This letter presents data from triaxial tests conducted as part of a research programme into the stress–strain behaviour of clays and silts at Cambridge University. To support findings from earlier research using databases of soil tests, eighteen CIU triaxial tests on speswhite kaolin were performed to confirm an assumed link between mobilisation strain (γ_{M = 2}) and overconsolidation ratio (OCR). In the moderate shear stress range (0·2c_{u} to 0·8c_{u}) the test data are essentially linear on log–log plots. Both the slopes and intercepts of these lines are simple functions of OCR.
Knowledge of soil stiffness and stress–strain behaviour is essential to the calculation of ground displacements that may damage structures. These serviceability considerations, termed SLS in Eurocode 7 (BSI, 2010), should be at the forefront of the geotechnical practitioner's mind. For example, deformations are important in the design of offshore wind turbines, both in terms of dynamic structural response under severe loads and due to the vulnerability of the drive and gearbox to tilting of the mast.
Research has been undertaken at Cambridge University to define and validate simplified mechanistic models in conjunction with soil stress–strain data to enable routine calculations of footing settlements (Osman & Bolton, 2005; Osman et al., 2007) and the displacement of braced retaining structures (Osman & Bolton, 2006; Lam & Bolton, 2011), for example. The calculation procedure is based on conservation of energy and is known as mobilisable strength design (MSD).
An important feature of MSD is the need to model the strength mobilisation of the soil. The shear stiffness of clays and silts at small strains has been shown to be empirically determinable using the maximum shear modulus (G_{0}) and a quasihyperbolic stress–strain relation in which the shear strain required to halve the stiffness was seen to vary with the liquid limit (w_{L}) (Vardanega & Bolton, 2011a). This letter presents measurements of stress versus strain for kaolin clay for various stress histories and for stress levels approaching failure, in undrained CIU triaxial compression tests. A summary of the basic kaolin parameters from the present study is shown in Table 1.

Plastic limit w_{P}: %  29·6 [4] 
Liquid limit w_{L}: %  62·6 [1] 
Slope of normal compression line λ  0·250 [4] 
Slope of unload–reload line κ  0·039 [6] 
During triaxial compression, the axial stress is increased while keeping the cell pressure constant. An undrained test maintains constant volume, allowing excess pore pressures to develop. Conventional triaxial testing methodology is outlined in Bishop and Henkel (1957).
In the triaxial tests, an external linear variable differential transformer (LVDT) measures the overall movement of the sample (used to capture the strain data) to an accuracy of 0·125 mm. A strain accuracy of 10^{−3} is sufficient to capture the influence of the OCR on the moderate stress region (defined in the next section), which is the aim of this letter.

Test ID  OCR  
180(1)  180  180  1 
180(2)  180  90  2 
180(5)  180  36  5 
180(10)  180  18  10 
180(20)  180  9  20 
300(2)  300  150  2 
300(5)  300  60  5 
300(10)  300  30  10 
300(15)  300  20  15 
300(20)  300  15  20 
500(2)  500  250  2 
500(10)  500  50  10 
500(15)  500  33·3  15 
500(20)  500  25  20 
500(1)_Xu  500  500  1 
500(2)_Xu  500  250  2 
500(5)_Xu  500  100  5 
500(10)_Xu  500  50  10 
The stress–strain curves from triaxial testing are sensibly linear over a range of moderate stresses when the data are plotted on log–log axes.

Test ID  A  b  R^{2}  n  c_{u}: kPa  e_{0}  γ_{M}_{ = 2} 
180(1)  5·861  0·484  0·850  59  51·7  1·15  0·00521 
180(2)  3·531  0·425  0·938  79  38·4  1·27  0·00880 
180(5)  2·825  0·443  0·999  102  32·0  1·12  0·01969 
180(10)  3·882  0·595  1·000  115  19·9  1·29  0·03201 
180(20)  3·572  0·530  0·999  139  15·9  1·21  0·02506 
300(2)  3·311  0·356  0·980  52  68·6  1·23  0·00445 
300(5)  3·034  0·389  0·998  110  45·6  1·17  0·00988 
300(10)  3·681  0·540  1·000  140  30·6  1·21  0·02488 
300(15)  3·733  0·489  0·991  194  34·2  1·08  0·01737 
300(20)  5·082  0·580  0·995  131  27·5  1·19  0·01924 
500(2)  4·227  0·423  0·926  50  105·1  1·13  0·00530 
500(10)  3·732  0·460  0·991  131  57·5  1·18  0·01337 
500(15)  3·750  0·584  0·998  159  36·4  1·14  0·03265 
500(20)  3·350  0·602  0·997  200  32·8  1·20  0·04389 
500(1)_Xu  3·236  0·311  0·987  45  93·4  n/a  0·00261 
500(2)_Xu  2·972  0·377  0·991  96  88·1  n/a  0·00815 
500(5)_Xu  1·879  0·291  0·998  96  69·7  n/a  0·01121 
500(10)_Xu  3·192  0·470  0·998  133  56·0  n/a  0·01978 
Based on data collected by Mayne (1980) and presented by Muir Wood (1990), Λ varies between 0·2 and 1·0 with a mean value of 0·63 and a standard deviation of 0·18. This is a significantly greater range than would be implied by equation (5).
Using the values from Table 1 in equation (6), it would be expected that Λ = 0·84, although Muir Wood cautions that it is difficult to determine a reliable value of κ from the mean slope of a swelling line. From Fig. 2, the value of Λ is shown to be 0·68 (when the regression is forced through the origin, as implied by equation (5)), which is slightly lower than the theoretical value; however, it is similar to the mean of previously collected experimental data (Mayne, 1980).
The general form of Ladd's relationship is shown to fit the test data well. This allows one to conclude that the c_{u} values computed from the test data are not unreasonable.
In the previously published database, b for the 115 tests in the database was shown to range from 0·3 to 1·2 with an average of 0·6.
for which R^{2} = 0·591, n = 18, SE = 0·064 and p < 0·001.
Figure 5 shows a plot of τ_{mob}/c_{u} as predicted using equation (8) for the exponent and equation (10) for the mobilised shear strength ratio, versus corresponding measurements. The R^{2} on the plot is 0·96 and the slope is very close to 1·0 (0·99), which validates the use of the two equations in tandem. This level of accuracy is only attained if the mobilisation strain (γ_{M = }_{2}) is known precisely.
Figure 7 shows the predicted versus measured plot when equations (8), (10) and (12) are used to predict τ_{mob}/c_{u}. The error bands widen to around ±40% due to the scatter about the trend line in Fig. 6. Using equations (8), (10) and (12), the predicted stress–strain curves are drawn in Fig. 8. Similar behaviour is shown in the data of Todi clay presented by Burland et al. (1996) and analysed in Vardanega & Bolton (2011b).
The implication for geotechnical design is that less strain is needed to mobilise the same proportion of shear strength the deeper a geotechnical structure is built. For a bored pile in overconsolidated clay, for example, the soil in contact with the shaft at the top of the pile is likely to be significantly more compliant than the soil in contact with the base.
If the pile head settlement at working load is to be calculated using a t–z analysis, one must make assumptions about the variation of t–z spring behaviour with depth. If a designer assigns a single value of G/c_{u} for the soil, this implies a single strain to failure at all depths. This letter has shown that such an assumption would be unwarranted.
This letter has focused on establishing a link between mobilisation strain and stress history. The following summary points and conclusions are made.

Data from 18 CIU triaxial tests on reconstituted kaolin samples confirm that the stress–strain curves (in the moderate stress region) are roughly linear on log–log plots.

The mobilisation strain framework presented by Vardanega & Bolton (2011b, 2011c) is verified for reconstituted kaolin. A simple stress–strain model for kaolin is
where b = 0·011(OCR) + 0·371. The average exponent b recorded by Vardanega & Bolton (2011b), for natural clays of unknown OCR, was 0·6 within a range of 0·3–1·2. This is not inconsistent with the current data of these tests on reconstituted kaolin.$${\tau}_{\text{mob}}\text{/}{c}_{\text{u}}=0\xb75{\left(\frac{\gamma}{{\gamma}_{M=2}}\right)}^{b}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{range}\text{\hspace{0.17em}}0\xb7\text{2}<{\tau}_{\text{mob}}\text{/}{c}_{\text{u}}<0\xb7\text{8}$$ 
The mobilisation strain γ_{M}_{ = 2} is shown to increase strongly with the logarithm of overconsolidation ratio via the following relationship for kaolin
$${\gamma}_{M=2}=0\xb700\text{4}0{\text{(OCR)}}^{0\xb7680}$$ 
Just as OCR has been previously found (equation (1)) to influence smallstrain stiffness, so it has now been demonstrated to influence both the position and slope of stress–strain curves of clay in the region of moderate strength mobilisations, when plotted on log–log axes.
Thanks are due to Mr Chris Knight for his technical services, to Miss X. Xu (former MEng student at Cambridge University) for providing her triaxial data for analysis, to the Cambridge Commonwealth Trust and Ove Arup and Partners for their financial support of the first author, and to the EPSRC for financial support from grant EP/H013857/1 ‘Cyclic behaviour of monopile foundations for offshore windfarms’.
A  regression coefficient 
b  exponent determined from regression analysis 
CIU  consolidated isotropic undrained 
c_{u}  undrained shear strength 
c_{u}  consolidated isotropic undrained 
d  exponent determined from regression analysis 
e_{0}  initial void ratio 
G  shear modulus 
G_{0}  maximum shear modulus 
I_{p}  plasticity index 
M  mobilisation factor c_{u}/τ_{mob} 
m  exponent determined from regression analysis 
n  number of data points used to generate a correlation 
OCR  overconsolidation ratio 
p  the smallest level of significance that would lead to the rejection of the null hypothesis, i.e. that the value of r = 0, in the case of determining the pvalue for a regression 
p_{a}  atmospheric pressure 
maximum effective consolidation pressure  
mean stress in the triaxial after swell back  
q  deviator stress 
R^{2}  coefficient of determination of a correlation (the square of the correlation coefficient r) 
S  regression coefficient 
SE  standard error in a regression, a quantification of deviation about the fitted line 
w_{L}  liquid limit 
γ  shear strain, taken as 1·5 times the axial strain (ε_{a}) in this letter 
γ_{M}_{ = 2}  mobilisation strain 
ε_{a}  axial strain 
κ  slope of unload–reload line 
Λ  exponent in the equation of Ladd et al. (1977) 
λ  slope of normal compression line 
vertical effective stress in the ground  
maximum past effective vertical stress in the ground  
τ_{mob}  mobilised shear stress 