**4. Novel material model based on Hall-Petch relationship in microforming**

Size effects in microforming cannot be conveyed by the classical theory of continuum plastic mechanics, which is scale-independent. The specimen size effects on the flow stress of polycrystalline Cu-Al alloy have been investigated, and the fact that the flow stress decreases with the dimensional reduction of specimen has been explained by the proposed affect zone model [24]. A flow stress model, a function of the ratio of the sheet thickness to grain size, has been established based on Hall-Petch relationship, dislocation pile-up theory, and affect zone model [25]. A mixed material model based on modified Hall-Petch relationship, surface layer model, and grained heterogeneity is proposed, and the 3D aggregate of polycrystalline is represented by a Voronoi tessellation. The effect of grain size on flow stress is an important aspect of polycrystalline metal plastic deformation. The simulation of microforming processes (micro cross wedge rolling (MCWR), micro flexible rolling and micro V-bending) have been conducted with the consideration of size effects from grain size and feature size. The validation of the proposed material model will be conducted by physical experiments through the comparison between experimental results and simulation ones.

Fundamentals have been developed to build up a FE model considering the occurrence of size effects at microscale by using the ANSYS/LS-DYNA program. The newly developed material model is implemented considering grained heterogeneity. As shown in **Figure 16**, two forming tools and a cylindrical workpiece of 0.831.2 mm<sup>2</sup> are meshed in solid element 164 with an 8-noded structure. In order to reduce computational time and ensure stability in large deformation, viscous hourglass control and one-point integration were applied for all elements. For each grain size, 10 different polycrystalline aggregates of workpiece were generated stochastically by the algorithm of 3D Voronoi tessellation. The simulation was performed by applying equal and opposite velocities to forming tools in the horizontal (x) direction. In whole process, the workpiece is left unconstrained, and the tools are held in the vertical (y) direction and in the out-of-plane (z) direction [12]. **Figure 17** shows the process of forging shape during micro cross wedge rolling.

mechanical properties [12–14]. The location of the maximum strain and stress cannot be deter-

**Figure 18.** Distribution of effective strain of the medial section in axial direction (a) uniform material properties and

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The stress and strain distribution on the profile for the halved 250 μm thick workpiece consisting of grains with the average grain size of 250 μm is illustrated in **Figure 19**. The stress-strain distribution is inhomogeneous because only some grains are in plastic regime while others

**Figure 19.** Stress-strain distribution on the profile after springback in micro flexible rolling: (a) von Mises stress

still undergo elastic strain regime during the flexible rolling process [15].

(b, c, d) inhomogeneous material properties with grain sizes of 6, 40, and 120 μm, respectively.

mined easily as that in the conventional CWR process.

distribution and (b) equivalent plastic strain distribution.

Laminar cricoid distribution of strain is typical in conventional CWR with homogeneous material properties and also exists in MCWR where billet material is homogeneous (**Figure 18a**). However, the grained heterogeneity effects on the metal deformability and strain distribution should be considered in microscale forming. It is shown in **Figure 18b**–**d** that the continuous laminar distribution of strain in the workpiece has been disturbed due to the inhomogeneous

**Figure 17.** Process of forging shape during MCWR.

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layer model, and grained heterogeneity is proposed, and the 3D aggregate of polycrystalline is represented by a Voronoi tessellation. The effect of grain size on flow stress is an important aspect of polycrystalline metal plastic deformation. The simulation of microforming processes (micro cross wedge rolling (MCWR), micro flexible rolling and micro V-bending) have been conducted with the consideration of size effects from grain size and feature size. The validation of the proposed material model will be conducted by physical experiments through the

Fundamentals have been developed to build up a FE model considering the occurrence of size effects at microscale by using the ANSYS/LS-DYNA program. The newly developed material model is implemented considering grained heterogeneity. As shown in **Figure 16**, two form-

8-noded structure. In order to reduce computational time and ensure stability in large deformation, viscous hourglass control and one-point integration were applied for all elements. For each grain size, 10 different polycrystalline aggregates of workpiece were generated stochastically by the algorithm of 3D Voronoi tessellation. The simulation was performed by applying equal and opposite velocities to forming tools in the horizontal (x) direction. In whole process, the workpiece is left unconstrained, and the tools are held in the vertical (y) direction and in the out-of-plane (z) direction [12]. **Figure 17** shows the process of forging

Laminar cricoid distribution of strain is typical in conventional CWR with homogeneous material properties and also exists in MCWR where billet material is homogeneous (**Figure 18a**). However, the grained heterogeneity effects on the metal deformability and strain distribution should be considered in microscale forming. It is shown in **Figure 18b**–**d** that the continuous laminar distribution of strain in the workpiece has been disturbed due to the inhomogeneous

are meshed in solid element 164 with an

comparison between experimental results and simulation ones.

110 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

ing tools and a cylindrical workpiece of 0.831.2 mm<sup>2</sup>

shape during micro cross wedge rolling.

**Figure 17.** Process of forging shape during MCWR.

**Figure 18.** Distribution of effective strain of the medial section in axial direction (a) uniform material properties and (b, c, d) inhomogeneous material properties with grain sizes of 6, 40, and 120 μm, respectively.

mechanical properties [12–14]. The location of the maximum strain and stress cannot be determined easily as that in the conventional CWR process.

The stress and strain distribution on the profile for the halved 250 μm thick workpiece consisting of grains with the average grain size of 250 μm is illustrated in **Figure 19**. The stress-strain distribution is inhomogeneous because only some grains are in plastic regime while others still undergo elastic strain regime during the flexible rolling process [15].

**Figure 19.** Stress-strain distribution on the profile after springback in micro flexible rolling: (a) von Mises stress distribution and (b) equivalent plastic strain distribution.

**Figure 20** shows the tension effect on the average springback from the proposed models. Regardless of the initial thickness and pass reduction, the springback decreases moderately when the front and back tensions increase in increments of 25 MPa from 0 to 100 MPa. For thicker workpiece, front and back tensions have a significant influence on eliminating the springback due to that front and back tensions are able to improve metal flow and relax residual stresses and then increase the thickness precision of rolled workpiece.

In **Figure 21**, it can be seen that the thickness springback increases as the initial workpiece thickness decreases. For a certain grain size, the grain number decreases in thickness direction for less thick workpiece. Therefore, the effect of each single grain plays a very significant role on the springback resulting in larger springback value. For each thickness, the curves are in similar trends under different reductions, and the springback difference is below 10.5% for each grain size, which is close to the simulation result.

Micro V-bending process is simulated with an implicit FEM package: ABAQUS/Standard. The processing parameters in the simulation are the same as those in physical experiments, and the value of coefficient of friction is set to be 0.02. The FE model of micro V-bending with Voronoized specimen is shown in **Figure 22a**. **Figure 22b** illustrates the grain heterogeneity in Voronoized specimen, among which different colors represent different mechanical properties of grains. It is shown in **Figure 23** that the upper bound grain plastic property is illustrated by dark blue (six grains), while light blue (six grains) is for the lower bound grain plastic property [16]. The FE model is close to real physical test condition as the right and the left sides of the sample are not equal in terms of grain size and the scatter of mechanical properties of grains, rather than set up as a traditional asymmetrical one.

**Figure 24** shows the simulation result of micro V-bending. The inhomogeneous deformation occurs significantly during bending process. The different colors in middle deformation zone represent that different grains have undergone different deformation because of grain heterogeneity. In the bending process, some grains first reach their yield stress and undergo plastic deformation prior to other grains. Even the workpiece has started the plastic deformation, some grains with higher yield stress may still be under elastic stress condition. This sort of grain heterogeneous deformation could influence the springback significantly and should be taken into account in numerical simulation of microforming [16].

**Figure 21.** Springback in thickness direction versus gain size for initial workpiece thickness of 100 μm: (a) 20% reduction

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**Figure 22.** FEM simulation of micro V-bending with (a) Voronoi tessellations and (b) grain heterogeneity.

and (b) 50% reduction.

**Figure 20.** Relationship between average springback in thickness direction and front and back tensions for initial workpiece thickness of 100, 250, and 500 μm: (a) 20% reduction and (b) 50% reduction.

**Figure 20** shows the tension effect on the average springback from the proposed models. Regardless of the initial thickness and pass reduction, the springback decreases moderately when the front and back tensions increase in increments of 25 MPa from 0 to 100 MPa. For thicker workpiece, front and back tensions have a significant influence on eliminating the springback due to that front and back tensions are able to improve metal flow and relax resid-

In **Figure 21**, it can be seen that the thickness springback increases as the initial workpiece thickness decreases. For a certain grain size, the grain number decreases in thickness direction for less thick workpiece. Therefore, the effect of each single grain plays a very significant role on the springback resulting in larger springback value. For each thickness, the curves are in similar trends under different reductions, and the springback difference is below 10.5% for

Micro V-bending process is simulated with an implicit FEM package: ABAQUS/Standard. The processing parameters in the simulation are the same as those in physical experiments, and the value of coefficient of friction is set to be 0.02. The FE model of micro V-bending with Voronoized specimen is shown in **Figure 22a**. **Figure 22b** illustrates the grain heterogeneity in Voronoized specimen, among which different colors represent different mechanical properties of grains. It is shown in **Figure 23** that the upper bound grain plastic property is illustrated by dark blue (six grains), while light blue (six grains) is for the lower bound grain plastic property [16]. The FE model is close to real physical test condition as the right and the left sides of the sample are not equal in terms of grain size and the scatter of mechanical

**Figure 24** shows the simulation result of micro V-bending. The inhomogeneous deformation occurs significantly during bending process. The different colors in middle deformation zone represent that different grains have undergone different deformation because of grain heterogeneity. In the bending process, some grains first reach their yield stress and undergo plastic deformation prior to other grains. Even the workpiece has started the plastic deformation, some grains with higher yield stress may still be under elastic stress condition. This sort of grain heterogeneous deformation could influence the springback significantly and should be

**Figure 20.** Relationship between average springback in thickness direction and front and back tensions for initial

ual stresses and then increase the thickness precision of rolled workpiece.

properties of grains, rather than set up as a traditional asymmetrical one.

taken into account in numerical simulation of microforming [16].

workpiece thickness of 100, 250, and 500 μm: (a) 20% reduction and (b) 50% reduction.

each grain size, which is close to the simulation result.

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**Figure 21.** Springback in thickness direction versus gain size for initial workpiece thickness of 100 μm: (a) 20% reduction and (b) 50% reduction.

**Figure 22.** FEM simulation of micro V-bending with (a) Voronoi tessellations and (b) grain heterogeneity.

**Figure 23.** Grain properties randomly assigned on a Voronoized bending sample.

**5. Modified FEM with the consideration of material and lubrication** 

**Distribution group 1 2 3 4 5 6 7 Average** Springback angle 30.52 27.37 32.09 31.78 33.44 28.65 34.12 31.14

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**Figure 26** represents a typical EBSD microstructure. First, the EBSD image was input into the MATLAB software, and the binary image was obtained with black grain boundaries and white grains. Noise and small holes were eliminated in the transformation. Then, the Moore-Neighbor tracing algorithm modified by Jacob's stopping criteria was applied in the binary image treatment. As shown in **Figure 26b**, the information of grains and individual closed subareas, including single grain's area, geometrical center and geometrical orientation, was detected and sorted in MATLAB. The blue ports in **Figure 26b** are the grain's geometrical centers [9, 20, 21]. **Figure 27** displays the Voronoi structures and their corresponding FE models with average

After annealed at 1100°C, the 50 μm thick blanks, with equiaxed crystals microstructure and average grain size of 40 μm, were drawn into micro cups. The drawn cup mouth is shown in **Figure 28a**, and the maximum thickness distributions of drawn cups are illustrated in **Figure 28b**–**d**, which represented the new developed model, a Voronoi model without the consideration of grain boundaries and a normal model in homogeneous material properties, respectively. The comparison of the maximum wall thickness between the simulation and the experimental results has been conducted. The localized deformation is ignored, and the maximum thickness was averaged with the lowest peak thickness values for all the simulation cases. It can be seen that the new model and the Voronoi model considered microscopic heterogeneity have higher maximum thickness than that in the normal model [9], where the largest thickness is obtained from the Voronoi model without grain boundaries buffer.

**characterization in MDD and MHDD**

**Table 2.** Springback angles from FE simulation (degree).

**Figure 25.** Bending specimens with different gain heterogeneity distributions.

grain sizes of 10, 20, and 40 μm, respectively.

**Figure 24.** (a) Final angle after springback and (b) von Mises stress distribution.

Seven plastic properties are obtained by experiment and calculation, and they are randomly distributed in bending specimens. Specimens with different random grain heterogeneity distributions are exhibited in **Figure 25**, which are called models "1," "2," "3," "4," "5," "6," and "7," respectively. Seven groups of micro V-bending FE simulations have been conducted with above-mentioned seven specimens individually. Springback angles of seven simulations and an average value are measured and calculated, as shown in **Table 2**.

**Figure 25.** Bending specimens with different gain heterogeneity distributions.


**Table 2.** Springback angles from FE simulation (degree).

Seven plastic properties are obtained by experiment and calculation, and they are randomly distributed in bending specimens. Specimens with different random grain heterogeneity distributions are exhibited in **Figure 25**, which are called models "1," "2," "3," "4," "5," "6," and "7," respectively. Seven groups of micro V-bending FE simulations have been conducted with above-mentioned seven specimens individually. Springback angles of seven simulations and

an average value are measured and calculated, as shown in **Table 2**.

**Figure 24.** (a) Final angle after springback and (b) von Mises stress distribution.

**Figure 23.** Grain properties randomly assigned on a Voronoized bending sample.

114 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques
