24.3.1 Deviatoric Behavior

The deviatoric behavior of the Bowl model is modeled with a modified Ramberg-Osgood model:

$$\gamma_{{xy}}^{} {\frac{{ \sigma _{xy} }}{{ G }}} \left(\vphantom{ 1 + \alpha \vert \sigma_{xy} \vert^{\beta} }\right. \alpha \sigma_{{xy}}^{} \beta \left.\vphantom{ 1 + \alpha \vert \sigma_{xy} \vert^{\beta} }\right)$$ (24.35)

where the shear strain in the xy direction is determined by the linear shear strain $\sigma_{{xy}}^{}$/G , and the actual shear stress level through the factor $\left(\vphantom{ 1 + \alpha \vert \sigma_{xy} \vert^{\beta} }\right.$1 + $\alpha$|$\sigma_{{xy}}^{}$|^$\beta$$\left.\vphantom{ 1 + \alpha \vert \sigma_{xy} \vert^{\beta} }\right)$ . The actual shear modulus G is given by

$$\left(\vphantom{ \frac{ \sigma '_{\mathrm{m}} }{ \sigma '_{\mathrm{m.ref}} } }\right. {\frac{{ \sigma '_{\mathrm{m}} }}{{ \sigma '_{\mathrm{m.ref}} }}} \left.\vphantom{ \frac{ \sigma '_{\mathrm{m}} }{ \sigma '_{\mathrm{m.ref}} } }\right)^{{\!\tfrac{1}{2}}}_{}$$ (24.36)

where G_ref is the reference shear modulus at the reference mean effective pressure $\sigma_{{\mathrm{m.ref}}}^{{\prime}}$ . The coefficients $\alpha$ and $\beta$ are given by

$$\alpha \left(\vphantom{ \frac{ 2 }{ \gamma_{0.5} \; G } }\right. {\frac{{ 2 }}{{ \gamma_{0.5} \; G }}} \left.\vphantom{ \frac{ 2 }{ \gamma_{0.5} \; G } }\right)^{{\!\beta}}_{} \beta {\frac{{ 2 \pi h_{\mathrm{max}} }}{{ 2 - \pi h_{\mathrm{max}} }}}$$ (24.37)

Where h_max is the maximum damping ratio of the soil, and $\gamma_{{0.5}}^{}$ the reference shear strain at the value G/G_ref = 0.5

Unloading and reloading is described by Masing's rule which states that if the original loading curve in the xy direction is given by (24.35) then the unloading and reloading curves are given by

$${\frac{{ \gamma_{xy} + \gamma_{\mathrm{rev}} }}{{ 2 }}} {\frac{{ \sigma _{xy} + \tau_{\mathrm{rev}} }}{{ 2 \: G }}} \left(\vphantom{ 1 + \alpha \left\vert \frac{ \sigma _{xy} + \tau_{\mathrm{rev}} }{ 2 } \right\vert^{\beta} }\right. \alpha \left\vert\vphantom{ \frac{ \sigma _{xy} + \tau_{\mathrm{rev}} }{ 2 } }\right. {\frac{{ \sigma _{xy} + \tau_{\mathrm{rev}} }}{{ 2 }}} \left.\vphantom{ \frac{ \sigma _{xy} + \tau_{\mathrm{rev}} }{ 2 } }\right\vert^{{\beta}}_{} \left.\vphantom{ 1 + \alpha \left\vert \frac{ \sigma _{xy} + \tau_{\mathrm{rev}} }{ 2 } \right\vert^{\beta} }\right)$$ (24.38)

in which ($\gamma_{{\mathrm{rev}}}^{}$,$\tau_{{\mathrm{rev}}}^{}$) are the coordinates of the current reversal point in the stress-strain curve [Fig.24.2].

Figure 24.2: Ramberg-Osgood model with Masing's rule