## 1 Introduction  
After earthquakes, detailed studies of the seismic ground responses are typically conducted, showing that (1) soil deposits exhibit strong dynamic nonlinear responses, and (2) characterization of the soil dynamic properties plays a crucial role in accurately estimating the site amplification characteristics [1; 2; 3].  
Strong ground motion causes the accumulation of plastic deformation during earthquake cycles, and elastic theory cannot be applied to describe the deformational characteristics of the soil deposits (i.e., shear modulus decreases and damping ratio increases with the amplitude of the shear strain). Site response analysis methods with nonlinear responses can be divided into the equivalent linear and nonlinear approaches. The first approach, the equivalent linear approach, approximates the nonlinear cyclic responses of soil samples. The effective shear strain (equal to approximately 65% of the peak strain in the time domain) is iteratively computed by updating the shear modulus and damping, after which the site response is simply estimated in the frequency domain. Given that the appropriate selection of the equivalent soil stiffness and damping for soil layers only represents a particular state in the stress-strain space and not the entire stress-strain evolution, this method fails to predict large plastic deformations of soil columns over the entire duration of a seismic event, and its use is strictly limited to relatively small shear strains or small nonlinearities [4; 5; 6].  
The second approach, in which the soil dynamic characteristics are captured through a nonlinear hysteretic constitutive relationship, can represent the strain-dependent shear modulus and damping ratio. The simplest constitutive relationship uses a model relating the shear stress to the shear strain, whereby the backbone curve is expressed by a hyperbolic function. Several functions with empirical fitting parameters have been proposed to describe the strain-dependent response defined by the backbone curve [7; 8; 9; 10; 11]. More sophisticated models for cyclic loading use more fitting parameters so as to precisely reproduce closed hysteresis loops that are influenced by basic soil parameters (plasticity, void ratio, confinement stress) and the imposed shear strain amplitude [12; 13; 14; 15; 16; 17; 18; 19; 20; 21]. While those models are capable of expressing the nonlinear hysteretic response or degree of strain-dependency during steady-state cyclic loading, their input parameters lack both a physical basis and robustness, and the use of such models is quite limited to certain soils. Therefore, a simple but robust model involving a small number of physically meaningful parameters is necessary for the characterization of the hysteretic nonlinear properties ([PERSON]'s criterion).  
This study proposes explicit formulas for capturing certain nonlinear hysteretic soil responses. These include a constitutive model, as well as the backbone curve, tangent shear modulus, secant shear modulus, and damping ratio. The proposed models are examined with resonant column test data (shear modulus and damping ratio). The validity of the linear characteristic equation is addressed during the process of data interpretation.