CreeSo 0.5.1
Demo
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Code / Equations
$$\sigma_i=\alpha \cdot M \cdot G \cdot b \cdot \left(\rho_m+c_s \cdot \rho_s\right)^\left(\frac{1}{2}\right)$$
sigma_i=alpha*M*G*b*(rho_m+c_s*rho_s)^(1/2)
$$h_b=\frac{\frac{1}{\rho_b+\rho_s}}{R_{sgb}}$$
h_b=1/(rho_b+rho_s)/R_sgb
$$P_{eff}=\frac{4}{3} \cdot G \cdot b^2 \cdot \rho_b \cdot \left(1-\frac{\pi}{2} \cdot \left(r_{prec,sgb}^2 \cdot N_{prec,sgb}\right) \cdot R_{sgb}\right)$$
P_eff=4/3*G*b^2*rho_b*(1-pi/2*(r_prec_sgb^2*N_prec_sgb)*R_sgb)
$$M_{sgb}=\frac{2 \cdot \pi \cdot \eta_v \cdot D_s \cdot \omega}{b \cdot k_B \cdot T \cdot \theta_{EBSD}}+\frac{2 \cdot \pi \cdot b \cdot D_{vp} \cdot \omega \cdot \theta_{EBSD}}{h_b^2 \cdot k_B \cdot T}$$
M_sgb=2*pi*eta_v*D_s*omega/(b*k_B*T*Winkel_EBSD)+2*pi*b*D_vp*omega*Winkel_EBSD/(h_b^2*k_B*T)
$$v_{g,T1}=exp\left(\frac{-Q}{k_B \cdot T}\right)$$
v_g_Term1=exp(-Q/(k_B*T))
$$v_{g,T2}=exp\left(\frac{-\sigma_i \cdot V_r}{k_B \cdot T}\right)$$
v_g_Term2=exp(-sigma_i*V_r/(k_B*T))
$$v_{g,T3}=sinh\left(\frac{\sigma_{app} \cdot V_r}{k_B \cdot T}\right)$$
v_g_Term3=sinh(sigma_app*V_r/(k_B*T))
$$v_g=a_1 \cdot v_{g,T1} \cdot v_{g,T2} \cdot 2 \cdot v_{g,T3}$$
v_g=a_1*v_g_Term1*v_g_Term2*2*v_g_Term3
$$\rho_t=\rho_m+\rho_s+\rho_b$$
rho_t=rho_m+rho_s+rho_b
$$L_{\alpha}=\frac{\left(1+\nu\right) \cdot G \cdot b \cdot \omega}{6 \cdot \pi \cdot \left(1-\nu\right) \cdot k_B \cdot T}$$
L_alpha=(1+nue)*G*b*omega/(6*pi*(1-nue)*k_B*T)
$$L_p=2^\left(\frac{1}{2}\right) \cdot a_g \cdot exp\left(\frac{\delta_W}{2 \cdot k_B \cdot T}\right)$$
L_p=2^(1/2)*a_g*exp(delta_W/(2*k_B*T))
$$v_{c,T1}=exp\left(\frac{-\sigma_i \cdot \omega}{k_B \cdot T}\right)$$
v_c_Term1=exp(-sigma_i*omega/(k_B*T))
$$v_{c,T2}=2 \cdot sinh\left(\frac{\sigma_{app} \cdot \omega}{k_B \cdot T}\right)$$
v_c_Term2=2*sinh(sigma_app*omega/(k_B*T))
$$v_{c,T3}=2 \cdot \pi \cdot \eta_v \cdot D_s$$
v_c_Term3=2*pi*eta_v*D_s
$$v_{c,T4}=L_{\alpha} \cdot \rho_t^\left(\frac{1}{2}\right)$$
v_c_Term4=L_alpha*rho_t^(1/2)
$$v_{c,T5}=b \cdot \left(1-\eta_v \cdot log\left(v_{c,T4}\right)\right)$$
v_c_Term5=b*(1-eta_v*log(v_c_Term4))
$$v_{cl}=\frac{v_{c,T3}}{v_{c,T5}} \cdot v_{c,T1} \cdot v_{c,T2}$$
v_cl=v_c_Term3/v_c_Term5*v_c_Term1*v_c_Term2
$$v_{cp}=\frac{2 \cdot \pi \cdot b \cdot D_{vp}}{L_p^2} \cdot v_{c,T1} \cdot v_{c,T2}$$
v_cp=2*pi*b*D_vp/L_p^2*v_c_Term1*v_c_Term2
$$v_c=v_{cl}+v_{cp}$$
v_c=v_cl+v_cp
$$v_{eff}=\frac{1}{\frac{1}{v_g}+\frac{\frac{\pi}{2} \cdot N_{prec,gi} \cdot r_{prec,gi}^3}{v_c}}$$
v_eff=1/(1/v_g+pi/2*N_prec_gi*r_prec_gi^3/v_c)
Orowan equation $$\dot{\varepsilon}=\frac{b}{M} \cdot v_{eff} \cdot \rho_m$$
eps_punkt=b/M*v_eff*rho_m
$$\rho_{m,T1}=\rho_m^\left(\frac{3}{2}\right)$$
rho_m_Term1=rho_m^(3/2)
$$\rho_{m,T2}=\frac{\beta \cdot \rho_s \cdot R_{sgb}}{h_b^2}$$
rho_m_Term2=beta*rho_s*R_sgb/h_b^2
$$\rho_{m,T3}=\frac{-\rho_m}{2 \cdot R_{sgb}}$$
rho_m_Term3=-rho_m/(2*R_sgb)
$$\rho_{m,T4}=\frac{-8 \cdot \rho_m^\left(\frac{3}{2}\right) \cdot v_c}{v_{eff}}$$
rho_m_Term4=-8*rho_m^(3/2)*v_c/v_eff
$$\rho_{m,T5}=-d_{anh} \cdot \left(\rho_m+\rho_s\right) \cdot \rho_m$$
rho_m_Term5=-d_anh*(rho_m+rho_s)*rho_m
$$\rho_{m,Tges}=\rho_{m,T1}+\rho_{m,T2}+\rho_{m,T3}+\rho_{m,T4}+\rho_{m,T5}$$
rho_m_Term_ges=rho_m_Term1+rho_m_Term2+rho_m_Term3+rho_m_Term4+rho_m_Term5
$$\dot{\rho_m}=v_{eff} \cdot \rho_{m,Tges}$$
rho_m_punkt=v_eff*rho_m_Term_ges
$$\rho_{s,T1}=\frac{\rho_m}{2 \cdot R_{sgb}}$$
rho_s_Term1=rho_m/(2*R_sgb)
$$\rho_{s,T2}=\frac{-8 \cdot v_c \cdot \rho_s}{v_{eff} \cdot h_b}$$
rho_s_Term2=-8*v_c*rho_s/(v_eff*h_b)
$$\rho_{s,T3}=-d_{anh} \cdot \rho_s \cdot \rho_m$$
rho_s_Term3=-d_anh*rho_s*rho_m
$$\rho_{s,Tges}=\rho_{s,T1}+\rho_{s,T2}+\rho_{s,T3}$$
rho_s_Term_ges=rho_s_Term1+rho_s_Term2+rho_s_Term3
$$\dot{\rho_s}=v_{eff} \cdot \rho_{s,Tges}$$
rho_s_punkt=v_eff*rho_s_Term_ges
$$\dot{\rho_b}=\frac{8 \cdot \left(1-2 \cdot \zeta\right) \cdot \rho_s \cdot v_c}{h_b}-\frac{M_{sgb} \cdot P_{eff} \cdot \rho_b}{R_{sgb}}$$
rho_b_punkt=8*(1-2*zeta)*rho_s*v_c/h_b-M_sgb*P_eff*rho_b/R_sgb
$$R_{sgb,T1}=M_{sgb} \cdot P_{eff}$$
R_sgb_Term1=M_sgb*P_eff
$$R_{sgb,T2}=\frac{G \cdot \eta_v \cdot K_c^2 \cdot \left(\left(\rho_m+\rho_s\right)^\left(\frac{1}{2}\right)-\frac{K_c}{2 \cdot R_{sgb}}\right) \cdot D_s \cdot \omega}{2 \cdot k_B \cdot T}$$
R_sgb_Term2=G*eta_v*K_c^2*((rho_m+rho_s)^(1/2)-K_c/(2*R_sgb))*D_s*omega/(2*k_B*T)
$$\dot{R_{sgb}}=R_{sgb,T1}+R_{sgb,T2}$$
R_sgb_punkt=R_sgb_Term1+R_sgb_Term2
Show / hide all variables
Variables
| Pos | Name | Symbol | Descr | Init | Deriv |
|---|---|---|---|---|---|
| 1 | i | \(i\) | Loop counter | 0 | |
| 2 | t | \(t\) | time in seconds | 0 | |
| 3 | dt | \(dt\) | time increment | 100 | |
| 4 | max_loop_count | \({max\_loop\_count}\) | maximal number of iterations | 3.6e+06 | |
| 5 | T | \(T\) | Temperature in Kelvin | 923 | |
| 6 | sigma_app | \(\sigma_{app}\) | Applied stress in megapascal | ||
| 7 | eps | \(\varepsilon\) | Creep strain | 0 | eps_punkt |
| 8 | a_g | \(a_g\) | Gitterkonstante Fe-bcc. [m] | 2.866e-10 | |
| 9 | alpha | \(\alpha\) | [-] | 0.02 | |
| 10 | b | \(b\) | Burgers - Vektor [m] | 2.48e-10 | |
| 11 | c_s | \(c_s\) | [-] | 0.3 | |
| 12 | d_anh | \(d_{anh}\) | edge dislocations [m] | -999 | |
| 13 | D_s | \(D_s\) | in 9Cr steel at 650°C [m²/s] | 2e-19 | |
| 14 | D_vp | \(D_{vp}\) | in P91, 650°C [m²/s] | 4.75e-19 | |
| 15 | eta_v | \(\eta_v\) | ferritic steel | 0.0002 | |
| 16 | G | \(G\) | 650°C, [Pa] | 6.2e+10 | |
| 17 | k_B | \(k_B\) | [J/K] | 1.38065e-23 | |
| 18 | K_c | \(K_c\) | for P91 | 2.1 | |
| 19 | M | \(M\) | Taylor Faktor | 3 | |
| 20 | omega | \(\omega\) | Volume per Atom [m³] | -999 | |
| 21 | Q | \(Q\) | Activation energy for dislocation glide | 4.01e-19 | |
| 22 | Winkel_EBSD | \(\theta_{EBSD}\) | [rad] | 0.0524 | |
| 23 | V_r | \(V_r\) | Activation volume for dislocation glide | -999 | |
| 24 | nue | \(\nu\) | 0.317 | ||
| 25 | delta_W | \(\delta_W\) | [J] | 1.26e-19 | |
| 26 | zeta | \(\zeta\) | [-] | 0.034 | |
| 27 | a_1 | \(a_1\) | 3.9 | ||
| 28 | A | \(A\) | 0.035 | ||
| 29 | beta | \(\beta\) | 0.0375 | ||
| 30 | rho_m | \(\rho_m\) | [m/m3] | 4.5e+14 | rho_m_punkt |
| 31 | rho_s | \(\rho_s\) | [m/m3] | 4.5e+11 | rho_s_punkt |
| 32 | rho_b | \(\rho_b\) | [m/m3] | 5.9e+14 | rho_b_punkt |
| 33 | R_sgb | \(R_{sgb}\) | 4e-07 | R_sgb_punkt | |
| 34 | r_prec_gi | \(r_{prec,gi}\) | 1.6e-08 | r_prec_gi_punkt | |
| 35 | N_prec_gi | \(N_{prec,gi}\) | 2.36e+20 | N_prec_gi_punkt | |
| 36 | k_prec_gi | \(k_{prec,gi}\) | 1e-31 | ||
| 37 | r_prec_sgb | \(r_{prec,sgb}\) | 7e-08 | r_prec_sgb_punkt | |
| 38 | N_prec_sgb | \(N_{prec,sgb}\) | 1.17e+19 | N_prec_sgb_punkt | |
| 39 | k_prec_sgb | \(k_{prec,sgb}\) | 1e-28 | ||
| 40 | rho_b_punkt | \(\dot{\rho_b}\) | 0 | ||
| 41 | rho_m_punkt | \(\dot{\rho_m}\) | 0 | ||
| 42 | rho_s_punkt | \(\dot{\rho_s}\) | 0 | ||
| 43 | R_sgb_punkt | \(\dot{R_{sgb}}\) | 0 | ||
| 44 | r_prec_gi_punkt | \(\dot{r_{prec,gi}}\) | 0 | ||
| 45 | r_prec_sgb_punkt | \(\dot{r_{prec,sgb}}\) | 0 | ||
| 46 | N_prec_gi_punkt | \(\dot{N_{prec,gi}}\) | 0 | ||
| 47 | N_prec_sgb_punkt | \(\dot{N_{prec,sgb}}\) | 0 | ||
| 48 | eps_punkt | \(\dot{\varepsilon}\) | 0 | ||
| 49 | pi | \(\pi\) | 3.14159 | ||
| 50 | sigma_i | \(\sigma_i\) | Parameter for Microstructure | 0 | |
| 51 | h_b | \(h_b\) | Parameter for Microstructure | 0 | |
| 52 | P_eff | \(P_{eff}\) | Parameter for Microstructure | 0 | |
| 53 | M_sgb | \(M_{sgb}\) | Parameter for Microstructure | 0 |
Show / hide the code for additional initialisation
Start code
Equations / code evaluated at start
$$d_{anh}=5 \cdot b$$
d_anh=5*b
$$\omega=\frac{a_g^3}{2}$$
omega=a_g^3/2
$$V_r=35 \cdot \omega$$
V_r=35*omega