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Plasticity

Plastic strain

In plastic analysis, the materi_strain_elasti rate follows by subtracting from the materi_strain_total rate the materi_strain_plasti rate


\begin{displaymath}
\dot{\epsilon_{ij}}^{\rm elas} = \dot{\epsilon_{ij}} -
\dot{\epsilon_{ij}}^{\rm plas}
\end{displaymath}

where the materi_strain_total rate is


\begin{displaymath}
\dot{\epsilon_{ij}} = 0.5 ( \frac{\partial v_i}{\partial x_j} +
\frac{\partial v_j}{\partial x_i} )
\end{displaymath}

The materi_strain_plasti rate follows from the condition that the stress cannot exceed the yield surface. This condition is specified by a yield function $f^{\rm yield}(\sigma_{ij})=0$. The direction of the plastic strain rate is specified by the stress gradient of a flow function $\frac{\partial f^{\rm flow}}{\partial \sigma_{ij}}$. If the yield function and flow function are chosen to be the same, the plasticity is called associative, else it is non-associative.

Von-Mises is typically used for metal plasticity. Mohr-Coulomb and Drucker-Prager are typically used for soils and other frictional materials. The plasticity models can freely be combined; the combination of the plasticity surfaces defines the total plasticity surface.

First some stress quantities which are used in most of the plasticity models are listed.

Equivalent Von-Mises stress:

\begin{displaymath}
\bar{\sigma} = \sqrt{ \frac{ s_{ij}s_{ij} } {2} }
\end{displaymath}

Mean stress:

\begin{displaymath}
\sigma_m = \frac{ \sigma_{11} + \sigma_{22} + \sigma_{33} } {3}
\end{displaymath}

Deviatoric stress:

\begin{displaymath}
s_{ij} = \sigma_{ij} - \sigma_m \delta_{ij}
\end{displaymath}

Modified CamClay plasticity model

Here we give the equations of the Modified Cam Clay model (Roscoe and Burland, 1968, summarized e.g. by Wood, 1990, see [7]). All stresses are effective (geotechnical) stresses, i.e.compression is positive! Definitions of variables:


\begin{displaymath}
p = (\sigma_{11}+\sigma_{22}+\sigma_{33})/3
\end{displaymath}


\begin{displaymath}
q = \{ \frac{1}{2} [ (\sigma_{11}-\sigma_{22})^2 +
(\sigm...
...+ 3 ( \sigma_{12}^2 + \sigma_{23}^2 + \sigma_{31}^2 ) \}^{1/2}
\end{displaymath}

The CamClay yield rule, which is also the flow rule, reads:


\begin{displaymath}
f = g = q^2 - M^2 [ p (p_0-p) ] = 0
\end{displaymath}

$M$ is a soil constant and $p_0$ is a history (hidden) variable which corresponds to the preconsolidation mean pressure. The hardening function, evolution, of $p_0$ reads:


\begin{displaymath}
d p_0 = \frac{ p_0 (1+e) d\varepsilon_v^p }{ \lambda-\kappa }
\end{displaymath}

in which


\begin{displaymath}
d\varepsilon_v^p = d\varepsilon_{11}^p+d\varepsilon_{22}^p+d\varepsilon_{33}^p
\end{displaymath}

and $\lambda$ and $\kappa$ are user specified soil constants. Further $e$ is the void ratio with the evolution equation:


\begin{displaymath}
de = -d\varepsilon_v (1+e)
\end{displaymath}

in which


\begin{displaymath}
d\varepsilon_v = d\varepsilon_{11}+d\varepsilon_{22}+d\varepsilon_{33}
\end{displaymath}

The poisson ratio $\nu$ reads:


\begin{displaymath}
\nu = \frac{3K - 2G}{2G+6K}
\end{displaymath}

in which the elastic bulk modulus $K$ is given by:


\begin{displaymath}
K = (1+e) p / \kappa
\end{displaymath}

and the Young's modulus $E$:


\begin{displaymath}
E = 2.*G*(1+\nu)
\end{displaymath}

in which $G$ is a user specified soil constant, By using this $\nu$ and $E$ the classical isotropic stress-strain law is used to calculate the stresses.

The soil constants $M$, $\kappa$, $\lambda$ need to be specified in group_materi_plasti_camclay. The soil constant $G$, need to be specified in group_materi_elasti_camclay. The history variables $e$, $p_0$ need to be initialized by materi_history_variables 2 record (and given initial values in node_dof records).

Remark 1: An additional parameter $N$ can be often found in textbooks on the Cam Clay model. We don't include it since it is linked to other model parameters via:

\begin{displaymath}
1+e = N - \lambda \ln p_0 + \kappa \ln (p/p_0)
\end{displaymath}

Remark 2: If you apply a geometrical linear analysis, see section 8.4, then also the calculation of de void ratio development is linearized, and so will contain some error as compared to the exact void ratio change. Hence for very large deformations, say above 10 percent or so, don't use such geometrical linear analysis.

Cap plasticity model

Warning: this cap model is still being tested.

This model accounts for permanent plastic deformations under high pressures for granular materials. It is intended to be used in combination with shear plasticity models like Drucker-Prager, Mohr-Coulomb, etc.

First a deviatoric stress measure $t$ and hydrostatic stress measure $p$ are defined

\begin{displaymath}
t = \sqrt{3} \bar{\sigma}
\end{displaymath}


\begin{displaymath}
p = \frac{\sigma_m}{3}
\end{displaymath}

See above for $\bar{\sigma}$ and $\sigma_m$. The yield rule for the group_materi_plasti_cap model reads:


\begin{displaymath}
f = \sqrt{ (p-p_a)^2 +
\left[ \frac{R t}{(1+\alpha-\frac{\alpha}{cos{\phi}}} \right] ^2
}
- R ( c + p_a tan{\phi} )
\end{displaymath}

Here $c$ is the cohesion and $\phi$ is the friction angle which should be taken equal to the values in the shear flow rule which you use. The parameter $p_a$ follows from


\begin{displaymath}
p_a = \frac{ p_b - Rc }{ 1 + R ~ tan{\phi}}
\end{displaymath}

where the hydrostatic compression yield stress is to be defined with an table of volumetric plastic strains $\epsilon_v^p$ versus $p_b$ with $\epsilon_v^p = \epsilon_{11} + \epsilon_{22} + \epsilon_{33}$.

Associative flow is used, so the flow rule is taken equal to the yield rule.

Summarizing the group_materi_plasti_cap model needs as input the cohesion $c$, the friction angle $\phi$, the parameter $\alpha$ (typically $1.~ 10^{-2}$ up to $5. ~ 10^{-2}$), and a table $\epsilon_v^p$ versus $p_b$.

Compression limiting plasticity model

This group_materi_plasti_compression model uses a special definition for the equivalent stress


\begin{displaymath}
\bar{\sigma} =
\sqrt{ {\sigma_1}^2 + {\sigma_2}^2 + {\sigma_3}^2 }
\end{displaymath}

where $\sigma_1$, $\sigma_2$ and $\sigma_3$ are the first, second and third principal stress respectively. Each of these is only incorporated if it is a compression stress. The model now reads


\begin{displaymath}
\bar{\sigma} - \sigma_y = 0
\end{displaymath}

This plasticity surface limits the allowed compression stresses.

di Prisco plasticity model

The di Prisco model is an non-associative plastic model for soils, which can be typically combined with the 'Lade elastic model'. This di Prisco model is a rather advanced soil model, which is explained in more detail in [3] and [4]. The yield rule reads:


\begin{displaymath}
f = 3 \beta_f (\gamma - 3) \ln \left( \frac{r}{r_c} \right)...
...gamma J_{3 \eta^*} +
\frac{9}{4} ( \gamma - 1 ) J_{2 \eta^*}
\end{displaymath}

and the flow rule yields:


\begin{displaymath}
g = 9 ( \gamma - 3 ) \ln \left( \frac{r}{r_g} \right) - \gamma J_{3 \eta^*} +
\frac{9}{4} ( \gamma - 1 ) J_{2 \eta^*}
\end{displaymath}

This is an anisotropic model in which the first and second invariant of the stress rate $\eta^*$ are defined relative to the rotation axes $\chi$.


\begin{displaymath}
r = \sigma_{ij} \chi_{ij}
\end{displaymath}


\begin{displaymath}
J_{3\eta^*} = \eta_{ij}^* \eta_{jk}^* \eta_{ki}^*
\end{displaymath}


\begin{displaymath}
J_{2\eta^*} = \eta_{ij}^* \eta_{ij}^*
\end{displaymath}


\begin{displaymath}
\eta_{hk}^* = \sqrt{3} \frac{ s_{hk}^* }{ r }
\end{displaymath}

where $s^*$ follows from


\begin{displaymath}
s_{hk}^* = \sigma_{hk}^* - r \chi_{hk}
\end{displaymath}

Further $r_g=1$.

The history variables are $\chi_{ij}$ ( rotation axes, 9 values), $\beta$ (yield surface form factor), and $r_c$ (preconsolidation mean pressure). The evolution laws for these history variables can be found in the papers listed above. The history variables $\chi_{ij}$ (9 values), $\beta$, $r_c$ need to be initialized by the materi_history_variables 11 record (and should be given initial values in node_dof records). In a normally consolidated sand with isotropic initial conditions $\chi_{ij} = \frac{ \delta_{ij} }{ \sqrt{3} }$, $\beta=0.0001$ and $r_c$ equals $\sqrt{3}$ times the means pressure.

The total model, yield rule and flow rule and evolution laws for history variables, contains a set of soil specific constants. In group_materi_plasti_diprisco you need to specify these constants. These constants are explained in more detail in the papers mentioned above, but here we give a short explanation. The constants $\hat{\theta}_c$, $\hat{\theta}_e$, $\xi_c$ and $\xi_e$ are linked to the dilitancy and the stress state during failure (standard triaxial compression and extension test in drained conditions). The constants $\gamma$, $c_p$, $\beta_f$ and $\beta_f^0$ are defined by means of the experimental curves ( $q$- $\epsilon_{axial}$, $\epsilon_{vol}$- $\epsilon_{axial}$) obtained by performing a standard compression test in drained conditions. Moreover, $\beta_f$, $\beta_f^0$ and $t_p$ can also be determined by means of the effective-stress path obtained by performing a standard triaxial compression test in undrained conditions. Finally $b_p$ can determined from an isotropic compression test. For a loose sand $\hat{\theta}_c=0.253$, $\hat{\theta}_e=0.0398$, $\xi_c=-0.2585$, $\xi_e=-0.0394$, $\gamma=3.7$, $c_p=18.$, $\beta_f=0.5$, $\beta_f^0=1.1$, $t_p=10.$, and $b_p=0.0049$.

Drucker-Prager plasticity model

The group_materi_plasti_druckprag model reads


\begin{displaymath}
3 \alpha \sigma_m + \bar{\sigma} - K = 0
\end{displaymath}


\begin{displaymath}
\alpha = \frac{2 \sin( \phi )}{\sqrt{3} ( 3 - \sin(\phi) )}
\end{displaymath}


\begin{displaymath}
K = \frac{ 6 c \cos( \phi )}{\sqrt{3} ( 3 - \sin(\phi) )}
\end{displaymath}

Here $c$ is the cohesion, which needs to be specified both for the yield function and the flow rule; by choosing different values non-associative plasticity is obtained.

Gurson plasticity model

The group_materi_plasti_gurson model reads

\begin{displaymath}
\frac{3 \bar{\sigma}^2}{\sigma_y^2} +
2 q_1 f^* \cosh ( q_2 \frac{3 \sigma_m}{2 \sigma_y} ) -
(1 + ( q_3 f^* ) ^2 ) = 0
\end{displaymath}

Here $f^*$ is the volume fraction of voids. The rate equation

\begin{displaymath}
\dot{f^*} = ( 1 - f^*) f^* \epsilon_{kk}^{\rm plas}
\end{displaymath}

defines the evolution of $f^*$ if the start value for $f^*$ is specified. Furthermore, $q_1$, $q_2$ and $q_3$ are model parameters.

Tension limiting plasticity model

This group_materi_plasti_tension model uses a special definition for the equivalent stress


\begin{displaymath}
\bar{\sigma} =
\sqrt{ {\sigma_1}^2 + {\sigma_2}^2 + {\sigma_3}^2 }
\end{displaymath}

where $\sigma_1$, $\sigma_2$ and $\sigma_3$ are the first, second and third principal stress respectively. Each of these is only incorporated if it is a tension stress. The model now reads


\begin{displaymath}
\bar{\sigma} - \sigma_y = 0
\end{displaymath}

This plasticity surface limits the allowed tension stresses.

A simple model for concrete can be obtained as follows. Ue group_materi_plasti_tension to limit the tension strength ft. Use group_materi_plasti_compression to limit the compressive strength fc. The tension strength could be softened to zero over a plastic strain of, say, 1 percent. The compressive strength could be softened to zero over a plastic strain of, say, 10 percent.

Von-Mises plasticity model

The group_materi_plasti_vonmis model reads


\begin{displaymath}
\sqrt{3} ~ \bar{\sigma} - \sigma_y = 0
\end{displaymath}

where without hardening the yield value is fixed $ \sigma_y = \sigma_{y0} $.

If however the group_materi_plasti_vonmis_nadai hardening law for Von-Mises plasticity is specified then


\begin{displaymath}
\sigma_y = \sigma_{y0} + C { ( \kappa\_0 + \kappa ) } ^ n
\end{displaymath}

where $C$, $\kappa\_0$ and $n$ are parameters for the hardening law, and $\kappa$ is the isotropic hardening parameter (see later). The parameter $\sigma_{y0}$ is specified by group_materi_plasti_vonmis.

Mohr-Coulomb plasticity model

The group_materi_plasti_mohrcoul model reads


\begin{displaymath}
0.5 ( \sigma_1 - \sigma_3 ) + 0.5 ( \sigma_1 + \sigma_3 ) \sin ( \phi )
- c ~ \cos ( \phi ) = 0
\end{displaymath}

Here $c$ is the cohesion, $\sigma_1$ is the maximal principal stress and $\sigma_3$ is the minimal principal stress. The angle $\phi$ needs to be specified for both the yield condition and the flow rule; by choosing different values, non-associative plasticity is obtained.

Isotropic Hardening

The size of the plastic strains rate is measured by the materi_plasti_kappa parameter

\begin{displaymath}
\dot{\kappa} = \sqrt{ 0.5 \dot{\epsilon}_{ij}^{\rm plas} \dot{\epsilon}_{ij}^{\rm plas} }
\end{displaymath}

This parameter can be used for isotropic hardening. Use the dependency_diagram for this.

Kinematic Hardening

The materi_plasti_rho matrix $\rho_{ij}$, governs the kinematic hardening in the plasticity models. It is used in the yield rule and flow rule to get a new origin by using the argument $\sigma_{ij} - \rho_{ij}$:


\begin{displaymath}
f^{\rm yield} = f^{\rm yield}(\sigma_{ij} - \rho_{ij})
\end{displaymath}


\begin{displaymath}
f^{\rm flow} = f^{\rm flow}(\sigma_{ij} - \rho_{ij})
\end{displaymath}

where the rate of the matrix $\rho_{ij}$ is taken to be


\begin{displaymath}
\dot { \rho_{ij} } = a \;\; \dot{\epsilon_{ij}}^{\rm plas}
\end{displaymath}

where $a$ is a user specified factor (see group_materi_plasti_kinematic_hardening).

Plastic heat generation

The plastic energy loss can be partially turned into heat rate per unit volume $q$:


\begin{displaymath}
q = \eta \: \sigma_{ij} \: \dot{\epsilon_{ij}}^{\rm plas}
\end{displaymath}

where $\eta$ is a user specified parameter (between 0 and 1) specifying which part of the plastic energy loss is turned into heat (see group_materi_plasti_heat_generation).


next up previous contents
Next: Damage Up: Material deformation and flow Previous: Elasticity   Contents
root
1999-04-23