laws:chab
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laws:chab [2022/04/04 09:00] helene [The model] |
laws:chab [2022/09/28 16:23] (current) helene |
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| |NINTV| number of time sub-steps in the material law| | |NINTV| number of time sub-steps in the material law| | ||
| |IDAM|UNITS: \\ = 0 no mechanical damage computation\\ = 1 isotropic uncoupled damage computation \\ = 2 isotropic coupled damage computation [UNSTABLE] \\ = 3 isotropic semi-coupled damage computation (use of $D_{n-1}$ to compute the effective stress at time step $t_n$, $D$ updated at the end of the time step)| | |IDAM|UNITS: \\ = 0 no mechanical damage computation\\ = 1 isotropic uncoupled damage computation \\ = 2 isotropic coupled damage computation [UNSTABLE] \\ = 3 isotropic semi-coupled damage computation (use of $D_{n-1}$ to compute the effective stress at time step $t_n$, $D$ updated at the end of the time step)| | ||
| - | |:::| TENS: \\ = 0 no corrosion damage \\ = 1 linear corrosion damage: $\dot{D}_u=\frac{k_u}{L_E}$\\ = 2 parabolic corrosion damage: $\dot{D}_u=\frac{k_u}{D_u L_E^2}$| | + | |:::| TENS: \\ = 0 no corrosion damage \\ = 1 linear corrosion damage: $\dot{D}_u=\frac{k_u}{L_E}$\\ = 2 parabolic corrosion damage: $\dot{D}_u=\frac{k_u}{D_u L_E^2}$ \\ = 3 power law for corrosion damage: $D_u=\frac{k_u}{L_E} t^{m_u}$| |
| |IARRH| = 1 expression of static recovery parameters using Arrhenius law| | |IARRH| = 1 expression of static recovery parameters using Arrhenius law| | ||
| |:::| = 2 expression of all parameters as exponential function of temperature| | |:::| = 2 expression of all parameters as exponential function of temperature| | ||
| |:::| = 0 parameters are interpolated linearly between to defined temperatures| | |:::| = 0 parameters are interpolated linearly between to defined temperatures| | ||
| - | |ILCF| = 1 computation of stress amplitude for cyclic loading (for Optim)| | + | |ILCF| = 1 computation of stress amplitude, mean stress, relaxation stress for cyclic loading (for Optim) - only available for uniaxial loading in x direction.| |
| ==== Real parameters ==== | ==== Real parameters ==== | ||
| Line 132: | Line 132: | ||
| ^Line 1 (4G10)^^ | ^Line 1 (4G10)^^ | ||
| |ETA| strain memory rate| | |ETA| strain memory rate| | ||
| - | |PRECNR| precision for convergence of the Newton-Raphson algorithm (default=10<sub>-4</sub>)| | + | |PRECNR| precision for convergence of the Newton-Raphson algorithm (default=10<sup>-4</sup>)| |
| |PERIOD| period of cyclic loading (only if ILCF=1)| | |PERIOD| period of cyclic loading (only if ILCF=1)| | ||
| - | |tH|Hold time in the cyclic loading (only if ILCF=1)| | + | |$t_H$|Hold time in the cyclic loading (only if ILCF=1)| |
| ^If IANISOTH=1 - Line 2 (3G10)^^ | ^If IANISOTH=1 - Line 2 (3G10)^^ | ||
| - | |BDG|Rate parameter controlling the evolution of D<sub>γ</sub>| | + | |$b_{D\gamma}$|Rate parameter controlling the evolution of D<sub>γ</sub>| |
| - | |be| Rate of evolution of the weighted average factor fe| | + | |$b_E$| Rate of evolution of the weighted average factor fe| |
| - | |fes| Saturation value of the weighted average factor fe| | + | |$f_E^s$| Saturation value of the weighted average factor fe| |
| ^If IDAM≠0 - Line 3 (7G10)^^ | ^If IDAM≠0 - Line 3 (7G10)^^ | ||
| - | |h| micro-defects closure parameter (=0.2 in general for metals ; 1 if micro-defects closure not taken into account)| | + | |$h$| micro-defects closure parameter (=0.2 in general for metals ; 1 if micro-defects closure not taken into account)| |
| - | |Dc| Critical damage value (<1)| | + | |$D_{crit}$| Critical damage value (<1)| |
| - | |τ| Specific time for the appearance of creep| | + | |$\tau$| Specific time for the appearance of creep| |
| |$k_1$|Global safety coefficient on fatigue damage| | |$k_1$|Global safety coefficient on fatigue damage| | ||
| |$k_2$|Safety coefficient applied to stress level on fatigue damage| | |$k_2$|Safety coefficient applied to stress level on fatigue damage| | ||
| Line 148: | Line 148: | ||
| |$k_4$|Safety coefficient applied to stress level on creep damage| | |$k_4$|Safety coefficient applied to stress level on creep damage| | ||
| ^If IARRH=1 - Line 3+i (i=1:nAF) (2G10)*i^^ | ^If IARRH=1 - Line 3+i (i=1:nAF) (2G10)*i^^ | ||
| - | |Ai| coefficient for expression of bi using Arrhenius equation| | + | |$A_i$| coefficient for expression of $b_i$ using Arrhenius equation: $b_i=A_i \exp(-B_i/T)$| |
| - | |Bi| coefficient for expression of bi using Arrhenius equation| | + | |$B_i$| coefficient for expression of $b_i$ using Arrhenius equation: $b_i=A_i \exp(-B_i/T)$| |
| === Temperature-dependent parameters - Case where iarrh=0 or iarrh=1 === | === Temperature-dependent parameters - Case where iarrh=0 or iarrh=1 === | ||
| Line 156: | Line 156: | ||
| ^Line 1 (4G10)^^ | ^Line 1 (4G10)^^ | ||
| - | |T|Temperature| | + | |$T$|Temperature| |
| - | |E| Young's modulus at temperature T| | + | |$E$| Young's modulus at temperature T| |
| - | |NU| Poisson's ratio at temperature T| | + | |$\nu$| Poisson's ratio at temperature T| |
| - | |α|Thermal expansion coefficient at temperature T\\ NB: in the preprocessor, the thermal expansion coefficient is transformed in its enthalpic formulation| | + | |$\alpha$|Thermal expansion coefficient at temperature T\\ NB: in the preprocessor, the thermal expansion coefficient is transformed in its enthalpic formulation| |
| ^Line 2 (5G10)^^ | ^Line 2 (5G10)^^ | ||
| - | |σ<sub>Y</sub>|Yield stress at temperature T| | + | |$\sigma_y$|Yield stress at temperature T| |
| - | |b| Rate of isotropic hardening \\ NB: to avoid convergence issues, b should be constant with temperature| | + | |$b$| Rate of isotropic hardening \\ NB: to avoid convergence issues, b should be constant with temperature| |
| - | |Q| Total isotropic saturation size of the yield surface\\ NB: to avoid convergence issues, Q should be constant with temperature| | + | |$Q$| Total isotropic saturation size of the yield surface\\ NB: to avoid convergence issues, Q should be constant with temperature| |
| - | |K| Drag stress in Norton-Hoff law| | + | |$K$| Drag stress in Norton-Hoff law| |
| - | |n| Viscosity exponent for Norton-Hoff law| | + | |$n$| Viscosity exponent for Norton-Hoff law| |
| ^Line 2+i (4G10) repeated NAF times (i=1:NAF)^^ | ^Line 2+i (4G10) repeated NAF times (i=1:NAF)^^ | ||
| - | |C<sub>i</sub>|Prager's linear coefficient in the i<sup>th</sup> A-F equation| | + | |$C_i$|Prager's linear coefficient in the i<sup>th</sup> A-F equation| |
| - | |γ<sub>i</sub>| Dynamic recovery parameter in the i<sup>th</sup> A-F equation| | + | |$\gamma_i$| Dynamic recovery parameter in the i<sup>th</sup> A-F equation| |
| - | |b<sub>i</sub>| Static recovery parameter in the i<sup>th</sup> equation| | + | |$b_i$| Static recovery parameter in the i<sup>th</sup> equation| |
| - | |r<sub>i</sub>| Static recovery exponent in the i<sup>th</sup> equation| | + | |$r_i$| Static recovery exponent in the i<sup>th</sup> equation| |
| ^Line 2+NAF+i (4G10) repeated NAFcyc times (i=1:NAFcyc)^^ | ^Line 2+NAF+i (4G10) repeated NAFcyc times (i=1:NAFcyc)^^ | ||
| - | |D<sub>γi</sub>|Parameter controlling the evolution of γ<sub>i</sub> with increment of plastic strain norm| | + | |$D_{\gamma,i}$|Parameter controlling the evolution of γ<sub>i</sub> with increment of plastic strain norm| |
| - | |A<sub>γi</sub>|Parameter controlling the saturation value of γ<sub>i</sub>| | + | |$a_{\gamma,i}$|Parameter controlling the saturation value of γ<sub>i</sub>| |
| - | |B<sub>γi</sub>|Parameter controlling the saturation value of γ<sub>i</sub>| | + | |$b_{\gamma,i}$|Parameter controlling the saturation value of γ<sub>i</sub>| |
| - | |C<sub>γi</sub>|Parameter controlling the saturation value of γ<sub>i</sub>| | + | |$c_{\gamma,i}$|Parameter controlling the saturation value of γ<sub>i</sub>| |
| ^Line 2+NAF+NAFcyc+i (4G10) repeated NAFY times (i=1:NAFY)^^ | ^Line 2+NAF+NAFcyc+i (4G10) repeated NAFY times (i=1:NAFY)^^ | ||
| - | |α<sub>bi</sub>| Rate of evolution of the mean stress tensor Y<sub>i</sub>| | + | |$\alpha_{b,i}$| Rate of evolution of the mean stress tensor Y<sub>i</sub>| |
| - | |Y<sub>st,i</sub>| Saturation value of the mean stress tensor Y<sub>i</sub>| | + | |$Y_{st,i}$| Saturation value of the mean stress tensor Y<sub>i</sub>| |
| ^If 1≤IDAM<10 - Line 2+NAF+NAFcyc+NAFY (8G10)^^ | ^If 1≤IDAM<10 - Line 2+NAF+NAFcyc+NAFY (8G10)^^ | ||
| - | |A|Parameter of correction in the energy stored by hardening| | + | |$A$|Parameter of correction in the energy stored by hardening| |
| - | |m|Exponent parameter of correction in the energy stored by hardening| | + | |$m$|Exponent parameter of correction in the energy stored by hardening| |
| - | |wD|Stored energy threshold for damage initiation| | + | |$w_D$|Stored energy threshold for damage initiation| |
| - | |Sf|Fatigue damage parameter| | + | |$S_f$|Fatigue damage parameter| |
| - | |expsf|Fatigue damage exponent parameter| | + | |$s_f$|Fatigue damage exponent parameter| |
| - | |Sc|Creep damage parameter| | + | |$S_c$|Creep damage parameter| |
| - | |expsc|Creep damage exponent| | + | |$s_c$|Creep damage exponent| |
| - | |k|Kachanov creep damage exponent| | + | |$k_c$|Kachanov creep damage exponent| |
| - | ^If IDAM≥10 - Line 3+NAF+NAFcyc+NAFY (1G10)^^ | + | ^If IDAM≥10 - Line 3+NAF+NAFcyc+NAFY (1G10 to 3G10)^^ |
| - | |D<sub>u</sub>|Uniform corrosion parameter| | + | |$k_u$|Uniform corrosion parameter| |
| + | |$L_E$| [Optional] Characteristic length of the element - if blank or 0, $L_E$ is computed as the cubic root of the volume element| | ||
| + | |$m_u$| [ONLY IF DIDAM=3] power law parameter| | ||
| === Temperature-dependent parameters - Case where iarrh=2 === | === Temperature-dependent parameters - Case where iarrh=2 === | ||
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| \[P(T)=A_P\left(1-B_P*\exp\left(\frac{T}{C_P}\right)\right)\] | \[P(T)=A_P\left(1-B_P*\exp\left(\frac{T}{C_P}\right)\right)\] | ||
| ^Line 1 (6G10)^^ | ^Line 1 (6G10)^^ | ||
| - | |A<sub>E</sub>|Young modulus parameter| | + | |$A_E$|Young modulus parameter| |
| - | |B<sub>E</sub>|Young modulus parameter| | + | |$B_E$|Young modulus parameter| |
| - | |C<sub>E</sub>|Young modulus parameter| | + | |$C_E$|Young modulus parameter| |
| - | |A<sub>nu</sub>|Poisson ratio parameter| | + | |$A_\nu$|Poisson ratio parameter| |
| - | |B<sub>nu</sub>|Poisson ratio parameter| | + | |$B_\nu$|Poisson ratio parameter| |
| - | |C<sub>nu</sub>|Poisson ratio parameter| | + | |$C_\nu$|Poisson ratio parameter| |
| ^Line 2 (6G10)^^ | ^Line 2 (6G10)^^ | ||
| - | |A<sub>α</sub>|Dilatation coefficient parameter| | + | |$A_\alpha$|Thermal expansion coefficient parameter| |
| - | |B<sub>α</sub>|Dilatation coefficient parameter| | + | |$B_\alpha$|Dilatation coefficient parameter| |
| - | |C<sub>α</sub>|Dilatation coefficient parameter| | + | |$C_\alpha$|Dilatation coefficient parameter. \\ If $C_\alpha=0$, $\int_0^T\alpha(T).dT$ is computed as: $\int_0^T\alpha(T).dT=A_{\alpha}T^2+B_{\alpha}T$| |
| - | |A<sub>σY</sub>|Yield stress parameter| | + | |$A_{\sigma_y}$|Yield stress parameter| |
| - | |B<sub>σY</sub>|Yield stress parameter| | + | |$B_{\sigma_y}$|Yield stress parameter| |
| - | |C<sub>σY</sub>|Yield stress parameter| | + | |$C_{\sigma_y}$|Yield stress parameter| |
| ^Line 3 (6G10)^^ | ^Line 3 (6G10)^^ | ||
| - | |A<sub>K</sub>|Drag stress parameter| | + | |$A_K$|Drag stress parameter| |
| - | |B<sub>K</sub>|Drag stress parameter| | + | |$B_K$|Drag stress parameter| |
| - | |C<sub>K</sub>|Drag stress parameter| | + | |$C_K$|Drag stress parameter| |
| - | |A<sub>n</sub>|Norton coefficient parameter| | + | |$A_n$|Norton coefficient parameter| |
| - | |B<sub>n</sub>|Norton coefficient parameter| | + | |$B_n$|Norton coefficient parameter| |
| - | |C<sub>n</sub>|Norton coefficient parameter| | + | |$C_n$|Norton coefficient parameter| |
| ^Line 4 (2G10)^^ | ^Line 4 (2G10)^^ | ||
| - | |b| Rate of isotropic hardening| | + | |$b$| Rate of isotropic hardening| |
| - | |Q| Total isotropic saturation size of the yield surface| | + | |$Q$| Total isotropic saturation size of the yield surface| |
| ^Line 5 (6G10)^^ | ^Line 5 (6G10)^^ | ||
| |B<sub>Ci</sub>|Parameter for Ci ∀i| | |B<sub>Ci</sub>|Parameter for Ci ∀i| | ||
| Line 267: | Line 269: | ||
| |**Line 7+NAF+2NAFcyc+nAFY (4G10)**|| | |**Line 7+NAF+2NAFcyc+nAFY (4G10)**|| | ||
| |The creep damage parameter $S_c$ is calculated using a simpler exponential law: \[A_{S_c}\exp\left(\frac{T}{B_{S_c}}\right)\]|| | |The creep damage parameter $S_c$ is calculated using a simpler exponential law: \[A_{S_c}\exp\left(\frac{T}{B_{S_c}}\right)\]|| | ||
| - | |A<sub>Sc</sub>| creep damage parameter| | + | |$A_{S_c}$| creep damage parameter| |
| - | |B<sub>Sc</sub>| creep damage parameter| | + | |$B_{S_c}$| creep damage parameter| |
| - | |exp<sub>Sc</sub>|exponent parameter for creep damage| | + | |$s_c$|exponent parameter for creep damage| |
| - | |k|Kachanov creep damage exponent| | + | |$k$|Kachanov creep damage exponent| |
| ^If IDAM≥10 ^^ | ^If IDAM≥10 ^^ | ||
| - | |**Line 1+NAF+2NAFcyc+nAFY+H(IDAM)*7 (3G10)**|| | + | |**Line 1+NAF+2NAFcyc+nAFY+H(IDAM)*7 (3G10 to 7G10)**|| |
| - | |A<sub>Du</sub>| corrosion damage parameter| | + | |$A_{k_u}$| corrosion damage parameter| |
| - | |B<sub>Du</sub>| corrosion damage parameter| | + | |$B_{k_u}$| corrosion damage parameter| |
| - | |C<sub>Du</sub>| corrosion damage parameter| | + | |$C_{k_u}$| corrosion damage parameter| |
| + | |$L_E$| [Optional] Characteristic length of the element - if blank or 0, $L_E$ is computed as the cubic root of the volume element| | ||
| + | |$A_{m_u}$| [ONLY IF DIDAM=3] power law parameter| | ||
| + | |$B_{m_u}$| [ONLY IF DIDAM=3] power law parameter| | ||
| + | |$C_{m_u}$| [ONLY IF DIDAM=3] power law parameter| | ||
| ===== Stresses ===== | ===== Stresses ===== | ||
| ==== Number of stresses ==== | ==== Number of stresses ==== | ||
| Line 291: | Line 297: | ||
| ===== State variables ===== | ===== State variables ===== | ||
| ==== Number of state variables ==== | ==== Number of state variables ==== | ||
| - | 24+6*nAF+6*nAFY+H(UIDAM)*(2+2*ddim+6)+DIDAM+8*ILCF \\ | + | $24+6n_{AF}+6n_{AF_Y}+(8+2ddim)\mathscr{H}(u_{i_{dam}})+2\mathscr{H}(d_{i_{dam}})+8i_{LCF}$ |
| - | Where: UIDAM=IDAM mod 10 and DIDAM=IDAM-UIDAM \\ | + | \\ |
| - | H() is the Heaviside step function. | + | Where: $u_{i_{dam}}\equiv i_{dam} \mod 10$ \\ and $d_{i_{dam}}=i_{dam}-u_{i_{dam}}$. \\ |
| + | $\mathscr{H}(x)$ is the Heaviside step function: $\mathscr{H}(x)=1$ if and only if $x>0$, otherwise, $\mathscr{H}(x)=0$. | ||
| ==== List of state variables ==== | ==== List of state variables ==== | ||
| |Q(1)|plastic strain norm $p$| | |Q(1)|plastic strain norm $p$| | ||
| Line 306: | Line 313: | ||
| |Q(18+6nAF+6i:23+6nAF+6i)|Modification tensor $\underline{Y}_i$ (6 components) for i=1:nAFY| | |Q(18+6nAF+6i:23+6nAF+6i)|Modification tensor $\underline{Y}_i$ (6 components) for i=1:nAFY| | ||
| |Q(24+6nAF+6nAFY)|Maximum temperature in the loading history| | |Q(24+6nAF+6nAFY)|Maximum temperature in the loading history| | ||
| - | ===Only if 10>IDAM>0=== | + | ===Only if $u_{i_{dam}}>0$=== |
| - | In the following table, ddim=1 for isotropic damage (scalar damage variable D) and ddim=6 for anisotropic damage (not implemented). | + | In the following table, ddim=1 for isotropic damage (scalar damage variable $D$) and ddim=6 for anisotropic damage (not implemented). |
| |Q(25+6NAF+6NAFY)| Stored energy $w_s$| | |Q(25+6NAF+6NAFY)| Stored energy $w_s$| | ||
| |Q(26+6NAF+6NAFY)| Visco-plastic multiplicator with damage $r$| | |Q(26+6NAF+6NAFY)| Visco-plastic multiplicator with damage $r$| | ||
| - | | Q(27+6NAF+6NAFY) \\ Q(26+ddim+6nAF+6nAFY)|Fatigue damage variable $D_f$ (isotropic) or tensor $\underline{D}_f$ (anisotropic)| | + | | Q(27+6NAF+6NAFY) \\ Q(26+ddim+6nAF+6nAFY)|Fatigue damage variable $D_f$ (isotropic) or tensor $\underline{D}_f$ (anisotropic - not implemented)| |
| - | | Q(27+ddim+6NAF+6NAFY) \\ Q(26+2ddim+6nAF+6nAFY)|Creep damage variable $D_c$ (isotropic) or tensor $\underline{D}_c$ (anisotropic)| | + | | Q(27+ddim+6NAF+6NAFY) \\ Q(26+2ddim+6nAF+6nAFY)|Creep damage variable $D_c$ (isotropic) or tensor $\underline{D}_c$ (anisotropic - not implemented)| |
| | Q(27+2ddim+6NAF+6NAFY) \\ … \\ Q(32+2ddim+6nAF+6nAFY)|Delayed stress tensor $\sigma^d$| | | Q(27+2ddim+6NAF+6NAFY) \\ … \\ Q(32+2ddim+6nAF+6nAFY)|Delayed stress tensor $\sigma^d$| | ||
| - | ===Only if IDAM≥10=== | + | ===Only if $i_{dam}$≥10=== |
| - | NQDU=25+6nAF+6nAFY+(8+2ddim)<IDAM> | + | $N_{Q,D_u}=25+6n_{AF}+6n_{AF_Y}+(8+2ddim)*\mathscr{H}(u_{i_{dam}})$ |
| + | \\ with $\mathscr{H}(u_{i_{dam}})=1$ if and only if $u_{i_{dam}}>0$ | ||
| | Q(NQDU)| $D_u$ - Uniform corrosion damage| | | Q(NQDU)| $D_u$ - Uniform corrosion damage| | ||
| | Q(NQDU+1)| $L_E=\sqrt[3]{V_E}$ - Characteristic length of the element where $V_E$ is the volume of the element (:!: only works with BWD3T elements)| | | Q(NQDU+1)| $L_E=\sqrt[3]{V_E}$ - Characteristic length of the element where $V_E$ is the volume of the element (:!: only works with BWD3T elements)| | ||
| - | === Only if ILCF>0 === | + | === Only if $i_{LCF}>0$ === |
| - | NQLCF=25+6nAF+6nAFY+(8+2ddim)<IDAM>+DIDAM (where DIDAM=1 if IDAM≥10 and 0 otherwise) | + | $N_{Q,LCF}=25+6n_{AF}+6n_{AF_Y}+(8+2ddim)*\mathscr{H}(u_{i_{dam}})+2d_{i_{dam}}$ \\ where $d_{i_{dam}}=1$ if $i_{dam}$≥10 and 0 otherwise) |
| | Q(NQLCF)| t (time)| | | Q(NQLCF)| t (time)| | ||
| |Q(1+NQLCF) |N (cycle)| | |Q(1+NQLCF) |N (cycle)| | ||
laws/chab.1649055656.txt.gz · Last modified: 2022/04/04 09:00 by helene
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