===== VMVP =====
**V**on **M**ises **V**isco-**P**lastic.
==== Description ====
3D isotropic viscoplastic damage law that enables the modeling of non-classical creep responses via Graham-Walles and a modified activation function-Norton viscosity function.

=== Overview ===
This law was implemented in the context of C.Rojas-Ulloa's PhD. project (01/2021-12/2025) on the modeling of the non-classical long-term creep response of Incoloy 800H (see [[https://doi.org/10.1016/j.camwa.2023.12.002|(C.Rojas-Ulloa et al., 2024)]] for more details). This colaw is based in the work of Hélène Morch ([[laws:chab|CHAB]], a von-Mises yield function combined with a Norton-type viscosity function, Kachanov-Lemaitre creep-fatigue damage and high flexibility for introducing parameters as $f(T)$). **VMVP** incorporates two new viscoplastic functions (Graham-Walles & AFN), a simple //IfW// creep-fatiogue damage formulation, and new parameter interpolation methods available for material parameters.

==== VMVP: formulation generalities====
The yield surface is defined by the von-Mises yield criterion: an isotropic $J_{2}$-type function of the form:\\


$J_{2}(\underline{\tilde{\sigma}}-\underline{\mathbb{X}})=\Bigl[\frac{3}{2}(\underline{\tilde{S}}-\underline{\mathbb{X}}):(\underline{\tilde{S}}-\underline{\mathbb{X}})\Bigr]^{0.5}$\\

where $\underline{\tilde{\sigma}}$ is the effective stress tensor. It is calculated as function of the unitary damage $D$, $0 \leq D \leq 1$  as: $\underline{\tilde{\sigma}}=\underline{\sigma}\cdot (1-D)^{-1}$\\

$\underline{\tilde{S}}$ is the deviatoric stress tensor, calculated as:

$\underline{\tilde{S}} = \underline{\tilde{\sigma}}-\frac{1}{3}tr(\underline{\tilde{\sigma}})\underline{\mathbb{I}}$\\


The function $\Phi$ defining the yield criteiron is:\\
$\Phi = J_{2}(\underline{\tilde{\sigma}}-\underline{\mathbb{X}})-\sigma_{y}-R \leq 0$\\

>where:
>* $\sigma_{y}$ is the yield stress of the material.
>* $R$ is the isotorpic hardening, calculated as the Voce iso. hardening law:
>$\hspace{1cm} \dot{R} = -b\cdot (Q-R)\cdot \dot{p}  \hspace{1cm} -> \hspace{1cm} R(p)=Q\cdot (1-\exp(-b\cdot p))$\\ where $p$ is the equivalent plastic strain $(-)$, and $\dot{p}$ is the equivalent plastic strain rate $(s^{-1})$\\

Inheriting the classical Chaboche-type formulation, the backstress tensor $\underline{\mathbb{X}}$ is calculated as the sum of a total of $nAF$ user-defined Armstrong-Frederick (AF) backstress equations. \\
$\dot{\underline{\mathbb{X}}} = \displaystyle{\sum_{i=1}^{nAF}} \frac{2}{3}C_{i}\dot{\underline{p}} - \gamma_{i}\Bigl( \dot{\underline{\mathbb{X}}}_{i} - \dot{\underline{\mathbb{Y}}}_{i} \Bigr)\dot{p} - b_{i}J_{2}\Bigl( \underline{\mathbb{X}}_{i} \Bigr)^{r_{i}-1} \underline{\mathbb{X}}_{i} + \frac{1}{C_i}\frac{\partial C_{i}}{\partial T}\dot{T} \underline{\mathbb{X}}_{i} $ \\

>Within the $\sum$ term, from left to right:
>> **The first term** is a Swift-type kinematic hardening. \\ Here, $C_{i}$ is the only material constant, and $\dot{\underline{p}}$ is the plastic strain rate vector (Voigt notation in Lagamine).\\
>> **The second term** is intended to model the static recovery of the material.\\ Following the Chaboche formulation, it addresses terms for:\\
>>> __Mean stress evolution $\underline{\mathbb{Y}}$__. Following Chaboche formulation and H. Morch's work, the mean-stress is conceived as the summation of a total of a user-defined number $j$ of terms $0 \leq j\leq i$. In its time-dependent variational form, the $j^{\text{th}}$ equation is calculated as:
>>>$\dot{\underline{\mathbb{Y}}}_j=\alpha_{b_{j}}\cdot \Bigl( \frac{3}{2} Y_{\text{st}_{j}}\frac{\underline{\mathbb{X}}_{j}}{J_{2}(\underline{\mathbb{X}}_{j})} + \underline{\mathbb{Y}}_{j} \Bigr) J_{2}(\underline{\mathbb{X}}_{j})^{r_{j}-1} $
>>>···
>>> __Strain memory surface__ $\gamma_{i}$. This Chaboche formulation is intended to model the non-masing behavior observed in certain Ni-based and martensitic alloys. Similarly, the total strain memory surface is the result of a sum of a user-defined number $k$ equations $k\geq 0$. The form of the $h^{\text{th}}$ equation in its variational (time-dependent) is:
>>> $\dot{\gamma_{k}} = D_{\gamma_{k}}\Bigl( \gamma_{k}^{0} - \gamma_{k} \Bigr) \dot{p}$
>>> where $D_{\gamma_{k}}$ is a material parameter, and the term $\gamma_{k}^{0}$ is a function of the form:
>>>$\gamma_{k}^{0}=a_{\gamma_{k}} + b_{\gamma_{k}}\exp\bigl( -c_{\gamma_{k}}\text{q} \bigr)$\\ where $\text{q}$ is the norm of the equivalent plastic strain $p$ in the loading history $(\text{q} = \underline{p}:\underline{p})$.\\
>> **The third term** deals with the dynamic recovery of the material.
>> Here, $b_i$ and $r_i$ are user-defined material parameters.\\
>> **The fourth term** is made for non-isothermal plasticity effects.
>> This term is important in presence of thermal gradients $\dot{T}$, and if we introduce the parameter $C_i$ as function of temperature.\\

==== VMVP: viscoplasticity ====
As a viscoplastiv law, **VMVP** includes a total of 3 viscoplastic functions. 
> **option ivp=1: Norton law**
>> This is the classical Norton constitutive law, described as:
>> $\dot{p} = \langle \frac{\sigma_v}{K} \rangle ^{n}$
>> where:
>> · $K$ and $n$ are the drag stress and Norton exponent (material parameters).
>> · $\sigma_{v}$ is the overstress, defined as the stress resulting from the calculation of the yield function $\Phi=0$, i.e.: 
>> $\sigma_{v} = J_{2}(\underline{\tilde{\sigma}}-\underline{\mathbb{X}}) - \sigma_{y}-R$ \\
> **option ivp=2: Graham-Walles creep equations**
>> This set of equations are intended to describe the total creep response as a summation of a total of $nvp$ user-defined number of individual functions, each of them representing different hardening/softening creep regimes. Each $l$ function is of the form:
>> $\dot{p} = \displaystyle{\sum_{l=1}^{nvp}} \Bigl[ K_{l}\exp{\Bigl(\frac{T}{C_l}\Bigr)} J_{2}(\underline{\tilde{\sigma}}-\underline{\mathbb{X}})^{n_l} p^{m_l} \Bigr] + |\dot{T}| K_{T} J_{2}(\underline{\sigma}) p^{m_T}$
>> where $K_{l}, C_{l}, n_{l}, m_{l}$ are user-defined material parameters for each $g$ creep equation, and $K_D, m_T$ are user-defined parameters for an additional creep-fatigue interaction term.\\
>
> **option ivp=4: AFN** (Activation Function $\times$ Norton)
>> This equation is new, and was formulated after a thorough analysis on the creep response of Incoloy 800H under lows-tress and high-temperature loadings. It is is intended to follow the classical Norton behavior, where the material reaches a steady-state creep rate $p_{\text{ss}}$. However, the phenomena such as solid-solution hardening or particle-strengthening may induce an initial creep hardening effect where the material can reach a creep rate lower that that of the steady state. To address this initial hardening, a phenomenological temperature and stress-dependent activation function has been added. The form of AFN is:\\ $\dot{p}= \exp \Bigl[ -a(1.05 + p - 0.001b)^{-c} \Bigr] \times \langle \frac{\sigma_v}{K} \rangle ^{n}$ \\ Let us now walk you through the Activation Function particulars:\\
>>> __Activation Function__. The Activation Function is of the form:
>>> $\exp \Bigl[ -a(1.05 + p - 0.001b)^{-c} \Bigr]$
>>> where $a, b, c$ are material parameters. Strictly speaking however, these are temperature and stress dependent functions. With all experimental and theoretical information available so far, we have reached the following mathematical formulations for each parameter:
>>>> $a(\sigma_v,T) = \frac{a_1}{1+\exp \bigl( \frac{\sigma_v - p_2}{a_3} \bigr)}$
>>>> $b(\sigma_v,T) = \frac{x_1}{1+\exp \bigl( \frac{p_2 - \sigma_v}{b_3} \bigr)} + b_4$
>>>> $c(\sigma_v,T) = \frac{c_1}{1+\exp \bigl( \frac{\sigma_v - p_2}{c_3} \bigr)}$
>>>> where $a_1, a_3, b_1, b_3, b_4, c_1, c_3$ are material parameters, and $p_2$ is a temperature-dependent function defined as an Arrhenius-type function:
>>>> $p_{2}(T) = p_{21} \Big[ 1 - p_{22} \exp \Big( \frac{T}{p_{23}} \Big)\Big]$
>>>> where $p_{21}, p_{22}, p_{23}$ are user-defined material parameters.

==== VMVP: damage formulations ====
Damage computation is added to enable the prediction of the effects of microcracks and its evolution towards the final fracture of the material. To use this formulation, the user must first define the damage methodology (dam_type) to be used in the simulation. A brief explanation of the avalable possibilities is given below:\\
> **dam_type = 0** No damage
>In this case, no damage is considered within  the finite element simulations. Therefore
> · $\tilde{\underline{\sigma}}_{k+1}=\underline{\sigma}_{k+1}$.
> · $D=0$\\
> **dam_type = 1** uncoupled damage
> Here damage is calculated explicitly (e.g., at the end of every %k+1% viscoplastic loop), but the stress is not affected by damage. 
>> · $\underline{\tilde{\sigma}}_{k+1}=\underline{\sigma}_{k+1}$
>> · $D_{k+1}=f \Bigl( \underline{\sigma}_{k+1}, D_{k},... \Bigr)$\\
> **dam_type = 2** semi-coupled damage
>In this case, damage is also computed explicitly. However, the effective stress is also computed, as a function of the damage in the previous configuration. i.e.:
>> · $\underline{\tilde{\sigma}}_{k+1}=\underline{\sigma}_{k+1} \cdot (1-D_{k})^{-1}$
>> · $D_{k+1}=f \Bigl( \underline{\sigma}_{k+1}, D_{k},... \Bigr)$\\
> **dam_type = 2** Fully-coupled damage.
>In this final case, the damage equations are included within the big Jacobian unknown set of equations, and are solved together with all the system in an implicit manner. It is a time-consuming and computationally heavy approach, where:
>> · $\underline{\tilde{\sigma}}_{k+1}=\underline{\sigma}_{k+1} \cdot (1-D_{k+1})^{-1}$
>> · $D_{k+1}=f \Bigl( \underline{\sigma}_{k+1}, D_{k+1},... \Bigr)$\\

In the case that damage is considered within the simulation (i.e., **dam_type>0**), the user must subsequently define a damage intiation criteria (dam_init). A total of three options are given:
> **daminit=0**
> In this case, damage is set to be initiated together with plastic deformation. It is not advisable, as it often results in a huge damage overestimation.\\
> **daminit=1**: plastic work level criterion
> Following H. Morch's developments, this criteiron is based in an adjusted plastic work level calculation where damage is initiated after a critical $w_c$ plastic work level is reached. The equation describing the accumulated plastic work considers terms for both isotorpic and kinematic hardening, and is of the form:
>> $w(p)=\displaystyle\int_{0}^{t} \Big[ \frac{A}{m} p^{\frac{1-m}{m}} R(p)\dot{p} \Big]dt + \displaystyle\sum_{i=1}^{nAF}\frac{3}{2}\underline{\mathbb{X}}_i : \underline{\mathbb{X}}_i$
>>damage is initiated when $w(p)\geq w_c$
>>where $w_c$, $A$ and $m$ are adjustable material parameters. Particularly, $w_c$ and $A$ can be introduced as $f(T)$.\\
> **daminit=3** Plastic deformation level
> This last criterion consist simply in the activation of the damage evolution mechanism when a user-defined critical plastic deformation level $p_c$ is reached. It can be introduced as a constant or as a temperature-dependent Arrhenius function.\\

Once the previous constrains are defined, the user mus now define the damage law. In **VMVP**, two damage laws are given for the user to choose from:
> **idam=1**: Lemaitre-Kachanov Creep-Fatigue damage formulation
>Implemented after H. Morch's work, this formulation has proven high reliability when facing complex creep-fatigue loadings. It consiste in the direct sum of the contributions of creep (Kachanov) and fatigue (Lemaitre) damage formulations. These are described hereafter:
>> __Fatigue damage: Lemaitre law__
>> This fatigue law is independent of time, and depends more on the. It is of the form:
>> $\dot{D}_{f}=k_1 \Bigl[\frac{\mathcal{Y(k_{2}\cdot \underline{\sigma})}}{S_{f}} \Bigr]^{S_{fe}} \dot{p}$
>> where $S_{f}$, $S_{fe}$, $k_1$, $k_2$ are material constants. In particular, $S_f$ is conceived as $f(T)$.
>>Additionally, $\mathcal{Y(k_{2}\cdot \underline{\sigma})}$ is a scalar function described as:
>>> $\mathcal{Y(\underline{\sigma})} = \frac{1+\nu}{2E}\Bigl[ \frac{\langle \underline{\sigma}_{ij} \rangle^{+} :\langle \underline{\sigma}_{ij} \rangle^{+}}{ (1-D)^2 }  + h_{\text{mD}}\frac{\langle \underline{\sigma}_{ij} \rangle^{-} :\langle \underline{\sigma}_{ij} \rangle^{-}}{ (1-h_{\text{mD}}D)^2 } \Bigr] - \frac{\nu}{2E}\Bigl[ \frac{\langle \underline{\sigma}_{kk} \rangle^{2} }{ (1-D)^2 }  + h_{\text{mD}}\frac{\langle \underline{\sigma}_{kk} \rangle^{2} }{ (1-h_{\text{mD}}D)^2 } \Bigr]  $   \\
>> __Creep damage: Kachanov law__
>> Similar to the previous law, Kachanov formulation is introduced as:
>> $\dot{D}_{c}=k_3 \Bigl[\frac{\mathcal{Y(k_{4}\cdot \underline{\sigma})}}{S_{c}} \Bigr]^{S_{ce}} \frac{1}{(1-D)^{k_k}} $
>> where $S_{c}$, $S_{ce}$, $k_3$, $k_4$, $k_k$ are material constants. In particular, $S_c$ is conceived as $f(T)$.\\
> **idam=2**: IfW Creep-Fatigue damage formulation
>This creep-fatigue damage law was developed by IfW institute in Aachen University. It came to our knowledge after being used by N. K. Karthik in the context of his PhD. project on the life prediction of metallic components subjected to creep-fatigue loadings. It is inspired in the Graham-Walles formulation, and is of the form:
> $\dot{D}=K_{\text{D}} J_{2}\Bigl( \underline{\tilde{\sigma}} - \underline{\mathbb{X}} \Bigr) + |\dot{T}|K_{\text{Td}}p^{m_{\text{Td}}}$
> where $K_{\text{Td}}$, $K_{\text{Td}}$ and $m_{\text{Td}}$ are material parameters.
===== Input file =====
==== Parameters defining the type of constitutive law ====

^Line 01; format (2I5, 60A1)^^
|IL|Law number|
|ITYPE| 282|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|

==== Integer parameters ====
 
^Line 02; format (9I5)^^^
|1:5|TID|=1 if parameters are given for linear interpolation at different temperatures\\ = 0 if not |
|6:10|ntemp|if TID=1, ntemp defines the number of temperatures at which parameters are given |
|11:15|ianisoth|=1 if anisothermal loadings\\ = 0 otherwise |
|16:20|MAXIT|=maximum number of Newton-Raphson iterations within the colaw (25 default) |
|21:25|nintv|=Number of sub-intervals (substeps); if 0, no substeps |
|26:30|ilcf| indicator for low-cycle fatigue |
|31:35|mat_interp|indicator for interpolation method (1$\leq$mat_interp$\leq$5) for material parameters $(E, \nu)$ |
|36:40|tdilc_interp|indicator for interpolation method (1$\leq$mat_interp$\leq$5)  for temp. dilatation coefficient |
|41:45|iso_interp|indicator for interpolation method (1$\leq$mat_interp$\leq$5)  for isotropic hardening parameters $(b, Q)$ |
^Line 03; format (10I5)^^^
|1:5|nAFX|Number of Armstrong-Fredericks equations used to define the back-stress $\underline{\mathbb{X}}$ (minimum value=1) |
|6:10|NAFC|Number of Armstrong-Fredericks equations taking into account cyclic hardening |
|11:15|NAFY|Number of Armstrong-Fredericks equations taking into account evolution of the mean stress |
|16:20|iarrh|ID for interpolation method of AD equations |
|21:25|ivp|ID for type of viscoplastic law\\ 1=Norton law (after [[laws:chab|CHAB]]);\\ 2=Graham-Walles; 4=AFN |
|26:30|vpinterp| ID for interpolation method for viscoplastic law parameters.\\ if ivp=2, vpinterp defines the number of Graham-Walles equations |
|31:35|type_dam|ID for damage type.\\ 0=no damage;\\ 1=uncoupled damage;\\ 3=semi-coupled damage;\\ 4=fully-coupled damage(not working yet) |
|36:40|idam|ID for damage law.\\ 1 = Kachanov(creep) + Lemaitre(fatigue) (after [[laws:chab|CHAB]]);\\ 2 = IfW (PhD. thesis from Narayana K. Karthik, RWTH Aachen university) |
|41:45|dam_interp|ID for interpolation of damage law parameters |
|46:50|daminit|ID for damage initiation criterion |

==== Real parameters ====
=== case TID = 0 ===
^ Line 04 ^^^format^
|1:$x$|$E$|Young modulus (MPa)|after mat_interp|
^ Line 05 ^^^^
|1:$x$|$\nu$|Poisson ratio (-)| after mat_interp|
^ Line 06 ^^^^
|1:$x$|$\alpha$|Thermal dilatation coefficient (m/K)|after mat_interp|
^ Line 07 ^^^^
|1:$x$|$\sigma_y$|Yield stress (MPa)|after iso_interp|
|$x$+1:$x$+10|$b$|Hardening saturation rate (-)|**1G10.0** |
|$x$+11:$x$+20|$Q$|Hardening saturation value (MPa)|**1G10.0** |
^ Kinematic hardening (Armstrong-Frederick);·······Lines 08:$m$ ^^^^
|**--> case iarrh=0**; repeat for $i$=1,nAFX ||||
| 1:10|$C_i$|Swift-type hardening term (MPa) |**1G10.0** |
|11:20|$\gamma_i$|Static recovery term (-)|**1G10.0** |
|21:30|$b_i$|Dynamic recovery term (-) |**1G10.0** |
|31:40|$r_i$|Dynamic recovery exponent (-) |**1G10.0** |
|**--> case iarrh=1**; repeat for $i$=1,nAFX ||||
| 1:10|$C_i $ |Swift-type hardening term (MPa)|**1G10.0** |
|11:20|$A_{\gamma_{i}}$|Static recovery term; single Arrhenius fit: \\ $\gamma_i(T)=A_{\gamma_{i}}\cdot[1-B_{\gamma_{i}}\cdot exp(\frac{T}{C_{\gamma_{i}}})]$ |**1G10.0** |
|21:30|$B_{\gamma_{i}}$ |::: |**1G10.0** |
|31:40|$C_{\gamma_{i}}$ |::: |**1G10.0** |
|41:50|$b_i $|Dynamic recovery term (-)|**1G10.0** |
|51:60|$r_i $|Dynamic recovery exponent (-)|**1G10.0** |ç
|**--> case iarrh=2**; repeat for $i$=1,nFA ||||
|**Line** 07$+2 \times i-1$ ||| |
|1:10|$A_{C_{i}}$|Swift-type hardening term (MPa); single Arrhenius fit: \\ $C_i(T)=A_{C_{i}}\cdot[1-B_{C_{i}}\cdot exp(\frac{T}{C_{C_{i}}})]$ |**1G10.0** |
|11:20|$B_{C_{i}}$ |::: |**1G10.0** |
|21:30|$C_{C_{i}}$ |::: |**1G10.0** |
|31:40|$A_{\gamma_{i}}$|Static recovery term; single Arrhenius fit: \\ $\gamma_i(T)=A_{\gamma_{i}}\cdot[1-B_{\gamma_{i}}\cdot exp(\frac{T}{C_{\gamma_{i}}})]$ |**1G10.0** |
|41:50|$B_{\gamma_{i}}$ |::: |**1G10.0** |
|51:60|$C_{\gamma_{i}}$ |::: |**1G10.0** |
|**Line** 07$+2\times i$ |||
|1:10|$A_{b_{i}}$|Dynamic recovery term (-); single Arrhenius fit: \\ $b_i(T)=A_{b_{i}}\cdot[1-B_{b_{i}}\cdot exp(\frac{T}{C_{b_{i}}})]$ |**1G10.0** |
|11:20|$B_{b_{i}}$ |::: |**1G10.0** |
|21:30|$C_{b_{i}}$ |::: |**1G10.0** |
|31:40|$r_i$|Dynamic recovery exponent (-) |**1G10.0** |
^ Cyclic hardening;·······Lines $m+1$:$n$^^^^
|**--> case iarrh=0** or **iarrh=1**; repeat for $j$=1,nAFC ||||
|1:10|$D_{\gamma_{j}}$ |Cyclic hardening parameter (-)|**1G10.0** |
|11:20|$a_{\gamma_{j}}$ |Strain memory surface parameter (-)|**1G10.0** |
|21:30|$b_{\gamma_{j}}$ |Strain memory surface parameter (-)|**1G10.0** |
|31:40|$c_{\gamma_{j}}$ |Strain memory surface parameter (-)|**1G10.0** |
|**--> case iarrh=2**; Single Arrhenius; repeat for $j$=1,nAFC ||||
|**Line** $n+2\times j-1$                                      ||| |
|1:10|$A_{D_{\gamma_{j}}}$|Cyclic hardening parameter (-); single Arrhenius fit: \\ $D_{\gamma_{j}}(T)=A_{D_{\gamma_{j}}}\times[1-B_{D_{\gamma_{j}}}\cdot \exp(\frac{T}{C_{D_{\gamma_{j}}}})]$ |**1G10.0** |
|11:20|$B_{D_{\gamma_{j}}}$ |::: |**1G10.0** |
|21:30|$C_{D_{\gamma_{j}}}$ |::: |**1G10.0** |
|**Line** $n+2\times j$                                        ||| |
|1:10|$B_{\gamma_{j}}$|Double Arrhenius fit: \\ $(a_{\gamma_{j}} ,b_{\gamma_{j}} ,c_{\gamma_{j}})(T)=A_{(a,b,c)_{j}}\cdot[1-B_{\gamma_{j}}\cdot\exp(\frac{T}{C_{\gamma_{j}}} ) - D_{\gamma_{j}}\cdot \exp(\frac{T}{E_{\gamma_{j}}})]$ \\ \\ Parameter $A_{(a,b,c)_{j}}$ are different for each $(a_{\gamma_{j}} ,b_{\gamma_{j}} ,c_{\gamma_{j}})$ equation. \\ They are defined hereafter:  |**1G10.0** |
|11:20|$C_{\gamma_{j}}$|::: |**1G10.0** |
|21:30|$D_{\gamma_{j}}$|::: |**1G10.0** |
|31:40|$E_{\gamma_{j}}$|::: |**1G10.0** |
|41:50|$A_{a_{j}}$|Parameter $A_a$ for $a_{\gamma_{j}}$|**1G10.0** |
|51:60|$A_{b_{j}}$|Parameter $A_b$ for $b_{\gamma_{j}}$ |**1G10.0** |
|61:70|$A_{c_{j}}$|Parameter $A_c$ for $c_{\gamma_{j}}$ |**1G10.0** |
^ Mean stress evolution;·······Lines $n+1$:$n+$nAFY ^^^^
|**--> case iarrh=0** or **iarrh=1**; repeat for $k$=1,nAFY ||||
|1:10|$\alpha_{b,k}$|Ratio of evolution of mean stress tensor $\underline{Y}_{k}$ |**1G10.0** |
|11:20|$Y_{\text{st},k}$|Saturation value of mean stress tensor $\underline{Y}_{k}$ |**1G10.0** |
|**--> case iarrh=2**; Single Arrhenius; repeat for $k$=1,nAFY ||||
|1:10|$A_{\alpha_{b,k}}$|Cyclic hardening parameter (-); single Arrhenius fit: \\ $\alpha_{b,k}(T)=A_{\alpha_{b,k}}\cdot[1-B_{\alpha_{b,k}}\cdot \exp(\frac{T}{C_{\alpha_{b,k}}})]$ |**1G10.0** |
|11:20|$B_{\alpha_{\text{b},k}}$ |::: |**1G10.0** |
|21:30|$C_{\alpha_{\text{b},k}}$ |::: |**1G10.0** |
|31:40|$A_{Y_{\text{st},k}}$|Cyclic hardening parameter (-); single Arrhenius fit: \\ $Y_{\text{st},k}(T)=A_{Y_{\text{st},k}}\cdot[1-B_{Y_{\text{st},k}}\cdot \exp(\frac{T}{C_{Y_{\text{st},k}}})]$ |**1G10.0** |
|41:50|$B_{Y_{\text{st},k}}$ |::: |**1G10.0** |
|51:60|$C_{Y_{\text{st},k}}$ |::: |**1G10.0** |
^ Viscoplastic laws;·······Lines $n+$nAFY+1:$p$ ^^^^
|**--> case ivp=1**: Simple Norton law ||||
|if (vp_interp=1) then:                ||||
| 1:10|$K$|Drag stress $(\text{MPa})$ |**1G10.0** |
|11:20|$n$|Exponent $(\text{-})$ |**1G10.0** |
|if (vp_interp=2) then:                ||||
|1:10|$A_K$|Drag stress $(\text{MPa})$ ; single Arrhenius fit: \\ $K(T)=A_K\cdot[1-B_{K}\cdot \exp(\frac{T}{C_{K}})]$ |**1G10.0** |
|11:20|$B_K$ |::: |**1G10.0** |
|21:30|$C_K$ |::: |**1G10.0** |
|31:40|$A_n$|Exponent $(\text{-})$ ; single Arrhenius fit: \\ $n(T)=A_n\cdot[1-B_{n}\cdot \exp(\frac{T}{C_{n}})]$ |**1G10.0** |
|41:50|$B_n$ |::: |**1G10.0** |
|51:60|$C_n$ |::: |**1G10.0** |
|**--> case ivp=2**: Graham-Walles equations ||||
|for $l$=1,ivp:    (here, ivp=number of G-W equations) ||| |
| 1:10|$K_{l}$|Constant of G-W creep equation $l$ $(-)$  |**1G10.0** |
|11:20|$C_{l}$|Temperature term of G-W creep equation $l$ $(ºC)$ |**1G10.0** |
|21:30|$n_{l}$|Stress exponent of G-W creep equation $l$ $(-)$ |**1G10.0** |
|31:40|$m_{l}$|Creep strain exponent of G-W creep equation $l$ $(-)$ |**1G10.0** |
|finally, **Line** $n+$nAFY$+$ivp$+1$  ||| |
| 1:10|$C_{T}$|Temperature term of G-W creep-fatigue equation $(ºC)$ |**1G10.0** |
|11:20|$K_{T}$|Constant of G-W creep-fatigue equation $(-)$ |**1G10.0** |
|21:30|$m_{T}$|Creep strain exponent of G-W creep-fatigue equation $(-)$ |**1G10.0** |
|**--> case ivp=4**: AFN (Activation function $\times$ Norton equations) ||||
|if (vp_interp=1) then:                ||||
|**Line** $n+$nAFY$+1$ ||| |
| 1:10|$a$|$a$ parameter $(-)$ |**1G10.0** |
|11:20|$b$|$b$ parameter $(-)$ |**1G10.0** |
|21:30|$c$|$c$ parameter $(-)$ |**1G10.0** |
|**Line** $n+$nAFY$+2$ ||| |
| 1:10|$K$|Drag stress $(\text{MPa})$ |**1G10.0** |
|11:20|$n$|Exponent $(\text{-})$      |**1G10.0** |
|if (vp_interp=2) then:                ||||
|**Line** $n+$nAFY$+1$ ||| |
| 1:10|$a_{1}$|- |**1G10.0** |
|11:20|$p_{21}$| Parameter $p_{2}=f(T)$ common for all $a,b,c$ AF functions: \\ $p_{2}(T)=p_{21}\cdot [1-p_{22}\cdot \exp{(\frac{T}{p_{23}})}]$ |**1G10.0** |
|21:30|$p_{22}$|::: |**1G10.0** |
|31:40|$p_{23}$|::: |**1G10.0** |
|41:50|$a_{3}$|- |**1G10.0** |
|**Line** $n+$nAFY$+2$ ||| |
| 1:10|$b_{1}$|- |**1G10.0** |
|11:20|$b_{3}$|- |**1G10.0** |
|21:30|$b_{4}$|- |**1G10.0** |
|**Line** $n+$nAFY$+3$ ||| |
| 1:10|$c_{1}$|- |**1G10.0** |
|11:20|$c_{3}$|- |**1G10.0** |
|**Line** $n+$nAFY$+4$ ||| |
|1:10|$A_K$|Drag stress $(\text{MPa})$ ; single Arrhenius fit: \\ $K(T)=A_K\cdot[1-B_{K}\cdot \exp(\frac{T}{C_{K}})]$ |**1G10.0** |
|11:20|$B_K$ |::: |**1G10.0** |
|21:30|$C_K$ |::: |**1G10.0** |
|31:40|$A_n$|Exponent $(\text{-})$ ; single Arrhenius fit: \\ $n(T)=A_n\cdot[1-B_{n}\cdot \exp(\frac{T}{C_{n}})]$ |**1G10.0** |
|41:50|$B_n$ |::: |**1G10.0** |
|51:60|$C_n$ |::: |**1G10.0** |
^ Damage laws (only if dam_type$\geq$1);·······Lines $p+1$:$end$ ^^^^
|**Line** $p+1$ ||| |
|1:10|$h_\text{mD}$ |Compression damage parameter $(-)$ |**1G10.0** |
|11:20|$D_\text{Frac}$  |Critical fracture damage $(-)$ |**1G10.0** |
|21:30|$\tau$ |Damping coefficient for controlling creep-fatigue damage evolution $(s)$ |**1G10.0** |
|**Line** $p+2$                       ||| |
|if (daminit=1) then:                 ||||
| |if (daminterp=1) then:             |||
| 1:10|$w_{\text{D}}$ |Critical plastic work level for damage initiation $(\frac{J}{mm^3})$ |**1G10.0** |
|11:20|$A_{\text{D}}$ |Fitting parameter for plastic work function $(-)$ |**1G10.0** |
|21:30|$m_{\text{D}}$ |Fitting parameter for plastic work function $(-)$ |**1G10.0** |
| |if (daminterp=2) then:                |||
|1:10|$A_{w_{\text{D}}}$|Critical plastic work level for damage initiation $(\frac{J}{mm^3})$;\\ single Arrhenius fit: \\ $w_{\text{D}}(T)=A_{w_{\text{D}}}\cdot[1-B_{w_{\text{D}}}\cdot \exp(\frac{T}{C_{w_{\text{D}}}})]$ |**1G10.0** |
|11:20|$B_{w_{\text{D}}}$ |::: |**1G10.0** |
|21:30|$C_{w_{\text{D}}}$ |::: |**1G10.0** |
|31:40|$A_{A_{\text{D}}}$|Fitting parameter for plastic work function $(-)$;\\ single Arrhenius fit: \\ $A_{\text{D}}(T)=A_{A_{\text{D}}}\cdot[1-B_{A_{\text{D}}}\cdot \exp(\frac{T}{C_{A_{\text{D}}}})]$ |**1G10.0** |
|41:50|$B_{A_{\text{D}}}$ |::: |**1G10.0** |
|51:60|$C_{A_{\text{D}}}$ |::: |**1G10.0** |
|61:70|$\tau$ |Damping coefficient for controlling creep-fatigue damage evolution $(s)$ |**1G10.0** |
|if (daminit=2) then:                 ||||
| |if (daminterp=1) then:             |||
| 1:10|$p_{\text{D}}$ |Critical plastic strain level for damage initiation $(-)$ |**1G10.0** |
| |if (daminterp=2) then:             |||
|1:10|$A_{p_{\text{D}}}$|Drag stress $(\text{MPa})$ ; single Arrhenius fit: \\ $p_{\text{D}}(T)=A_{p_{\text{D}}}\cdot[1-B_{p_{\text{D}}}\cdot \exp(\frac{T}{C_{p_{\text{D}}}})]$ |**1G10.0** |
|11:20|$B_{p_{\text{D}}}$ |::: |**1G10.0** |
|21:30|$C_{p_{\text{D}}}$ |::: |**1G10.0** |
|**Line** $p+3$                  ||| |
|if (idam=1) then:               ||||
| |if (daminterp=1) then:        |||
| 1:10|$S_{\text{f}}$ |Lemaitre parameter $(-)$ |**1G10.0** |
|11:20|$S_{\text{fe}}$|Lemaitre parameter $(-)$ |**1G10.0** |
|21:30|$k_{1}$        |Lemaitre parameter $(-)$ |**1G10.0** |
|31:40|$k_{2}$        |Lemaitre parameter $(-)$ |**1G10.0** |
| |if (daminterp=2) then:        |||
|1:10|$A_{S_{\text{f}}}$|Lemaitre parameter $(-)$; single Arrhenius fit: \\ $S_{\text{f}}(T)=A_{S_{\text{f}}}\cdot[1-B_{S_{\text{f}}}\cdot \exp(\frac{T}{C_{S_{\text{f}}}})]$ |**1G10.0** |
|11:20|$B_{S_{\text{f}}}$ |::: |**1G10.0** |
|21:30|$C_{S_{\text{f}}}$ |::: |**1G10.0** |
|31:40|$S_{\text{fe}}$|Lemaitre parameter $(-)$ |**1G10.0** |
|41:50|$k_{1}$        |Lemaitre parameter $(-)$ |**1G10.0** |
|51:60|$k_{2}$        |Lemaitre parameter $(-)$ |**1G10.0** |
|if (idam=2) then:               ||||
|1:10|$K_{\text{D}}$   |IfW creep-fatigue damage parameter $(-)$ |**1G10.0** |
|11:20|$K_{\text{TD}}$ |IfW creep-fatigue damage parameter $(-)$ |**1G10.0** |
|21:30|$m_{\text{TD}}$ |IfW creep-fatigue damage parameter $(-)$ |**1G10.0** |
|**Line** $p+4$ (only if idam=1) ||| |
| |if (daminterp=1) then:        |||
| 1:10|$S_{\text{c}}$ |Lemaitre parameter $(-)$ |**1G10.0** |
|11:20|$S_{\text{ce}}$|Lemaitre parameter $(-)$ |**1G10.0** |
|21:30|$k_{3}$        |Lemaitre parameter $(-)$ |**1G10.0** |
|31:40|$k_{4}$        |Lemaitre parameter $(-)$ |**1G10.0** |
|41:50|$k_{\text{k}}$   |Lemaitre parameter $(-)$ |**1G10.0** |
| |if (daminterp=2) then:        |||
|1:10|$A_{S_{\text{c}}}$|Lemaitre parameter $(-)$; single Arrhenius fit: \\ $S_{\text{c}}(T)=A_{S_{\text{c}}}\cdot[1-B_{S_{\text{c}}}\cdot \exp(\frac{T}{C_{S_{\text{c}}}})]$ |**1G10.0** |
|11:20|$B_{S_{\text{c}}}$ |::: |**1G10.0** |
|21:30|$C_{S_{\text{c}}}$ |::: |**1G10.0** |
|31:40|$S_{\text{ce}}$|Lemaitre parameter $(-)$ |**1G10.0** |
|41:50|$k_{3}$        |Lemaitre parameter $(-)$ |**1G10.0** |
|51:60|$k_{4}$        |Lemaitre parameter $(-)$ |**1G10.0** |
|61:70|$k_{\text{k}}$  |Lemaitre parameter $(-)$ |**1G10.0** |


==== Other things ====
^Interpolation methods $f(T)$ included within **VMVP**: ^^
|1|Unique value, no interpolation |
|2|Single Arrhenius function \\ \\  $P(T)=A_P\cdot \Bigl[ 1-B_P\cdot \exp \Bigl(\frac{T}{C_P} \Bigr) \Bigr]$ \\ \\ Parameters $A_P$, $B_P$ & $C_P$ are introduced in format **3G10.0**|
|3|Double Arrhenius function\\ \\ $P(T)=A_P\cdot \Bigl[1-B_P\cdot \exp \Bigl(\frac{T}{C_P} \Bigr) - D_P\cdot \exp \Bigl(\frac{T}{E_P} \Bigr) \Bigr]$\\ \\ Parameters $A_P$, $B_P$, $C_P$, $D_P$ & $E_P$ are introduced in format **5G10.0**|
|4|3 deg. polynomial function\\ \\ $P(T)=A_P\cdot[1 - B_P\cdot(T-T_0) + C_P\cdot(T-T_0)^2 - D_P\cdot(T-T_0)^3]$ \\ \\ Parameters $A_P$, $B_P$, $C_P$, $D_P$ & $T_0$ are introduced in format **5G10.0**|
|5|Logarithmic function\\ \\ $P(T)=A_P + B_P \cdot \ln{(\frac{T_{abs}}{C_P})}$ \\ \\ Parameters $A_P$, $B_P$ & $C_P$ are introduced in format **3G10.0** \\ FYI, $T*$ is the temperature in $K$ |

^List of state variables^^
|Q(1)|Equiv. plastic (creep) strain (-)|
|Q(2)|Equiv. plastic (creep) strain rate $(s^{-1})$|
|Q(3)|Equiv. thermal strain (-)|
|Q(4:9)|Plastic strain vector (-)|
|Q(10:15)|Total strain vector (-)|
|Q(16)|Isotropic/cyclic hardening R (MPa)|
|Q(17)|Plastic memory surface radius (-)|
|Q(18:23)|Plastic memory surface center $\zeta$|
|Q(24)|Total accumulated damage|
|Q(25)|Accumulated **creep** damage|
|Q(26)|Accumulated **fatigue** damage|
|Q(27)|Accumulated plastic energy $w_D$|
|Q(28:28+6$\times$nAFX-1)|Kinematic hardening components $\underline{\mathbb{X}}$ |