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laws:vmvp
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Table of Contents
VMVP
Von Mises Visco-Plastic.
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.
Numerical model
This law was implemented in the context of C.Rojas-Ulloa's PhD. project on the modeling of the non-classical long-term creep response of Incoloy 800H (see (C.Rojas-Ulloa et al., 2024) for more details). It is based in the law implemented by Hélène Morch (CHAB), a viscoplastic damage law combines the von Mises yield criterion with a Norton viscosity function, a complex backstress formulation, and Kachanov-Lemaitre creep-fatigue damage laws). VMVP includes
Input file
Parameters defining the type of constitutive law
| Line 01; format (2I5, 60A1) | |
|---|---|
| IL | Law number |
| ITYPE | 271 |
| 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 $\underline{X}$ (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 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 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 |
| –> 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 $P(T)=A_P\cdot[1-B_P\cdot \exp(\frac{T}{C_P})]$ Parameters $A_P$, $B_P$ & $C_P$ are introduced in format 3G10.0 |
| 3 | Double Arrhenius function $P(T)=A_P\cdot[1-B_P\cdot \exp(\frac{T}{C_P} )] - D_P\cdot \exp(\frac{T}{E_P})]$ 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 4G10.0 |
| 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 $\underline{X}$ |
laws/vmvp.1731504942.txt.gz · Last modified: 2024/11/13 14:35 by carlos
