The batch of specimens by different annealing temperature and dwelling times generates different grain sizes, which can further affect material mechanical properties, such as the yield strength, the ultimate tensile stress and the elongation. The true stress–strain curves of the specimens with different grain sizes are shown in Figure 2. The values of true stress *σ*_{t} and true strain *ε*_{t} are calculated by the conventional empirical formulation which is expressed as ${\mathrm{\sigma}}_{\mathrm{t}}={\mathrm{\sigma}}_{\mathrm{e}}\left(1+{\mathrm{\epsilon}}_{\mathrm{e}}\right),\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}{\mathrm{\epsilon}}_{\mathrm{t}}=\mathrm{l}\mathrm{n}\left(1+{\mathrm{\epsilon}}_{\mathrm{e}}\right)$, where *σ*_{e} and *ε*_{e} are the engineering stress and engineering strain. It can be seen that the yield strength and flow stress for a given strain decrease with the increase of grain size, while the effect of grain size on elastic modulus is negligible. Figure 3 shows the variations of fracture stress and fracture strain obtained from the true stress–strain curves with grain size. It is found that fracture stress and fracture strain decrease with grain size.

Figure 2: The true stress–strain curves of the specimens with different grain sizes.

Figure 3: The variation of fracture stress and fracture strain with grain size (*d*).

The grain size dependence of flow stress can be modeled with the well-known H–P relationship denoted as eq. (1):
$\mathrm{\sigma}={\mathrm{\sigma}}_{\mathrm{h}\mathrm{p}}+{k}_{\mathrm{h}\mathrm{p}}{d}^{-1/2}$(1)

where *σ*_{hp} and *k*_{hp} are material constants that depend on strain. The H–P relationship is developed based on the pile-up of slip bands at grain boundary. It predicts well for flow stress changing with the inverse square root of grain size when the strain is less than a few percent, which is shown in Figure 4 and is also verified by quantities of researches via a variety of experiments, such as uniaxial tensile [4, 14], bending test [4, 5], extrusion test [12] and upsetting test [18]. It can been seen from Figure 4 that each set of experimental data can fit well with a dash line by the H–P relationship between true stress and the inverse square root of grain size for the low strain levels of 0.025, 0.05, 0.1, 0.15, which in depth indicates the validity of the H–P relationship. However, the fitting precision for large strain levels, such as 0.325, 0.35, is severely poor. The higher the strain level raises, the greater the deviation between the experimental data for a given strain, thus *d*^{–1/2} dependence of fracture stress should not be expected.

Figure 4: Hall–Petch curves of the test specimens for different strain levels.

Based on dislocation density theory, the well-known Bailey–Hirsch equation [27] expresses that the flow stress has a linear relation with the total dislocation density (*ρ*_{D}), which is expressed as eq. (2):
$\mathrm{\sigma}={\mathrm{\sigma}}_{0}+\mathrm{\alpha}\mathrm{\mu}b\sqrt{{\mathrm{\rho}}_{D}}={\mathrm{\sigma}}_{0}+\mathrm{\alpha}\mathrm{\mu}b\sqrt{{\mathrm{\rho}}^{\mathrm{s}}+{\mathrm{\rho}}^{\mathrm{g}}}$(2)

where *σ* and *σ*_{0} represent the flow stress and friction stress. *α*, *μ* and *b* are the constant, shear modulus and Burgers vector, respectively. Furthermore, the total dislocation density is composed of the statistically stored dislocation density (*ρ*^{s}) and the geometrically necessary dislocation density (*ρ*^{g}). The statistically stored dislocation density is generated by accumulation in grain interior and results in work hardening. The geometrically necessary dislocation density aims to maintain the geometric continuity in the vicinity of grain boundaries. In addition, combined with the above study and some hypotheses, a dislocation-based modeling concerning fracture stress and fracture strain changing with grain size was put forward by Fu [14] and described as eqs (3) and (4):
${\mathrm{\sigma}}_{\mathrm{c}}={\mathrm{\sigma}}_{\mathrm{\rho}}+\mathrm{\alpha}G\sqrt{{\mathrm{\epsilon}}_{\mathrm{c}}}\sqrt{\frac{{C}_{1}b}{L}+{C}_{2}b\left(g+\frac{hd}{t}\right)}$(3)
${\mathrm{\epsilon}}_{\mathrm{c}}=\frac{{b}^{2}}{\left(g+\frac{f}{t}d\right){\displaystyle \left(\frac{{C}_{1}b}{L}+{C}_{2}b\left(g+\frac{h}{t}d\right)\right)}}$(4)

where *σ*_{c} and *ε*_{c} are fracture stress and fracture strain. *T* and *d* are the thickness and grain size of the specimens. *G*, *b* and *L* are the shear modulus, Burgers vector and slip length, respectively. *σ*_{ρ}, *α*, *C*_{1}, *C*_{2}, *h* and *f* are constants. In order to analyze the relation among them directly, thus the above two equations are simplified using the mathematical method. When grain size is in micro-scale, supposing that *d* is considered to be negligible compared with *d*^{2}; hence eq. (4) can be simplified which is expressed as the following eq. (5):
${\mathrm{\epsilon}}_{\mathrm{c}}={\mathrm{\epsilon}}_{0}+{l}_{\mathrm{c}}{d}^{-1}$(5)

where *ε*_{0} and *l*_{c} are constants. Substituting the relationship into eq. (3) and neglecting some insignificant constants, it can be rearranged and simplified as eq. (6):
${\mathrm{\sigma}}_{\mathrm{c}}={\mathrm{\sigma}}_{\mathrm{\rho}}+{k}_{\mathrm{c}}{d}^{-1}$(6)

Based on the above analysis and deduction, it is revealed that the fracture stress and fracture strain have a proportional relation with the inverse of grain size. The relationship is consistent with the previous proposed equations proposed by Fan [20], which thinks *d*^{–1} dependence is more appropriate to stress with large strain instead of *d*^{–1/2}. Therefore, it is reasonable for the previous proposed hypothesis and simplification, thus the relationships that the fracture stress and fracture strain associated with the inverse of grain size can be established.

Figure 5 shows the changing relation of fracture stress and fracture strain versus *d*^{–1}. It is found that fracture stress and fracture strain have the same rule of variation with the inverse of grain size. A critical ratio of specimen thickness to grain size (*t*/*d* is 4.77) exists to result in the fitting curves dividing two distinct segmentations for fracture stress and fracture strain changing with *d*^{–1}. When *t*/*d* is less than 4.77 or grain size is relatively larger, fracture stress and fracture strain increase rapidly with the inverse of grain size, which the corresponding coefficients *σ*_{ρ} and *k*_{c} are 622.03 and 35,135.17 MPa μm for fracture stress, and *ε*_{ρ} and *l*_{c} are 0.3263 and 8.048 μm for fracture strain. In other words, fracture stress and fracture strain decrease rapidly with the increasing grain size under the condition of grain size more than 62.85 μm. Nevertheless, when *t*/*d* is more than 4.77 or grain size is relatively less, fracture stress increases slowly with the increase of *d*^{–1} compared with the former, which *σ*_{ρ} increases to 1073.19 MPa and *k*_{c} decreases to 5,918.43 MPa μm. The *k*_{c} value for smaller grain size is approximately one-sixth of that for larger grain size. It indicates that the variation of *k*_{c} is severely big and *k*_{c} value for smaller grain size is far less than compared with that for larger grain size. The less the *k*_{c} value for smaller grain size is, the less the variation of the effect of grain size on fracture stress is. Due to the relatively less *k*_{c} value, the variation of fracture stress is little in the case of smaller grain size. However, fracture strain increases inconspicuously with *d*^{–1} and the value of *l*_{c} decreases to 0.2309. In a word, fracture stress begins to decrease slowly with the increase of grain size, while grain size is more than 62.85 μm, subsequently decreasing rapidly. Furthermore, fracture strain begins to remain constant basically, which is different from fracture stress. However, when grain size is more than the critical value, fracture strain has the same changed rule with fracture stress, which decreases rapidly with the increasing grain size.

Figure 5: The variation of fracture stress (a) and fracture strain (b) with *d*^{–1}.

For the experimental specimens, the crystal microstructure of the specimens undergoes a transition from polycrystals, multicrystals to single crystal with increase of grain size, as reported by Hansen [28]. As to Figure 5, the polycrystals, multi-crystals and single-crystal are located in the right segmentations with low slope, the region around the critical *t*/*d* ratio, the left segmentations of high slope, respectively. Polycrystals with finer grain size have more grain boundary than multi-crystals and single crystal. It is well known that grain boundary impedes the motion of dislocations in the process of plastic deformation. The volume fraction of grain boundary gradually decreases with the increase of grain size. As reported by Fu et al. [14], when *t*/*d* is less than the critical ratio of 4–5, the volume fraction grain boundary increases rapidly with the increase of the ratio of specimen thickness to grain size. While the *t*/*d* ratio is more than the critical ratio, the volume fraction of grain boundary becomes to increase slowly. The change rule of fracture stress and fracture strain is consistent with that of the volume fraction of grain boundary and can be well interpreted by grain boundary theory.

Some other experimental data are extracted from prior literatures of Fu et al. [14], Klein et al. [21] and Diehl et al. [29] about the effect of specimen thickness and grain size on fracture behavior. The *t*/*d* is employed as the variable of *X*-axis with the purpose of facilitating comparing the experimental results with other data. To make these different data can be plotted in a picture, the data of fracture stress and fracture strain denoted as the variable of *Y*-axis are normalized, which are shown in Figure 6. It can be seen that fracture stress and fracture strain for different materials basically have the same varying rule that a critical *t*/*d* ratio exists to make the fitting curves divided into two distinct segmentations, which further verifies the validation of the foregoing analysis. For fracture stress, the *k*_{c} value for Inconel 718 is approximately equal to that for other material. It suggests that the variation of grain size effect on fracture stress is suitable to different material sheets. Nevertheless, the *l*_{c} value for Inconel 718 is quite different and less compared with that for other materials in the case of more than the critical *t*/*d* ratio. It may be because that nickel–chromium alloy is harder and more difficult deformation than copper alloy, resulting in that the variation of grain size has a little effect on fracture strain. The precision for fracture stress agreeing with the varying rule is higher than that for fracture strain. However, some of the data point is deviated from the fitting straight line, which is attributed to a series of diverse reasons, such as the preparation of the specimens, inaccurate operation and the error of measurement. In addition, the critical *t*/*d* ratio of 4–5 for Inconel 718 is higher than that of 2–3 for pure copper with different brands and heat treatment; the comprehensive reasons are probably caused by the variety of thickness, grain size and stacking fault energy. While the current data is not enough for analyzing the comprehensive effect of specimen thickness, grain size and the *t/d* ratio on fracture stress and fracture strain in detail, it is necessary to study in-depth the dependence of the *t/d* ratio on fracture stress and strain by more experimental results.

Figure 6: Normalized fracture stress (a) and fracture strain (b) versus *t*/*d*.

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