5.1.4 Egg Cam-clay

the use of a nonlinear elasticity model which exhibits an increasing bulk elastic stiffness as the material undergoes compression, and

the use of a plasticity model defined by an elliptically shaped yield surface with an elliptically shaped cap, associated flow and a hardening rule that allows the yield surface to grow or shrink.

**Figure 5.3:** Egg Cam-clay models
$\begin{figure}\vspace*{-1ex} \begin{footnotesize}\setlength{\unitlength}{1cm} ... ...enterline{\raise 4.5cm\box\graph} } \end{picture}\end{footnotesize} \end{figure}$

5.3

$\scriptstyle \prime$

is the isotropic effective pressure and q the deviatoric stress. See §17.1.6 for background theory.

The regular part of the input for the Egg Cam-clay model is sufficient in most cases [§5.1.4.1]. An extended input may be specified to overrule the regular default values of some parameters in the model [§5.1.4.2]. There is also an enhanced model which may be used for research purposes [§5.1.4.3].

5.1.4.1 Regular Input

The following input parameters are regular if the nonlinear analysis is started with the initial stress option [Vol. Analysis Procedures].

(syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ...] \>\>\texttt{POISON}\>\texttt{\textit{nu}}$_{r}\,$ \end{tabbing} \end{figure}$

YIELD

CLAY specifies that the Egg Cam-clay model must be used.

YLDVAL

mandatory input parameters for the yield contour.

Parameter sphi is sin $\phi$ of friction angle $\phi$ with M = 6 sin $\phi$ /(3 - sin $\phi$ ) . ( 0 < sin $\phi$ < 1 )Compare Drucker-Prager plasticity [§5.1.2].

For exponential nonlinear elasticity the hardening parameter lambda is the slope $\lambda$ = $\partial$ v/ $\partial$ ln p during compression, ( $\lambda$ > 0 )where $\lambda$ is linked to the compression index C_c by $\lambda$ = C_c/ln 10 , and where v is the total volumetric strain (total = elastic + plastic). For linear elasticity the hardening parameter lambda refers to the plastic strain only.

CAP

indicates the use of a cap shape factor $\alpha_{{K_{\mathrm{nc}}}}^{}$ for the wet side of the yield surface. DIANA derives this cap shape factor, which is specific for the Egg model, from the K_nc ratio between horizontal and vertical stress for normally consolidated soil [§17.1.6.5]. If CAP is not specified, then the Egg model reduces to the Modified Cam-clay model with a single elliptical yield surface ( $\alpha$ = 1 ).

OCR

ocr is the overconsolidation ratio $\mathcal {OCR}$ . [ $\mathcal {OCR}$ = 1 ] DIANA derives the maximum vertical effective stress, experienced by the soil element, from the in situ vertical stress by $\sigma^{{\prime}}_{{\mathrm{v_{max}}}}$ = $\mathcal {OCR}$ x $\sigma^{{\prime}}_{{\mathrm{v}}}$ .

ELAST

CLAY specifies Exponential nonlinear elasticity for clay.

ELAVAL

k is the slope $\kappa$ = $\partial$ v/ $\partial$ ln p ( $\kappa$ > 0 )during nonlinear elastic swelling or reloading. A rule of thumb is ${\frac{{1}}{{5}}}$ $\lambda$ $\leq$ $\kappa$ $\leq$ ${\frac{{1}}{{3}}}$ $\lambda$ where the small values (stiff behavior) appear during unloading at early stage.

POROSI

n is the initial porosity n₀ = e₀/(1 + e₀) = (v₀ -1)/v₀ ( 0 $\leq$ n₀ $\leq$ 1 )where e₀ is the initial voids ratio and v₀ is the specific volume.

POISON

nu is the constant Poisson's ratio $\nu$ . ( -1 < $\nu$ < 0.5 )

Note that for nonlinear analysis no explicit input of the bulk stiffness (for instance via the Young's modulus E ) is necessary since it is defined as a linear function of the mean stress p' , see the background theory in §17.1.6.2. However, for stress initialization via the INITIA STRESS CALCUL command [Vol. Analysis Procedures], as it is the case when applying a weight load and a K₀ procedure, a realistic value of the Young's modulus should be specified, see the example below. A non-zero value of the Young's modulus is required in the preliminary linear analysis for assessing the initial stress distribution. If not specified, DIANA will arbitrarily assume a value of 10 x 10¹⁰ which can lead to unexpected behavior, especially in the presence of interface elements or in the case of a non-horizontal free surface. Note that the YOUNG parameter input in combination with the Egg Cam-clay model is only used in linear analysis but is ignored in nonlinear analysis.
The CLAY yield criterion and the CLAY nonlinear elasticity must be specified both.
An additional log line with NLCRIT in the output file indicates the number of integration points in critical state.

(file.dat)

'MATERI'
 1    YOUNG  8.0E6
      DENSIT 2000.0
      YIELD  CLAY
:            sphi   lambda
      YLDVAL 0.4    0.37
      CAP
      OCR    1.2
      POROSI 0.56
      ELAST  CLAY
      ELAVAL 0.074
      POISON 0.2

5.1.4.2 Extended Input

With extended input you may overrule the regular defaults that DIANA uses to determine the preconsolidation stress p_c , the cap shape factor $\alpha$ and the K₀ ratio. DIANA derives the p_c for each integration point from the maximum stress experienced by the soil element. In its turn, this maximum stress is determined from the in situ stress and the parameters $\mathcal {OCR}$ , K_nc and $\mathcal {OCR}$ _p .

(syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ...m}{[}\>\texttt{K0}\>\texttt{\textit{k0}}$_{r}\,${]} \end{tabbing} \end{figure}$

CAP

indicates the Egg Cam-clay model, where alpha is the explicitly specified cap shape factor $\alpha$ . [ $\alpha$ = 1 ]

K0

k0 is the ratio K₀ to determine the in situ horizontal stresses from the initial stress [§9.6]. The default for Cam-clay plasticity is

K₀ = $\displaystyle \mathcal {OCR}$ K_nc - $\displaystyle {\frac{{\nu}}{{1 - \nu}}}$ ( $\displaystyle \mathcal {OCR}$ - 1)

(5.4)

with $\nu$ the Poisson's ratio.

Initial stress (syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ...}\>\texttt{OCRP}\>\texttt{\textit{ocrp}}$_{r}\,${]} \end{tabbing} \end{figure}$

OCR

ocr is the overconsolidation ratio $\mathcal {OCR}$ [§5.1.4.1]. [ $\mathcal {OCR}$ = 1 ]

KNC

knc is the K -ratio for normally consolidated soil K_nc . [ K_nc = 1 - sin $\phi$ ] The horizontal effective stress, acting when the maximum vertical stress was present, is calculated from

$\displaystyle \sigma^{{\prime}}_{{\mathrm{h_{max}}}}$ = K_nc x $\displaystyle \sigma^{{\prime}}_{{\mathrm{v_{max}}}}$ (5.5)

OCRP

ocrp is an extra multiplication factor $\mathcal {OCR}$ _p . [ $\mathcal {OCR}$ _p = 1 ] The preconsolidation stress p_c based on the maximum stresses is post-multiplied with $\mathcal {OCR}$ _p .

As an alternative to the previous three parameters for the initial stress, the preconsolidation stress may be specified explicitly.

Explicit preconsolidation stress (syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ...}\>\texttt{PRECON}\>\texttt{\textit{pc}}$_{r}\,${]} \end{tabbing} \end{figure}$

PRECON: pc is the preconsolidation stress p_c .

5.1.4.3 Enhanced Model

Merely for research purposes, the Cam-clay model may be enhanced with three additional parameters [Fig.5.4].

**Figure 5.4:** Egg Cam-clay - enhanced yield contour
$\begin{figure}\begin{footnotesize}\setlength{\unitlength}{1cm} \begin{picture... ...enterline{\raise 3.1cm\box\graph} } \end{picture}\end{footnotesize} \end{figure}$

(syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ...{k}}$_{r}\,$ {[}\,\texttt{\textit{pt}}$_{r}\,${]} \end{tabbing} \end{figure}$

YLDVAL: with two extra parameters: pshift is a pressure shift $\Delta$ p ( $\Delta$ p $\geq$ 0 ) [ $\Delta$ p = 0 ] and gamma is a shape factor $\gamma$ ( $\gamma$ $\geq$ 1 ) [ $\gamma$ = 1 ] for the dry side of the yield surface. These parameters are rather numerical values which ensure a strength at the origin of the stress diagram.
ELAVAL: with an extra parameter pt ( pt $\geq$ 0 )to enhance the elasticity model [ pt = 0 ] with a numerical pressure shift which moves the elastic properties along the hydrostatic p axis, i.e., p becomes p + pt .

Next: 5.1.5 Modified Mohr-Coulomb Up: 5.1 Isotropic Plasticity Previous: 5.1.3 Rankine Principal Stress Contents Index

DIANA-9.3 User's Manual - Material Library
First ed.

Copyright (c) 2008 by TNO DIANA BV.