24.3.2 Volumetric Behavior

In the Bowl model the total volumetric strain $\varepsilon_{{\mathrm{vol}}}^{}$ is decomposed into a dilatancy component $\varepsilon_{{\mathrm{vol}}}^{{\mathrm{s}}}$ due to cyclic shear loading, and an elastic component $\varepsilon_{{\mathrm{vol}}}^{{\mathrm{c}}}$ due to changes in the effective mean stress:

$$\varepsilon_{{\mathrm{vol}}}^{} \varepsilon_{{\mathrm{vol}}}^{{\mathrm{c}}} \varepsilon_{{\mathrm{vol}}}^{{\mathrm{s}}}$$ (24.39)

The elasticity component $\varepsilon_{{\mathrm{vol}}}^{{\mathrm{c}}}$ is given by the compressive relationship

$$\varepsilon_{{\mathrm{vol}}}^{{\mathrm{c}}} {\frac{{ C_{\mathrm{s}} }}{{ 1 + e_{0} }}} {\frac{{ \sigma '_{\mathrm{m}} }}{{ \sigma '_{\mathrm{m}.0} }}}$$ (24.40)

in which e₀ is the initial void ratio and C_s the swelling index. The initial effective mean stress is denoted $\sigma{^\prime}_{{\mathrm{m}.0}}$ . The dilatancy component, $\varepsilon_{{\mathrm{vol}}}^{{\mathrm{s}}}$ , comprises two components:

$$\varepsilon_{{\mathrm{vol}}}^{{\mathrm{s}}} \varepsilon_{{\mathrm{vol}}}^{{\Gamma}} \varepsilon_{{\mathrm{vol}}}^{{G^{*}}}$$ (24.41)

The dilatancy component $\varepsilon_{{\mathrm{vol}}}^{{\Gamma}}$ models the increasing volume due to shear loading as a function of the equivalent shear strain $\Gamma$ according to

$$\varepsilon_{{\mathrm{vol}}}^{{\Gamma}} \Gamma^{{B}}_{}$$ (24.42)

The dilatancy component $\varepsilon_{{\mathrm{vol}}}^{{G^{*}}}$ describes the compaction of the material due to shear loading as a function of the cumulative shear strain G^* according to

$$\varepsilon_{{\mathrm{vol}}}^{{G^{*}}} {\frac{{ G^{*} }}{{ C + D G^{*}}}}$$ (24.43)

where

$$\int_{{0}}^{{t}} \dot{{G}}^{{*}}_{} \tau$$ (24.44)

with $\dot{{G}}^{{*}}_{}$ the rate of the internal variable G^* . The governing rate equation is given by

$$\dot{{G}}^{{*}}_{} \begin{cases} \vert\vert \dot{\boldsymbol{\Gamma}} \vert\ver... ...l{\Gamma} \vert\vert _{L_{2}} \leq R_{\mathrm{e}}$} \end{cases}$$ (24.45)

Where O^T = $\left\{\vphantom{ O_{zx} , O_{zy} }\right.$O_zx, O_zy$\left.\vphantom{ O_{zx} , O_{zy} }\right\}$ is the origin of the elastic area, and R_e is the threshold value of the shear increment at which no increase in the negative dilatancy can occur. This threshold value reduces the dilatancy generation for small amplitudes of shear strain and is defined as

$$\left\vert\vphantom{ \frac{X_{\mathrm{lim}} \: \sigma '_{\mathrm{... ...{\mathrm{l}} \: \sigma '_{\mathrm{m}.0} \right\vert^{\beta} \right) \: }\right. {\frac{{X_{\mathrm{lim}} \: \sigma '_{\mathrm{m}.0} }}{{ G_{0} }}} \left(\vphantom{ 1 + \alpha_{0} \left\vert X_{\mathrm{l}} \: \sigma '_{\mathrm{m}.0} \right\vert^{\beta} }\right. \alpha_{{0}}^{} \left\vert\vphantom{ X_{\mathrm{l}} \: \sigma '_{\mathrm{m}.0} }\right. \sigma{^\prime}_{{\mathrm{m}.0}} \left.\vphantom{ X_{\mathrm{l}} \: \sigma '_{\mathrm{m}.0} }\right\vert^{{\beta}}_{} \left.\vphantom{ 1 + \alpha_{0} \left\vert X_{\mathrm{l}} \: \sigma '_{\mathrm{m}.0} \right\vert^{\beta} }\right) \left.\vphantom{ \frac{X_{\mathrm{lim}} \: \sigma '_{\mathrm{m}.0... ...thrm{l}} \: \sigma '_{\mathrm{m}.0} \right\vert^{\beta} \right) \: }\right\vert$$ (24.46)

X_lim = $\sigma_{{xy}}^{}$/$\sigma_{{\mathrm{m}.0}}^{}$ is lower limit of the liquefaction resistance which is considered a material parameter. The radius of the elastic area is dependent on the initial effective mean stress level through the parameter $\alpha_{{0}}^{}$ and the initial shear stiffness G₀ . Figure 24.3 illustrates the definition of some strain related quantities for a two dimensional shear loading.

Figure 24.3: Shear strain related variables for Bowl model