17.1.6 Egg Cam-clay

In order to simulate the behavior of clay and clay-like materials, elastoplastic modeling is a well established concept, see for instance Britto & Gunn [13, Ch.5]. DIANA offers an extension to the Modified Cam-clay model including features such as nonlinear elasticity, hardening and softening, dilatation and contraction, a critical state and a modification of the original formulation in order to describe K₀ -consolidation more accurately. The implementation of the Egg Cam-clay model is mainly based on the work of Groen [36]. See also Van Eekelen & Van Den Berg [105].

17.1.6.1 Strains and Stresses

The hydrostatic pressure is defined in terms of effective stresses as

$${\tfrac{{1}}{{3}}} \sigma_{{xx}}{^\prime} \sigma_{{yy}}{^\prime} \sigma_{{zz}}{^\prime} {\tfrac{{1}}{{3}}}$$ (17.84)

and the deviatoric-like stress as

$$\boldsymbol\xi \boldsymbol\sigma$$ (17.85)

in which

$${\frac{{1}}{{3}}} \left[\vphantom{ \negthickspace \begin{array}{cccccc} 2 & -1 & -1... ... 0 & 6 & 0\\ [0.8ex] 0 & 0 & 0 & 0 & 0 & 6 \end{array} \negthickspace }\right. \begin{array}{cccccc} 2 & -1 & -1 & 0 & 0 & 0\\ [0.8ex] -1 & 2 &... ...0\\ [0.8ex] 0 & 0 & 0 & 0 & 6 & 0\\ [0.8ex] 0 & 0 & 0 & 0 & 0 & 6 \end{array} \left.\vphantom{ \negthickspace \begin{array}{cccccc} 2 & -1 & -1... ... 0 & 6 & 0\\ [0.8ex] 0 & 0 & 0 & 0 & 0 & 6 \end{array} \negthickspace }\right]$$ (17.86)

The effective deviatoric stress is then defined according to

$$\sqrt{{ \tfrac{3}{2}\boldsymbol{\xi}^{\mathrm{\scriptscriptstyle{T}}}\mathbf{R} \, \boldsymbol{\xi}}} \sqrt{{ 3 J_{2} }}$$ (17.87)

with the diagonal matrix

$$\left[\vphantom{ \negthickspace \begin{array}{cccccc} 1 & 0 & 0 &... ...\ [0.8ex] 0 & 0 & 0 & 0 & 0 & \tfrac{1}{2} \end{array} \negthickspace }\right. \begin{array}{cccccc} 1 & 0 & 0 & 0 & 0 & 0\\ [0.8ex] 0 & 1 & 0 ... ...0 & 0 & \tfrac{1}{2}& 0\\ [0.8ex] 0 & 0 & 0 & 0 & 0 & \tfrac{1}{2} \end{array} \left.\vphantom{ \negthickspace \begin{array}{cccccc} 1 & 0 & 0 &... ...\ [0.8ex] 0 & 0 & 0 & 0 & 0 & \tfrac{1}{2} \end{array} \negthickspace }\right]$$ (17.88)

The strains are defined in a slightly different manner. The volumetric strain is

$$\varepsilon_{{\mathrm{v}}}^{} \varepsilon_{{xx}}^{} \varepsilon_{{yy}}^{} \varepsilon_{{zz}}^{}$$ (17.89)

and the deviatoric strain is

$$\boldsymbol\gamma \boldsymbol\varepsilon$$ (17.90)

in which the matrix Q = RP = PR .

Actually, the Cam-clay model is a combination of a nonlinear elastic and a plasticity model. Both for the volumetric strain and the deviatoric strain we use the basic additive decomposition into an elastic part and a plastic part

$$\varepsilon_{{\mathrm{v}}}^{} \varepsilon_{{\mathrm{v}}}^{{\mathrm{e}}} \varepsilon_{{\mathrm{v}}}^{{\mathrm{p}}} \boldsymbol\gamma \boldsymbol\gamma \boldsymbol\gamma$$ (17.91)

17.1.6.2 Nonlinear Elasticity

It is observed experimentally in clays that during elastic swelling or reloading

$${\frac{{ 1 + e }}{{\kappa}}}$$ (17.92)

in which e is the void ratio, $\kappa$ is a material parameter and p' is the current hydrostatic pressure. The void ratio e is defined as the ratio between pore volume V_p and material volume V_m :

$${\frac{{ V_{\mathrm{p}} }}{{ V_{\mathrm{m}} }}} {\frac{{ V - V_{\mathrm{m}} }}{{ V_{\mathrm{m}} }}} {\frac{{ V }}{{ V_{\mathrm{m}} }}}$$ (17.93)

Figure 17.10 shows the

Figure 17.10: Soil response in compression

assumed elastic behavior, indicating constant $\kappa$ . K_t sets the rate relation between the hydrostatic pressure and the elastic volumetric strain

$$\dot{{p}}{^\prime} \dot{{\varepsilon }}_{{\mathrm{v}}}^{{\mathrm{e}}}$$ (17.94)

In this form, the soil has no tensile strength. Therefore, (17.92) is most conveniently modified to

$${\frac{{ 1 + e }}{{\kappa}}}$$ (17.95)

In which p_t is the so-called tensile pressure. This tensile pressure is merely a numerical artifice to take tensile stresses into account when the initial pressure is assumed to be equal to zero. In practice however, Cam-clay analyses will nearly always start with non-zero initial stresses. It is assumed that the void ratio does not change significantly, even for large loading steps. Therefore, the void ratio e is updated explicitly during the loading process. The change of volume is determined by the original volume V and the volumetric strain $\dot{{\varepsilon }}_{{\mathrm{v}}}^{}$

$$\dot{{V}} \dot{{\varepsilon }}_{{\mathrm{v}}}^{}$$ (17.96)

Assume that the material volume is constant, i.e., $\dot{{V}}_{{\mathrm{m}}}^{}$ = 0 , then the void rate is defined by

$$\dot{{e}} {\frac{{ \dot{V}_{\mathrm{p}} }}{{ {V}_{\mathrm{m}} }}} {\frac{{ \dot{V} - \dot{V}_{\mathrm{m}} }}{{ {V}_{\mathrm{m}} }}} {\frac{{ \dot{V} }}{{ {V}_{\mathrm{m}} }}} \dot{{\varepsilon }}_{{\mathrm{v}}}^{} {\frac{{ V }}{{ V_{\mathrm{m}} }}}$$ (17.97)

and with (17.93) this yields

$$\dot{{e}} \dot{{\varepsilon }}_{{\mathrm{v}}}^{} \left(\vphantom{ 1 + e }\right. \left.\vphantom{ 1 + e }\right) \therefore \dot{{\varepsilon }}_{{\mathrm{v}}}^{} {\frac{{ \dot{e} }}{{ 1 + e }}}$$ (17.98)

This can be integrated over a time increment $\Delta$t as

$$\int_{{t}}^{{t + \Delta t}} \dot{{\varepsilon }}_{{\mathrm{v}}}^{} \tau \int_{{t}}^{{t + \Delta t}} {\frac{{ \dot{e} }}{{1 + e}}} \tau$$ (17.99)

Hence

$$\Delta \varepsilon_{{\mathrm{v}}}^{} \left\vert\vphantom{ \frac{ 1 + e^{t+\Delta t} }{ 1 + e^{t} } }\right. {\frac{{ 1 + e^{t+\Delta t} }}{{ 1 + e^{t} }}} \left.\vphantom{ \frac{ 1 + e^{t+\Delta t} }{ 1 + e^{t} } }\right\vert$$ (17.100)

and therefore

$$\Delta \left(\vphantom{ 1 + e^{t} }\right. \left.\vphantom{ 1 + e^{t} }\right) \left(\vphantom{ \Delta \varepsilon _{\mathrm{v}} }\right. \Delta \varepsilon_{{\mathrm{v}}}^{} \left.\vphantom{ \Delta \varepsilon _{\mathrm{v}} }\right)$$ (17.101)

See also Borja [10].

Equation (17.95) can be recast in a nonlinear relation between the volumetric strain and the pressure as

$${\frac{{ \,\mathrm{d}p' }}{{ p' + p_{\mathrm{t}} }}} {\frac{{ 1 + e }}{{ \kappa }}} \varepsilon_{{\mathrm{v}}}^{{\mathrm{e}}}$$ (17.102)

After integration over an increment and rearranging, the following relation between the pressure and the volumetric strain increment ensues

$$\left(\vphantom{ -\frac{ 1 + e_{0} }{\kappa} \Delta\varepsilon _{\mathrm{v}}^{\mathrm{e}} }\right. {\frac{{ 1 + e_{0} }}{{\kappa}}} \Delta \varepsilon_{{\mathrm{v}}}^{{\mathrm{e}}} \left.\vphantom{ -\frac{ 1 + e_{0} }{\kappa} \Delta\varepsilon _{\mathrm{v}}^{\mathrm{e}} }\right)$$ (17.103)

In this equation, p'₀ is the pressure at the beginning of a loading step, and p' is the current pressure. This relation defines the pressure as a function of the volumetric strain increment, the pressure at the beginning of a loading step and the void ratio at the beginning of a loading step e₀ . From relation (17.102) the tangential compression modulus can be obtained via

$${\frac{{\,\mathrm{d}p^{\prime}}}{{\,\mathrm{d}\Delta \varepsilon _{\mathrm{v}}^{\mathrm{e}} }}} {\frac{{1+e_{0}}}{{\kappa}}} \prime$$ (17.104)

which differs from (17.95) in the sense that the void ratio is assumed to remain constant over an increment.

The nonlinear elasticity of the model is defined in terms of volumetric components only. Assuming isotropy, one can specify either a constant Poisson's ratio $\nu$ , with the (variable) tangent shear modulus defined as

$${\tfrac{{3}}{{2}}} {\dfrac{{ 1 - 2 \nu }}{{ 1 + \nu }}}$$ (17.105)

Alternatively, one can consider an approach in which the shear modulus G is kept constant and the Poisson's ratio $\nu$ varies according to

$$\nu_{{\mathrm{t}}}^{} {\frac{{ 3 K_{\mathrm{t}} - 2 G }}{{ 6 K_{\mathrm{t}} + 2 G }}}$$ (17.106)

17.1.6.3 Yield Function

The yield function for the Modified Cam-clay model can be written as

f = q² + M²p'(p' - 2a) (17.107)

in which

$${\frac{{ 6 \sin\phi }}{{ ( 3 - \sin\phi ) }}}$$ (17.108)

with $\phi$ the angle of internal friction. Yield function parameter a is a measure for the current overconsolidation and is related to the preconsolidation stress by a = ${\frac{{1}}{{2}}}$p'_c . It is known that the Modified Cam-clay model overestimates the initial overconsolidation pressure. Furthermore, the original version of the Modified Cam-clay model does not have a tensile strength. Therefore, the yield surface of the Modified Cam-clay model is changed into

$${\frac{{ M^{2} }}{{ \beta^{2} }}} \left(\vphantom{ ( p' + \Delta p ) ( p' + \Delta p - 2 a ) + a^{2} ( 1 - \beta^{2} ) }\right. \Delta \Delta \beta^{{2}}_{} \left.\vphantom{ ( p' + \Delta p ) ( p' + \Delta p - 2 a ) + a^{2} ( 1 - \beta^{2} ) }\right)$$ (17.109)

with $\Delta$p a reference pressure to model cohesive behavior. Due to parameter $\beta$ , the actual yield surface is now composed of two yield surfaces

$$\beta \begin{cases} \gamma \qquad \text{if} \quad p' + \Delta p \l... ...1}{\alpha} \qquad \text{if} \quad p' + \Delta p > a \end{cases}$$ (17.110)

with $\alpha$ a defined material parameter, which is in the range from 1 to 2 for Modified Cam-clay. Constant $\gamma$ is an optional shape factor for the dry side of the yield surface. Figure 17.11

Figure 17.11: Cam-clay models

shows the form of the yield surface in p -q space. Yield function parameter a is now related to the preconsolidation stress by p'_c${\frac{{\alpha}}{{1 + \alpha}}}$ .

Optionally DIANA derives the cap shape factor $\alpha$ , which is specific for the Egg model, automatically from the K_nc ratio between horizontal and vertical stress for normally consolidated soil. See Van Eekelen & Van Den Berg [105] for details. Alternatively $\alpha$ is input by the user or a regular Modified Cam-clay model is used.

It is assumed that the Egg Cam-clay model is an associated plasticity model. Thus the yield surface (17.109) also defines the plastic potential g . Using the additive decomposition of the strain rates, this leads for the volumetric strain rate to

$$\varepsilon_{{\mathrm{v}}}^{} \varepsilon_{{\mathrm{v}}}^{{\mathrm{e}}} \lambda {\frac{{\partial f }}{{ \partial p }}} \varepsilon_{{\mathrm{v}}}^{{\mathrm{e}}} \lambda {\frac{{ M^{2} }}{{ \beta^{2} }}} \Delta$$ (17.111)

and for the deviatoric strain rate to

$$\boldsymbol\gamma \boldsymbol\gamma \lambda {\frac{{\partial f}}{{\partial q}}} {\frac{{\partial q}}{{ \partial\boldsymbol{\xi}}}} \boldsymbol\gamma \lambda \boldsymbol\xi$$ (17.112)

Where $\lambda$ is the plastic multiplier.

17.1.6.4 Hardening Behavior

The final ingredient for a plasticity model is the evolution of the hardening parameter. It is observed experimentally that

$${\frac{{ \alpha }}{{ 1 + \alpha }}} \prime \therefore {\dfrac{{ 1 + e }}{{ \lambda - \kappa }}} \varepsilon_{{\mathrm{v}}}^{{\mathrm{p}}}$$ (17.113)

in which $\lambda$ is a material constant [Fig.17.10]. Like in the elastic behavior, we will assume that the void ratio is updated explicitly so that it is kept constant during an increment. Similar to the integration of the elastic stress-strain relation, the hardening rule (17.113) can be integrated to

$$\left(\vphantom{ -\frac{ 1 + e_{0} }{ \lambda - \kappa } \Delta \varepsilon _{\mathrm{v}}^{\mathrm{p}} }\right. {\frac{{ 1 + e_{0} }}{{ \lambda - \kappa }}} \Delta \varepsilon_{{\mathrm{v}}}^{{\mathrm{p}}} \left.\vphantom{ -\frac{ 1 + e_{0} }{ \lambda - \kappa } \Delta \varepsilon _{\mathrm{v}}^{\mathrm{p}} }\right)$$ (17.114)

with a₀ being the hardening parameter at the beginning of an increment.

By default, DIANA derives the preconsolidation stress p'_c , or actually the initial hardening parameter a , from the maximum stress experienced by the soil element, see Britto & Gunn [13, Ch.5]. Alternatively, p'_c is input by the user.

17.1.6.5 Cap Shape Factor

In a one-dimensional normal compression situation, that corresponds to an axisymmetric stress state, one can easily find that the ratio of stress increment equals:

$${\frac{{ \Delta q }}{{ \Delta p' }}} {\frac{{ \Delta \sigma _{z} - \Delta \sigma _{r} }}{{ \tfrac{1}{3}( \Delta \sigma _{z} + 2 \Delta\sigma _{r} ) }}} {\frac{{ 3 ( 1 - K_{\mathrm{NC}} ) }}{{ 1 + 2 K_{\mathrm{NC}} }}} \eta_{{\mathrm{NC}}}^{}$$ (17.115)

If you choose for an automatic determination of the cap shape factor, DIANA will calculate $\alpha$ in such a way that for a one-dimensional compression situation, the stress ratio $\Delta$q/$\Delta$p' equals $\eta_{{\mathrm{NC}}}^{}$ .

Through the definitions of $\Delta$$\varepsilon_{{\mathrm{v}}}^{}$ = $\Delta$$\varepsilon_{{a}}^{}$ +2$\Delta$$\varepsilon_{{r}}^{}$ and $\Delta$$\varepsilon_{{\mathrm{s}}}^{}$ = ${\tfrac{{2}}{{3}}}$($\Delta$$\varepsilon_{{a}}^{}$ - $\Delta$$\varepsilon_{{r}}^{}$) , and from the fact that for one-dimensional compression $\Delta$$\varepsilon_{{r}}^{}$ = 0 , one may find the following strain ratio:

$${\frac{{ \Delta \varepsilon _{\mathrm{v}} }}{{ \Delta \varepsilon _{\mathrm{s}} }}} {\frac{{ \Delta \varepsilon _{\mathrm{v}}^{\mathrm{e}} + \Delta \... ... _{\mathrm{s}}^{\mathrm{e}} + \Delta \varepsilon _{\mathrm{s}}^{\mathrm{p}} }}} {\frac{{ 3 }}{{ 2 }}}$$ (17.116)

We can develop the four strain components from (17.116). For the two elastic components $\Delta$$\varepsilon_{{\mathrm{v}}}^{{\mathrm{e}}}$ and $\Delta$$\varepsilon_{{\mathrm{s}}}^{{\mathrm{e}}}$ we have the following relationships:

$$\begin{split}\Delta \varepsilon _{\mathrm{v}}^{\mathrm{e}} &=... ...c{ \kappa }{ 1 + e } \cdot \frac{ \Delta p' }{ p' } \end{split}$$ (17.117)

The two plastic components $\Delta$$\varepsilon_{{\mathrm{v}}}^{{\mathrm{p}}}$ and $\Delta$$\varepsilon_{{\mathrm{s}}}^{{\mathrm{p}}}$ can be expressed in the following way (see Muir Wood [72]):

$$\begin{split}\left\{ \negthickspace \begin{array}{c} \Delta \... ... [3ex] \Delta q \end{array} \negthickspace \right\} \end{split}$$ (17.118)

For the Egg Cam-clay model we have:

$$\begin{split}\frac{ \partial f }{ \partial p'_{\mathrm{c}} } ... ...tial q } &= \frac{ \partial g }{ \partial q } = 2 q \end{split}$$ (17.119)

Inserting (17.119) into (17.118) leads, after some algebraic manipulations, to:

$$\begin{split}\Delta \varepsilon _{\mathrm{v}}^{\mathrm{p}} &=... ...+ 1 } { \alpha^{2} } \cdot \frac{ \Delta p' }{ p' } \end{split}$$ (17.120)

Finally, substitution of (17.120) and (17.117) into (17.116) gives the following implicit equation for $\alpha$ :

$$\begin{split}\alpha^{2} \left( \left( \frac{ M }{ \eta_{\math... ...{\mathrm{NC}} }{ M } \right)^{\!2} } +1 \right) = 0 \end{split}$$ (17.121)

DIANA calculates the root of (17.121) to assess the cap shape factor $\alpha$ .