1 Introduction

With the non-renewable resources such as coal, oil, and natural gas being consumed constantly over the decades and its concomitant ever-growing severity of energy crisis and environmental problems, the development and utilization of new energy are promoted. Nuclear power generation does not emit greenhouse gases and has the advantages of being low-cost and efficient compared to coal-burning generating, thus becoming one of the most widely used new energy sources in the world nowadays. However, the safety of nuclear power plants cannot be ignored with the rapid development of nuclear power. Structural materials, as the cornerstone of a nuclear reactor, will be irradiated by about 14 MeV neutrons dose during the operation, the synergistic effect of displacement damage and nuclear transmutation products would induce the deterioration of mechanical properties and impaired the integrity of structural components [1,2,3,4,5,6].

To monitor the operation status of a nuclear reactor, the structural materials need to be periodically inspected to know the real-time material properties. Under such a situation, standard specimens may be undesirable when testing irradiated materials, as insufficient materials can be available from in-service structures to make standard specimens, also, large volumes of irradiated materials can create security problems. Thus, small specimen testing is becoming increasingly necessary in identifying mechanical properties of irradiation metals or alloys [7,8,9,10,11]. More importantly, small specimens are also expected to assess the mechanical properties of metal additive manufacturing (AM) parts. AM can fabricate geometrically complex parts in days rather than months. But any variation of AM processing parameters such as laser energy density and building thickness, etc. would eventually lead to the location-specific distribution of defects and microstructure heterogeneities as well as anisotropic mechanical properties [12, 13]. Thus, the mechanical properties of AM parts are difficult to evaluate using the standard specimens due to the dimension restriction and anisotropic mechanical properties [12]. Therefore, the small specimen is more effective than the standard specimen in rapidly evaluating the mechanical properties of AM parts. What is more, besides in the field of life management of operating equipment in harsh operating conditions like radiation and the development of new materials such as AM, the small specimen is also gradually applied to other in-service components, such as jet engine fan blades [14], industrial gas turbine blades [15], power plant applications [16] and steam turbine [17].

Nowadays, small specimen test technology (SSTT) is becoming increasingly relevant in fusion nuclear reactors for identifying a wide range of post-irradiation material properties [18,19,20]. To ensure the safe operation of a fusion nuclear reactor, the detailed behavior of post-irradiation materials must be known. In this situation, researchers from various institutions and universities in Germany and Japan have set up the SSTT program in IFMIF (the international fusion materials irradiation facility) since 2013. IFMIF, as a deuterium–lithium neutron source with high intensity, can radiate about 1000 small specimens at the same time [21, 22]. These small specimens will provide data on the mechanical properties of post-irradiation materials such as tensile properties, fatigue properties, and fracture toughness. The main subjects of the SSTT program in IFMIF are to extrapolate the experimental data with the small specimen to one with the standard specimen and standardization or modification of test methodologies and specimen sizes [21, 22]. Taking the fatigue test as an example, there are fixed experimental procedures to conduct a test [21]. Firstly, the stress distribution of two types of small specimens (hourglass type specimen and round-bar type specimen) surface needs analysis by finite element (FE) to get a stress distribution similar to that of standard specimens. This step contributes to the optimization of specimen dimensions. Secondly, preliminary experiments are carried out on the specimens designed by FE to define the test methodology at one certain strain amplitude. Temperature, load and total strain range, and other factors are taken into account. The specimen with a fatigue life of more than 104 cycles would be selected for the following experiment. Finally, systematic fatigue experiments are carried out under full strain amplitude to evaluate the difference in fatigue performance between the standard specimen designed according to ASTM E466 and the small specimen. The final results show that the fatigue performance of the small round-bar type specimen is comparable to that of the standard specimen. This standardized experimental method not only provides ideas for optimizing small specimen dimensions but also makes the results representative and repeatable.

Although SSTT has various advantages in identifying mechanical properties of operating equipment, the miniaturization of specimens inevitably brings about what is called “size effect”. The sizes and geometries of the specimens have a great effect on the mechanical properties of materials, such as strength, ductility, and fatigue performance. Therefore, a precise understanding of the size effect is an important prerequisite in interpreting mechanical behavior. The effects of specimen dimensions (gauge length, width, and thickness) on the tensile properties of various materials have been investigated extensively. Sergueeva et al. [23] found that Ti–6Al–4 V sheet material with a fixed thickness of 115 μm and a width of 1 mm but varying gauge length from 2 to 40 mm exhibited a consistent tensile strength of about 1000 MPa. However, there is an evident influence of gauge length on the total elongation and Young’s Modulus, the former decreased with increasing gauge length, and the latter showed the opposite trend. Yang et al. [24] systematically investigated the influence of specimen thickness on the tensile properties of pure Cu with different grain sizes. Tensile strength, total elongation, and work-hardening coefficient started to decrease with decreasing thickness once the ratio (T/d) of specimen thickness (T) to grain size (d) is below the critical value. Yuan et al. [25] found that both the yield strength and tensile strength keep stable when the thickness is larger than 1 mm using single-factor analysis. Hamada et al. [26] investigated the ductility of high-Mn TWIP steel under quasi-static and high-speed deformation conditions and found that the increasing elongation in the high-speed tensile test was attributed to the decrease in the gauge length of the specimens. Regarding the fatigue properties, size effects on fatigue properties of pure metals and engineering alloys were paid attention to in recent years. Dai et al. [27] investigated the size effects on fatigue properties of Cu foils with varying thicknesses (20, 40, 80, 160, and 420 μm). The results demonstrated that the fatigue performance of the Cu foils could be influenced by the tensile plasticity of the foils and coupling between T and d. Ma et al. [28] showed that the fatigue strength of the T = 40 μm CA6NM martensite stainless steel specimens was lower than that of the standard specimen in the low-cycle fatigue (LCF) regime, and the difference in fatigue performance gradually reduced in the high-cycle fatigue (HCF) regime, and the fatigue limits between the two types of specimens tended to be the same. Zhang et al. [29] found the fatigue strength of small Ti alloy specimens (T = 0.1 and 0.2 mm) is lower than that of the thicker specimens in the LCF regime, while this trend became reversed in the HCF regime. As mentioned above, both the external geometrical size (gauge length, width, thickness) and its interplay with microstructure (T/d) have a strong influence on the mechanical properties of materials.

Although many studies have been carried out on the size effect of tensile and fatigue performance, nearly no systematic analyses of tensile and fatigue performance on the relation between the standard specimens designed according to ASTM and small specimens. In this study, we present an experimental investigation of the tensile and fatigue behavior of pure Cu with different thicknesses ranging from 3 to 0.2 mm. Obvious size effects on tensile plasticity, yield strength, and fatigue performance are found, and the corresponding mechanism is investigated then. Furthermore, considering the great potential of small specimens in the rapid measurement of mechanical properties and residual life assessment of operating components, our present work is expected to offer guidance on how to evaluate the mechanical properties of standard specimens by small specimens from the practical point.

2 Experimental

2.1 Material and Specimen Preparation

Commercial pure Cu with a purity of better than 99.99% was selected for this study. Cu is extensively used in the structural component of various industries, especially in the condenser tube for coastal power plants [30]. What is more, such a model material does not contain any phases, which can effectively rule out potential size effects, and facilitate the establishment of theoretical models. In this study, three kinds of specimen dimensions were studied (Fig. 1). Standard specimens (Fig. 1a) were designed according to ASTM E466 with a gauge length (L) of 15 mm, a width (W) of 6 mm and a thickness (T) of 3 mm. The other two small specimens (Fig. 1b, c) had the same gauge length (L = 3 mm) and width (W = 2 mm) but different thicknesses ranging from 1.5 to 0.2 mm. EBSD orientation maps of the Cu specimens in Fig. 2a, b show that coarse grains are decorated with a high density of annealing twins for T = 3 mm and T = 1.5 specimens. Complete grain structure no longer exists in the thickness direction for T = 0.2 mm specimens, as shown in Fig. 2c. The average grain size of the pure Cu is about 259 μm. Flat dog-boned specimens were sectioned by electro-discharge machining (EDM). Then, all the specimen surfaces in the gauge regions were grounded and polished using SiC paper followed by chemically polishing to minimize the surface defects.

Fig. 1
figure 1

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

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Fig. 2
figure 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)

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2.2 Tensile and Fatigue Tests

Tensile tests of standard specimens (T = 3 mm) were performed using a Shimadzu Ag–X machine with a strain rate of \(\dot{\varepsilon }\)= 1 × 10–3 s−1, and the tensile strains of standard specimens were detected by a non-contact optical measurement system. Tensile tests of small specimens (T = 1.5 mm, T = 0.2 mm) were performed on a INSTRON 5848 microforce testing system with a strain rate of \(\dot{\varepsilon }\)= 1 × 10–3 s−1. At least three specimens with the same dimensions were tested to ensure data reproducibility. Axial force-controlled fatigue tests of the standard specimens were performed on an INSTRON E10000 with a frequency of 40 Hz and a stress ratio of -1. Axial force-controlled fatigue tests of T = 1.5 mm specimens were performed on an INSTRON E1000 using a stress ratio of -1 and a frequency of 40 Hz. Meanwhile, axial force-controlled fatigue tests of T = 0.2 mm specimens were performed on an INSTRON E1000 using a stress ratio of 0.1 and a frequency of 40 Hz. The reasons for the different stress ratios between T = 0.2 mm specimens and other specimens are as follow. The fatigue limit is defined at R = − 1, and the fitting of the stress amplitudes versus the number of cycles to failure (S–N) curve by Basquin equation may meet the condition of fully reversed loading, thus the stress ratio is set at − 1 for standard specimen and T = 1.5 mm specimen. As for the T = 0.2 mm specimen, it has little resistance to buckling under fully reversed cyclic loading, thus the stress ratio is set at 0.1. The stress ratio indeed has an influence on fatigue performance, which is why the Gerber relation (Subsection 3.2) is used to convert the stress amplitude of the T = 0.2 mm specimen at R = 0.1 to that at R = − 1 in the performance comparison that follows. The microstructure was characterized by using a scanning electron microscope (SEM, SUPRA 55) equipped with an electron backscatter diffraction (EBSD) system. Fracture morphology was examined by a scanning electron microscope (SEM, LEO SUPRA 35).

3 Results

3.1 Tensile Properties

Figure 3a presents tensile engineering stress–strain curves of pure Cu specimens with T varying from 3 to 0.2 mm. Variations of strength (yield strength and tensile strength) and ductility as a function of thickness are summarized in Fig. 3b, c, respectively. As shown in Fig. 3b, there are no obvious changes in the yield strength (YS = 72.9 MPa) and tensile strength (UTS = 184.1 MPa) of Cu specimens with decreasing T from 3 to 1.5 mm, but a sudden reduction in the yield strength (YS = 59.5 MPa) and tensile strength (UTS = 131.5 MPa) is observed in T = 0.2 mm specimens. A similar thickness effect on uniform elongation is also observed in the pure Cu specimen (Fig. 3c). There is no significant effect on uniform elongation (27%) with decreasing T from 3 to 1.5 mm, the uniform elongation decreases to 19.3% when T = 0.2 mm. However, the T = 1.5 mm specimen shows the largest total elongation of 52.3%. The total elongation of T = 0.2 mm and 3 mm specimens are 20.8% and 32.5%, respectively.

Fig. 3
figure 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

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3.2 Fatigue Properties

Considering the mean stress effect due to non-fully reversed loading (R = 0.1), the applied stress amplitude of T = 0.2 mm specimens at R =  − 1 is evaluated by the Gerber relation [31].

$${\sigma }_{\mathrm{a}}={\sigma }_{\mathrm{a}}{|}_{{\sigma }_{\mathrm{m}}=0}\left\{1-{\left({\sigma }_{\mathrm{m}}{/\sigma }_{\mathrm{UTS}}\right)}^{2}\right\},$$
(1)

where \({\sigma }_{\mathrm{m}}\) is the mean stress, \({\sigma }_{\mathrm{a}}\) is the stress amplitude for nonzero mean stress, \({\upsigma }_{\mathrm{a}}{|}_{{\upsigma }_{\mathrm{m}}=0}\) is the stress amplitude for a zero mean stress, \({\upsigma }_{\mathrm{UTS}}\) is the tensile strength of the T = 0.2 mm specimen.

Figure 4 shows the S–N curves of pure Cu specimens with different thicknesses. As can be seen from Fig. 4, the S–N curves of T = 1.5 mm and 3 mm specimens are almost overlapped, and the average fatigue limit (at 1 × 107 cycles) is about 60.2 MPa. The fatigue life decreases remarkably whether in the LCF regime or the HCF regime for the T = 0.2 mm specimens. The fatigue limit of the T = 0.2 mm specimens is about 42.3 MPa, which dropped by about a third compared with the thicker specimens.

Fig. 4
figure 4

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

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The relationship between \({\sigma }_{\mathrm{a}}\) at R =  − 1 and \({N}_{\mathrm{f}}\) is fitted based on the classic Basquin equation.

$${\sigma }_{\mathrm{a}}={\sigma }_{\mathrm{f}}^{{\prime}}{(2{N}_{\mathrm{f}})}^{b},$$
(2)

where \({\sigma }_{\mathrm{f}}^{{\prime}}\) and b are the fatigue strength coefficient and the fatigue strength exponent, respectively. Because the S–N curves of T = 1.5 mm and 3 mm specimens are almost overlapped, the two curves are fitted as a whole, and the S–N curve of the T = 0.2 mm specimens is fitted separately. Obviously, a strong linear relationship can be found between \({\sigma }_{\mathrm{a}}\) and \({N}_{\mathrm{f}}\) in double logarithm coordinate as shown in Fig. 4. The parameters \({\sigma }_{\mathrm{f}}^{{\prime}}\) and b values of the T = 0.2 mm specimens and thicker specimens in Eq. (2) can be determined by data fitting, as listed in Table 1.

Table 1 Fitting parameters of pure Cu specimens with varying thickness
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Actually, \({\sigma }_{\mathrm{f}}^{{\prime}}\) is nearly equal to the true tensile strength after modification with necking [31]. Clearly, the T = 0.2 mm specimen exhibited a much lower \({\sigma }_{\text{f}}^{{\prime}}\) compared to the thicker specimens due to its relatively low tensile strength shown in Fig. 3b. b is a negative value, which reflects the damage mechanism associated with cyclic deformation [32]. The value of b of the T = 0.2 mm specimen increases slightly compared with the thicker specimens.

3.3 Microstructure Characterization

Figure 5 presents SEM observations of fatigue damage morphologies of Cu specimens with different T at low (Fig. 5a, c, e) and high (Fig. 5b, d, f) stress amplitudes. These results indicate that fatigue cracks initiated and then grew mainly along slip bands and grain boundaries (GBs) impinged by extensive slip bands. Meanwhile, more slip bands and cracks appeared on the surface of the specimens at high-stress amplitudes when compared with low-stress amplitudes. This means that more severe fatigue damage led to the degradation of fatigue properties, which is consistent with the corresponding \({N}_{\mathrm{f}}\) in Fig. 4.

Fig. 5
figure 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

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However, the fatigue crack initiation behavior changed rarely with the stress amplitude and the specimen thickness. Therefore, for the reason of the remarkable decrease of fatigue properties of the T = 0.2 mm specimen, the possibility of cracking mechanism transition can be eliminated.

4 Discussion

4.1 Influence of Grain Boundaries and Surface Effect on Elongation

Figure 3c clearly shows that the uniform elongation of the T = 0.2 mm specimen decreases but the uniform elongation of thicker specimens keeps stable, while the T = 1.5 mm specimen shows the largest total elongation. To evaluate the tensile ductility of pure Cu, the total elongation is then fitted by Oliver formula [33]:

$$E=\alpha {\left(L/{S}^{1/2}\right)}^{\beta },$$
(3)

where E is the total elongation, L is the gauge length and S is the initial cross-section area. α and β are parameters obtained by fitting. Meanwhile, a ratio of L to S1/2 is defined as the geometry coefficient K, which links the dimensions of the gauge region of the specimen. As shown in Fig. 6a, the total elongation of T = 1.5 mm and T = 3 mm specimens can be fitted well by the Oliver formula with α and β are 72 and − 0.615, respectively. However, the total elongation of the T = 0.2 mm specimen significantly deviates from the fitting curve. Figure 6b shows the true stress–strain curves and the corresponding strain hardening curves of pure Cu specimens. At the initial stage of plastic deformation, the strain hardening rate of the T = 0.2 mm specimen is lower than that of the thicker ones, resulting in decreasing uniform elongation. Thus, the total elongation deviates from the fitting curve.

Fig. 6
figure 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

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Then the reasons for the decreasing strain hardening rate of the T = 0.2 mm specimen are analyzed. According to Taylor formula, the relationship between flow stress (σ) and dislocation density (ρ) can be expressed as [34, 35]:

$$\sigma ={\sigma }_{0}+\alpha \mu bM{\rho }^{1/2},$$
(4)

where \({\sigma }_{0}\) is the friction stress, \(\mu\) and b are shear modulus and Burgers vector respectively. M is the Taylor factor and \(\alpha\) is a material parameter. Furthermore, an equation related to dislocation density and flow stress based on Eq. (4) can be derived:

$$\partial \rho /\partial \sigma =2{\rho }^{1/2}/\alpha M\mu b.$$
(5)

The evolution rate of the dislocation density with plastic strain (ε) can be written as [34, 36,37,38,39,]–[40]:

$$\partial \rho /\partial \varepsilon =M {\rho }^{1/2}/\beta b+M {k}_{\mathrm{g}}/bd+M/b{l}_{0}-\lambda \rho -M/bs,$$
(6)

where \(\beta\) represents the ratio between the mean free path of gliding dislocations and the average dislocation distance, \({k}_{\mathrm{g}}\) is a constant, \({l}_{0}\) represents the dimension characteristic of the initial dislocation structures, \(\lambda\) is the rate of the annihilation process and s is the average distance between free surfaces. The first term on the right side of Eq. (6) is related to the athermal storage of dislocations. The second term represents the contribution of grain boundaries to the strain hardening. The third term is related to the presence of initial dislocation structures generated during previous forming processes. The last two terms represent annihilation of dislocations due to cross-slip or climb during dynamic recovery and the rate of dislocation annihilation due to free surfaces, respectively. Then, an expression for the strain hardening rate can be obtained by combining Eqs. (5) and (6).

$$\partial \sigma /\partial \varepsilon =\alpha M\mu b/2\left(M /\beta b+M{k}_{\mathrm{g}}/bd{\rho }^{1/2}+M/b{l}_{0}{\rho }^{1/2}-\lambda {\rho }^{1/2}-M/bs{\rho }^{1/2}\right).$$
(7)

As shown in Eq. (7), the strain hardening rate can be influenced by the contribution of GBs and dislocation annihilation due to free surfaces. The contribution of GBs to the strain hardening rate is due to the geometrically necessary dislocations (GNDs) caused by the strain incompatibilities between grains. For the T = 0.2 mm specimen, few GBs exist in the vertical direction of the loading axis. For this kind of specimen, the contribution of GBs strongly reduced compared with the thicker specimens, resulting in a decrease in the strain hardening rate. At the same time, dislocation annihilation processes should play an important role in the modification of the strain mechanisms of thinner specimens due to the increasing effect of free surfaces [41, 42]. As a consequence, for the T = 0.2 mm specimen, the dislocation annihilation rate increases due to the increasing effect of the free surface. The density of forest dislocation is decreased accordingly, which results in a decrease in strain hardening rate. In summary, the decreasing strain hardening rate of the T = 0.2 mm specimen is caused by the lack of GBs in the thickness direction of the specimen and the free surface effect, the combination of these two factors ultimately leads to a decrease in elongation.

On the other hand, the change of stress state with decreasing thickness also results in a decrease in elongation. The stress state of the T = 0.2 mm specimen is transformed into the planar stress state, thus the failure mode changes from the normal fracture mode of thicker specimens to the shear failure mode. In the shear failure mode, necking only occurs along the thickness direction, but necking occurs in both the thickness and width directions in the normal fracture mode [43]. Small necking of T = 0.2 mm specimen leads to a small necking length in the tensile direction, which results in a decrease in elongation.

4.2 Influence of Surface Effect on Yield Strength

Our results show that the yield strength of the T = 0.2 mm specimen decreases notably, but that of T = 1.5 mm and T = 3 mm specimens keeps stable. This softening behavior of pure metals has been extensively studied [42, 44,45,46]. The whole specimen contains two different parts: surface grains and interior grains, the decrease in the yield strength is due to the competition between the H-P strengthening effect of the interior grain layer and the softening effect of the surface grain layer.

To further understand the variation of yield strength with the specimen thickness, the yield strengths of specimens with different thicknesses are normalized by that of the thickest specimen. Figure 7a, b present the normalized yield strengths of pure metals and engineering alloys as a function of T/d, respectively [28, 29, 46,47,48,49,50,51,52]. The yield strength greatly decreases as the value of T/d reduces below a critical value shown in Fig. 7. However, the critical T/d values cannot be determined as a fixed value, there is a certain correlation between the critical T/d value and the grain size.

Fig. 7
figure 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

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The critical T/d values of the specimens as shown in Fig. 7 are summarized in Fig. 8a as a function of grain size. It is noted in Fig. 8a that the critical T/d value decreases with increasing grain size. For large grains, dislocation tangles would be expected to form at the region far from the free surface for the long glide distance of dislocations to the free surface, thus the surface grain layer occupies a smaller proportion of the grain than that of the small grains [53]. As illustrated in Fig. 8b, x, a ratio of the thickness of the surface grain layer (h) to the grain size (d), decreases with increasing grain size. As a result, the softening effect of the surface grains of the specimen with the smaller grain size is stronger than that of the specimen with the larger one, leading to the decrease in the critical T/d value with increasing grain size.

Fig. 8
figure 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)

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In order to explore the relationship between x and d, the theoretical model proposed by Dai et al. and the modified model proposed by Wan et al. are further modified [46, 54] based on the following assumption,

  1. (1)

    \(h = x \cdot d\)

  2. (2)

    The strength of entire surface grain layers is estimated by the yielding of a single crystal. Then, the formula of yield strength can be rewritten as

    $${\sigma }_{\mathrm{y}}=\left({\sigma }_{0}+k{\cdot d}^{-1/2}\right)\cdot \left(1-2\cdot x/\left(T/d\right)\right)+({\tau }_{\mathrm{CRSS}}/\Omega )\cdot [2\cdot x/(T/d)],$$
    (8)

    where \({\sigma }_{0}\) and k are the Hall–Petch coefficients, \({\tau }_{\mathrm{CRSS}}\) is the critical resolved shear stress. \(\Omega\) is the average Schmid factor (\(\Omega\)=1/3). Normalization by \({\sigma }_{\mathrm{bulk}}\) leads to

    $${\sigma }_{\mathrm{y}}/{\sigma }_{\mathrm{bulk}}=\left\{\left({\sigma }_{0}+k{\cdot d}^{-1/2}\right)\cdot \left[1-(2\cdot x)/\left(T/d\right)\right]+{(\tau }_{\mathrm{CRSS}}/\Omega )\cdot (2\cdot x)/\left(T/d\right)\right\}/{(\sigma }_{0}+k{\cdot d}^{-1/2}),$$
    (9)

By setting the ratio of \({\sigma }_{\mathrm{y}}/{\sigma }_{\mathrm{bulk}}\) equal to 0.95 as the critical point for yield strength to decrease, x can be computed by the following formula:

$$x=0.025\cdot \left(T/d\right)/\{1-[1/{(\sigma }_{0}+k{\cdot d}^{-1/2})]\cdot {(\tau }_{\mathrm{CRSS}}/\Omega )\},$$
(10)

where T/d is the critical value when yield strengths greatly decrease. Here, taking the critical values of Cu (16–140 μm) shown in Fig. 8a as an example, x can be computed exactly by plugging in the correlation values (\({\sigma }_{0}\)=25.5 MPa, k = 110 MPa \({\cdot \mathrm{\mu m}}^{-1/2}\), \({\tau }_{\mathrm{CRSS}}\)=0.98 MPa [55]) into the Eq. (10). The relationship between the calculated results of x and d is represented in Fig. 9a. It can be seen from Fig. 9a that the x value decreases with increasing d, as schematically illustrated in Fig. 8b. Based on the strong correlation between x and d, the relationship was obtained through fitting data shown in Fig. 9a as,

Fig. 9
figure 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

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$$x=\alpha \cdot {d}^{\beta }.$$
(11)

As shown in Fig. 9a, such an empirical formula can well describe the relationship between x and d with parameters \(\mathrm{\alpha }\) and \(\beta\) being fixed at 1 and − 0.44, respectively. Apparently, the theoretical yield strength of polycrystalline Cu could be directly obtained by substituting Eq. (11) into Eq. (8), and the 3D image as a function of d and T/d is shown in Fig. 9b.

4.3 Mechanism for the Size-Dependent Fatigue Performance

In general, the present study reveals that the fatigue life of the T = 0.2 mm specimen is lower than that of the thicker specimens either in the LCF regime or in the HCF regime. It is well known that HCF life consists of two parts: crack initiation life and crack growth life. Therefore, the reasons for the reduction of fatigue life of T = 0.2 mm specimens can be discussed from two aspects of crack initiation and crack growth.

In the LCF regime, the crack initiation lifetime tends to be consistent, the main difference lies in crack growth life. In the plane strain condition, a cyclic plastic zone size (\({\gamma }_{\mathrm{p}}\)) at the crack tip is given by [56]:

$${\gamma }_{\mathrm{p}}=(1/3\pi )\cdot \left(\Delta {K}_{\mathrm{th}}/2{\sigma }_{\mathrm{y}}\right),$$
(12)

where \(\Delta {K}_{\mathrm{th}}\) is the threshold stress intensity factor range (\(\Delta {K}_{\mathrm{th}}=3\mathrm{MPa }{\mathrm{m}}^{1/2}\) measured in bulk CG Cu [57]). Using Eq. (12), \({\gamma }_{\mathrm{p}}\) can be estimated to be 2.67 μm for the T = 0.2 mm specimen and 2.18 μm for the thicker specimen, which is significantly smaller than the grain size, leading to the fact that the cyclic plastic zone at the crack tip is restricted to only one grain.

The contribution of GBs is to act as an obstacle to crack growth to increase fatigue resistance. For the T = 0.2 mm specimen, there is only an individual grain in the thickness direction. For this kind of specimen, the contribution of GBs significantly reduces compared with that for the thicker specimens, leading to a decrease in the fatigue lifetime.

In the HCF regime, the crack initiation lifetime plays a decisive role in the fatigue life. According to the modified slip band-grain boundary (BSD) model [58, 59], the threshold stress for crack growth can be expressed as:

$${\sigma }_{\mathrm{th}}={\sigma }_{\mathrm{fr}}^{*}+{K}_{\mathrm{c}}^{\mathrm{m}}/{(\pi {w}_{0})}^{1/2}={\sigma }_{\mathrm{fr}}^{*}+{K}_{\mathrm{c}}^{\mathrm{m}}/{[\left(\pi /2\right)d]}^{1/2},$$
(13)

where \({w}_{0}\) is the size of the blocked slip band zone, which equals one-half of the grain size d. \({K}_{\mathrm{c}}^{\mathrm{m}}\) is the microscopic stress intensity factor at the tip of the slip band, and \({\sigma }_{\mathrm{fr}}^{*}\) is the friction stress (which equals the yield strength) for dislocation movement in the slip band. In this study, d and \({K}_{\mathrm{c}}^{\mathrm{m}}\) stay the same for the specimen with varying thickness. The corresponding change in friction stress is significant. As shown in Fig. 3b, a sudden reduction in the yield strength can be observed in the T = 0.2 mm specimen. Thus the threshold stress for crack growth decreases for the T = 0.2 mm specimen compared with that for the thicker specimen, which results in the decreasing fatigue life of T/d < 1 specimens in the HCF regime.

4.4 On the Evaluation of Fatigue Performance with Small Specimens

It is universally accepted that the Basquin equation is widely used to describe stress-controlled fatigue properties. The relevant results of our S–N curves fitted by the Basquin equation are shown in Fig. 4 and Table 1. Furthermore, to establish the conversion relationship between the small specimen and the standard specimen, the Basquin equations for the small specimen and the standard specimen are respectively expressed as:

$${\sigma }_{{a}_{1}}={\sigma }_{{f}_{1}}^{{\prime}}{(2{N}_{{f}_{1}})}^{{b}_{1}},$$
(14)
$${\sigma }_{{a}_{2}}={\sigma }_{{f}_{2}}^{{\prime}}{(2{N}_{{f}_{2}})}^{{b}_{2}}.$$
(15)

When two types of specimens are under the same stress amplitude, an equation between \({N}_{{f}_{1}}\) and \({N}_{{f}_{2}}\) can be obtained by combining Eqs. (14) and (15):

$${N}_{{f}_{2}}=1/2{\cdot \left({\sigma }_{{f}_{1}}^{{\prime}}/{\sigma }_{{f}_{2}}^{{\prime}}\right)}^{1/{b}_{2}}\cdot {\left(2{N}_{f1}\right)}^{{b}_{1}/{b}_{2}}.$$
(16)

According to Eq. (16), the fatigue life of the standard specimen can be determined when the fatigue parameters of both two types of specimens and the fatigue life of the small specimen are known. Here, 1/\({b}_{2}\) can be rewritten as \({(b}_{1}/{b}_{2})\cdot (1/{b}_{1})\). In other words, it is easy to evaluate the fatigue performance of standard specimens using small specimens when the values of \({\sigma }_{f1}^{^{\prime}}/{\sigma }_{f2}^{^{\prime}}\) and \({b}_{1}/{b}_{2}\) are certain.

Taking the fatigue data of pure Cu in this experiment as an example, the values of b between the small specimen and the standard specimen do not differ much, the value of \({b}_{1}/{b}_{2}\) is approximately 1. And the value of \({\sigma }_{f1}^{^{\prime}}/{\sigma }_{f2}^{^{\prime}}\) is approximate to the ratio of true tensile strengths of the small specimen to the standard specimen. Then, the fatigue life of the pure Cu specimen can be gotten by tensile tests of two types of specimens and fatigue tests of small specimens.

For Ti alloy reported by Zhang et al. [29], the variation trend of \({\sigma }_{\mathrm{f}}^{^{\prime}}\) is consistent with that of the tensile strength, but the value of b increases when the specimen thickness reduces to 0.1–0.2 mm. The local orientation of the α/β lamellae may lead to the high fatigue cracking resistance of small specimens, which is a possible reason for the change of the b value. The value of \({b}_{1}/{b}_{2}\) is about 0.33. As for CA6NM martensite stainless steel [28], the fatigue life of the standard specimen can be evaluated by the T = 0.23 mm specimen because the fatigue data of the two types of specimens are located in the same dispersion zone. As the CA6NM specimen thickness continues to decrease below 0.1 mm, the correspondence between \({\sigma }_{\mathrm{f}}^{^{\prime}}\) and tensile strength no longer applied. The small specimen is not suitable for the fatigue life evaluation in this case. Equation (16) has the potential for fatigue life prediction. The value of \({\sigma }_{f1}^{^{\prime}}/{\sigma }_{f2}^{^{\prime}}\) in the equation can be determined by tensile tests when T > 0.2 mm. However, the value of \({b}_{1}/{b}_{2}\) seems to be strongly influenced by the microstructure, and this influence needs to be explored in future research work. Furthermore, with the ever-increasing popularity of small specimens in quickly evaluating and monitoring the key nuclear components in-service, the present work is expected to provide a potential strategy for quick evaluation of fatigue performance using small specimens.

5 Conclusions

The tensile and fatigue behaviors of pure Cu were investigated by using small specimens with thicknesses ranging from 3 to 0.2 mm. The findings are summarized below.

  1. 1.

    The elongation of the T = 0.2 mm specimen is lower than the T = 1.5 mm and T = 3 mm specimens due to the decrease in the strain hardening rate. Both the free surface effect and the lack of contribution of grain boundaries lead to such a decrease in the strain hardening rate.

  2. 2.

    A theoretical formula for calculating the ratio of the surface grain layer thickness to the grain size is established. Combining this formula, the theoretical model proposed to predict yield strength could be further modified.

  3. 3.

    The decreasing fatigue lifetime of T = 0.2 mm specimens in the LCF regime is attributed to the loss of the GB resistance to crack growth. The threshold stress for crack growth decreases for the T = 0.2 mm specimen compared with that for the thicker specimens, leading to the decreasing fatigue life of the T = 0.2 mm specimens in the HCF regime.

  4. 4.

    The Basquin equation-based model is proposed as a potential way for fatigue life prediction by using small specimens. Nevertheless, the relationship between the relevant parameters in the model and the microstructure of the specimen remains to be further explored.