by SC5S5R20, which slightly condensed about 5 kPa. Therefore, it can be expected that the samples with the addition of both rubber and stabilisers have a higher amount of shear stress than other mixtures at the constant volume state. Hence,
0.E+00 2.E+03 4.E+03 6.E+03 8.E+03 1.E+04 1.E+04
50 100 250 500 750 1000
Gcv,KPa
ϭ,KPa
1-SC0R0 2-SC0R10 3-SC0R20 4-SC5R0
44-SS5R0 5-SC10R0 55-SS10R0 555-SC5S5R0
6-SC5R10 66-SS5R10 7-SC10R10 77-SS10R10
777-SC5S5R10 8-SC5R20 88-SS5R20 9-SC10R20
99-SS10R20 999-SC5S5R20
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by considering the higher strain resistance properties of the composites of group STR10 and STR20, and the minimum reduction of shear stress after failure, the result of Figure4- 16 has been verified.
Figure4- 16. The difference between shear stress at the failure and shear stress at the constant volume state.
From all the obtained results to this point, it can be hypothesised that the relationship between the minimum void ratio of composites and its elastic modulus parameters might be corrected. This hypothesis was found to be true because of the creation of a denser composite resulting from the reduction of the matrix void ratio.
Also, the decreasing of the matrix void ratio was also observed, when the shear modulus increased with the increment of normal stress. This phenomenon is observed again owing to increasing the inter-particles friction when applying higher vertical loads.
The composites with the addition of both rubber and stabilisers tended to have a stabilised behaviour. This behaviour is obvious in the STR20 groups, which generated
0 20 40 60 80 100 120 140 160 180 200
The difference of shear Stress (kPa)
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the most flexible composites and the minimum change after the failure, with noticeable strength properties.
To create an accurate study about the constant volume state, the vertical strain of composites needs to be analysed.