11.6. Nomenclature¶
Subscript |
Description |
|---|---|
\(\text{l}\) |
Fully annealed material |
\(\theta\) |
Angular direction in cylindrical coordinates |
\(\theta,\text{o}\) |
Initial, reference |
\(\text{c}\) |
Cladding |
\(\text{ci}\) |
Cladding inner radius |
\(\text{e}\) |
Equivalent |
\(\text{f}\) |
Fuel |
\(\text{f}\) |
Failure |
\(\text{fo}\) |
Fuel out radius |
\(\text{g}\) |
Gas |
\(\text{gb}\) |
Grain boundary |
\(\text{i}\) |
Time step index |
\(\text{i}\) |
Radial element index |
\(\text{I, II}\) |
Element types |
\(\text{j}\) |
Axial segment index |
\(\text{m}\) |
Mean, average |
\(\text{n}\) |
Iteration index |
\(\text{N}\) |
Iteration index |
\(\text{Na}\) |
Sodium |
\(\text{pu}\) |
True plastic (strain) |
\(\text{r}\) |
Radial direction in cylindrical coordinates |
\(\text{r}\) |
Rupture |
\(\text{s}\) |
Saturation |
\(\text{th}\) |
Threshold |
\(\text{u}\) |
True/ultimate (stress) |
\(\text{z}\) |
Axial direction in cylindrical coordinates |
Superscript |
Description |
\(\theta\) |
Thermal (strain) |
\(a\) |
Apparent (strain) |
\(c\) |
Crack (strain) |
\(C\) |
Cavity |
\(e\) |
Elastic (strain) |
\(ext\) |
External loads |
\(G\) |
Fuel-cladding gap |
\(i\) |
Iteration counter |
\(int\) |
Internal stress field |
\(IGL\) |
Ideal gas law |
\(MECH\) |
Mechanical analysis |
\(p\) |
Plastic (strain) |
\(ps\) |
Pseudo (stress) |
\(P\) |
Plenum |
\(ref\) |
Reference |
\(s\) |
Swelling (strain) |
\(t\) |
Total (strain) |
\(T\) |
Total |
Notation |
Description |
¯ (overline) |
Equivalent |
_ (underline) |
Indicates a vector |
|
Indicates derivative with respect to time |
ˆ (hat) |
Indicates a unit vector |
[ ] |
Indicates a matrix |
{}T |
Transpose of a vector |
\(\nabla\) |
Gradient |
Symbol |
Description |
\(\alpha\) |
Thermal expansion coefficient |
\(\delta\) |
Nodal displacement |
\(\Delta L/L_{0}\) |
Linear thermal expansion |
\(\Delta t\) |
Time step size |
\(\varepsilon\) |
Strain |
\(\hat{\varepsilon}\) |
Cladding hardness parameter |
\(\phi\) |
Overpressure in grain boundary bubble growth model |
\(\phi\) |
Finite element displacement function |
\(\gamma\) |
Specific surface free energy in surface tension restraint term |
\(\nu\) |
Poisson’s ratio |
\(\Omega\) |
Atomic volume in grain boundary bubble growth model |
\(\psi\) |
Constant in finite element displacement function in axial direction |
\(\rho\) |
Radius of curvature in surface tension restraint term |
\(\rho\) |
Density |
\(\sigma\) |
Stress |
\(\theta\) |
Intersection angle between the bubble surface and grain boundary |
\(\theta\) |
Initial hardening rate |
\(a\) |
Finite element inner radius |
\(A\) |
Area of intersection between the finite elements |
\(b\) |
Finite element outer radius |
\(B\) |
Matrix in equilibrium equation |
\(C\) |
Curve bounding A |
\(C\) |
Matrix in generalized Hooke’s law |
\(C\) |
Material functions in secondary creep equation |
\(\text{d}\varepsilon\) |
Incremental strain |
\(\text{d}\lambda\) |
Proportionality factor for plastic strain increments |
\(\text{d}\sigma\) |
Incremental stress |
\(\text{dp}\) |
Pressure increment |
\(D\) |
Grain size |
\(D\) |
Diffusion coefficient in grain boundary bubble growth model |
\(E\) |
Young’s modulus |
\(F\) |
Constitutive function for plastic strains |
\(F\) |
Force |
\(F\) |
Axial force |
\(\dot{F}\) |
Fission rate |
\(G\) |
Constitutive function for swelling strains |
\(G\) |
Shear modulus |
\(\hat{i}\) |
Unit vector in coordinate directions |
\(K\) |
Stiffness matrix |
\(K\) |
Bulk modulus |
\(L\) |
Finite element radial thickness |
\(M\) |
Rate sensitivity parameter in cladding yield and saturation flow stresses |
\(M\) |
Mass |
\(\dot{M}\) |
Eutectic penetration rate |
\(\hat{n}\) |
Outward normal to the finite element surface, S |
\(N\) |
Finite element shape function |
\(N\) |
Number density of gas bubbles in grain boundary bubble growth model |
\(P\) |
Pressure |
\(Q\) |
Activation energy in secondary creep equation |
\(Q\) |
Thermal modulus |
\(R\) |
Residual in iterative solution of equilibrium equations |
\(R\) |
Radius |
\(R\) |
Gas constant |
\(S\) |
Deviatoric stress |
\(S\) |
Surface area of finite element |
\(T\) |
Time |
\(T\) |
Elements of load vector in equilibrium equation expressing external forces |
\(T\) |
Temperature |
\(U\) |
Displacement in radial coordinate direction |
\(V\) |
Displacement in θ coordinate direction |
\(V\) |
Volume of finite element |
\(W\) |
Grain boundary thickness |
\(W\) |
Displacement in axial coordinate direction |
\(W\) |
Gradient vector for plastic strain increments |
\(Z\) |
Zener-Holloman parameter |