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, otherwise 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.

Typically, if you use Mohr-Coulomb or Drucker-Prager to model shear failure for soils, you should use the tension limiting model to limit tension stresses, preferably group_materi_plasti_tension_direct.

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:

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

CamClay plasticity model

Here we provide the equations of the Cam Clay model (Roscoe and Burland, 1968, summarized e.g. by Wood, 1990, see [19]). All stresses are effective (geotechnical) stresses, i.e.compression is positive! Definitions of variables:

$$p = (\sigma_{1}+\sigma_{2}+\sigma_{3})/3$$

$$q = \{ \frac{1}{2} [ (\sigma_{1}-\sigma_{2})^2 + (\sigma_{2}-\sigma_{3})^2 + (\sigma_{3}-\sigma_{1})^2 ] \}^{1/2}$$

in the principal stress axes. 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_plasti_camclay_history 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.

Cap1 plasticity model

This group_materi_plasti_cap1 model is the first cap model that 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 the average stress $p$ and the equivalent shear stress $q$ are introduced:

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

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

These are used to define the cap plastic yield function:

$$f = \frac{q^2}{M^2} + p^* ( p^* - p_c^*)$$

where

$$p^* = p + c \cot \phi ~~~~~~~~~~~ p_c^* = p_c + c \cot \phi$$

The parameter $p_c$ is a history variable of this model. The parameter $\phi$ is the coulomb friction angle, and $c$ is the cohesion. The parameter $M$ denotes the tangent of the Critical State Line in the model, Typically you can use:

$$M = \frac{6 \sin \phi}{3 - \sin \phi}$$

The history parameter $p_c$ is assumed to harden with the cap plastic volume strain rate according to the rate form:

$$\dot{\epsilon}_{cv}^p = \frac{\lambda^* / \kappa^* - 1}{K^{ref}} \left( \frac{p^{ref}}{p_c^*} \right)^m \dot{p}_c$$

Here $\kappa^*$ is the swelling index (e.g. 0.03), $\lambda^*$ is the compression index (e.g. 0.15), $K^{ref}$ is the bulk modulus at stress $p^{ref}$ (typically $100 kPa$), which typically can be taken as: $K^{ref} = \frac{E^{ref}}{3 (1-2 \nu)}$, and finally $m$ is an exponent (e.g. 0.6).

Initialise materi_plasti_cap1_history in the initialisation part. The state variable $p_c$ for this hardening soil model enters the node_dof records. You need to give an initial value for it in the node_dof records. See also [2].

Cap2 plasticity model

This is the second cap model that 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_cap2 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_cap2 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

$$\bar{\sigma} = \sqrt{ {\sigma_{min}}^2 }$$

where $\sigma_{min}$ is the largest compressive principal stress. The model now reads

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

This plasticity surface limits the allowed compressive 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 [7]. 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 group_plasti_diprisco_history 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 dilatancy 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 $b_p=0.0049$.

di Prisco plasticity model with varying density

This essentially is the same as the normal di Prisco model, but instead of one set of parameters you need to specify two sets of parameters, one of loose soil and one for dense soil. The actual applied parameters will then be interpolated from the loose parameters and dense parameters depending on the actual density of the soil. The parameters need to be specified in group_materi_plasti_diprisco_density.

The history variables are those of group_materi_plasti_diprisco and finally extra the relative density (by example 20 or 40). So there are 12 history variables in total.

Drucker-Prager plasticity model

The group_materi_plasti_druck_prag 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.

You should also include tension cut-off, preferably with group_materi_plasti_tension_direct.

Generalised Non Associate CamClay for Bonded Soils plasticity model

The group_materi_plasti_generalised_non_associate_cam_clay_for_bonded_soils is presently available for selected customers only. It is a modification of the 'Milan' model of Prof. Roberto Nova.

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.

Hardening-Soil model

In this section, the principal stresses are ordered such that

$$\sigma_3 > \sigma_2 > \sigma_1$$

so that $\sigma_1$ is the largest compressive stress. Likewise for the principal plastic strains:

$$\epsilon^p_3 > \epsilon^p_2 > \epsilon^p_1$$

First the elasticity parameters are defined. The elasticity parameters for the first loading are:

$${\rm Young's ~ modulus} = E_{50} = E_{50}^{ref} { \left( \frac{... ...+ c ~ \cot \phi } \right) }^m ~~~~ {\rm and ~ Poisson's ~ ratio} = \nu_{50}$$

The elasticity parameters for the elastic unloading and reloading are:

$${\rm Young's ~ modulus} = E_{ur} = E_{ur}^{ref} { \left( \frac{... ...~ \cot \phi } \right) }^m ~~~~ {\rm and ~ Poisson's ~ ratio} ~ = ~ \nu_{ur}$$

The yield function reads:

$$f = \frac{ 1 }{ E_{50} } \frac{ q }{ 1 - q/q_a } - \frac{2 q}{E_{ur}} - \gamma^p$$

where $q$ is the equivalent shear stress and $\gamma^p$ is the equivalent plastic shear strain.

The equivalent asymptotic shear stress reads

$$q_a = \frac{ q_f }{ R_f }$$

in which $q_f$ is the shear failure stress, and $R_f$ is the failure ratio.

Specify all elasticity parameters in group_materi_elasti_hardsoil. Typically you have:

• $E_{50}^{ref}$ from experiment at stress $\sigma_{50}^{ref}$
• $\nu_{ur}$ from experiment or the typical undrained value 0.495 or the typical drained value 0.3
• $m$ from experiment or the typical value 0.5
• $E_{ur}^{ref}$ from experiment at stress $\sigma_{ur}^{ref}$, or the typical value $3 E_{50}^{ref}$
• $\nu_{ur}$ from experiment or the typical undrained value 0.495 or the typical drained value 0.2

Specify all plasticity parameters in group_materi_plasti_hardsoil.

• $\phi$ from experiment (maximum friction angle)
• $c$ from experiment (cohesion)
• $\psi$ from experiment (maximum dilatancy angle)
• $R_f$ from experiment or the typical value 0.9 (failure ratio)

Initialise materi_strain_plasti_hardsoil in the initialisation part. This causes that the node_dof records will be filled with the shear plastic strains. Also initialise materi_plasti_hardsoil_history.

See also [17] for some details. Especially notice that the model is more suited for monotonic loading than for load cycling (since it violates thermodynamics and tends to generate energy).

Matsuoka-Nakai model plasticity model

The Matsuoka-Nakai model [12] 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

$$I_1 = (\sigma_{ij}) = \sigma_{11}+\sigma_{22}+\sigma_{33} = \sigma_1 + \sigma_2 + \sigma_3 = 3 \sigma_m$$

$$I_2 = \frac{1}{2} \left( \mbox{tr} ( \sigma_{ik}\sigma_{kj} ) - I_1^2 \right) = -\sigma_1 \sigma_2 - \sigma_2 \sigma_3 - \sigma_3 \sigma_1$$

$$I_3 = (\sigma_{ij}) = \sigma_1 \sigma_2 \sigma_3$$

$\sigma_1$, $\sigma_2$ 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 [18].

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 [13]

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

with

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

You should also include tension cut-off, preferably with group_materi_plasti_tension_direct.

Matsuoka-Nakai hardening-softening plasticity model

The group_materi_plasti_matsuoka_nakai_hardening_softening model is the same as the standard Matsuoka-Nakai model. However, the parameters $c$ and $\phi$ (both for the yield rule and for the flow rule) are softened on the effective plastic strain $\kappa^{shear}$.

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

You should also include tension cut-off, preferably with group_materi_plasti_tension_direct.

Mohr-Coulomb plasticity model

The group_materi_plasti_mohr_coul 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 largest principal stress and $\sigma_3$ is the smallest 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.

As an alternative consider using group_materi_plasti_mohr_coul_direct, which is more stable and fast.

You should also include tension cut-off, preferably with group_materi_plasti_tension_direct.

Mohr-Coulomb hardening-softening plasticity model

The group_materi_plasti_mohr_coul_hardening_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 effective plastic strain $\kappa^{shear}$.

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

You should also include tension cut-off, preferably with group_materi_plasti_tension_direct.

Multilaminate plasticity model

Plastic yield function.

The multi-laminate model predefines a number of weak planes, which have reduced plasticity parameters as compared to the bulk material. The numerical model will thus have the tendency to start slipping on the weak planes first, just like physical reality with weak planes. In fact, the yield function for each laminate amounts to a standard mohr-coulomb slip condition with predefined slip plane. The model reads

$$f_k = ( \vert \sigma_{pq} \vert + \sigma_{qq} \tan ( \phi ) - c )_k$$

where $p$ denotes the in-plane direction of a laminate, $q$ denotes the normal direction of the laminate, $\phi$ denotes the friction angle of the laminate, $c$ is the cohesion in the laminate, and finally $k$ is the laminate number. The direction $p$ is taken such in the plane of the laminate, that $\sigma_{pq}$ is the maximum shear stress in the laminate plane. The stress $\sigma_{qq}$ is normal to the laminate plane. The user needs to specify a normal vector $n_{qk}$ to the plane of laminate $k$, so that the plane of the laminate is precisely defined.

Plastic flow rule.

To allow for non-associated plastic flow, a dilatancy angle $\psi$ is used:

$$g_k = ( \vert \sigma_{pq} \vert + \sigma_{qq} \tan ( \psi ) - c )_k$$

where again $k$ denotes the number of the Multilaminate.

Elasto-plastic versus elasto-viscoplastic.

The multi-laminate plasticity model can be used elasto-plastic, but can also be used with viscoplasticity (time-dependent plasticity). In the latter case, you can apply the input data
group_materi_plasti_visco_power_name and
group_materi_plasti_visco_power_value.

Tension cutoff in laminates

To allow for laminate crack opening, you can specify a tension cutoff limit as yield function:

$$f_k = ( \sigma_{qq} - \sigma_t )_k$$

where $sigma_t$ is the maximum allowable tension stress, and $k$ is again the laminate number. Specify this model with the input data group_materi_plasti_laminate0_tension.

Initialisation multi-laminate model

You always need to initialise materi_plasti_laminate with the number of required laminates.
Optionally initialise materi_strain_plasti_laminate_mohr_coul etc. if you want to
see the mohr-coulomb slip strains in the laminates.
Optionally initialise materi_strain_plasti_laminate_tension etc. if you want to
see tension cutoff strains in the laminates,

Status of laminates

The status of the mohr-coulomb yield condition in the integration points of elements can be found after a calculation in element_intpnt_plasti_laminate0_mohr_coul_status etc. Likewise, the status of the tension yield condition can be found in element_intpnt_plasti_laminate0_tension_status etc.

Tension limiting plasticity model

This group_materi_plasti_tension model uses a special definition for the equivalent stress

$$\bar{\sigma} = \sqrt{ {\sigma_{max}}^2 }$$

where $\sigma_{max}$ is the largest principal tension stress.

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

This plasticity surface limits the allowable 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_vonmises 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.

Von-Mises plasticity model

The group_materi_plasti_vonmises 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_vonmises_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_vonmises.

Isotropic Hardening and softening

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

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

The size of the shear plastic strains rate is measured by the materi_plasti_kappa_shear parameter

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

where the plastic shear strains are defined by

$$\dot{\epsilon}_{ij}^{\rm shear,plas} = \dot{\epsilon}_{ij}^{\rm p... ... plas} + \dot{\epsilon}_{22}^{\rm plas} + \dot{\epsilon}_{33}^{\rm plas} ) / 3$$

These parameters $\kappa$ and $\kappa^{shear}$ 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_heat_generation).