Elasto-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

$$\dot{\epsilon_{ij}}^{\rm elas} = \dot{\epsilon_{ij}} - \dot{\epsilon_{ij}}^{\rm plas}$$

where the materi_strain_total rate is

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

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:

$$\bar{\sigma} = \sqrt{ \frac{ s_{ij}s_{ij} } {2} }$$

Mean stress:

$$\sigma_m = \frac{ \sigma_{11} + \sigma_{22} + \sigma_{33} } {3}$$

Deviatoric stress:

$$s_{ij} = \sigma_{ij} - \sigma_m \delta_{ij}$$

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 [12]). All stresses are effective (geotechnical) stresses, i.e.compression is positive! Definitions of variables:

$$p = (\sigma_{11}+\sigma_{22}+\sigma_{33})/3$$

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

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

$$f = g = q^2 - M^2 [ p (p_0-p) ] = 0$$

$M$ 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:

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

in which

$$d\varepsilon_v^p = d\varepsilon_{11}^p+d\varepsilon_{22}^p+d\varepsilon_{33}^p$$

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

$$de = -d\varepsilon_v (1+e)$$

in which

$$d\varepsilon_v = d\varepsilon_{11}+d\varepsilon_{22}+d\varepsilon_{33}$$

The poisson ratio $\nu$ reads:

$$\nu = \frac{3K - 2G}{2G+6K}$$

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

$$K = (1+e) p / \kappa$$

and the Young's modulus $E$:

$$E = 2.*G*(1+\nu)$$

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_g. For an alternative see group_materi_elasti_camclay_poisson. 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:

$$1+e = N - \lambda \ln p_0 + \kappa \ln (p_0/p)$$

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

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

$$t = \sqrt{3} \bar{\sigma}$$

$$p = - \sigma_m$$

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

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

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

$$p_a = \frac{ p_b - Rc }{ 1 + R ~ tan{\phi}}$$

where the hydrostatic compression yield stress $p_b$ is to be defined with an table of volumetric plastic strains $\epsilon _v^p$ versus $p_b$ with $\epsilon_v^p = \epsilon_{11}^p + \epsilon_{22}^p + \epsilon_{33}^p$. As always, positive strain denote extension whereas negative strains denote compression.

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

$$\bar{\sigma} = \sqrt{ {\sigma_1}^2 + {\sigma_2}^2 + {\sigma_3}^2 }$$

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

$$\bar{\sigma} - \sigma_y = 0$$

This plasticity surface limits the allowed compression stresses.

Tension limiting plasticity model

This group_materi_plasti_tension model uses a special definition for the equivalent stress

$$\bar{\sigma} = \sqrt{ {\sigma_1}^2 + {\sigma_2}^2 + {\sigma_3}^2 }$$

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

$$\bar{\sigma} - \sigma_y = 0$$

This plasticity surface limits the allowed tension stresses.

A simple model for concrete can be obtained as follows. Use 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 an effective plastic strain $\kappa$ of, say, 1 percent. The compressive strength could be softened to zero over an effective plastic strain $\kappa$ of, say, 10 percent.

Another possibility for concrete is to combine group_materi_plasti_tension to limit the tension strength ft, and use the group_materi_plasti_vonmis model to limit the compressive strength fc.

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 [4] and [5]. The yield rule reads:

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

and the flow rule yields:

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

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

$$r = \sigma_{ij} \chi_{ij}$$

$$J_{3\eta^*} = \eta_{ij}^* \eta_{jk}^* \eta_{ki}^*$$

$$J_{2\eta^*} = \eta_{ij}^* \eta_{ij}^*$$

$$\eta_{hk}^* = \sqrt{3} \frac{ s_{hk}^* }{ r }$$

where $s^*$ follows from

$$s_{hk}^* = \sigma_{hk}^* - r \chi_{hk}$$

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.

A cohesion $C$ can be also be introduced if required.

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

$$3 \alpha \sigma_m + \bar{\sigma} - K = 0$$

$$\alpha = \frac{2 \sin( \phi )}{\sqrt{3} ( 3 - \sin(\phi) )}$$

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

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

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

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

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

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

HLC plasticity model

The group_materi_plasti_hlc Hau-Liu-Chang model is nearly similar to the Gurson's equation, and it is reads

$$\frac{3 \bar{\sigma}^2}{\sigma_y^2} + f^* ( + \frac{1}{m_{20}}) m_1 \exp ( \frac{3 \sigma_m}{2 \sigma_y} ) - 1 = 0$$

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

The rate equation

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

defines the evolution of $f^*$ if the start value for $f^*$ is specified. The variable $m_{20}$ is a function of the porosity:

$$m_{20} = ( m_{21} - m_{22} f^*) \exp ( m_{21} \frac{\sigma_m}{\bar{\sigma}} )$$

Furthermore, $\sigma _y$, $m_{21}$, $m_{22}$, $m_{23}$ and $m_1$ are model parameters. The parameters can be calculated numerically by simple numerical simulations.

See http://www.tam.nwu.edu/wkl/paper/suhao-paper2.html for further info about the model.

Von-Mises plasticity model

The group_materi_plasti_vonmis model reads

$$\sqrt{3} ~ \bar{\sigma} - \sigma_y = 0$$

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

$$\sigma_y = \sigma_{y0} + C { ( \kappa\_0 + \kappa ) } ^ n$$

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.

Modified Matsuoka-Nakai model plasticity model

The Matsuoka-Nakai model [8] is a perfectly plastic model thus the fixed yield surface represents the failure surface as well. The model is based on experimental results with soils and can be formulated in terms of three stress invariants

$$f = I_3 + \frac{\cos^2 \phi}{9-\sin^2 \phi} ~ {I_1 \, I_2} = 0$$

where

$$\begin{eqnarray*} I_1 &=& \mbox{tr}(\sigma_{ij}) = \sigma_{11}+\sigma_{22}+\sigm... ...\\ I_3 &=& \mbox{det}(\sigma_{ij}) = \sigma_1 \sigma_2 \sigma_3 \end{eqnarray*}$$

$\sigma_1$, $\sigma_2$ and $\sigma_3$ are the principal stresses (all stresses are effective; compressive stresses are negative). The parameter $\phi$ is equal to the angle of internal friction in axisymmetric (triaxial) compression [11].

For axisymmetric stress states the Matsuoka-Nakai model corresponds to the Mohr-Coulomb model. Nevertheless, the Matsuoka-Nakai model is described by a smooth surface in the stress space and thus it is more suitable from the computational aspect.

When the cohesion $c$ is considered in the model, the yield condition is formulated for a modified stress [9]

$$\bar{\sigma}_{ij} = \sigma_{ij}-\sigma_0\delta_{ij}$$

with

$$\sigma_0 = c \cot \phi ~.$$

Notice that straightforward usage of the function $f$ above also as flow rule $g$ would lead to a loss of the information on $\phi$ in the derivative of $g$, since the $\phi$ only appears in a constant.

For that reason we apply as flow rule $g$ the Drucker-Prager function (see elsewehere in this theoretical part). A separate $\phi_{flow}$ can be specified which enters this Drucker-Prager $g$. Since we use the Drucker-Prager function as flow rule $g$, the present plasticity model has been named Modified Matsuoka-Nakai.

Mohr-Coulomb plasticity model

The group_materi_plasti_mohrcoul model reads

$$0.5 ( \sigma_1 - \sigma_3 ) + 0.5 ( \sigma_1 + \sigma_3 ) \sin ( \phi ) - c ~ \cos ( \phi ) = 0$$

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.

For a numerically more stable solution, consider using Matsuoka-Nakai plasticity in stead of Mohr-Coulomb.

Mohr-Coulomb softening plasticity model

The group_materi_plasti_mohrcoul_softening model is the same as the standard Mohr-Coulomb model. Now, however, the parameters $c$ and $\phi$ (both for the yield rule and for the flow rule) are softened on the the effective plastic strain $\kappa$.

By example, for the cohesion a linear variation is taken between the initial value $c_0$ at $\kappa=0$, up to $c_1$ at a specified critical value of $\kappa$, and constant $c_1$ for larger values of $\kappa$. The same is done for $\phi$ for the yield rule and for the flow rule.

Isotropic Hardening

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

$$\dot{\kappa} = \sqrt{ 0.5 \dot{\epsilon}_{ij}^{\rm plas} \dot{\epsilon}_{ij}^{\rm plas} }$$

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 $\sigma_{ij} - \rho_{ij}$:

$$f^{\rm yield} = f^{\rm yield}(\sigma_{ij} - \rho_{ij})$$

$$f^{\rm flow} = f^{\rm flow}(\sigma_{ij} - \rho_{ij})$$

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

$$\dot { \rho_{ij} } = a \;\; \dot{\epsilon_{ij}}^{\rm plas}$$

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 $q$:

$$q = \eta \: \sigma_{ij} \: \dot{\epsilon_{ij}}^{\rm plas}$$

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_heatgeneration).