In the last decades, products miniaturization has been a trend of production and manufacturing with the development of science and processing technology, due to the increasing demand on microparts and microproducts. Nevertheless, when the part dimension is scaled down from macro- to micro-scale, the mechanism properties, such as yield strength and flow stress and fracture behavior, cannot be deduced from macroscopic theory due to size effect. Thus, it is necessary to understand in-depth deformation mechanism and fracture behavior in micro-forming processes.
As for grain size dependence of material mechanism, such as deformation behaviors, fracture stress and strain, ductile fracture toughness, etc., numerous researches have been conducted to explore micro-scale behavior of used metal and alloy materials. The grain size dependence of strength of polycrystalline metal and alloy was firstly described by the empirical Hall–Petch (H–P) relationship [1, 2], namely flow stress had a positive proportion to the inverse square root of grain size. Armstrong  first proposed that the effect between grain size and specimen size on material properties had different bases. They pointed to a critical (t/d)2 value of more than 20 for an equiaxed specimen cross section. Raulea et al. [4, 5] found that the yield strength decreased with the decreasing number of grains across specimen thickness in the polycrystalline materials. The effect of reducing specimen thickness on yield strength was the same as that of increasing grain size. While grain size was over specimen thickness, the repeatability of experimental results reduced and the yield strength and average load of bending increased with grain size. Geiger et al. [6–8] conducted the bending and upsetting experiments to study the size effect in micro-scale plastic deformation. They proposed a numerical mesoscopic model based on the theory of metal physics to simulate micro-forming process. The results of simulation were consistent with the experiment results in cylinder compression tests. Keller et al. [9–11] carried out the research about grain size effect on mechanism properties of Ni polycrystalline alloy, such as working hardening, flow stress decrease and dislocation pattern. They found that a modification of the hardening mechanisms was induced for low ratio of specimen thickness to grain size and the coefficients of the H–P relationship were modified. Chan et al. [12–14] investigated material deformation behavior in micro-forming through various tests. They found that the flow stress–strain curve obtained from micro-compression test was not applicable in modeling of micro-extrusion. The effect of specimen thickness and grain size of pure copper on fracture stress and strain was investigated and a comprehensive relation between fracture stress and strain and the ratio of specimen thickness to grain size was formulated based on dislocation density theory. Zhao et al. [15, 16] found that fracture strain and strain hardening rate increased with the increasing specimen thickness and decreasing gauge length via the tensile test of ultrafine and coarse-grained copper foils. Furushima et al.  focused on the ductile fracture and free surface roughening behaviors for pure copper foils and found that fracture strain of metal foils is extremely low in spite of a large work-hardening exponent compared with that of metal sheets. Ran et al.  investigated the influence of size effect on the ductile fracture in micro-scaled plastic deformation. They argued that the step-like elongated shear dimples exist in the two fracture surfaces of the broken flanged part, and the dimple size on the transgranular fracture surfaces was affected by different scaling factors. Curry and Knott  conducted the experiments of mild steel to investigate the relationship between fracture toughness and material microstructure in the cleavage fracture. They concluded the fracture toughness and cleavage fracture stress were determined as a function of grain size. Fan  described the grain size dependence of ductile fracture toughness in terms of a composite model with the consideration of grain boundary and grain interior based on dislocation density theory. Through fitting the data of both d–1 and d–1/2 dependence on ductile fracture toughness by the least square method and comparing the linear regression coefficients, it was concluded that the regression coefficients fitted by d–1 and d–1/2 were relatively close. Klein et al.  studied the size effect on the fracture behavior of thin metallic foils. They detected that the effect of grain size on fracture strain was explained on the basis of texture differences, the number of activated gliding systems as a dependence on the ratio of grain size to foil thickness. Hadrboletz et al.  investigated the fatigue and fracture behavior of foils with varying thickness. They concluded that a reduction of the number of activated slip systems was corroborated by the decrease of fracture strain, and the results from Mo and Al foils indicated that similar considerations might be applied to explain the size effect of metallic foils. Kumar et al.  investigated the effect of material microstructure and grain size on the fracture toughness. It was concluded that low-temperature austenitization followed by quenching resulted in better mechanical properties compared to high-temperature austenitization. Meng et al. [24–26] investigated the material deformation behavior and ductile fracture in progressive micro-forming process and believed that the material fracture behavior is affected by free surface roughening in addition to size effect.
Although the investigations about grain size effect on mechanical behavior were conducted by the researchers over the past decades, less attention has been paid to the studies of grain size effect on deformation mechanism and fracture behavior of difficult-to-deformation material, such as nickel-based alloy. In the paper, the uniaxial tensile tests of Inconel 718 alloy sheet were carried out to study fracture behavior. The experimental results were analyzed and compared. In addition, fracture morphologies were observed by the scanning electron microscope (SEM).
Inconel 718 alloy is selected as the test material due to its good comprehensive properties of high strength, resistant to corrosion and oxidation and good welding performance. The material is widely applied in aeronautics and astronautics, nuclear power and oil industry. The chemical concentration in mass % of the Inconel 718 polycrystalline sheet was shown in Table 1. The specimens with the thickness of 300 μm (t) were cut to dog bone-shaped specimens with the gauge length of 31 mm and the width of 6 mm by laser-cutting machine. The specimens were annealed at temperature ranging from 950 to 1,150 °C for 25 min to obtain different grain sizes. Through grinding by different sandpapers from coarse to fine, mechanical polishing, etching chemically with a solution of 5 ml of HCl and 5 ml of H2O2 for 5–10 s, the microstructures of the heat-treated specimens were observed by the optical microscope. The average grain sizes were estimated by the intercept method, as shown in Table 2. The higher the heat treatment temperature is, the larger the obtained grain sizes become. Figure 1 shows the microstructures of some of the test specimens with different grain sizes, which indicates an obvious variation across specimen thickness.
The tensile tests were conducted on an MTS (MTS: Mechanical Testing & Simulation, the MTS880 machine is attached to MTS System Corporation) testing machine at ambient temperature with the load cell capacities of 5 kN. All the specimens were stretched at a low tensile speed of 2 mm/min. The elongation of the specimens was measured by an extensometer under the safe critical strain of 0.1. When the strain of the specimen was over the critical value, removing the extensometer, then the elongation was estimated by the mean value of the displacement to the gauge length. Furthermore, to observe grain size effect on fracture characteristics, the fracture morphologies were observed and recorded by the SEM.
Results and discussion
Effect of grain size on fracture stress and strain
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 σt and true strain εt are calculated by the conventional empirical formulation which is expressed as , where σe and εe 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.
The grain size dependence of flow stress can be modeled with the well-known H–P relationship denoted as eq. (1): (1)
where σhp and khp 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  and upsetting test . 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.
Based on dislocation density theory, the well-known Bailey–Hirsch equation  expresses that the flow stress has a linear relation with the total dislocation density (ρD), which is expressed as eq. (2): (2)
where σ and σ0 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 (ρs) and the geometrically necessary dislocation density (ρg). 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  and described as eqs (3) and (4): (3) (4)
where σc and εc 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. σρ, α, C1, C2, 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 d2; hence eq. (4) can be simplified which is expressed as the following eq. (5): (5)
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 , 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 σρ and kc are 622.03 and 35,135.17 MPa μm for fracture stress, and ερ and lc 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 σρ increases to 1073.19 MPa and kc decreases to 5,918.43 MPa μm. The kc value for smaller grain size is approximately one-sixth of that for larger grain size. It indicates that the variation of kc is severely big and kc value for smaller grain size is far less than compared with that for larger grain size. The less the kc 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 kc 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 lc 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.
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 . 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. , 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. , Klein et al.  and Diehl et al.  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 kc 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 lc 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.
Effect of grain size on fracture morphology
The variation of grain size across specimen thickness has effect not only on fracture stress and fracture strain in sheet metal forming, but also on microscopic characteristics. To understand in depth grain size effect on fracture mechanisms in microscopic scale, the morphologies of fracture surfaces should be observed and researched by the SEM.
Figure 7 shows the fracture morphologies of some of the deformed specimens with different grain sizes; it is revealed a typical ductile fracture model of the test specimens, which contains a mass of dimples and various sizes of microvoids. As shown in Figure 7(a), the fracture surface consists of large density of tearing edges. In the case of small grain size, the volume fraction of grain boundary is relatively larger, and the deformation process tends to generate more small dimples, rather than enlarge the previously-generated small ones. It leaded to quantities of small dimples and microvoids appeared on the fracture surfaces. The serpentine sliding features that appeared on the walls of dimples are due to the new sliding process on the free surface of dimples. It indicates that the specimens still undergo a certain degree of plastic deformation after the formation of dimples. Figure 7(b) and (c) shows that some big dimples and microvoids that are covered with several small dimples and microvoids distributed in the bottom of them appear on the fracture surfaces. It is the reason that the coalescence of dimples and microvoids occurs during the process of plastic deformation. The number of microvoids becomes relatively less compared with that of the former, while the mean sizes of them become larger. It is caused by the nucleated microvoids occurring to the growth and coalescence, which in turn make them larger. In addition, Figure 7(d) shows that the less number of microvoids and the larger sizes of microvoids appear on the fracture surface of the deformed specimens with grain sizes of 126.60 µm, which also contains apparently sliding bands.
As a result, it can be seen that the number of dimples and microvoids decreases and the mean size of microvoids becomes larger with the increase of grain size. The grain boundary acts as an obstacle to dislocation motion in the deformation process, thus the stress concentrates on the grain boundary zone, resulting in nucleation, growth, distortion and coalescence of microvoids. In addition, the content of the second δ phase within the specimen decreases and grain size increases with the increase of annealing temperature. It is because δ phase starts to melt when the temperature is over 980 °C and grain occurs to recrystallization and growth. The microvoids tend to nucleate around the δ phase, hence δ phase have an important effect on the development of microvoids. With the increase of grain size, the content of δ phase and volume fraction of grain boundary become less, thus the number of microvoids is decreased. Due to the less grain boundary and δ phase in the case of larger grain size, small dimples and microvoids start to grow and coalesce, rather than generating the new small one; therefore, the mean size of microvoids increases with grain size. With the continuous deformation and increasing stress, the specimens will occur to crack on the fracture region when the localized stress reaches the critical value of the crack tips.
The conventional theory and knowledge of material mechanism in macro-scale are not suitable for micro-forming due to size effect. As the investigation of effects of grain size on fracture behaviors of Inconel 718 is relatively less, the tests of axis tensile of the Inconel 718 sheet with different grain size across the same thickness are conducted. Some conclusions can be summarized as follows:
(1) The variation of grain size has a significant effect on mechanical behavior in the test, such as yield strength, fracture stress and fracture strain. The experimental results for flow stress in the low strain levels have a satisfactory agreement with the curves fitted by the H–P relationship. Furthermore, a critical t/d ratio of 4.77 exists for fracture stress and fracture strain changing with grain size. When t/d is more than the critical ratio, fracture stress decreases slowly with the increase of grain size, while fracture strain basically remains constant and just decreases slightly. When t/d is less than the critical ratio, fracture stress and fracture strain decrease rapidly with the increasing grain size. The range of change is relatively large compared with the former. Fracture stress and fracture strain for Inconel 718 thin sheet and pure copper sheet have a satisfactory agreement with the segmented varying rule.
(2) The fracture feature of the specimens presents a typical ductile fracture model, containing a mass of dimples and various sizes of microvoids when the specimen contains a large amount of grains across specimen thickness. The number of dimples and microvoids on the fractured specimen surface decreases with the increase of grain size. Additionally, the mean size of microvoids becomes large with the increasing grain size
This work was supported by the graduate innovation fund YCSJ-03-2014-09 of Beihang University in 2014 and Beihang University of aeronautics and astronautics manufacturing engineering. The authors would be very grateful to PhD B. Meng for suggestions and comments.
 E.O. Hall, Proc. Phys. Soc., 64 (1951) 747–753. Google Scholar
 N.J. Petch, J. Iron Steel Inst., 174 (1953) 25–28. Google Scholar
 R.W. Armstrong, J. Mech. Phys. Sol., 9 (1961) 196–199. Google Scholar
 L.V. Raulea, A.M. Goijaerts and L.E. Govaet, J. Mater. Process. Technol., 115 (2001) 44–48. Google Scholar
 M. Geiger, F. Vollersten and R. Kals, CIRP Annu. Manufact. Technol., 45 (1996) 277–282. Google Scholar
 M. Geiger, M. Kleiner, R. Eckstein, N. Tiesler and U. Engel, CIRP Annu. Manufact. Technol., 50 (2001) 445–462.Google Scholar
 U. Engel and R. Eckstein, J. Mater. Proc. Technol., 125 (2002) 35–44. Google Scholar
 C. Keller and E. Hug, Mater. Lett., 62 (2008) 1718–1720. Google Scholar
 M. Rudloff, M. Risbet, C. Keller and E. Hug, Mater. Lett., 62 (2008) 3591–3593.Google Scholar
 C. Keller, E. Hug and D. Chateigner, Mater. Sci. Eng. A, 500 (2009) 207–215. Google Scholar
 W.L. Chan, M.W. Fu and J. Lu, Mater. Des., 32 (2011) 525–534. Google Scholar
 W.L. Chan, M.W. Fu and J. Lu, Mater. Des., 32 (2011) 198–206.Google Scholar
 M.W. Fu and W.L. Chan, Mater. Des., 32 (2011) 4738–4746. Google Scholar
 Y.H. Zhao, Y.Z. Guo, Q. Wei and A.M. Dangelewice, Scripta Mater., 59 (2008) 627–630. Google Scholar
 Y.H. Zhao, Y.Z. Guo, Q. Wei and A.M. Dangelewice, Mater. Sci. Eng. A, 525 (2009) 68–77. Google Scholar
 T. Furushima, H. Tsunezaki, K. Manabe and S. Alexsandrovet, Int. J. Mach. Tool. Manufact., 76 (2014) 34–48. Google Scholar
 J.Q. Ran, M.W. Fu and W.L. Chan, Int. J. Plast., 41 (2013) 65–81. Google Scholar
 D.A. Curry and J.F. Knott, Met. Soc., 10 (1976) 1–6. Google Scholar
 Z.Y. Fan, Mater. Sci. Eng. A, 191 (1995) 73–83. Google Scholar
 M. Klein, A. Hadrboletz, B. Weiss and G. Khatibi, Mater. Sci. Eng. A, 319 (2001) 924–928. Google Scholar
 A. Hadrboletz, B. Weiss and G. Khatibi, Int. J. Fract., 109 (2001) 69–89. Google Scholar
 A.S. Kumar, B.R. Kumar, G.L. Datta and V.R. Ranganath, Mater. Sci. Eng. A, 527 (2010) 954–960. Google Scholar
 B. Meng, M.W. Fu, C.M. Fu and J.L. Wang, Int. J. Mech. Sci., 93 (2015) 191–203. Google Scholar
 B. Meng, M.W. Fu, C.M. Fu and K.S. Chen, Mater. Des., 83 (2015) 14–25.Google Scholar
 B. Meng and M.W. Fu, Mater. Des., 83 (2015) 400–412. Google Scholar
 M.F. Ashby, Philos. Mag., 21 (1970) 399. Google Scholar
 N. Hansen, Acta Metall. Sin., 25 (1977) 863–869. Google Scholar
 A. Diehl, U. Engel and M. Geiger, Int. J. Adv. Manufact. Technol., 47 (2010) 53–61. Google Scholar
About the article
Published Online: 2016-01-26
Published in Print: 2016-11-01