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
• (dot)	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