Fractional viscoelastic solution of stratum displacement of a shallow tunnel under the surface slope condition

The unified displacement function(UDF)is presented to describe the deformation behaviours of the tunnel profile along with time under the surface slope condition.Based on the discrete Fourier method,the third-order UDF in the physical plane is expanded to the Laurent series in the complex variable plane.The complex variable method is employed to derive the elastic analytical solution of stra-tum displacement,when the third-order UDF is taken as the displacement boundary condition of tunnel cross-section(DBCTC).The proposed elastic solution agrees well with the results of the finite element method for the consistent model,which verifies the correctness of the proposed analytical solution.Combining the corresponding principle and fractional Generalized Kelvin viscoelastic constitutive model,the fractional viscoelastic solution under the surface slope condition is determined.The time effect of stratum displacement is presented in two aspects:time-dependent DBCTC and time-dependent material parameters.The parameter analysis is performed to investigate influences of deformation modes of the third-order UDF,slope angle,tunnel radius and fractional order on the time effect of stratum vertical and horizontal displacement.

3-D rheologic model of earthquake preparation (Ⅱ): Strain field and its applications

On the basis of the three-dimensional elastic inclusion model, the analytic expression of viscoelastic strain field is derived, i.e., the analytic expression of viscoelastic strain at an arbitrary point (x, y, z) in x-axis, y-axis and z-axis produced by three-dimension inclusion in the semi-infinite rheologic medium defined by the standard linear rheologic model, namely the normal strains exx(r, t), eyy(r, t) and ezz(r, t), the shear strains exy(r, t) and eyx(r, t), eyz(r, t) and ezy(r, t), exz(r, t) and ezx(r, t), and the bulk-strain q (r, t). By computing the spatial-temporal variation of bulk strain on the ground produced by a spherical rheologic inclusion in a semi-infinite rheologic medium, we obtained some significant results that the bulk-strain variation with time produced by a hard inclusion has three stages (a, b, g) with different characteristics, which are similar to those of most geodetic deformation curves, but not the case for those by a soft inclusion. It is meaningful that these theoretical results have been applied to explain preliminarily the characteristics of stage variation of spatial-temporal evolution, the pattern and quadrant distribution of earthquake precursors, the changeability, spontaneity and complexity of short-term and imminent-term precursors. It offers a theoretical base to found the physical model of earthquake precursors and a reference to predict physically the earthquakes.

Three-dimensional Numerical Simulation of Viscoelastic Phase Separation under Shear: the Roles of Bulk and Shear Relaxation Moduli

The morphological, dynamic and rheological characteristics in the viscoelastic phase separation（VPS） of sheared polymer solutions are investigated by three-dimensional（3D） numerical simulations of viscoelastic model. The simulations are accelerated by graphic process unit（GPU） to break through the limitation of computation power. Firstly, the morphological and dynamic characteristics of VPS under shear are presented by comparing with those in classic phase separation（CPS）. The results show that the phase inversion and phase shrink take place in VPS under shear. Then, the roles of bulk and shear relaxation moduli in VPS are investigated in details. The bulk relaxation modulus slows down the phase separation process under shear, but not affects the dynamic path of VPS. The dynamic path can be divided into three stages： freezing stage, growth stage and stable stage. The second overshoot phenomenon in the shear stress is observed, and explained by the breakdown and reform of string structures. The shear modulus affects morphology evolution in the late stage of VPS under shear.