s inconsistent with the basic features of soil stress-strain relationship curves. Hence, a piecewise function is adopted to describe the soil skeleton curve in this article.
Based on differences in soil properties, each soil type has an upper limit for the shear strain . When the value of the soil reference shear strain amplitude exceeds the upper limit , the soil is yielded. When the value of the soil reference shear strain amplitude increases further, the shear stress in the soil no longer increases, and even tends to decrease, which are the characteristics of a geotechnical material first hardening and second softening. The skeleton curve of the Davidenkov model is amended as follows:
(7)
(8)
Per the Mashing rule, based on the modified Davidenkov skeleton curve, the one-dimensional loading and unloading stress-strain relationship curve of the soil is established and can be expressed as follows:
(9)
where and , respectively, are the values of the shear stress and shear strain amplitude on the load and unload turning point of the shear strain hysteresis curve.
Compared with the unmodified Davidenkov model, when the shear strain amplitude is , the skeleton curves of the soil under the action of cyclic loading are consistent with the loading and unloading stress-strain relationship curves. When the shear strain amplitude is , the stress-strain relationship curves of the soil under the action of cyclic loading are still confirmed by rules (2) and (3) of the Mashing rule; however, when the loading and unloading curve meets the upper-limit shear stress horizon line, the loading and unloading curve develop along the horizontal, as expressed in Fig. 2-2.
Fig.2-2 Dynamic shear stress-strain curves given by the corrected Davidenkov model
When the shear strain amplitude is , the equation for the damping ratio is (6). When the shear strain amplitude is , the area of the hysteresis loop is the sum of the areas of arches GHI and EDC and the parallel quadrilateral ECIG. The area of the two arches is equal to the area of hysteresis loop ABA'F when . Therefore, when the shear strain amplitude is , the area of the hysteresis loop is calculated as follows:
(10)
Imaginary elastic strain energy is the area of the triangle OCJ, which is calculated as follows.
(11)
Therefore, the damping ratio equation is as follows:
(12)
When analyzing the nonlinear dynamic response of underground structures, the stress-strain relationship of the soil and rock usually uses an octahedral. Assuming that the shear stress increment on the octahedral is and the shear strain variable increment is , the shear deformation modulus can be expressed as
(13)
The one-dimensional dynamic constitutive relationship of the soil is generalized to three dimensions. Per (9), the relationship between shear stress and shear strain on the octahedral can be approximated as
(14)
where and are, respectively, the values of the shear stress and shear strain amplitude on the load and unload turning point of the shear strain hysteresis curve on the octahedral. The incremental form for the above formula
(15)
where is the tangent shear modulus of the soil. According to (13), the equations for the initial loading segment shear modulus is as follows:
(16)
When the soil i
