## 1 Introduction  
Soft silty soils are widespread in deltas, estuaries and coastal zones, and are prone to liquefaction under cyclic wave loading because of their high natural water content and weak permeability [1, 2], which may result in serious damage to marine structures, such as the instability of gravity platforms [3], large horizontal displacements of immersed tunnels [4], tilting of caisons [5], shear failure of breakwater slopes [6], shear and tensile failures of bolts [7], and floating up of pipelines [8]. Such silty soils are also characterized by high anisotropy of strength properties, mainly manifested as stress path dependence. In nature, the wave-induced cyclic loading is non-proportional and multiaxial in a saturated seabed of finite thickness, accompanied by simultaneous changeover in the magnitude and direction of the principal stress. It has been recognized that the rotation of the principal stress axis plays a crucial role in wave-induced liquefaction of sand beds; for example, the resistance to liquefaction under progressive waves is considerably smaller than the resistance exhibited under standing waves for sand beds [9]. However, the effects of fabric anisotropy and non-proportional loading on silt beds still require further study. Soil anisotropy has been widely examined in numerous laboratory tests, under proportional and non-proportional loading, and has also been considered in a variety of soil constitutive models [10, 11, 12, 13, 14, 15, 16]. It also has been found that sandy and silty soils exhibit more complex undrained behavior under cyclic loading than under monotonic loading [17]; for example, ref. [18] carried out a string of triaxial tests and concluded that with the increase in the degree of soil anisotropy, the shear strength of sand increases in the monotonic triaxial test, while the liquefaction resistance decreases in the cyclic triaxial test. This can be interpreted by the fact that under axial stress-controlled cyclic loading, sand exhibits a considerably higher contraction on the triaxial extension side, which is definitive evidence of soil anisotropy being related to the orientation of the major principal stress direction. However, the cyclic triaxial test is a typical case of proportional loading, which cannot involve either continuous principal stress rotations or their orientation with regard to the axis of anisotropy. In most practical cases, the stress path tends to deviate from the proportional loads significantly, such as wave loads on seabed sediments [19], multi-directional seismic loads on the ground [20], and train moving loads in railway track foundations, involving continuous rotation of the principal stress axes during cyclic shear deformation. At the same time, the use of hollow cylinder torsional shear tests offers an effective way to investigate soil anisotropic behavior, considering that the orientation and magnitude of the principal stress can be independently controlled in these tests. Most of the previous studies on the torsional shear testing of hollow cylinders have concentrated on situations where the major principal stress direction was fixed at specific values, or the deviatoric stress was continuously rotated during the cyclic shear process. However, when allowing for soil that is highly anisotropic and nonlinear, the study of the coupling effect of the principal stress magnitude and principal stress direction is still in its infancy. Indeed, the stress path involving simultaneous changeover in the magnitude and directions of the principal stress is an even more general case of non-proportional loading, as compared with rotational shear, defined as the shear stress that is perpendicular to the mirror image generated by a thin-walled hollow cylindrical specimen, caused by torsion. This condition can be best demonstrated by the fact that the stress path of finite seabed deposits is related to wave loading [21].