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Numerical simulation of flexible micro-bending processes with consideration of grain structure
Computational Materials Science 110 (2015) 134–143
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Numerical simulation of flexible micro-bending processes with consideration of grain structure Xiao Wang ⇑, Qing Qian, Zongbao Shen, JianWen Li, Hongfeng Zhang, Huixia Liu School of Mechanical Engineering, Jiangsu University, Zhenjiang 212013, China
a r t i c l e
i n f o
Article history: Received 12 May 2015 Received in revised form 11 August 2015 Accepted 14 August 2015 Available online 27 August 2015 Keywords: Micro-bending Size effects FE-simulation Constitutive model Voronoi tessellation
a b s t r a c t A finite element model of the flexible micro-bending process based on various grain sizes of pure copper is developed. The geometrical model of grain structure is established with Voronoi tessellation, which is employed to describe the polycrystalline aggregation. A model based on dislocation density is adopted to describe the flow stress of grain interior (GI) and grain boundary (GB) quantitatively. In this paper, silicon rubber is used as the flexible punch and four annealing conditions of pure copper as the workpieces, respectively. The influence of grain structure and grain size is discussed. It is observed that as the ratio of workpiece thickness (t) to grain size (d) decreases, the forming depth increases. The inhomogeneous deformation occurs in the coarse-grained micro-parts. Furthermore, the results indicate that the surface asperity increases with grain size. The numerical simulation results agree well with the tendency of experimental results. During the micro-bending process, the phenomenon of stress concentration occurs at the grain boundary of the micro-parts. The maximum von mises stress appears at the grain boundary located at the fillet position. The maximum von mises plastic strain primarily concentrates on the junction of the grain interior and grain boundary in the fine-grained parts, while it concentrates at the surface of the grain interior in the coarse-grained parts. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction In recent years, with the development of the electronic industry, micro–electro–mechanical systems and medical devices, the demand for micro-parts has increased. Micro-forming is one of the most promising manufacturing processes for fabricating micro-parts with high productivity, low cost and good mechanical properties [1]. With the geometrical dimension of micro-parts decreasing to the microscale, affected by the micro-structure and feature size of the micro-parts, the deformation mechanism, the material behaviour and the friction condition are different from those of the macro-parts [2]. These properties are the so-called size effects. Many experimental studies have been conducted to explore the impact of size effects on the material mechanical properties of micro-parts. A variety of mathematical models have been proposed. One of the most classic models is the Hall–Petch model, which was proposed by Hall and Petch [3]. The Hall–Petch model assumes that the material yield stress has a linear relation with the reciprocal of the square root of the grain size. Shan [4]
⇑ Corresponding author. Tel.: +86 051188797998; fax: +86 051188780276. E-mail address: [email protected] (X. Wang). http://dx.doi.org/10.1016/j.commatsci.2015.08.030 0927-0256/Ó 2015 Elsevier B.V. All rights reserved.
explained that the yield stress decreased with the increase of the grain size during the micro-bending forming process of the Hall–Petch model. However, upon further study, it was discovered that the material yield strength no longer followed the Hall–Petch relationship [5] when the workpiece thickness was reduced to a constant grain size. The influence of the ratio of the workpiece thickness (t) to grain size (d), i.e., the N value, became the primary impact factor in material flow stress [6,7]. Leu [8] focused on the flow stress affected by the ratio of workpiece thickness to grain size via sheet material tensile tests. Therefore, the surface layer model, which divides a specimen into two portions, the inner layer portion and the surface layer portion, was proposed [9]. Flow stress in the surface layer is smaller than that in the inner layer because fewer grain boundaries and dislocation can slip to the surface of the specimen freely in the surface layer. The volume fraction of the surface layer increases as the N value decreases. Therefore, the flow stress decreases as the N value decreases. Based on this model, many phenomena can be well explained. Chan [10] studied the flow stress and spring back angle affected by size effects based on the surface layer model. Lin [11] used the surface layer model to explain the reduction of flow stress with the decrease of workpiece thickness when the grain size was a constant. Wang [12] investigated the size effects
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on deformation behaviour in the laser dynamic microforming of copper foil based on the surface layer model. A methodology based on the composite model to study the effects of grain size on material behaviour was developed [13,14]. In the composite model, a grain was considered as being composed of both a grain boundary and grain interior. The flow stress of workpieces was determined by the volume fraction and the flow stress of each grain portion. By using the composite model, Liu [15] successfully predicted flow stress affected by grain size and workpiece thickness and studied the springback angle of the micro-bending process. Fu [16] investigated the micro blanking and deep drawing compound process to study the effects of grain size, the thickness of workpiece and the radius of punch on the deformation mechanism. To reveal the variation of the micro structure during the micro forming process, the constitutive model based on the mechanism of plastic deformation has been rapidly developed. Dislocation slip is the main mechanism of plastic deformation. Some researchers have established the flow stress model based on the evolution equations of dislocation density [17]. Therefore, the plastic deformation behaviour of polycrystalline materials is described by the microscopic mechanism. Fu [18] explained the grain boundary strengthening mechanism with the model based on the dislocation density during the micro-forming process. Based on the above models, several numerical simulations to study the influence of size effects on deformation behaviour have been conducted. Geißdörfer [19] simulated the size effects on the forming force during the micro-upsetting process by using the Hall–Petch model and the surface layer model. Wang [20] divided a workpiece into three portions, namely, the free surface portion, transition portion and internal portion, and used a numerical simulation method to study the change of flow stress with the workpiece thickness. Based on the dislocation density model and the surface layer model, Molotnikov [21] calculated the flow stress with different grain sizes and workpiece thicknesses, and the limited drawing ratio, influenced by grain size, was studied using the finite element (FE) simulation. A FE model based on the surface layer model was established by Lu [22] to study the effect of grained heterogeneity on the microcross wedge rolling process. Overall, many experiments have been performed to study the influence of size effects on material behaviour, but the number of numerical simulation studies is relatively fewer. Moreover, those numerical simulations mainly focused on the tensile tests and compression processes. However, little work has been performed with respect to the micro-bending process. Conversely, many reports have ignored the influences of grain boundary strengthening and the grain structure. In this paper, Voronoi tessellation is introduced to generate the polycrystalline structure. The grain structure is divided into two portions: the grain boundary (GB) portion and the grain interior (GI) portion. Based on the dislocation density model, the flow stress of grain boundary and grain interior is established. The deformation behaviour of micro-bending influenced by grain size and grain structure is discussed.
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location density is affected by the equivalent plastic strain and plastic strain rate. The dislocation accumulation in the grain boundary and grain interior can be expressed as [24]
pffiffiffiffiffiffiffi dqGI ¼ kGI qGI kqGI deGI
ð1Þ
pffiffiffiffiffiffiffiffi dqGB A 2 ¼ kGB qGB kqGB q deGB e_ GB GB
ð2Þ
where qGI is the dislocation density in GI, qGB is the dislocation density in GB, kGI ¼ aGI =b, kGB ¼ aGB =b, aGI and aGB are the numerical constants (aGI ¼ 0:1, aGB ¼ 0:4 [25]), and b = 0.3 nm is the Burgers vector [26]. eGI and eGB are equivalent plastic strain in GI and GB, respectively, e_ GB is the equivalent plastic strain rate in GB, 1=n k ¼ k0 ðe_ =e_ ref Þ is a factor with k0 ¼ 4:6 [27], and e_ is the plastic
strain rate. e_ ref ¼ 1001 s is the reference plastic strain rate [28], the exponent n ¼ 100 [29], and the diffusion coefficient [30] is
A¼
2 4c lX v ðcj bÞDc pð1 v Þ kT lb
ð3Þ
where c ¼ 103 is a constant, v ¼ 0:34 is Poisson’s ratio, b = 0.3 nm is the Burgers vector, cj is the extended jog density along the dislocation line, with cj b 102 [30], Dc ¼ 2:2 1019 m2 s1 is the grain boundary diffusion coefficient [31], X ¼ 1:27 1029 m3 is the atomic volume for copper, T = 276 K is room temperature, v ¼ 78 mJ m2 is the stacking fault energy [32], and l = 42 GPa is the shear modulus [26]. The initial dislocation densities in GB and GI are considered to have values of qGB ¼ 1014 m2 and qGI ¼ 1013 m2 [21]. The plastic strain rate of the workpiece is assumed as 0.02 s1. Based on Eqs. (1) and (2), the evolution of dislocation density with the strain in GB and GI is shown in Fig. 1. With the increase of strain, more dislocation accumulates in the grain boundary. The dislocation density in GB increases much faster than that in GI. The flow stress of GB and GI are determined by the local dislocation density from Taylor’s relational expression:
pffiffiffiffi
rf ¼ Mabl q
ð4Þ
where M ¼ 3:06 is the Taylor factor [26], a = 0.3 is a constant [26], b = 0.3 nm is the Burgers vector [26], l = 42 GPa is the shear modulus [26], q is the dislocation density. Fig. 2 shows the true strain– stress curves of GB and GI from Eq. (4). The true stress in GB is larger than that in GI due to the higher dislocation density. In the finite element model, the material model of multilinear isotropic hardening, which is suitable for simulating the lager deformation of metal materials, is used to describe the material
2. FE simulation 2.1. Dislocation density-based model for pure copper In the process of grain plastic deformation, dislocation slip is the main plastic deformation mechanism. Due to the proliferation of dislocation sources, the number of dislocations in crystal will increase during the plastic deformation process. The flow stress model, using dislocation density as the internal variable, can really describe the deformation behaviour of polycrystalline material [23]. Pure copper is chosen as the workpiece in this paper. The dis-
Fig. 1. Dislocation density evolution with increasing strain for pure copper.
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Fig. 2. True stress–strain curves for GB and GI.
of GB and GI. The stress–strain points of the multilinear isotropic hardening material model are obtained from Fig. 2. 2.2. Hyperplastic material model for silicon rubber materials
0:7
tGB ¼ 0:133d
In the flexible micro forming process, silicon rubber is chosen as the flexible mould. In this study, the Mooney–Rivlin model is used to describe the silicon rubber material [33]. The stress component is obtained by the partial derivation of the strain energy function (W). The function of the Mooney–Rivlin strain energy follows with: n X
diagram to simulate the deformation behaviour in the microforming process [36,37]. Pure copper foil with a 50 lm thickness is chosen as the experimental material. To investigate the grain size effect on deformation behaviour, the as-received copper foils are respectively heated in a vacuum furnace at temperatures of 450 °C, 500 °C, 550 °C and 600 °C for 1 h to obtain the different grain size structures. The annealing process is shown in Fig. 3. The annealed pure copper foils are etched with a solution of 10 g FeCl3, 30 ml HCl and 120 ml H2O to reveal the grain structures. The grain sizes are calculated by using the transversal method according to the standard ASTM-E112. The average grain sizes (d) are 10.8 lm, 15.5 lm, 21.6 lm and 37.8 lm, respectively. In this study, the grain structure is established by Voronoi tessellation to simulate the effect of grain structure on the deformation behaviour in the micro-bending process. The number of nucleation points is calculated according to the average grain size in the annealed workpieces. The virtual grain structure is calculated using the Voronoi algorithm, which is integrated in the Matlab software. The grain boundary is obtained by offsetting the lines in the Voronoi polygon. The grain boundary thickness (tGB) is [18]:
W¼
1 C km ðI1 3Þk þ ðI2 3Þm þ kðI3 1Þ2 2 kþm¼1
ð5Þ
rij ¼
@W @ eij
ð6Þ
where W is the strain energy per unit of the reference volume, I1, I2 and I3 are the strain invariants, k is the bulk modulus, the material is incompressible when I3 ¼ 1, and Ckm represents the hyperplastic constants. Generally, two parameters (C10 and C01) are used to describe the deformation behaviour of hyperplastic rubber. In this study, rubber with a hardness of 55° is used as the flexible mould to study the deformation behaviour of the micro-bending process. The mechanical properties of the silicon rubber are listed in Table 1. 2.3. FE model 2.3.1. Grain structure In the field of micro-forming, the geometric model of polycrystal is particularly important for micro-forming numerical simulations. Usually, the grain structure is established by a regular hexagon to study the deformation of polycrystalline in the FE model. However, the hexagon is unable to describe the behaviour of grain boundary strengthening and the inhomogeneous deformation of the grain interior in the micro-forming process. The influence of grain structure is also neglected. The Voronoi polygon, which is similar to the morphogenetic process of nucleation and growth from random seeds, is widely used to establish the grain structure in the field of material science [34,35]. Many researchers have built the microstructure of a workpiece by using the Voronoi
where d is the grain size. Information regarding the points and lines is saved in the dxf format. By using the information in the dxf file, the geometric model of grain boundary and grain interior is established in ANSYS using the ANSYS parametric design language (APDL). Through the method mentioned before, real and virtual grain structures of different grain sizes are shown in Fig. 4. Based on Voronoi tessellation, the grain structure can be described. 2.3.2. Boundary conditions Fig. 5 shows the geometrical model of flexible micro-bending. The numerical model, from top to bottom, is as follows: rigid punch, silicon rubber, pure copper foil, and micro die. The geometrical model is meshed by using the PLANE 182 element, a twodimensional 8-node element used to simulate stress and strain. The contacts between the surface of the rigid punch, silicon rubber, pure copper foil and micro die are defined as line to line contacts. To ensure the convergence of the solution, the surface that has the relative higher hardness is defined as the contact surface, while that with the lower hardness is defined as the target surface. The rigid punch is loaded with a pressure of 22.5 Mpa. Additionally, the friction coefficient between the micro-die and the copper foil is set at 0.19 in the micro-bending process. The preconditioned conjugate gradient iterative equation solver, which can be run in a shared memory parallel or distributed memory parallel program, is chosen as the equation solver to reduce solving time. The engineering drawing of the rigid die is shown in Fig. 6.
Table 1 Mechanical properties of silicon rubber materials. Hardness shore A (°)
M–R constant C 01 (Mpa)
M–R constant C 10 (Mpa)
Poisson’s ratio l
55
0.382
0.096
0.49997
ð7Þ
Fig. 3. Different recrystallization annealing processes for pure copper.
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Fig. 4. Real and virtual grain structures under different annealing temperatures. (a) Real grain structure at 450 C. (b) Virtual grain structure at 450 C. (c) Real grain structure at 500 C. (d) Virtual grain structure at 500 C. (e) Real grain structure at 550 C. (f) Virtual grain structure at 550 C. (g) Real grain structure at 600 C. (h) Virtual grain structure at 600 C.
3. Results and discussion 3.1. Grain boundary strengthening
Fig. 5. The geometric model of rubber deformation. (a) Punch. (b) Rubber. (c) Micro-die. (d) Pure copper foil.
Fig. 6. The engineering drawing of the rigid die.
Fig. 7 shows the micro-bending process step by step. Firstly, under the punch force, rubber self-deformation occurs, as shown in Fig. 7(a). Secondly, as the punch force increases, the deformation of rubber increases and bending deformation occurs in the centre of the pure copper foil, as shown in Fig. 7(b). Then, the foil generates deformation at the fillets of the micro die. The forming depth increases due to the force generated by the rubber self-deformation, as shown in Fig. 7(c). Finally, as shown in Fig. 7 (d), it can be observed that the maximum forming depth occurs at the centre of the foil. Fig. 8 shows the von mises stress distribution of the formed parts annealed at 550 °C. It is shown that the maximum von mises stress reaches 1280 Mpa in the grain boundary, while it approximates to 300 Mpa in the grain interior. The von mises stress in the grain boundary is rather higher than that in the grain interior. Stress concentration occurs in the grain boundary during the micro bending process. During plastic deformation, dislocation pile-up appears in the grain boundary with the increasing strain due to the dislocation motion. Due to Taylor’s relationship, the flow stress increases with the dislocation density. The flow stress of the grain boundary is higher than that of the grain interior. The grain boundary, with higher flow stress, constitutes the rigid meshing structure in the workpiece. This results in the stress concentration in the grain boundary. Fig. 9 reveals the von mises stress changes with time at different locations in Fig. 8. In the initial stage of the forming process (0–2 s), the von mises stress in the bottom location is higher than that in the fillet location because the formation of bending firstly
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Fig. 7. The typical stages of deformation with contours of Y-displacement distributions (d = 21.6 lm). (a) First stage. (b) Second stage. (c) Third stage. (d) Final deforming.
Fig. 8. The contours of von mises stress distribution (d = 21.6 lm).
appears at the centre of the copper foil. Similarly, plastic deformation at the centre is larger than that in the fillet of the micro parts. Consequently, the stress concentration firstly appears at the bottom of the workpiece. As the load continues to increase, deformation occurs in the fillets. The von mises stress of the grain boundary and grain interior at the fillets is higher than that at the bottom. The larger bending and tensile deformation at the fillets leads to
Fig. 9. The change of von mises stress with time.
higher von mises stress. The fluctuation of the stress in the fillet may be caused by the change of force direction in the fillets during deformation. It is obvious that there is quite a difference between fillet-1 and fillet-2 in the grain boundary in terms of stress. The von mises stress of the grain boundary and grain interior in fillet-2 is rather higher than that in fillet-1. This may be due to the different volume fractions of the grain boundary at the locations of the fillets. From Fig. 8, it can be observed that compared with fillet-1, fillet-2 has fewer grain boundary structures. Due to the grain boundary with higher flow stress, more grain boundary structures would improve the structural rigidity. As a result, the plastic deformation that occurred in fillet-2 is larger than that in fillet-1. The von mises stress in fillet-2 is higher than that in fillet-1. The von mises stress and von plastic strain along the path in Fig. 8 are shown in Fig. 10. It is clear that the grain boundary stress is significantly higher than the grain interior stress. The grain boundary strain is a little lower than that of the grain interior. The grain boundary with higher flow stress may cause the behaviour of grain boundary strengthening. Moreover, it is found that the stress and strain rise along the path. In the micro forming process, the tensile and bending deformations are generated at the bottom of the micro-parts. The tensile stress generated by the tensile deformation is compensated by the compressive stress generated by the bending deformation in the upside. Therefore, the
Fig. 10. Von mises plastic strain and von mises stress distributions along the path.
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stress located in the underside is higher than that in the upside at the bottom of the micro-parts.
reduction of grain boundary constraints when the grain size increases.
3.2. The distribution of von mises plastic strain and von mises stress
3.3. Grain size effect on the forming depth
Fig. 11 shows the contours of von mises plastic strain distribution for different grain sizes. It is shown that the strain at the fillets is higher than at other locations. The plastic strain in the grain interior is higher than the grain boundary plastic strain. Compared with the grain interior, the formability of the grain boundary is relatively difficult. In the grain boundary, the dislocation slip is restricted because the atomic arrangement is irregular and the lattice has a serious distortion in the grain boundary. More slip systems are needed to meet the demand of compatibility of the deformation. Therefore, formability is difficult in the grain boundary. In the fine-grained micro-parts (d = 10.8 lm, d = 15.5 lm, d
A series of numerical simulations are conducted to investigate the grain size effect on the forming depth in flexible microbending processes. In the field of micro-forming, the ratio of workpiece thickness to grain size (N = t/d) is the critical evaluating criterion that affects the deformation behaviour of a workpiece [6]. When the grain size and workpiece are close to the same scale, the deformation behaviour of the material is determined by several grains at the forming location. The flow stress would change with the grain size due to the volume fraction of the grain boundary and grain interior. In this paper, the outline of the micro-parts with different grain sizes is shown in Fig. 13. It is clear that the N value is the significant factor affecting the forming depth. The forming depth increases as the N value decreases. The increasing trend of the forming depth is related to the decrease of the flow stress of the workpiece. As the N value decreases, the grain boundary strengthening decreases because of the decrease of the GB volume fraction [38]. Therefore, the flow stress of workpieces would decrease as the N value decreases. In this paper, the results agree well with other research findings. Lai [39] found that a grain had smaller flow stress on the surface of a workpiece because surface grains have fewer grain boundary constraints. As the N value decreases, the increase of volume fraction of the surface layer results in the decrease of flow stress of the workpiece. By using the Hall–Petch model, the decrease of flow stress with the increase of grain sizes can be clearly revealed. An experiment is conducted to verify the rationality of the numerical simulation. The experimental equipment used for micro-bending is shown in Fig. 14. In this paper, the universal testing machine is used as the loader. The outline of the micro-parts in
= 21.6 lm), the maximum plastic strain concentrates at the junction area of the grain boundary and grain interior. However, in the coarse-grained micro-parts (d = 37.8 lm), the maximum plastic strain is not only concentrated at the junction area of the grain boundary and grain interior but also at the surface of the grain interior. The cracked location of the workpiece can be predicated by the von mises strain distribution. Therefore, the fracture of the fine-grained workpiece may occur at the junction area of the grain boundary and grain interior. The coarse-grained workpiece may be more likely to produce a transgranular fracture. The von mises stress at the bottom of micro-parts with different grain sizes is shown in Fig. 12. From Fig. 12, the behaviour of grain boundary strengthening is also obviously shown. The von mises stress in the grain boundary is higher than that in the grain interior. Moreover, with the increase of grain size, the minimum stress increases from 8.84 Mpa to 47.66 Mpa. This finding indicates that the deformation increases as the grain size increases. The increasing plastic deformation of the grain interior may be caused by the
Fig. 11. The contours of von mises plastic strain distribution for different grain sizes. (a) d = 10.8 lm, (b) d = 15.5 lm, (c) d = 21.6 lm, and (d) d = 37.8 lm.
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Fig. 12. The contours of von mises stress distribution at the bottom of forming parts. (a) d = 10.8 lm, (b) d = 15.5 lm, (c) d = 21.6 lm, and (d) d = 37.8 lm.
Fig. 13. The simulated deformation of different grain sizes.
Fig. 15. The experimental deformation of different grain sizes.
of the forming depth in the numerical simulation agrees well with the experimental study. However, the forming depth in the numerical simulation is slightly smaller than that in the experimental study. In the study of numerical simulation, only the dislocation slip is discussed in terms of the plastic deformation. The subgrain formation, grain boundary sliding and grain rotation have been ignored. Furthermore, the initial dislocation density may have errors in the grain boundary and grain interior. Therefore, the forming depth in the experiment is higher than that in the numerical simulation.
3.4. Grain size effect on surface asperity
Fig. 14. Experimental equipment used for micro-bending. 1. Rigid punch. 2. Container. 3. Base. 4. Rubber. 5. Pure copper foil. 6. Micro-die.
the experimental micro-bending process is shown in Fig. 15. It is found that the forming depth increases as the N value decreases. By comparing Fig. 13 with Fig. 15, it can be found that the trend
The properties of each grain have an effect on the deformation behaviour when there are only a few grains along the workpiece thickness direction. Homogeneous deformation is affected by the random characteristics of grain size. It is found that inhomogeneous deformation occurs in micro-parts with larger grain size due to the grained heterogeneity (e.g., grain size, shape and deformability) [22]. In this study, the homogeneous deformation behaviours affected by grain size are studied by using the numerical simulation method. Information of the node coordinates is extracted from the results of the simulation to obtain the surface roughness profile of the micro parts. The bottom surface node coordinates are shown in Fig. 16. For the purpose of quantitatively assessing the homogeneous
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Fig. 16. The simulated roughness profile at the bottom of micro parts with different grain sizes. (a) d = 10.8 lm, (b) d = 15.5 lm, (c) d = 21.6 lm, and (d) d = 37.8 lm.
deformation, it is necessary to define the centreline, which could describe the geometric shape. In this study, the contour is obtained by using the least squares principle. The least squares centreline is shown in Fig. 16. It can be observed that with the increase of grain size, the fluctuation of nodes on the surface around the centreline becomes obvious and the inhomogeneous deformation could result in the increase of surface asperity. To assess the surface roughness profile quantitatively, we must adopt the parameters and their values to indicate the degree of roughness. The arithmetical mean deviation (Ra) of the profile is used to evaluate the roughness. The value of Ra is approximately calculated by:
Ra ¼
n 1X jzi j n 1
ð8Þ
where n is the number of collected nodes and zi means the ordinate value of the ith node to the centreline. The surface asperity of the micro parts with different grain sizes is shown in Fig. 17. It can be observed that the asperity increases as the grain size increases. The increase of asperity may be caused by the surface grains with
Fig. 17. The simulated arithmetical mean deviation (Ra) of the profile.
fewer grain boundary constraints as the grain sizes increase. At the same time, compared with the grain boundary, the grain interior deformation contributes more to the overall deformation in the micro-bending process. The increase of volume fraction of the grain interior caused by the increase of grain sizes may lead to inhomogeneous deformation and then the increase of surface asperity. An experiment is conducted to study the size effect on surface asperity of the formed micro parts. A high resolution true colour confocal microscope (Axio CSM 700) is employed to obtain the 3D morphology of the formed parts. The rough surface at the bottom of the micro parts is shown in Fig. 18. From the surface topography, it can be observed that the surface becomes rougher as the grain size increases. Inhomogeneous deformation could be caused by the decrease of slip systems and grain boundary constraints in the coarse-grained workpiece. With the increase of grain size, constraints with adjacent grains may weaken due to the decrease of the grain boundary structure. Therefore, grains could be more easily moved out of the surface of the coarse-grained workpiece. On the other hand, the total slip systems increase as the grain size decreases. For the fine-grained workpiece, each grain only undergoes a relatively small plastic deformation to maintain the compatibility of the strain [40]. In the coarse-grained workpiece, grains need additional shear deformation and rotational deformation to maintain the compatibility of the strain [12]. In conclusion, inhomogeneous deformation could occur in coarse-grained micro parts. The arithmetical mean deviation (Ra) of the bottom region is measured to evaluate the effects of grain size effect on the surface roughness. The surface asperity of micro parts with different grain sizes is shown in Fig. 19. The surface asperity has a tendency to increase with grain size. Figs. 17 and 19 show that the trends of surface asperity in the simulated results agree well with the experimental results. However, it is found that the surface roughness of the experimental results is higher than that in the simulated results because the numerical simulation only considers the grain size and constraints of the grain boundary. Furthermore, the surface roughness of the as-received material has not been discussed in the numerical simulation.
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Fig. 18. Surface topography of micro parts with different grain sizes. (a) d = 10.8 lm, (b) d = 15.5 lm, (c) d = 21.6 lm, and (d) d = 37.8 lm.
Fig. 19. The experimental arithmetical mean deviation (Ra) of the profile.
4. Conclusions In this study, the effects of size on deformation behaviour in the flexible micro-bending process are investigated using the numerical simulation method. The micro structures of the grain boundary and grain interior are established via Voronoi tessellation, which is employed to describe the polycrystalline aggregation. The flow stress of the grain boundary and grain interior are established by using the dislocation density-based model, which can better reflect the behaviour of grain boundary strengthening. The following conclusions can be made based on the numerical simulated results: (1) In the flexible micro-bending process, the von mises stress in the grain boundary is significantly higher than that in the
grain boundary. The behaviour of grain boundary strengthening occurs obviously in the micro forming process. (2) The maximum von mises plastic strain mainly concentrates at the junction of the grain interior and grain boundary in fine-grained micro-parts, while it concentrates at the surface of the grain interior in coarse-grained micro-parts. (3) It is found that the forming depth increases as the ratio of workpiece thickness to grain size decreases. The results of numerical simulations agree well with the experimental results. (4) Inhomogeneous deformation occurs in coarse-grained micro-parts according to the numerical simulation and experiment. Furthermore, the surface roughness increases as the grain sizes increase.
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Acknowledgments This work is supported by the National Natural Science Foundation of China (NO. 51175235) and the Natural Science Foundation of Jiangsu Province (NO. BK2012712). References [1] K.F. Zhang, L. Kun, J. Mater. Process. Technol. 209 (11) (2009) 4949–4953. [2] F. Vollertsen, D. Biermann, H.N. Hansen, et al., CIRP Annals-Manuf. Technol. 58 (2) (2009) 566–587. [3] T.G. Nieh, J. Wadsworth, Scr. Metall. Mater. 25 (4) (1991) 955–958. [4] D. Shan, C. Wang, B. Guo, et al., Trans. Nonferr. Metals Soc. China 19 (2009) 507–510. [5] L.V. Raulea, A.M. Goijaerts, L.E. Govaert, et al., J. Mater. Process. Technol. 115 (1) (2001) 44–48. [6] C. Keller, E. Hug, X. Feaugas, Int. J. Plast. 27 (4) (2011) 635–654. [7] P. Janssen, T.H. De Keijser, M. Geers, Mater. Sci. Eng., A 419 (1) (2006) 238–248. [8] D. Leu, J. Manuf. Sci. Eng. 131 (1) (2009) 011002. [9] S. Miyazaki, K. Shibata, H. Fujita, Acta Metall. 27 (5) (1979) 855–862. [10] W.L. Chan, M.W. Fu, J. Lu, Mater. Des. 32 (1) (2011) 198–206. [11] X. Lin, G. Wang, W. Zheng, et al., Trans. Nonferr. Metals Soc. China 24 (2) (2014) 470–476. [12] X. Wang, Y. Ma, Z. Shen, et al., J. Mater. Process. Technol. 220 (2015) 173–183. [13] U.F. Kocks, Metall. Mater. Trans. 1 (5) (1970) 1121–1143. [14] M.A. Meyersm, E. Ashworth, Philos. Mag. A 46 (5) (1982) 737–759. [15] J.G. Liu, M.W. Fu, J. Lu, et al., Comput. Mater. Sci. 50 (9) (2011) 2604–2614. [16] M.W. Fu, B. Yang, W.L. Chan, J. Mater. Process. Technol. 213 (1) (2013) 101–110. [17] U.F. Kocks, J. Eng. Mater. Technol. 98 (1) (1976) 76–85. [18] H. Fu, D.J. Benson, M.A. Meyers, Acta Mater. 49 (13) (2001) 2567–2582.
143
[19] S. Geißdörfer, U. Engel, M. Geiger, Int. J. Mach. Tools Manuf 46 (11) (2006) 1222–1226. [20] C. Wang, B. Guo, D. Shan, Trans. Nonferr. Metals Soc. China 21 (6) (2011) 1362–1366. [21] A. Molotnikov, R. Lapovok, C.F. Gu, et al., Mater. Sci. Eng., A 550 (2012) 312–319. [22] H.N. Lu, D.B. Wei, Z.Y. Jiang, et al., Comput. Mater. Sci. 77 (2013) 44–52. [23] H. Zhang, X. Dong, Q. Wang, et al., A dislocation density based crystal plasticity model of 3D complex sheet metal forming, The 8th international conference and workshop on numerical simulation of 3D sheet metal forming processes, 2011. [24] H. Liu, W. Pantleon, L. Mishnaevsky Jr., Comput. Mater. Sci. 83 (2014) 318–330. [25] H. Mecking, U.F. Kocks, Acta Metall. 29 (11) (1981) 1865–1875. [26] C.W. Sinclair, W.J. Poole, Y. Bréchet, Scripta Mater. 55 (8) (2006) 739–742. [27] Y. Estrin, L.S. Toth, A. Molinari, et al., Acta Mater. 46 (15) (1998) 5509–5522. [28] E.I. Galindo-Nava, J. Sietsma, P. Rivera-Díaz-Del-Castillo, Acta Mater. 60 (6) (2012) 2615–2624. [29] Y. Estrin, Dislocation-density-related constitutive modeling, in: A.S. Krausz, K. Krausz (Eds.), Unified constitutive laws of plastic deformation, vol. 1, Academic Press Inc., New York City, 1996, pp. 69–106. [30] F. Prinz, A.S. Argon, W.C. Moffatt, Acta Metall. 30 (4) (1982) 821–830. [31] J.C. Fisher, J. Appl. Phys. 22 (1) (1951) 74–77. [32] Y.H. Zhao, X.Z. Liao, Y.T. Zhu, et al., Mater. Sci. Eng., A 410 (2005) 188–193. [33] M.H. Dirikolu, E. Akdemir, J. Mater. Process. Technol. 148 (3) (2004) 376–381. [34] B.N. Boots, D.J. Murdoch, Comput. Geosci. 9 (3) (1983) 351–365. [35] G. Cailletaud, S. Forest, D. Jeulin, et al., Comput. Mater. Sci. 27 (3) (2003) 351–374. [36] P. Zhang, M. Karimpour, D. Balint, et al., Comput. Mater. Sci. 64 (2012) 84–89. [37] H. Liu, L. Mishnaevsky, Acta Mater. 71 (2014) 220–233. [38] M.W. Fu, W.L. Chan, Mater. Des. 32 (10) (2011) 4738–4746. [39] X. Lai, L. Peng, P. Hu, et al., Comput. Mater. Sci. 43 (4) (2008) 1003–1009. [40] Z. Gao, L. Peng, P. Yi, et al., J. Mater. Process. Technol. 219 (2015) 28–41.
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Numerical simulation of flexible micro-bending processes with consideration of grain structure Xiao Wang ⇑, Qing Qian, Zongbao Shen, JianWen Li, Hongfeng Zhang, Huixia Liu School of Mechanical Engineering, Jiangsu University, Zhenjiang 212013, China
a r t i c l e
i n f o
Article history: Received 12 May 2015 Received in revised form 11 August 2015 Accepted 14 August 2015 Available online 27 August 2015 Keywords: Micro-bending Size effects FE-simulation Constitutive model Voronoi tessellation
a b s t r a c t A finite element model of the flexible micro-bending process based on various grain sizes of pure copper is developed. The geometrical model of grain structure is established with Voronoi tessellation, which is employed to describe the polycrystalline aggregation. A model based on dislocation density is adopted to describe the flow stress of grain interior (GI) and grain boundary (GB) quantitatively. In this paper, silicon rubber is used as the flexible punch and four annealing conditions of pure copper as the workpieces, respectively. The influence of grain structure and grain size is discussed. It is observed that as the ratio of workpiece thickness (t) to grain size (d) decreases, the forming depth increases. The inhomogeneous deformation occurs in the coarse-grained micro-parts. Furthermore, the results indicate that the surface asperity increases with grain size. The numerical simulation results agree well with the tendency of experimental results. During the micro-bending process, the phenomenon of stress concentration occurs at the grain boundary of the micro-parts. The maximum von mises stress appears at the grain boundary located at the fillet position. The maximum von mises plastic strain primarily concentrates on the junction of the grain interior and grain boundary in the fine-grained parts, while it concentrates at the surface of the grain interior in the coarse-grained parts. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction In recent years, with the development of the electronic industry, micro–electro–mechanical systems and medical devices, the demand for micro-parts has increased. Micro-forming is one of the most promising manufacturing processes for fabricating micro-parts with high productivity, low cost and good mechanical properties [1]. With the geometrical dimension of micro-parts decreasing to the microscale, affected by the micro-structure and feature size of the micro-parts, the deformation mechanism, the material behaviour and the friction condition are different from those of the macro-parts [2]. These properties are the so-called size effects. Many experimental studies have been conducted to explore the impact of size effects on the material mechanical properties of micro-parts. A variety of mathematical models have been proposed. One of the most classic models is the Hall–Petch model, which was proposed by Hall and Petch [3]. The Hall–Petch model assumes that the material yield stress has a linear relation with the reciprocal of the square root of the grain size. Shan [4]
⇑ Corresponding author. Tel.: +86 051188797998; fax: +86 051188780276. E-mail address: [email protected] (X. Wang). http://dx.doi.org/10.1016/j.commatsci.2015.08.030 0927-0256/Ó 2015 Elsevier B.V. All rights reserved.
explained that the yield stress decreased with the increase of the grain size during the micro-bending forming process of the Hall–Petch model. However, upon further study, it was discovered that the material yield strength no longer followed the Hall–Petch relationship [5] when the workpiece thickness was reduced to a constant grain size. The influence of the ratio of the workpiece thickness (t) to grain size (d), i.e., the N value, became the primary impact factor in material flow stress [6,7]. Leu [8] focused on the flow stress affected by the ratio of workpiece thickness to grain size via sheet material tensile tests. Therefore, the surface layer model, which divides a specimen into two portions, the inner layer portion and the surface layer portion, was proposed [9]. Flow stress in the surface layer is smaller than that in the inner layer because fewer grain boundaries and dislocation can slip to the surface of the specimen freely in the surface layer. The volume fraction of the surface layer increases as the N value decreases. Therefore, the flow stress decreases as the N value decreases. Based on this model, many phenomena can be well explained. Chan [10] studied the flow stress and spring back angle affected by size effects based on the surface layer model. Lin [11] used the surface layer model to explain the reduction of flow stress with the decrease of workpiece thickness when the grain size was a constant. Wang [12] investigated the size effects
X. Wang et al. / Computational Materials Science 110 (2015) 134–143
on deformation behaviour in the laser dynamic microforming of copper foil based on the surface layer model. A methodology based on the composite model to study the effects of grain size on material behaviour was developed [13,14]. In the composite model, a grain was considered as being composed of both a grain boundary and grain interior. The flow stress of workpieces was determined by the volume fraction and the flow stress of each grain portion. By using the composite model, Liu [15] successfully predicted flow stress affected by grain size and workpiece thickness and studied the springback angle of the micro-bending process. Fu [16] investigated the micro blanking and deep drawing compound process to study the effects of grain size, the thickness of workpiece and the radius of punch on the deformation mechanism. To reveal the variation of the micro structure during the micro forming process, the constitutive model based on the mechanism of plastic deformation has been rapidly developed. Dislocation slip is the main mechanism of plastic deformation. Some researchers have established the flow stress model based on the evolution equations of dislocation density [17]. Therefore, the plastic deformation behaviour of polycrystalline materials is described by the microscopic mechanism. Fu [18] explained the grain boundary strengthening mechanism with the model based on the dislocation density during the micro-forming process. Based on the above models, several numerical simulations to study the influence of size effects on deformation behaviour have been conducted. Geißdörfer [19] simulated the size effects on the forming force during the micro-upsetting process by using the Hall–Petch model and the surface layer model. Wang [20] divided a workpiece into three portions, namely, the free surface portion, transition portion and internal portion, and used a numerical simulation method to study the change of flow stress with the workpiece thickness. Based on the dislocation density model and the surface layer model, Molotnikov [21] calculated the flow stress with different grain sizes and workpiece thicknesses, and the limited drawing ratio, influenced by grain size, was studied using the finite element (FE) simulation. A FE model based on the surface layer model was established by Lu [22] to study the effect of grained heterogeneity on the microcross wedge rolling process. Overall, many experiments have been performed to study the influence of size effects on material behaviour, but the number of numerical simulation studies is relatively fewer. Moreover, those numerical simulations mainly focused on the tensile tests and compression processes. However, little work has been performed with respect to the micro-bending process. Conversely, many reports have ignored the influences of grain boundary strengthening and the grain structure. In this paper, Voronoi tessellation is introduced to generate the polycrystalline structure. The grain structure is divided into two portions: the grain boundary (GB) portion and the grain interior (GI) portion. Based on the dislocation density model, the flow stress of grain boundary and grain interior is established. The deformation behaviour of micro-bending influenced by grain size and grain structure is discussed.
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location density is affected by the equivalent plastic strain and plastic strain rate. The dislocation accumulation in the grain boundary and grain interior can be expressed as [24]
pffiffiffiffiffiffiffi dqGI ¼ kGI qGI kqGI deGI
ð1Þ
pffiffiffiffiffiffiffiffi dqGB A 2 ¼ kGB qGB kqGB q deGB e_ GB GB
ð2Þ
where qGI is the dislocation density in GI, qGB is the dislocation density in GB, kGI ¼ aGI =b, kGB ¼ aGB =b, aGI and aGB are the numerical constants (aGI ¼ 0:1, aGB ¼ 0:4 [25]), and b = 0.3 nm is the Burgers vector [26]. eGI and eGB are equivalent plastic strain in GI and GB, respectively, e_ GB is the equivalent plastic strain rate in GB, 1=n k ¼ k0 ðe_ =e_ ref Þ is a factor with k0 ¼ 4:6 [27], and e_ is the plastic
strain rate. e_ ref ¼ 1001 s is the reference plastic strain rate [28], the exponent n ¼ 100 [29], and the diffusion coefficient [30] is
A¼
2 4c lX v ðcj bÞDc pð1 v Þ kT lb
ð3Þ
where c ¼ 103 is a constant, v ¼ 0:34 is Poisson’s ratio, b = 0.3 nm is the Burgers vector, cj is the extended jog density along the dislocation line, with cj b 102 [30], Dc ¼ 2:2 1019 m2 s1 is the grain boundary diffusion coefficient [31], X ¼ 1:27 1029 m3 is the atomic volume for copper, T = 276 K is room temperature, v ¼ 78 mJ m2 is the stacking fault energy [32], and l = 42 GPa is the shear modulus [26]. The initial dislocation densities in GB and GI are considered to have values of qGB ¼ 1014 m2 and qGI ¼ 1013 m2 [21]. The plastic strain rate of the workpiece is assumed as 0.02 s1. Based on Eqs. (1) and (2), the evolution of dislocation density with the strain in GB and GI is shown in Fig. 1. With the increase of strain, more dislocation accumulates in the grain boundary. The dislocation density in GB increases much faster than that in GI. The flow stress of GB and GI are determined by the local dislocation density from Taylor’s relational expression:
pffiffiffiffi
rf ¼ Mabl q
ð4Þ
where M ¼ 3:06 is the Taylor factor [26], a = 0.3 is a constant [26], b = 0.3 nm is the Burgers vector [26], l = 42 GPa is the shear modulus [26], q is the dislocation density. Fig. 2 shows the true strain– stress curves of GB and GI from Eq. (4). The true stress in GB is larger than that in GI due to the higher dislocation density. In the finite element model, the material model of multilinear isotropic hardening, which is suitable for simulating the lager deformation of metal materials, is used to describe the material
2. FE simulation 2.1. Dislocation density-based model for pure copper In the process of grain plastic deformation, dislocation slip is the main plastic deformation mechanism. Due to the proliferation of dislocation sources, the number of dislocations in crystal will increase during the plastic deformation process. The flow stress model, using dislocation density as the internal variable, can really describe the deformation behaviour of polycrystalline material [23]. Pure copper is chosen as the workpiece in this paper. The dis-
Fig. 1. Dislocation density evolution with increasing strain for pure copper.
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Fig. 2. True stress–strain curves for GB and GI.
of GB and GI. The stress–strain points of the multilinear isotropic hardening material model are obtained from Fig. 2. 2.2. Hyperplastic material model for silicon rubber materials
0:7
tGB ¼ 0:133d
In the flexible micro forming process, silicon rubber is chosen as the flexible mould. In this study, the Mooney–Rivlin model is used to describe the silicon rubber material [33]. The stress component is obtained by the partial derivation of the strain energy function (W). The function of the Mooney–Rivlin strain energy follows with: n X
diagram to simulate the deformation behaviour in the microforming process [36,37]. Pure copper foil with a 50 lm thickness is chosen as the experimental material. To investigate the grain size effect on deformation behaviour, the as-received copper foils are respectively heated in a vacuum furnace at temperatures of 450 °C, 500 °C, 550 °C and 600 °C for 1 h to obtain the different grain size structures. The annealing process is shown in Fig. 3. The annealed pure copper foils are etched with a solution of 10 g FeCl3, 30 ml HCl and 120 ml H2O to reveal the grain structures. The grain sizes are calculated by using the transversal method according to the standard ASTM-E112. The average grain sizes (d) are 10.8 lm, 15.5 lm, 21.6 lm and 37.8 lm, respectively. In this study, the grain structure is established by Voronoi tessellation to simulate the effect of grain structure on the deformation behaviour in the micro-bending process. The number of nucleation points is calculated according to the average grain size in the annealed workpieces. The virtual grain structure is calculated using the Voronoi algorithm, which is integrated in the Matlab software. The grain boundary is obtained by offsetting the lines in the Voronoi polygon. The grain boundary thickness (tGB) is [18]:
W¼
1 C km ðI1 3Þk þ ðI2 3Þm þ kðI3 1Þ2 2 kþm¼1
ð5Þ
rij ¼
@W @ eij
ð6Þ
where W is the strain energy per unit of the reference volume, I1, I2 and I3 are the strain invariants, k is the bulk modulus, the material is incompressible when I3 ¼ 1, and Ckm represents the hyperplastic constants. Generally, two parameters (C10 and C01) are used to describe the deformation behaviour of hyperplastic rubber. In this study, rubber with a hardness of 55° is used as the flexible mould to study the deformation behaviour of the micro-bending process. The mechanical properties of the silicon rubber are listed in Table 1. 2.3. FE model 2.3.1. Grain structure In the field of micro-forming, the geometric model of polycrystal is particularly important for micro-forming numerical simulations. Usually, the grain structure is established by a regular hexagon to study the deformation of polycrystalline in the FE model. However, the hexagon is unable to describe the behaviour of grain boundary strengthening and the inhomogeneous deformation of the grain interior in the micro-forming process. The influence of grain structure is also neglected. The Voronoi polygon, which is similar to the morphogenetic process of nucleation and growth from random seeds, is widely used to establish the grain structure in the field of material science [34,35]. Many researchers have built the microstructure of a workpiece by using the Voronoi
where d is the grain size. Information regarding the points and lines is saved in the dxf format. By using the information in the dxf file, the geometric model of grain boundary and grain interior is established in ANSYS using the ANSYS parametric design language (APDL). Through the method mentioned before, real and virtual grain structures of different grain sizes are shown in Fig. 4. Based on Voronoi tessellation, the grain structure can be described. 2.3.2. Boundary conditions Fig. 5 shows the geometrical model of flexible micro-bending. The numerical model, from top to bottom, is as follows: rigid punch, silicon rubber, pure copper foil, and micro die. The geometrical model is meshed by using the PLANE 182 element, a twodimensional 8-node element used to simulate stress and strain. The contacts between the surface of the rigid punch, silicon rubber, pure copper foil and micro die are defined as line to line contacts. To ensure the convergence of the solution, the surface that has the relative higher hardness is defined as the contact surface, while that with the lower hardness is defined as the target surface. The rigid punch is loaded with a pressure of 22.5 Mpa. Additionally, the friction coefficient between the micro-die and the copper foil is set at 0.19 in the micro-bending process. The preconditioned conjugate gradient iterative equation solver, which can be run in a shared memory parallel or distributed memory parallel program, is chosen as the equation solver to reduce solving time. The engineering drawing of the rigid die is shown in Fig. 6.
Table 1 Mechanical properties of silicon rubber materials. Hardness shore A (°)
M–R constant C 01 (Mpa)
M–R constant C 10 (Mpa)
Poisson’s ratio l
55
0.382
0.096
0.49997
ð7Þ
Fig. 3. Different recrystallization annealing processes for pure copper.
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Fig. 4. Real and virtual grain structures under different annealing temperatures. (a) Real grain structure at 450 C. (b) Virtual grain structure at 450 C. (c) Real grain structure at 500 C. (d) Virtual grain structure at 500 C. (e) Real grain structure at 550 C. (f) Virtual grain structure at 550 C. (g) Real grain structure at 600 C. (h) Virtual grain structure at 600 C.
3. Results and discussion 3.1. Grain boundary strengthening
Fig. 5. The geometric model of rubber deformation. (a) Punch. (b) Rubber. (c) Micro-die. (d) Pure copper foil.
Fig. 6. The engineering drawing of the rigid die.
Fig. 7 shows the micro-bending process step by step. Firstly, under the punch force, rubber self-deformation occurs, as shown in Fig. 7(a). Secondly, as the punch force increases, the deformation of rubber increases and bending deformation occurs in the centre of the pure copper foil, as shown in Fig. 7(b). Then, the foil generates deformation at the fillets of the micro die. The forming depth increases due to the force generated by the rubber self-deformation, as shown in Fig. 7(c). Finally, as shown in Fig. 7 (d), it can be observed that the maximum forming depth occurs at the centre of the foil. Fig. 8 shows the von mises stress distribution of the formed parts annealed at 550 °C. It is shown that the maximum von mises stress reaches 1280 Mpa in the grain boundary, while it approximates to 300 Mpa in the grain interior. The von mises stress in the grain boundary is rather higher than that in the grain interior. Stress concentration occurs in the grain boundary during the micro bending process. During plastic deformation, dislocation pile-up appears in the grain boundary with the increasing strain due to the dislocation motion. Due to Taylor’s relationship, the flow stress increases with the dislocation density. The flow stress of the grain boundary is higher than that of the grain interior. The grain boundary, with higher flow stress, constitutes the rigid meshing structure in the workpiece. This results in the stress concentration in the grain boundary. Fig. 9 reveals the von mises stress changes with time at different locations in Fig. 8. In the initial stage of the forming process (0–2 s), the von mises stress in the bottom location is higher than that in the fillet location because the formation of bending firstly
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Fig. 7. The typical stages of deformation with contours of Y-displacement distributions (d = 21.6 lm). (a) First stage. (b) Second stage. (c) Third stage. (d) Final deforming.
Fig. 8. The contours of von mises stress distribution (d = 21.6 lm).
appears at the centre of the copper foil. Similarly, plastic deformation at the centre is larger than that in the fillet of the micro parts. Consequently, the stress concentration firstly appears at the bottom of the workpiece. As the load continues to increase, deformation occurs in the fillets. The von mises stress of the grain boundary and grain interior at the fillets is higher than that at the bottom. The larger bending and tensile deformation at the fillets leads to
Fig. 9. The change of von mises stress with time.
higher von mises stress. The fluctuation of the stress in the fillet may be caused by the change of force direction in the fillets during deformation. It is obvious that there is quite a difference between fillet-1 and fillet-2 in the grain boundary in terms of stress. The von mises stress of the grain boundary and grain interior in fillet-2 is rather higher than that in fillet-1. This may be due to the different volume fractions of the grain boundary at the locations of the fillets. From Fig. 8, it can be observed that compared with fillet-1, fillet-2 has fewer grain boundary structures. Due to the grain boundary with higher flow stress, more grain boundary structures would improve the structural rigidity. As a result, the plastic deformation that occurred in fillet-2 is larger than that in fillet-1. The von mises stress in fillet-2 is higher than that in fillet-1. The von mises stress and von plastic strain along the path in Fig. 8 are shown in Fig. 10. It is clear that the grain boundary stress is significantly higher than the grain interior stress. The grain boundary strain is a little lower than that of the grain interior. The grain boundary with higher flow stress may cause the behaviour of grain boundary strengthening. Moreover, it is found that the stress and strain rise along the path. In the micro forming process, the tensile and bending deformations are generated at the bottom of the micro-parts. The tensile stress generated by the tensile deformation is compensated by the compressive stress generated by the bending deformation in the upside. Therefore, the
Fig. 10. Von mises plastic strain and von mises stress distributions along the path.
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stress located in the underside is higher than that in the upside at the bottom of the micro-parts.
reduction of grain boundary constraints when the grain size increases.
3.2. The distribution of von mises plastic strain and von mises stress
3.3. Grain size effect on the forming depth
Fig. 11 shows the contours of von mises plastic strain distribution for different grain sizes. It is shown that the strain at the fillets is higher than at other locations. The plastic strain in the grain interior is higher than the grain boundary plastic strain. Compared with the grain interior, the formability of the grain boundary is relatively difficult. In the grain boundary, the dislocation slip is restricted because the atomic arrangement is irregular and the lattice has a serious distortion in the grain boundary. More slip systems are needed to meet the demand of compatibility of the deformation. Therefore, formability is difficult in the grain boundary. In the fine-grained micro-parts (d = 10.8 lm, d = 15.5 lm, d
A series of numerical simulations are conducted to investigate the grain size effect on the forming depth in flexible microbending processes. In the field of micro-forming, the ratio of workpiece thickness to grain size (N = t/d) is the critical evaluating criterion that affects the deformation behaviour of a workpiece [6]. When the grain size and workpiece are close to the same scale, the deformation behaviour of the material is determined by several grains at the forming location. The flow stress would change with the grain size due to the volume fraction of the grain boundary and grain interior. In this paper, the outline of the micro-parts with different grain sizes is shown in Fig. 13. It is clear that the N value is the significant factor affecting the forming depth. The forming depth increases as the N value decreases. The increasing trend of the forming depth is related to the decrease of the flow stress of the workpiece. As the N value decreases, the grain boundary strengthening decreases because of the decrease of the GB volume fraction [38]. Therefore, the flow stress of workpieces would decrease as the N value decreases. In this paper, the results agree well with other research findings. Lai [39] found that a grain had smaller flow stress on the surface of a workpiece because surface grains have fewer grain boundary constraints. As the N value decreases, the increase of volume fraction of the surface layer results in the decrease of flow stress of the workpiece. By using the Hall–Petch model, the decrease of flow stress with the increase of grain sizes can be clearly revealed. An experiment is conducted to verify the rationality of the numerical simulation. The experimental equipment used for micro-bending is shown in Fig. 14. In this paper, the universal testing machine is used as the loader. The outline of the micro-parts in
= 21.6 lm), the maximum plastic strain concentrates at the junction area of the grain boundary and grain interior. However, in the coarse-grained micro-parts (d = 37.8 lm), the maximum plastic strain is not only concentrated at the junction area of the grain boundary and grain interior but also at the surface of the grain interior. The cracked location of the workpiece can be predicated by the von mises strain distribution. Therefore, the fracture of the fine-grained workpiece may occur at the junction area of the grain boundary and grain interior. The coarse-grained workpiece may be more likely to produce a transgranular fracture. The von mises stress at the bottom of micro-parts with different grain sizes is shown in Fig. 12. From Fig. 12, the behaviour of grain boundary strengthening is also obviously shown. The von mises stress in the grain boundary is higher than that in the grain interior. Moreover, with the increase of grain size, the minimum stress increases from 8.84 Mpa to 47.66 Mpa. This finding indicates that the deformation increases as the grain size increases. The increasing plastic deformation of the grain interior may be caused by the
Fig. 11. The contours of von mises plastic strain distribution for different grain sizes. (a) d = 10.8 lm, (b) d = 15.5 lm, (c) d = 21.6 lm, and (d) d = 37.8 lm.
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Fig. 12. The contours of von mises stress distribution at the bottom of forming parts. (a) d = 10.8 lm, (b) d = 15.5 lm, (c) d = 21.6 lm, and (d) d = 37.8 lm.
Fig. 13. The simulated deformation of different grain sizes.
Fig. 15. The experimental deformation of different grain sizes.
of the forming depth in the numerical simulation agrees well with the experimental study. However, the forming depth in the numerical simulation is slightly smaller than that in the experimental study. In the study of numerical simulation, only the dislocation slip is discussed in terms of the plastic deformation. The subgrain formation, grain boundary sliding and grain rotation have been ignored. Furthermore, the initial dislocation density may have errors in the grain boundary and grain interior. Therefore, the forming depth in the experiment is higher than that in the numerical simulation.
3.4. Grain size effect on surface asperity
Fig. 14. Experimental equipment used for micro-bending. 1. Rigid punch. 2. Container. 3. Base. 4. Rubber. 5. Pure copper foil. 6. Micro-die.
the experimental micro-bending process is shown in Fig. 15. It is found that the forming depth increases as the N value decreases. By comparing Fig. 13 with Fig. 15, it can be found that the trend
The properties of each grain have an effect on the deformation behaviour when there are only a few grains along the workpiece thickness direction. Homogeneous deformation is affected by the random characteristics of grain size. It is found that inhomogeneous deformation occurs in micro-parts with larger grain size due to the grained heterogeneity (e.g., grain size, shape and deformability) [22]. In this study, the homogeneous deformation behaviours affected by grain size are studied by using the numerical simulation method. Information of the node coordinates is extracted from the results of the simulation to obtain the surface roughness profile of the micro parts. The bottom surface node coordinates are shown in Fig. 16. For the purpose of quantitatively assessing the homogeneous
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Fig. 16. The simulated roughness profile at the bottom of micro parts with different grain sizes. (a) d = 10.8 lm, (b) d = 15.5 lm, (c) d = 21.6 lm, and (d) d = 37.8 lm.
deformation, it is necessary to define the centreline, which could describe the geometric shape. In this study, the contour is obtained by using the least squares principle. The least squares centreline is shown in Fig. 16. It can be observed that with the increase of grain size, the fluctuation of nodes on the surface around the centreline becomes obvious and the inhomogeneous deformation could result in the increase of surface asperity. To assess the surface roughness profile quantitatively, we must adopt the parameters and their values to indicate the degree of roughness. The arithmetical mean deviation (Ra) of the profile is used to evaluate the roughness. The value of Ra is approximately calculated by:
Ra ¼
n 1X jzi j n 1
ð8Þ
where n is the number of collected nodes and zi means the ordinate value of the ith node to the centreline. The surface asperity of the micro parts with different grain sizes is shown in Fig. 17. It can be observed that the asperity increases as the grain size increases. The increase of asperity may be caused by the surface grains with
Fig. 17. The simulated arithmetical mean deviation (Ra) of the profile.
fewer grain boundary constraints as the grain sizes increase. At the same time, compared with the grain boundary, the grain interior deformation contributes more to the overall deformation in the micro-bending process. The increase of volume fraction of the grain interior caused by the increase of grain sizes may lead to inhomogeneous deformation and then the increase of surface asperity. An experiment is conducted to study the size effect on surface asperity of the formed micro parts. A high resolution true colour confocal microscope (Axio CSM 700) is employed to obtain the 3D morphology of the formed parts. The rough surface at the bottom of the micro parts is shown in Fig. 18. From the surface topography, it can be observed that the surface becomes rougher as the grain size increases. Inhomogeneous deformation could be caused by the decrease of slip systems and grain boundary constraints in the coarse-grained workpiece. With the increase of grain size, constraints with adjacent grains may weaken due to the decrease of the grain boundary structure. Therefore, grains could be more easily moved out of the surface of the coarse-grained workpiece. On the other hand, the total slip systems increase as the grain size decreases. For the fine-grained workpiece, each grain only undergoes a relatively small plastic deformation to maintain the compatibility of the strain [40]. In the coarse-grained workpiece, grains need additional shear deformation and rotational deformation to maintain the compatibility of the strain [12]. In conclusion, inhomogeneous deformation could occur in coarse-grained micro parts. The arithmetical mean deviation (Ra) of the bottom region is measured to evaluate the effects of grain size effect on the surface roughness. The surface asperity of micro parts with different grain sizes is shown in Fig. 19. The surface asperity has a tendency to increase with grain size. Figs. 17 and 19 show that the trends of surface asperity in the simulated results agree well with the experimental results. However, it is found that the surface roughness of the experimental results is higher than that in the simulated results because the numerical simulation only considers the grain size and constraints of the grain boundary. Furthermore, the surface roughness of the as-received material has not been discussed in the numerical simulation.
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Fig. 18. Surface topography of micro parts with different grain sizes. (a) d = 10.8 lm, (b) d = 15.5 lm, (c) d = 21.6 lm, and (d) d = 37.8 lm.
Fig. 19. The experimental arithmetical mean deviation (Ra) of the profile.
4. Conclusions In this study, the effects of size on deformation behaviour in the flexible micro-bending process are investigated using the numerical simulation method. The micro structures of the grain boundary and grain interior are established via Voronoi tessellation, which is employed to describe the polycrystalline aggregation. The flow stress of the grain boundary and grain interior are established by using the dislocation density-based model, which can better reflect the behaviour of grain boundary strengthening. The following conclusions can be made based on the numerical simulated results: (1) In the flexible micro-bending process, the von mises stress in the grain boundary is significantly higher than that in the
grain boundary. The behaviour of grain boundary strengthening occurs obviously in the micro forming process. (2) The maximum von mises plastic strain mainly concentrates at the junction of the grain interior and grain boundary in fine-grained micro-parts, while it concentrates at the surface of the grain interior in coarse-grained micro-parts. (3) It is found that the forming depth increases as the ratio of workpiece thickness to grain size decreases. The results of numerical simulations agree well with the experimental results. (4) Inhomogeneous deformation occurs in coarse-grained micro-parts according to the numerical simulation and experiment. Furthermore, the surface roughness increases as the grain sizes increase.
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Acknowledgments This work is supported by the National Natural Science Foundation of China (NO. 51175235) and the Natural Science Foundation of Jiangsu Province (NO. BK2012712). References [1] K.F. Zhang, L. Kun, J. Mater. Process. Technol. 209 (11) (2009) 4949–4953. [2] F. Vollertsen, D. Biermann, H.N. Hansen, et al., CIRP Annals-Manuf. Technol. 58 (2) (2009) 566–587. [3] T.G. Nieh, J. Wadsworth, Scr. Metall. Mater. 25 (4) (1991) 955–958. [4] D. Shan, C. Wang, B. Guo, et al., Trans. Nonferr. Metals Soc. China 19 (2009) 507–510. [5] L.V. Raulea, A.M. Goijaerts, L.E. Govaert, et al., J. Mater. Process. Technol. 115 (1) (2001) 44–48. [6] C. Keller, E. Hug, X. Feaugas, Int. J. Plast. 27 (4) (2011) 635–654. [7] P. Janssen, T.H. De Keijser, M. Geers, Mater. Sci. Eng., A 419 (1) (2006) 238–248. [8] D. Leu, J. Manuf. Sci. Eng. 131 (1) (2009) 011002. [9] S. Miyazaki, K. Shibata, H. Fujita, Acta Metall. 27 (5) (1979) 855–862. [10] W.L. Chan, M.W. Fu, J. Lu, Mater. Des. 32 (1) (2011) 198–206. [11] X. Lin, G. Wang, W. Zheng, et al., Trans. Nonferr. Metals Soc. China 24 (2) (2014) 470–476. [12] X. Wang, Y. Ma, Z. Shen, et al., J. Mater. Process. Technol. 220 (2015) 173–183. [13] U.F. Kocks, Metall. Mater. Trans. 1 (5) (1970) 1121–1143. [14] M.A. Meyersm, E. Ashworth, Philos. Mag. A 46 (5) (1982) 737–759. [15] J.G. Liu, M.W. Fu, J. Lu, et al., Comput. Mater. Sci. 50 (9) (2011) 2604–2614. [16] M.W. Fu, B. Yang, W.L. Chan, J. Mater. Process. Technol. 213 (1) (2013) 101–110. [17] U.F. Kocks, J. Eng. Mater. Technol. 98 (1) (1976) 76–85. [18] H. Fu, D.J. Benson, M.A. Meyers, Acta Mater. 49 (13) (2001) 2567–2582.
143
[19] S. Geißdörfer, U. Engel, M. Geiger, Int. J. Mach. Tools Manuf 46 (11) (2006) 1222–1226. [20] C. Wang, B. Guo, D. Shan, Trans. Nonferr. Metals Soc. China 21 (6) (2011) 1362–1366. [21] A. Molotnikov, R. Lapovok, C.F. Gu, et al., Mater. Sci. Eng., A 550 (2012) 312–319. [22] H.N. Lu, D.B. Wei, Z.Y. Jiang, et al., Comput. Mater. Sci. 77 (2013) 44–52. [23] H. Zhang, X. Dong, Q. Wang, et al., A dislocation density based crystal plasticity model of 3D complex sheet metal forming, The 8th international conference and workshop on numerical simulation of 3D sheet metal forming processes, 2011. [24] H. Liu, W. Pantleon, L. Mishnaevsky Jr., Comput. Mater. Sci. 83 (2014) 318–330. [25] H. Mecking, U.F. Kocks, Acta Metall. 29 (11) (1981) 1865–1875. [26] C.W. Sinclair, W.J. Poole, Y. Bréchet, Scripta Mater. 55 (8) (2006) 739–742. [27] Y. Estrin, L.S. Toth, A. Molinari, et al., Acta Mater. 46 (15) (1998) 5509–5522. [28] E.I. Galindo-Nava, J. Sietsma, P. Rivera-Díaz-Del-Castillo, Acta Mater. 60 (6) (2012) 2615–2624. [29] Y. Estrin, Dislocation-density-related constitutive modeling, in: A.S. Krausz, K. Krausz (Eds.), Unified constitutive laws of plastic deformation, vol. 1, Academic Press Inc., New York City, 1996, pp. 69–106. [30] F. Prinz, A.S. Argon, W.C. Moffatt, Acta Metall. 30 (4) (1982) 821–830. [31] J.C. Fisher, J. Appl. Phys. 22 (1) (1951) 74–77. [32] Y.H. Zhao, X.Z. Liao, Y.T. Zhu, et al., Mater. Sci. Eng., A 410 (2005) 188–193. [33] M.H. Dirikolu, E. Akdemir, J. Mater. Process. Technol. 148 (3) (2004) 376–381. [34] B.N. Boots, D.J. Murdoch, Comput. Geosci. 9 (3) (1983) 351–365. [35] G. Cailletaud, S. Forest, D. Jeulin, et al., Comput. Mater. Sci. 27 (3) (2003) 351–374. [36] P. Zhang, M. Karimpour, D. Balint, et al., Comput. Mater. Sci. 64 (2012) 84–89. [37] H. Liu, L. Mishnaevsky, Acta Mater. 71 (2014) 220–233. [38] M.W. Fu, W.L. Chan, Mater. Des. 32 (10) (2011) 4738–4746. [39] X. Lai, L. Peng, P. Hu, et al., Comput. Mater. Sci. 43 (4) (2008) 1003–1009. [40] Z. Gao, L. Peng, P. Yi, et al., J. Mater. Process. Technol. 219 (2015) 28–41.