Elastic-Plastic Model and Correlation with ${\bf G/G_{max}}$

The general form of three-dimensional Drucker-Prager elastic-plastic constitutive relationship Eq. (18) can be simplified to any direction of interest. Thus, knowing the current state of the stress tensor and strain-increment in that particular direction, the stress increment in that direction can be solved for incrementally to get the elastic-plastic stress-strain relationship.

For the case of shear wave propagation in the 3-direction with horizontal polarization Eq. (18) can be simplified to the following form by substituting $i=3$; $j=1$; $m=3$ and $n=1$, which after some tensor algebra becomes:

$$d\sigma_{31}=\left[2G_{31}-\frac{\frac{G_{31}^2}{J_2}}{G_{31}+9K\alpha^2}S_{31}S_{31}\right]d\epsilon_{31}$$ (21)

The bulk modulus, K, is calculated from $E_{11}$, $E_{33}$, $\nu_{12}$, $\nu_{31}$, using the relationship,

$$K = \frac{1}{3 \displaystyle{\left( \frac{1}{E_{11}}-\frac{\nu_{12}}{E_{11}}-\frac{\nu_{31}}{E_{33}}\right)} }$$ (22)

The Drucker-Prager material constant, $\alpha$, is related to the friction angle, $\phi$, as,

$$\alpha=\frac{2\sin \phi}{\sqrt{3}(3-\sin \phi)}$$ (23)

Knowing $E_{11}$, $E_{33}$,$\nu_{12}$, $\nu_{31}$ (Table 3) and using $\phi=20^0$ [9] Eq. (21) was solved incrementally and the predicted elastic-plastic stress ($\tau_{31}$)-strain ( $\gamma_{31}$) relationship is shown in Fig. 3(a).

Eq. (21) also represents the variation of $G/G_{max}$ which is usually found in traditional analysis of dynamics of soils. The term inside the bracket in Eq. (21) is the elastic-plastic shear modulus, which is dependent on the stress deviator and reduces with increase in stress/strain. Fig. 3(b) shows the predicted reduction of shear modulus in terms of dimensionless parameters $G_{31}/G_{31 max}$ versus $\gamma_{31}$.

Figure 3: (a) Predicted Elastic and Elastic-Plastic Stress-Strain Relationship; (b) Predicted Variation of Shear Modulus with Strain