Size effect on deformation behavior and ductile fracture in microforming of pure copper sheets considering free surface roughening

Materials & Design 83 (2015) 400–412

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Size effect on deformation behavior and ductile fracture in microforming of pure copper sheets considering free surface roughening B. Meng, M.W. Fu ⇑ Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

a r t i c l e

i n f o

Article history: Received 2 April 2015 Revised 29 May 2015 Accepted 6 June 2015 Available online 19 June 2015 Keywords: Micro-scaled plastic deformation Flow behavior Ductile fracture Size effect Free surface roughening

a b s t r a c t In meso/micro-scaled plastic deformation, material deformation and ductile fracture are quite different from those in macro-scale. The roughness of the free surfaces of workpiece increases with deformation and the decrease of grain number in the sample thickness direction, leading to the nonuniformity of specimen thickness. The so-called size effect and free surface roughening may in turn affect the deformation behavior, ductility and fracture morphology of the samples. To explore the coupled effect of workpiece geometry and grain size on material flow behavior in meso/micro-scaled plastic deformation, uniaxial tensile test of pure copper sheets with different thicknesses and comparable microstructure was performed. The experimental results reveal that the material flow stress, fracture stress and strain, and the number of microvoids on fracture surface are getting smaller with the decreasing ratio of specimen thickness to grain size. In addition, the modified Swift’s equation and the corrected uniform strain are closer to the experimental ones considering the thickness nonuniform coefficient induced by the free surface roughening. Furthermore, the observation of fracture morphologies confirms that the local deformation caused by the free surface roughening leads to strain localization and a decreased fracture strain when there are only a few grains involved in plastic deformation. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction With the ubiquitous trend of product miniaturization, meso/micro-scaled parts have been widely used in many industry clusters such as electronics, automobiles, healthcare, aerospace, and biomedicine [1–3]. Particularly, the copper alloy microparts are widely used in electronic and communication industries due to the excellent conductivity and good ductility of copper alloy. To fabricate copper alloy microparts efficiently, microforming is used due to its high productivity, low cost, and the good mechanical properties of microformed parts [4–6]. However, when the workpiece geometry is scaled down from macro-scale to meso/micro-scale, the material deformation behavior, fracture toughness and surface roughening may also change accordingly. The deformation behavior and ductile fracture can further affect the forming performance and defect formation due to the fewer grains in the deformation zone and the random orientation and property distributions of individual grains. Thus, understanding of the size effect and its affected material deformation behavior

⇑ Corresponding author. E-mail address: [email protected] (M.W. Fu). http://dx.doi.org/10.1016/j.matdes.2015.06.067 0264-1275/Ó 2015 Elsevier Ltd. All rights reserved.

is crucial to product quality and defect avoidance in meso/micro-scaled sheet metal forming. To explore the size-dependent material flow and fracture behavior, many attempts have been carried out. Fan [7] investigated the grain size effect on the ductile fracture toughness of polycrystalline metals and alloys, and proposed a semi-empirical equation to describe the dependence of fracture toughness on grain size. Michel and Picart [8] conducted the tensile and hydraulic bulging tests to evaluate the thickness-dependent flow stress of brass sheets, and established a constitutive model by taking into account the size effect. Kim et al. [9] studied the feature/specimen size effect and quantified it via relating the size effect to the fundamental properties of single and polycrystalline deformation. Sinclair et al. [10] explored the grain size dependent work-hardening of copper polycrystals. They found that the grain size effect is related to the interaction between dislocation and grain boundary, and this effect on work hardening disappears at a large strain owing to the dynamic recovery. Peng et al. [11,12] developed different material models considering the size effect in microforming process. They believed that the flow stress in meso/micro-scale falls in between those of single crystal and polycrystal. In addition, Molotnikov et al. [13,14] established a physically based constitutive model to represent the size effect on tensile strength considering both material microstructure and

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sheet thickness. They studied the effect of sheet thickness in micro deep drawing process of coarse-grained and ultrafine-grained copper sheets. Vollertsen [15] summarized the size effect on flow stress, tribology, sheet formability and forming processes in meso/micro-scaled plastic deformation. Chan and Fu [16,17] examined the geometry and grain size effects on the deformation behavior and ductile fracture of copper sheet via uniaxial tensile test. They found that flow stress, fracture stress and strain, and the number of microvoids decrease with the ratio of sheet thickness (t0) to grain size (d0). Chen and Ngan [18] investigated the coupled effect of grain and specimen sizes on the tensile strength of Ag wires. They revealed that the strengthening effect depends on the specimen shape as the ratio of t0/d0 decreases from about 3. Lu et al. [19] presented a new material model for micro-scaled plastic deformation based on the grained heterogeneity and specimen dimension, and conducted finite element (FE) simulation to validate this model by using Voronoi tessellation to describe the polycrystalline microstructure. Ran et al. [20,21] reported that the ductile fracture on the flanged surface of the flanged cylindrical parts is easier to occur in macro-scale flanged upsetting process. Furthermore, they proposed a hybrid model to characterize the size effect on fracture surface morphology and fracture formation by considering the phase composition and distribution of material. Wang et al. [22] performed the uniaxial compression test of pure nickel polycrystals with a constant thickness and various grain sizes to investigate the size effect on flow stress. They found that the conventional Hall–Petch law is not applicable when the specimens have only a few grains across the thickness direction. Xu et al. [23,24] explored the effect of geometry and grain sizes on the forming limit of sheet metals, and developed a coupled damage model based on the Gurson–Tvergaard–Needleman and the Thomason models via considering size effect on void evolution. Meng et al. [25] studied the grain size effect on deformation behavior, dimensional accuracy, defect formation and surface quality in progressive microforming by directly using sheet metals. Meanwhile, some studies are focused on size-dependent free surface roughening phenomenon and its affected deformation behavior and ductile fracture. Mahmudi and Mehdizadeh [26] investigated the surface roughening behavior of brass sheets under uniaxial and equi-biaxial stress states. They found the roughness increment is proportional to the applied strain and the grain size irrespective of stress state. Wittridge and Knutsen [27] reported that the ribbing profile of sheet metal during uniaxial deformation is attributed to the grain anisotropy roughening, which further produces strain localization. Simons et al. [28] systematically studied the size effect on the tensile properties of copper sheets, and emphasized that the increase in surface roughness is moderate for as-received samples and very apparent for the annealed specimens. This is because there are only a few grains across the thickness of the annealed sample. These grains are less constrained and can rotate out of their initial position by gliding along grain boundaries, leading to a remarkable increase of surface roughness. Furushima et al. [29–31] conducted a series of studies to explore the effect of free surface roughening on the ductile fracture of copper sheet and foil via experiment, modeling, and FE simulation. They found that fracture strain and the number of dimple on the fracture surface decrease for the metal foil, and the ductile fracture criteria are invalid in prediction of the fracture of metal foil with the thickness of 50 lm. Suh et al. [32] studied the combined effect of sample thickness and surface roughness on the uniaxial tensile property of aluminum sheet, and found the effect of surface roughness increases with the decrease of sheet thickness. Romanova et al. [33] numerically investigated the effects of grain shape, loading state and boundary condition on the meso-scaled surface roughening of polycrystalline aluminum alloy under uniaxial tension deformation. They pointed out that the surface ridge and

401

valley are caused by the developed micro-scale normal and shear stresses. Abe [34] proposed a surface roughening model to describe the plastic deformation of polycrystalline metal via introducing the anisotropic coefficient of individual grain, and analyzed the formability of sheet metal under biaxial stretching based on the free surface roughening phenomenon caused by different grain orientations. Khoddam et al. [35] estimated the surface wrinkling during the uniaxial straining of TWIP steel with an initial grain size of 160 lm. Shimizu et al. [36] investigated the surface roughening at the corner of a drawn cup made of ultrathin metal foil, and indicated that the thickness nonuniformity exacerbates with the reducing thickness, which in turn affects the instability and limiting strain of metal foils. Based on the above brief review, it can be seen that the specimen and grain size effects on the micro-scaled plastic deformation behavior and ductile fracture of sheet metal have been sufficiently investigated. However, there is still a lack of knowledge on the comprehensive influence of the geometry size and material microstructure on deformation behavior, ductile fracture and free surface roughening. Actually, the size-dependent free surface roughening could in turn affect the material deformation behavior, ductile fracture, and further the dimensional accuracy and surface quality of the microformed parts. Therefore, this effect should be also considered in addition to the size effect. This research is thus aimed at addressing the former issues and exploring how the geometry and grain size effects affect flow behavior, fracture stress and strain, surface roughness, and fracture morphologies via uniaxial tensile tests. Meanwhile, the surface roughness of the deformed specimens under different material conditions was measured and characterized. The factor of free surface roughening is introduced into the constitutive model and the plastic instability criterion to explore its effect on material flow stress, uniform strain, and fracture strain.

2. Experimental details 2.1. Sample preparation Pure copper sheets with the thicknesses of 0.2, 0.4 and 0.6 mm were used as the testing material in this research. The copper sheets with the thicknesses of 0.2 and 0.4 mm were heat treated to release the internal residual stress after rolling, whereas the sheets with the thickness of 0.6 mm were not heat treated. The test specimen was cut along the rolling direction by electrical discharge machining method. The dimension of the test sample and the surface roughness requirement are presented in Fig. 1(a). After cutting, the samples were ready for tensile test, which are termed as the ‘‘as-received’’ in this paper. Meanwhile, some machined samples were heat treated to explore the effect of material microstructure on deformation behavior and free surface roughening phenomenon. Since the recrystallization temperature of pure copper is about 380 °C, the annealing temperatures and dwelling times are 500 °C and 2 h, 600 °C and 2 h, and 750 °C and 3 h in the vacuum environment, respectively. These samples are thus identified as the ‘‘annealed’’ in this research. The flow chart of sample preparation process is presented in Fig. 1(b). The metallographic observation was conducted on the microscope (Epiphot 200, Nikon) after the specimens were etched with a solution of 5 g of FeCl3, 15 ml of HCl and 85 ml of H2O for 10 s. The microstructure along the thickness direction and the average grain size under different thicknesses and heat treatment conditions are shown in Fig. 2 and Table 1, respectively. It is observed that the grain size increases with the annealing temperature. There are only 2–3 grains in the thickness of the samples with the thickness of 0.2 mm annealed at 750 °C. In addition, the grain distribution of the as-received

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Fig. 1. Preparation of the tensile sample: (a) Dimension of tensile sample and (b) flow chart of sample preparation process.

Fig. 2. Microstructures of the used copper sheets.

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Table 1 Average grain sizes of the testing material under different conditions. Thickness, t0 (lm)

Average grain size, d0 (lm) ± deviation (lm) Asreceived

500 °C annealed

600 °C annealed

750 °C annealed

200 400 600

4.5 ± 0.4 14.5 ± 0.7 9.7 ± 0.5

13.9 ± 1.0 36.9 ± 2.3 21.3 ± 2.3

30.8 ± 3.1 44.0 ± 2.5 23.8 ± 2.9

80.2 ± 8.7 67.7 ± 3.9 34.9 ± 4.0

samples with the thickness of 0.6 mm is uneven due to the prior rolling process. 2.2. Tensile test The tensile tests were carried out at room temperature in a programmable MTS testing machine. The elongation of the testing specimen was measured by an extensometer with the gauge length of 25 mm under a constant strain rate of 0.004 s1. The test samples were elongated until fracture. For each condition with a given average grain size (d0) and thickness (t0), five specimens were tested. Since the measurement of strain is inexact outside the gauge length, the experimental data was excluded from the subsequent analysis when the sample was fractured beyond the gauge range. The obtained true stress–strain curves under different conditions are shown in Fig. 3. It was found that the samples with a small grain size exhibit a relatively high yield stress and a large amount of work-hardening followed by a high fracture strain. However, the as-received sample with the thickness of 0.6 mm shows an extremely high flow stress and premature fracture caused by stress concentration during rolling process. After the tensile tests, the fracture surfaces were observed via scanning electron microscope (JSM-6490, Jeol). Furthermore, the cross-sectional morphologies of the specimens across the thickness direction were also observed. 3. Results and discussion 3.1. Size effect on free surface roughening In microforming, the roughening appearance of free surface is easy to occur when the deformation body contains several grains. The so-called free surface roughening phenomenon is resulted from the difference of hardness level and crystal orientation between the neighboring grains. The variation of thickness and surface irregularity can be attributed to the difference of Schmid factor of individual grains, which links the external applied stress to the resolved shear stress onto a specific slip system [37]. Moreover, each grain has its characteristic Schmid factor, and the grain with the lowest Schmid factor deforms last. The grain with high Schmid factor will be stretched easily without many hindrances, which causes local depression on the sample surface under tensile deformation. While in the region containing many hard grains with low Schmid factor, the opposite scenario occurs. The grain deforms only little and corresponds to the ridge on the sample surface. As a result, the discrepancy in the Schmid factor of neighboring grains leads to the accompanying roughening of the sample surface [38,39]. Fig. 4 shows the surface roughness of both the original and deformed samples along the tensile direction measured by the surface profiler (Tencor P10). The scanning length was measured as 1.2 mm. It is found from Fig. 4 that the surface roughness of the deformed sample increases significantly with the decrease of t0/d0, and the ribbed appearance occurs on the profile of the fractured sample. This is because the surface grains are less constrained and easier to deform on the free surface with a small ratio of t0/d0 [11,12,18,23,24,28]. Thus, free surface roughening phenomenon often occurs in

Fig. 3. True strain–stress curves of pure copper sheets with different thicknesses: (a) t0 = 0.2 mm, (b) t0 = 0.4 mm and (c) t0 = 0.6 mm.

meso/micro-scaled plastic deformation, which seriously affects the material formability, surface quality and dimensional accuracy of the fabricated microparts.

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the change in the average property throughout the specimen thickness. The local deformation aggravating with plastic deformation has a strong effect on the plastic instability, material flow behavior and ductile fracture of the specimens with only a few grains in the thickness direction. As mentioned, the free surface roughening can cause inhomogeneity from the deformation perspective and irregularity from the point of view of geometry and shape. Regarding the effect of such surface heterogeneity on the material formability in sheet metal forming, Marciniak and Kuczynski [40] proposed the so-called M–K model. Its fundamental assumption is that a long band of groove is initially located on the sheet surface, and eventually develops into a local neck. Likewise, the M–K model is applicable with respect to the surface imperfections stemming from the free surface roughening as the surface roughness develops to a certain degree. In this research, to quantitatively describe the size-dependent free surface roughening, the thickness irregularity n is defined by the following equation: Fig. 4. Free surface roughening behavior of copper sheets after uniaxial tension deformation.

To further explore the free surface roughening phenomenon, the cross-sectional profile of the deformed samples across thickness direction was observed and shown in Fig. 5. It can be observed that the sample profile becomes more fluctuant with the increase of grain size when the initial sample thickness is fixed, and some obvious depressions can be found on the samples annealed at 750 °C. In addition, the grains on the valley were elongated obviously along the tensile direction, whereas there is no significant variation in the microstructure of the neighboring peaks. In this way, the strain localization and instability were developed due to

n¼

t0 2Rt ¼1 t t

ð1Þ

In Eq. (1), t and t0 denote the thicknesses of the ridge and valley regions in the sample during the tension deformation respectively, as illustrated in Fig. 6. Rt is the maximum roughness height, which represents the sum of the maximum value of the peak height and the maximum value of the valley depth in the evaluation length. The smaller value of n represents the larger thickness inhomogeneity. Furthermore, the instantaneous thickness t can be obtained by using the volume constancy law and the assumption of isotropic material as follows [31,41]:

  e t ¼ t 0 exp  2

Fig. 5. Cross-sectional morphologies of the samples under different material conditions.

ð2Þ

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405

Fig. 6. Heterogeneous surface model and its illustration.

In Eq. (2), e is the equivalent strain. Prior studies [26,31] have experimentally manifested that the increased surface roughness Rt induced by the free surface roughening is proportional to the grain size and the equivalent strain, which is defined as:

Rt ¼ cd0 e þ R0

ð3Þ

where c is material constant, and R0 is the initial surface roughness. The values of c and R0 in Eq. (3) can be determined via linear fitting of the measured roughness, as shown in Table 2. It can be seen from Table 2 that the initial thickness affects the roughening behavior, and then the value of Rt/t0 is focused:

Rt c R0 R0 ¼ e þ ¼ ae þ t0 g t0 t0

ð4Þ

In Eq. (4), g = t0/d0, which is the number of grains across the sheet thickness, a = c/g, which represents the rate of increase in Rt/t0. Substituting Eqs. (2) and (4) into Eq. (1), the relationship between the nonuniform coefficient n and the equivalent strain can be illustrated by the following equation:

n¼1

2ðcd0 e þ R0 Þ 2ðae þ R0 =t 0 Þ ¼1 t 0 expðe=2Þ expðe=2Þ

ð5Þ

Fig. 7 shows the relationship between the rate of increase in Rt/t0 and the value of g obtained experimentally. It is found that the increasing rate of Rt/t0 grows significantly as the number of grains in the thickness is less than 10. Conversely, the value of a does not change obviously in the case of a large number of grains in the thickness. It is observed from Eq. (5) that the increase of a leads to a considerable thickness inhomogeneity. Fig. 8 presents the change of thickness nonuniformity with the equivalent strain under different material conditions. It is found that the value of n Table 2 Computed values of c and R0. t0 (lm)

d0 (lm)

g = t0/d0

c

R0 (lm)

200

4.5 13.9 30.8 80.2

44.4 14.4 6.5 2.5

0.424 0.890 0.569 0.255

1.134 1.158 0.497 0.752

400

14.5 36.9 44.0 67.7

27.6 10.8 9.1 5.9

0.723 0.352 0.870 1.178

1.105 1.710 2.264 0.766

600

9.7 21.3 23.8 34.9

61.8 28.2 25.2 17.2

0.173 0.678 0.798 0.674

1.241 1.340 0.510 0.560

Fig. 7. Relationship between increasing rate a and t0/d0.

reduces with the decrease of t0/d0 and the increase of plastic strain. The value of n equals 0.88 at the strain of about 0.25 for the sheets with the thickness of 0.4 mm and the grain size of 67.7 lm. In other words, the heterogeneity of sheet thickness reaches about 12% in this scenario. Particularly, the sample with the thickness of 0.6 mm and the grain size of 9.7 lm fractured at a low strain of 0.25 in spite of the high value of n. This is because the residual stress in the as-rolled sheet results in the premature fracture. It is inferred that the significant thickness inhomogeneity for the samples with a small t0/d0 is attributed to the high roughening rate during the plastic deformation. In addition, it is also found that the tensile ability is reduced by the surface roughening phenomenon via premature fracture. Such influence becomes more significant with the decrease of t0/d0. With the small value of n, the localized strain may occur at the thinnest zone in the thickness direction. Thus, the value of n correlates strongly with both the material flow behavior and the fracture mechanism. In this case, the effect of the additional surface roughness on the flow behavior and ductile fracture is of tantalized interest. 3.2. Size effect on material flow behavior Fig. 9 summarizes the size-dependent mechanical behaviors including yield stress and ultimate tensile stress. The effect of grain size on yield stress can be extrapolated by Hall–Petch relationship, which reveals that the yield stress is inversely proportional to the square root of the grain size. Nevertheless, the tensile properties depend on not only on grain size, but also on the sample thickness. The tensile strength is likely to be affected by the combination of grain size, sample thickness and the value of t0/d0. It can be seen from Fig. 9 that the yield stress and tensile stress increases significantly with t0/d0, and then reach a stabilized stage when the value of t0/d0 is larger than a critical value. A clear trend could be observed that the flow stresses of the samples with only a small number of grains in the thickness direction show a strong size dependence. For the samples with the thickness of 0.4 mm, the yield stress and tensile stress increase by approximately 36% and 34% respectively, when the number of grains is increased from 9.1 to 10.8. However, no such unambiguous size effect on the tensile stress and yield stress is discovered in the samples with the thickness of 0.6 mm. The softening in flow stress with the decrease of t0/d0 is caused by the fact that the grains are not strongly confined during their deformation as there are only a few grains in the cross-section.

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Fig. 9. Variations of yield stress and ultimate tensile stress with t0/d0.

r ¼ Kðe þ e0 Þn

ð6Þ

where K, n and e0 are the strength coefficient, strain hardening exponent, and initial strain respectively, which can be obtained by fitting the curves as shown in Fig. 3. In Fig. 6, A and A0 are the assumed theoretical cross-sectional area and the real value of the loaded samples, respectively. Then, A0 = nA, and the balance of force in tensile direction can be expressed as:

P ¼ rAn

ð7Þ

According to Eq. (7), the effective cross-sectional area of the test samples is reduced by the free surface roughening. Thus, the flow stress is concentrated upon the imperfections of the samples caused by the additional surface roughness. The correction for the Swift’s model is obtained via introducing the thickness nonuniform coefficient n: 0

r¼

Fig. 8. Variation of thickness nonuniform coefficient with equivalent strain under different sample thicknesses: (a) t0 = 0.2 mm, (b) t0 = 0.4 mm and (c) t0 = 0.6 mm.

To further investigate the size effect on flow behavior of copper sheets, Swift type equation is utilized, which is widely used to describe the stress–strain curves of metals in macroscopic scale in the following:

K 0 ðe þ e00 Þn n

ð8Þ

Fig. 10 shows the obtained material parameters in Eqs. (6) and (8) for each material condition. It is found that the strength coefficient and strain-hardening exponent are shifted down with increasing ratio of t0/d0. Additionally, all the material parameters are reduced via considering the thickness heterogeneity caused by the free surface roughening. Moreover, the discrepancy of material constant between the Swift’s equation and the corrected model is decreased with the increase of t0/d0. This is because the increasing rate of surface roughening is decreased with the increase of t0/d0, as shown in Fig. 7. The thickness nonuniform coefficient n for the samples involving a great number of grains in thickness direction tends to be 1, and the corrected model is equivalent to Swift’s model in this scenario. In the tensile test, the work-hardening is generally accompanied by a large plastic deformation. The samples have the ability to harden substantially with the high value of strain-hardening index in the working process. That is to say, the uniform elongation is positively related to the strain-hardening exponent. On this basis, it is inferred from Fig. 10(b) that the uniform elongation of the samples is reduced owing to the free surface roughening. In contrast to the as-received samples, the annealed specimens show no prior plastic strain, as shown in Fig. 10(c). Particularly, the sample with the thickness of 0.6 mm exhibits an extremely high initial strain. This indicates that the annealing treatment eliminates the prior deformation and stress concentration during the rolling process.

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and strain-hardening exponent in Eqs. (6) and (8) can be fitted as follows:

8 > K ¼ a1 gb1 > > > < n ¼ a gb2 2 0 0 b01 > K ¼ a > 1g > > : 0 0 b02 n ¼ a2 g

ð9Þ

The fitted results are shown in Table 3. It is necessary to point out that the initial strain was fitted by a step function. For the as-received samples, the initial strain equals to the computed value, as shown in Fig. 10(c). However, the initial strain is equivalent to zero for the annealed ones. The comparison between the fitted curves and the experimental data is presented in Fig. 11. It is seen that the fitting precision of Swift’s model is significantly improved by considering thickness inhomogeneity, which further demonstrates the necessity for the consideration of the thickness nonuniformity arising from the free surface roughening in meso/micro-scaled plastic deformation. 3.3. Size effect on ductile fracture behavior

Fig. 10. Size effects on the material constants: (a) strength coefficient, (b) strainhardening exponent, and (c) initial strain.

The macroscopic model is not applicable to micro-scaled deformation as the ratio of surface and internal grains is large and size effect exists [12,20,31]. To use the Swift’s equation in micro-scaled plastic deformation, the size factor should be considered. According to the relation between the material parameters and the ratio of t0/d0, material constants including strength coefficient

Fig. 12 presents the top-view surface morphologies of the fractured samples under different conditions after tensile tests. In the figure, RD and TD denote the rolling and transverse directions, respectively. It is found that the number of dimples and microvoids on the fracture surface decreases with t0/d0 due to the fact that the voids tend to nucleate at grain boundaries or inclusions. When there are only several grains across the thickness direction, the share of grain boundaries is reduced, and thus the voids formation seldom takes place in micro-scaled plastic forming [24]. Furthermore, some clear patches can be observed on the fracture surface of the annealed samples because the grains near the sample surface are less constrained and oriented differently, which can rotate out of the initial position via gliding along the grain boundaries [28]. This is also the reason why the free surface roughening intensifies with the decrease of t0/d0. In addition, a general phenomenon can be seen that the samples with only a few grains in the thickness direction fail with the preferred corrugated fracture surface, while the as-received samples exhibit normal tensile mode by generating a necked region in the thickness direction. The fracture appearance becomes irregular with the decrease of t0/d0, which is closely related to the statistical distribution of thickness reduction caused by the free surface roughening. Fig. 13 shows the cross-sectional morphologies of the deformed samples along the tensile direction. It is found that the fracture morphology varies with material condition. For the sheets with the thickness of 0.2 mm, the material at the fracture point becomes more slenderized with the increase of annealing temperature. For the samples with the thickness of 0.4 and 0.6 mm, however, the corrugated fracture morphology in the thickness direction is observed due to the nucleation and growth of microvoids. The variation of fracture morphology shown in Figs. 12 and 13 indicates that the fracture mode is transformed from the pure shear deformation to normal tensile failure with the increase of t0/d0. To confirm the change of ductile fracture mechanism with t0/d0, the ratio of uniform strain eu obtained from the point of the maximum tensile strength to fracture strain ef is focused. In general, the sample deforms uniformly under the uniaxial tensile condition until the flow stress achieves the maximum. Then the diffused necking occurs, and the microvoids begin to nucleate and grow. After that, the flow stress drops until the fracture occurs locally at the necked region. Thus, the closer the ratio of eu to ef is to 1, the less diffused necking is. Accordingly, a smaller value of eu/ef represents more diffused deformation and substantial necked region. Fig. 14 shows the variation of eu/ef under different material

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Table 3 The fitted results of Swift’s model and the corrected Swift’s model. t0 (lm)

a1

a01

b1

b1

0

a2

a02

b2

b2

0

200 400 600

826.97 793.52 1122.80

622.01 578.03 883.57

0.137 0.118 0.232

0.049 0.031 0.175

0.851 1.143 8.936

0.726 1.022 5.947

0.167 0.266 0.912

0.124 0.247 0.819

conditions. It is found that the samples with the thickness of 0.2 mm and annealed samples with the thickness of 0.4 mm fractured almost without diffused necking. Conversely, the samples with the thickness of 0.6 mm underwent sufficiently diffused necking before the fracture occurred at the necked region. In addition, the fraction of diffused necking is decreased with the increase of grain size for the samples with a given thickness. The results presented in Fig. 14 agree well with the observations shown in Fig. 12. Fig. 15 shows the experimental fracture strain ef under various material conditions. It is observed that the fracture strain depends highly on sample thickness and material microstructure. The fracture strain decreases with the decrease of t0/d0 for the annealed samples. In addition, the scatter of fracture strain intensifies with the decrease of sample thickness and the increase of grain size. Regarding the reasons for the size dependence of fracture strain, Simons et al. [28] revealed that there is a combined effect of various factors including sample microstructure, the number of active gliding systems and the original surface roughness. In this study, the combination of the ratio of t0/d0 and the additional surface roughness leads to the size-dependent fracture strain. The importance of the effect of t0/d0 was extensively explored in prior arts [15,17,18,20,21,23,24]. Nevertheless, the influence of the additional surface roughness in plastic deformation was not considered in these studies. Based on the observations of free surface roughening phenomenon, the additional surface roughness Rt is comparable to the thickness t0 as the samples have only a few grains in thickness direction. In this scenario, the effect of additional surface roughness on the ductile fracture behavior needs to be considered. To explore the effect of free surface roughening on fracture toughness, the influence of increasing rate of surface roughening as well as its induced thickness nonuniformity on fracture strain is presented in Fig. 16. The increasing rate of surface roughening a is closely related to fracture strain for the samples with different thicknesses. For the specimen with a given thickness, the larger value of a leads to a smaller fracture strain. This is because the evoked surface roughness is easier to reach the same order of specimen thickness with a high roughening rate. Fig. 16(b) shows the relation between the thickness nonuniform coefficient at fracture point nf and fracture strain ef. Generally, the fracture strain grows with the nonuniform coefficient at fracture point for the specimens with a certain thickness. According to Eq. (5), the large value of a causes a considerable thickness inhomogeneity, and the samples consequently failed at a smaller macroscopic strain. In fact, a large variation of the fracture strain shown in Fig. 15 under the small value of t0/d0 is the result of the stochastic distribution of the surface flaws arising from the free surface roughening. To evaluate the effect of free surface roughening on the fracture toughness quantitatively, the thickness nonuniform coefficient is introduced into the plastic instability condition to revise the formulation of the theoretical limiting uniform strain. By substituting Swift’s equation, viz., Eq. (6) into the plastic instability equation at the maximum stress presented in Eq. (10), the theoretical uniform strain e0u upon the nucleation of microvoids can be determined by Eq. (11) without considering the thickness heterogeneity: Fig. 11. Comparison of the true strain–stress curves of the specimens with different thicknesses: (a) t0 = 0.2 mm, (b) t0 = 0.4 mm and (c) t0 = 0.6 mm.

@r ¼r @e

ð10Þ

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Fig. 12. Top-view of fracture surface morphologies after uniaxial tensile tests.

Fig. 13. Cross-sectional morphologies at fracture point after uniaxial tensile tests.

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Fig. 14. Changes of the ratio of limiting uniform strain to fracture strain under different material conditions.

e0u ¼ n  e0

ð11Þ

Considering the influence of the additional surface roughness, the plastic instability condition can be obtained via differentiating Eq. (7) and letting dP = 0 [42]:

  dr dn ¼r 1 nde de

The revised theoretical uniform strain e00u can be determined by Eq. (14). It can be found from Eq. (11) that the larger strain-hardening exponent n is, the larger the limiting uniform strain can be achieved. However, a small fracture strain eu of 0.22 is observed in the sample with the thickness of 0.2 mm and annealed at 750 °C, in spite of the large strain-hardening exponent of 0.73. Thus, the macro-scaled empirical equation is not applicable in meso/micro-scaled plastic deformation. Fig. 17 presents the comparison of the uniform strain obtained via experiment, Eq. (11) and Eq. (14). It is observed that the theoretical value of the uniform strain e0u is very close to the experimental data eu for the samples with the large thickness and annealed at low temperature. However, the difference between e0u and eu becomes larger and larger with the increase of annealing temperature and the decrease of sample thickness. Nevertheless, the discrepancy between the corrected value e00u and the experimental data eu is reduced via considering the additional thickness heterogeneity caused by the free surface roughening. However, there is still a deviation between e00u and eu with the decrease of t0/d0. It is regarded as the consequence of the size effects of sample thickness and material microstructure. It is thus believed that the combination of size effect and free surface roughening leads to the discrepancy between e0u and eu, as well as the variation of fracture strain with t0/d0. When the evoked surface roughness is comparable to the

ð12Þ

By substituting Eq. (6) into Eq. (12), the corrected theoretical uniform strain e00u is calculated by the following equation:

  dn n ¼ ðe00u þ e0 Þ 1  nde

ð13Þ

Substituting Eq. (5) into Eq. (13), the following equation is established:

n¼





"

e00u þ e0 1 þ

2a exp e00u =2  2ðae00u þ R0 =t 0 Þ 



# ð14Þ

Fig. 15. Fracture strain determined by experiment under different material conditions.

Fig. 16. Effect of the free roughening behavior on fracture strain: (a) relative roughening rate a, and (b) the thickness nonuniform coefficient at fracture point nf.

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Fig. 17. Comparison of the uniform strain between experiments and theoretical calculation with or without considering free surface roughening.

sample thickness, the thickness inhomogeneity needs to be considered to describe the deformation behavior and ductile fracture, in addition to size effect. 4. Conclusions In this paper, the uniaxial tensile tests of pure copper sheets with different thicknesses and comparable grain sizes were conducted. The size effect on flow behavior, free surface roughening and ductile fracture was studied. The thickness nonuniformity arising from the free surface roughening was defined and introduced into the Swift’s model and plastic instability condition to characterize the combined effects of the additional surface roughness and size effect. The following findings are thus drawn. (1) The free surface roughening in the gauge range was observed after tensile test, which is resulted from the variation of the orientation and deformation properties of surface grains. The surface nonuniformity originated from the free surface roughening is increased with the decrease of t0/d0, and its effect on flow behavior and material ductility is considerable when the additional surface roughness reaches the same order of sample thickness. (2) Obvious specimen thickness and grain size effects on tensile properties including yield stress, ultimate tensile stress, and strain-hardening exponent were found. The yield stress and ultimate tensile stress increase with t0/d0. This is because the volume fraction of surface grains grows with the decrease of t0/d0, the surface grains are less constrained and deformed more easily. In addition, Swift’s constitutive model was employed to describe the flow behavior of copper sheets, and its accuracy is improved via introducing the additional thickness inhomogeneity caused by free surface roughening. (3) The samples containing only a few grains across the thickness direction fail with corrugated fracture, whereas the as-received samples exhibit normal tensile mode. The specimens with the thickness of 0.2 mm and the annealed samples with the thickness of 0.4 mm fractured without diffused necking, while the samples with the thickness of 0.6 mm underwent sufficiently diffused necking before

fracture. The variations of fracture morphology and diffused necking reveal that the fracture mode is transformed from pure shear deformation to normal tensile failure with the increase of t0/d0. (4) The scatter of fracture strain is increased with the decrease of t0/d0. Furthermore, the size dependence of fracture strain is closely related to the increasing rate of surface roughening a and the thickness nonuniformity at fracture point nf. As there are only a few grains in the thickness direction, the number of activated slipping systems is reduced, and the additional surface roughness is intensified, which leads to the strain localization and the decreased fracture strain with the decrease of t0/d0. (5) The discrepancy between the theoretical uniform strain and experimental data worsens with the decrease of t0/d0, and the macro-scale plastic instability condition is not valid to predict the uniform strain for the sheets containing only a few grains across the thickness direction. Such difference is reduced via considering the thickness nonuniformity due to the free surface roughening. The residual deviation between the corrected uniform strain and experimental result further indicates that the combination of size effect and free surface roughening leads to the change of ductile fracture mechanism with the varying t0/d0.

Acknowledgements The authors would like to acknowledge the funding support to this research from the General Research Fund of Hong Kong Government under the Project of 515012 (B-Q33F) and the projects of G-YBDM, G-YM93, G-U923 and G-UB59 from The Hong Kong Polytechnic University. References [1] T. Masuzawa, State of the art of micromachining, CIRP Ann. Manuf. Technol. 49 (2000) 473–488. [2] F. Vollertsen, H. Schulze Niehoff, Z. Hu, State of the art in micro forming, Int. J. Mach. Tools Manuf. 46 (2006) 1172–1179. [3] A.R. Razali, Y. Qin, A review on micro-manufacturing, micro-forming and their key issues, Proc. Eng. 53 (2013) 665–672.

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