Ground stress-strain behavior becomes nonlinear as it approaches the yield criterion; nonlinear elastic models simulate such ground behavior by modifying the foundation modulus. The function proposed by Duncan and Chang (1970) is used to calculate the foundation modulus. The stress-strain curve of the function is a hyperbola and the foundation modulus is a function of confining stress and shear stress. This nonlinear elastic material model is very useful because it only needs material properties that can be easily obtained from the triaxial compression test or literature. The Duncan and Change nonlinear stress-strain curve represents a hyperbola between the axial strain space generated by shear stress ( ) and it can be defined according to stress state and stress path by 3 foundation moduli (Initial modulus , Tangent modulus , Unloading-reloading modulus ).

The main nonlinear parameters of the hyperbolic model are as follows.

The results of the triaxial compression test can be plotted on a vertical axis of  or   and a horizontal axis of . Set each axis to a log scale and the vertical axis value at =1 is the Initial Loading Modulus (K). The Initial Stiffness Exp1nt(n) can be found from the slope when the vertical axis is  and the Bulk Modulus Exp1nt(m) can be found from the slope when the vertical axis is . Here, the Bulk modulus Bm is defined by the following equation and can be predicted using the relationship with Poisson’s ratio. Here, the Poisson’s ratio is limited to values within 0 to 0.5.

  :  Amount of principal stress change,

  :  Amount of volume strain change

<Set material property>

The Duncan and Change nonlinear stress-strain curve can be defined according to stress state and stress path by 3 foundation moduli (Initial modulus , Tangent modulus , Unloading-reloading modulus ).

<Nonlinear stress-strain behavior>

Here, the Failure Ratio (Rf) can be found by the relationship between the Initial modulus and Tangent modulus . The ratio of failure is the ratio between the asymptote of the hyperbola and the maximum shear strength and has a value between 0.75~1. A convergence problem can occur when the Tangent modulus  is too small and so the minimum Tangent modulus value is set as the atmospheric pressure (Pa). Bulk modulus number (Kb) can be calculated from the Bulk modulus (Bm) and Bulk modulus index (m).

Here,

  :   Bulk modulus,

  :   Bulk modulus number,

   :   Bulk modulus index

Unloading-reloading modulus number  can be calculated from the unloading-reloading ratio .

If the confining stress is '0(zero)' or negative (tensile state) when calculating the initial moduli, the moduli can be '0(zero)' or a negative value. Hence, a lower bound needs to be set for the confining stress and the set Minimum confining pressure is 0.01Pa.

The suggested parameter values depending on the density of sandy soils are as follows. (Duncan, J. M. and Chan, C. Y. (1970))

                                  ,: Dry state friction angle and cohesion

<Table. Summary of stress-strain parameters for uniform fine silica sand>