A semi-empirical relationship for the small-strain shear modulus of soft clays

The small-strain shear modulus (Gmax) is a soil property that has many practical applications. The authors compiled a database of Gmax measurements for 40 normally consolidated to slightly overconsolidated low to high plasticity clays. Using these data, the authors propose a semiempirical relationship between Gmax, effective stress ('v or 'c), preconsolidation stress ('p) and insitu void ratio (e0) for four ranges of plasticity index (Ip): Ip < 30%, 30% ≤ Ip < 50%, 50% ≤ Ip < 80% and 80% ≤ Ip < 120%. With results from bender element tests on a Gulf of Mexico clay subjected to multiple load-unload consolidation loops, the authors were able to validate the proposed relationships for 30% ≤ Ip < 50% and 50% ≤ Ip < 80%. The proposed relationship for 30% ≤ Ip < 50% and 50% ≤ Ip < 80% captures changes in laboratory Gmax resulting from variations in effective stress ('c), maximum past stress ('v,max), and void ratio. The proposed relationships are a simple and efficient tool that can provide independent insight on Gmax if the stress history of a clay is known, or on stress history if Gmax is known.


Introduction
The small-strain shear modulus, Gmax, refers to the ratio of shear stress to shear strain at strains < 10 -3 %. It is a fundamental soil property that has many practical implications. For instance, the proper evaluation of Gmax is a pre-requisite for reliably assessing stress-strain relationships of natural soils (e.g., [1]), specimen disturbance (e.g., [2]), ground movement (e.g., [3], [4]), site response analysis (e.g., [5]) and numerical modelling (e.g., [6], [7]) Owing to the development of techniques to measure Gmax in the laboratory (e.g., resonant column, torsional shear, bender elements integrated in oedometer and triaxial setups) and in the field (e.g., sCPTu, cross-hole, MASW), a significant number of research works focusing on the Gmax of both coarse-grained and fine-grained soils has been undertaken in the last couple of decades. This paper focuses on the Gmax of soft clays.
Based on the extensive works done on the Gmax of soft clays, spanning almost 60 years ( [8] - [19]), our understanding of the controlling factors has been sharpened. These include effective stress, void ratio, plasticity index, preconsolidation stress, anisotropy (inherent and stress-induced), age of deposit, degree of saturation, cementation, thixotropy, and mineralogy.
Some researchers have attempted to provide a unifying framework pertaining to the evaluation of Gmax of clays ( [20] - [22]). However, it is quite challenging to propose a framework that considers all the abovementioned factors. This paper provides a framework that takes into account effective stress, 'c, preconsolidation stress, 'p, void ratio, e0 and plasticity index, Ip. The framework is based on a database consisting of 40 different clays having a range of Ip and tested both in-situ and in the laboratory. The resulting semi-empirical correlations are compared with the results of bender element tests run on a Gulf of Mexico (GOM) clay mounted in a triaxial apparatus.
2 Relating G max , ', ' p , e, and I p in clay The small-strain shear modulus of a saturated clayey soil is controlled by the stiffness of the matrix, which depends on effective stress, preconsolidation stress, void ratio, stress history, strength anisotropy, plasticity index and mineralogy (among others). Quantifying the Gmax of a clay matrix is a complicated task for two reasons: 1. Gmax is dependent to varying degrees on a number of factors; 2. A significant number of those factors are interdependent. For example, higher consolidation stress corresponds to smaller void ratio; higher preconsolidation stress corresponds to smaller void ratio; higher plasticity index corresponds to smaller particle size and larger initial void ratio; and so on. Clay structure is primarily influenced by effective stress, preconsolidation stress, plasticity index, and void ratio (e.g., [23]), and from a practical point of view, it is relatively easy to quantify these properties. Therefore, a relationship between Gmax, ', 'p, e, and Ip is both justified and practically useful.

Why is Gmax a function of both ' and void ratio?
Small-strain stiffness depends on how densely packed the soil particles are. The closer the particles, the denser the fabric, the less likely it is to deform in shear, and the larger the stiffness. For a clay, both the effective stress and the void ratio control the fabric. The larger the effective stress or the smaller the void ratio, the closer the particles are packed in the soil matrix and the higher the stiffness ( [9]). Therefore, the relationship shown in Equation 1 is valid:

Why is Gmax a function of 'p?
The undrained shear strength su of a clayey soil depends on its 'p irrespective of Ip and confining pressure (e.g., [24]). The undrained shear strength of a clayey soil often is related to its Young's modulus, E, (e.g., [25]) because the higher the shear strength, the steeper the initial stressstrain response and hence the larger the value of E. The shear stiffness being a function of E, it follows that Gmax is a function of su and hence 'p. Conceptually, the higher the preconsolidation pressure, the more contacts exist in a clay matrix and the stiffer it is. Therefore, the relationship shown in Equation 2 is valid: From an extensive study of Norwegian clays, [22] showed that 'p correlated well with the square of the shear wave velocity, Vs 2 , with the coefficient of variance R 2 being 0.80. Thus, 'p correlates well with Gmax.

Why is Gmax a function of Ip?
Clayey soils with high plasticity are predominantly composed of thin, flexible plate-shaped particles with larger relative surface areas, to which a large volume of water is adsorbed. As a result of inter-particle forces, those clays tend to have a flocculated open structure characterized by many face-to-edge contacts between particles and/or aggregated particles. The open fabric of those clays implies a higher compressibility and hence a smaller stiffness. Therefore, a higher plasticity index implies a smaller shear stiffness (e.g., [11], [13]) as suggested by Equation 3.
Therefore, it follows that the functional form of Gmax given by Equation 4 should be valid.
where k, m, n, o and p are fitting parameters. Based on the above, and with further reasoning (outside the scope of this paper), the functional form of Gmax shown in Equation 5 was deduced. The parameter A is a function of plasticity index and will be discussed later.
3 Description of the tests on GOM clay

Apparatus and bender elements
The testing equipment, manufactured by Wykeham Farrance/ControlsGroup, consisted of a computercontrolled servo-pneumatic system, designed to perform both monotonic and cyclic triaxial tests. The main components of the equipment are a triaxial loading frame, a cell pressure source, a back-pressure source, a PWP transducer, a data acquisition system, a computer and top and bottom caps equipped with bender elements, which are powered by wave form signal generator. A bender element is a piezoceramic electromechanical transducer, which can convert mechanical energy (movement) either to or from electrical energy. It consists of two thin piezoceramic plates, which are rigidly bonded together with conducting surfaces between them and on the outsides. The polarization of the ceramic material in each plate and the electrical connections are such that when a driving voltage is applied to the element, one plate elongates and the other shortens. The net result is a bending displacement, which is greater in magnitude than the length changes in either of the two layers. This bending displacement produces a shear wave that propagates parallel to the length of the element into the soil specimen. On the other end of the specimen, upon receiving the shear wave, the bender element is forced to bend as one piezoceramic plate goes into tension and the other into compression. This results in an electrical signal that can be measured.
For the setup adopted for the tests described herein, the bender element integrated in the top cap was the sender and the bender element in the bottom cap was the receiver. The time interval between sending a signal (tsent) and receiving it (treceived) and the distance between the bender elements (L) are used to calculate Vs and subsequently Gmax, as shown in Equations 6 and 7, respectively. Note that L is the distance between the bender elements, i.e., the length of the specimen minus the protrusions of the bender elements from the caps (protrusionBE ≈ 17.60 mm). The density of a saturated specimen, T, can be obtained from the void ratio, e, specific gravity, Gs and density of water, w, as shown in Equation 8.
The apparatus also consists of a cell pressure source and a back-pressure source, which are two air-pressurized cylinders that rely on an air-water interface to generate the desired pressure (up to 1000 kPa). The data acquisition system is capable of monitoring displacement, cell pressure and back pressure and PWP independently, and measurements are recorded by a Windows-based computer, which is used to send signals to the equipment via servo-valves. The PWP transducer records data at the base of the specimen with an accuracy of 0.1 kPa.
Specimens of GOM clay were prepared, equipped with bender elements and subjected to equal all-around consolidation stress beyond their in-situ 'p. At various consolidation stresses, with the instantaneous length of the specimen and with the instantaneous void ratio known, measurements of Vs were taken. Thus values Gmax corresponding to a given set of 'c, 'v,max and e and involving at least one unload-reload loop could be obtained.

GOM clay and specimen preparation
The GOM clay used was sampled during a May 2012 commercial cruise with a Jumbo Piston Core (JPC) sampler from about 3000 m below sea level in the Walker Ridge area of the Gulf of Mexico basin. Details about the JPC sampling technique can be found in [26]. JPC sampling yields about 30 m of soil enclosed in a PVC tube having an internal diameter of 4 inches (102 mm) and an external diameter of 4.5 inches (114 mm). After sampling, the tube is cut into smaller 0.9 m-long tubes, sealed at the ends and transported to the laboratory for testing.
The JPC tubes selected for testing, originating from a depth of 2.5 m to 15.5 m below the seabed, were carefully cut at two locations to the desired length using a rotating tube cutter. The tube cutter was turned slowly while applying gentle pressure such that there was no visual evidence of damage. A wire saw was passed through the two cuts and the clay was extruded in the same direction as sampling. Following extrusion, the GOM clay samples were trimmed with a wire-saw to an average diameter of 70.3 mm and an average height of 145.0 mm (average H/D = 2.1). Each specimen was mounted on the bottom cap (equipped with a bender element) and equipped with a top cap (with bender element), vertical filter paper side drains and a latex membrane (thickness = 0.30 mm). The specimens then were carefully mounted in the cyclic triaxial apparatus, back-pressure saturated (B-value > 0.97) and consolidated under equal all-around stress to the virgin compression zone. Table 1summarizes the index properties of the GOM clay and some details about the testing program. Specimen GOM BE1 was first consolidated under equal all-around stress to 34 kPa ('c = 34 kPa, 'v,max = 34 kPa). The confining stress 'c was increased in steps to 155 kPa ('c = 155 kPa, 'v,max = 155 kPa), then decreased to 73 kPa ('c = 73 kPa, 'v,max = 155 kPa), then increased to 314 kPa ('c = 314 kPa, 'v,max = 314 kPa), following which it was decreased to 81 kPa ('c = 81 kPa, 'v,max = 314 kPa). Thus, GOM BE1 had 2 unload-reload consolidation loops, during which 18 measurements of Vs were taken.

Bender element test results
The results of the 4 bender element tests are shown in Fig.  1 in a Gmax vs. 'v,max x ('c/e) space. It can be seen that all 4 test results fall within a relatively narrow band. This is so because all the specimens had relatively similar plasticity index. Based on the plot, a case can be made for the validity of Equation 5; for the range of Ip tested, the data support the form of Equation 5. It can be seen that Gmax of GOM clay is a function of 'v,max x ('c/e) regardless of whether the clay is normally consolidated or overconsolidated.

Database and proposed relationships
To validate Equation 5 for a range of Ip (note that the BE tests on GOM clay involved similar plasticity index), the authors collected a database of several clays from around the world with field and laboratory measurements of Vs or Gmax on intact and reconstituted specimens having a range of Ip and overconsolidation ratio (OCR = 'p/'v). Table 2, through Table 5, respectively, provide details about the clays in the database having Ip < 30%, 30% ≤ Ip < 50%, 50% ≤ Ip < 80% and 80% ≤ Ip < 120%. (Note that RC = resonant column, BE = bender elements, CH = cross-hole, DH = down-hole, SC = seismic cone, CTX = cyclic triaxial, cDSS = cyclic direct simple shear, TS = torsional shear, SASW = spectral analysis of surface waves, MASW = multi-channel analysis of surface waves.)      Fig 2 plots the data collected in the 4 given plasticity bins (<30%, 30 -50%, 50 -80%, 80 -120%) in Gmax -'p x ('v/e) space. The field data on 'p come from laboratory oedometer tests on undisturbed samples, and 'v were estimated considering the soil profile at a given site and the position of the water table. Note that 'p is the equivalent of 'v,max of a specimen consolidated to the virgin compression zone and 'v is the equivalent of 'c for a specimen consolidated under equal all-around pressure. For specimens consolidated under anisotropic conditions, 'c was used to imply mean effective stress. Also shown in the plots are the trendlines and the predictive equations for Gmax.

 
314 exp 0.008 p A I    (9) Although effective mean stress 'mean is generally a more powerful tool to quantify Gmax, knowledge of the coefficient of earth pressure at rest k0 is required because it influences Gmax (e.g., [17], [18]). However, information about k0 is not always readily available. Therefore, it was decided to adopt 'v or 'c in proposing the relationships for Gmax because Gmax hv ≈ Gmax hh ≈ Gmax vv for normally to moderately overconsolidated soft clays (e.g., [27]) and it is easier to define 'v than 'h. The influence of k0 on Gmax is probably why the bender element tests run on GOM clay lie slightly below the predictive trendlines for 30% ≤ Ip < 50% and 50% ≤ Ip < 80%. All the BE tests had k0 = 1, whereas soft clays in the field typically have k0 = 0.55 -1.0. By using Equation 5 and Equation 9, and provided 'v,'p, e and Ip are known, Gmax of a clay can be effectively estimated. Likewise, if Gmax is known, an estimate of the preconsolidation pressure can be obtained.

Conclusions
Bender element tests performed on a Gulf of Mexico clay consolidated under equal all-around stresses and subjected to multiple load-unload loops were supplemented with a database of 40 clays of different plasticity index to illustrate that the small-strain shear modulus of clays depends on consolidation stress, plasticity index, preconsolidation stress, and void ratio.