UFG AA	Ultrafine-grained aluminum alloy
CPFE	Crystal plasticity finite element
FIP	Fatigue indicator parameter
RVE	Representative volume element
EBSD	Electron backscatter diffraction
LCF	Low cycle fatigue
HCF	High cycle fatigue
F–S	Fatemi and Socie
NPCAH	Non-proportional cyclic additional hardening
$N_N$	Fatigue nucleation life
$N_P$	Small crack propagation life
$FI{P}_N$	FIP for the whole RVE
$FI{P}_N^{Surf}$	FIP on the free surface
$FI{P}_p$	FIP for small crack propagation
$R_{P,UFG},R_P$	Effective yield strength of UFG and its CG counterparts
$\widehat{f},f^{Surf}$	The dot product of normal stress and mesoscopic longitudinal strain for the RVE and its free surface
$n^{\prime },{K^{\prime }}_{UFG}$	Cyclic stress–strain hardening exponent and coefficient
${\epsilon }_{\mathrm{n}}^{(\alpha )}$	Normal strain on the $\alpha$ slip system
$\Delta {\epsilon }_{eq,N}/2$	Equivalent multiaxial strain reflecting non-proportional cyclic additional hardening
$\Delta {\epsilon }_{eq,m}/2$	Mean equivalent normal strain amplitude
$\Delta {\gamma }_m/2$	Mean shear strain amplitude
$\Delta {\epsilon }_{n,m}/2$	Mean normal strain amplitude
V int	Interaction energy
${\tilde{\epsilon }}_{FDR}$	The accumulated mean multiaxial damage strain in FDR
${\epsilon }_{CPZ}$	The strain in cyclic plastic region
${\mathrm{r}}_{FDR}$	The radius of the fatigue damage region
C11, C12, C44	Elastic constants
$\alpha ,r_i,r_f$	Crack length, initial crack length and critical crack length
${\chi }^{(\alpha )}$	Back stress on the αth slip system
${\tau }^{(\alpha )}$	Resolved shear stress on the αth slip system
$\Delta \sigma /2, \Delta {\sigma }_{eff}/2$	Cyclic stress amplitude, effective stress amplitude
${\rho }_0$	Initial dislocation density
$b$	Burgers vector
$\Delta G_0$	Activation energy
m	Activation length parameter
${\mathrm{k}}_B$	Boltzmann constant
p, q	Exponential constants
T	Tested temperature
dmean	Mean grain size
G	Shear modulus
${\omega }_1, {\omega }_2$	Latent hardening ratio
$v_0$	Attempt frequency
$A^{\alpha \beta }$	Hardening matrix
$y_c$	Material constant
$r_D$	Dynamic recovery parameter
$h_b$	Hardening constant
$\Delta GB$	Activation energy of grain boundary diffusion
$\Delta G_0$	Activation energy
u	Material parameter
${\epsilon }_{l\underset{\_}{l}},{\epsilon }_{l\underset{\_}{l}}^{Surf}$	Mesoscopic longitudinal strain for the RVE and its free surface
${\sigma }_n,{\sigma }_n^{Surf}$	Normal stress for the RVE and its free surface
$\Delta {\gamma }_{\max }/2$	Maximum plastic shear strain amplitude
$N_f$	Total fatigue initiation life
$N_N$	Fatigue nucleation life
${\widehat{\tau }}_{UFG}$	Lattice friction stress at the current temperature
${\mathrm{r}}_{FDR}$	Fatigue damage region
$K_{GBS}$	Grain boundary strengthening factor
$K_{GBC}$	Grain boundary constraint factor
$\Delta CTOD$	Crack tip opening displacement
${\tau }_{ath}^{(\alpha )}$	Slip resistance on the αth slip system
$N_P$	Small crack propagation life
FCP	Fatigue crack propagation
FN	Fatigue nucleation