Fig. 1 Dimensions of tensile and fatigue specimens with a thickness of a 3 mm, b 1.5 mm, c 0.2 mm, respectively

Fig. 2 Comparison of the EBSD orientation maps of the a T?=?3 mm, b T?=?1.5 mm, c T?=?0.2 mm specimens. X axis is along the gauge length direction. Z axis is along the thickness direction (see Fig. 1a)

Fig. 3 a Engineering stress-strain curves of pure Cu specimens with thicknesses ranging from 0.2 to 3 mm, b variation of the engineering strength with reducing the specimen thickness, c variation of the engineering strain with reducing the specimen thickness

Fig. 4 Applied stress amplitude-fatigue life curves of pure Cu specimens with thicknesses ranging from 3 to 0.2 mm

Fig. 5 SEM observations of fatigue damage morphologies of Cu specimens with a T?=?0.2 mm at ${\sigma }_{\mathrm{a}}$=47.2 MPa, Nf?=?1,340,116 cycles, b T?=?0.2 mm at ${\sigma }_{a}$=54.5 MPa, Nf?=?228,202 cycles, c T?=?1.5 mm at ${\sigma }_{\mathrm{a}}$=70 MPa, Nf?=?2,978,695 cycles, d T?=?1.5 mm at ${\sigma }_{\mathrm{a}}$=80 MPa, Nf?=?185795cycles, e T?=?3 mm at ${\sigma }_{a}$=70 MPa, Nf?=?1,597,996 cycles, f T?=?3 mm at ${\sigma }_{\mathrm{a}}$=80 MPa, Nf?=?263,185 cycles

Fig. 6 a Total elongation plotted against K and fitted by Oliver’s formula, b strain hardening rate curves, and the corresponding true stress-true strain curves of pure Cu specimens

Fig. 7 Summary of the normalized yield strength for a pure metals [46,47,48], b engineering alloys [28,29,48,49,50,51,52] as a function of T/d

Fig. 8 a Effect of grain size on the critical T/d value [28,29,46,47,48,49,50,51,52], b schematic illustrations of grain size dependence of the x value (ratio of the thickness of surface grain layer h to the grain size d)

Fig. 9 a Grain size effect on x and the fitting formula, b the predicted yield strength of polycrystalline Cu as a function of d and T/d