**2. FEM analysis applied to thin strip rolling**

The three-dimensional (3D) finite element method (FEM) has been used in the analysis of strip rolling, shape rolling, and slab rolling, and Jiang et al. [4–6, 22] used this finite element method to solve special-shaped strip rolling. This is a major drawback to producing accurate and reliable models for the cold rolling of thin strip due to the lack of well-defined friction boundary conditions. The 3D rigid plastic FEM has been proposed to solve the thin strip rolling considering friction variation in the deformation zone, and the comparison between the computed results and measured values has also been made.

In the friction variation model, the friction varies along the contact length of the deformation zone. The frictional shear stress model is modified as [5]:

$$\tau\_{\prime} = K\_{\prime} \frac{m\_{\ast} \sigma\_{\ast}}{\sqrt{3}} \left( \frac{2}{\pi} \tan^{-1} \left\{ \frac{V\_{\varepsilon}}{k\_{\ast}} \right\} \right) \tag{1}$$

where *m*<sup>1</sup> is the friction factor; *σ<sup>s</sup>* yields stress; *Ki* is a coefficient of the friction shear stress changes with *K*<sup>1</sup> and *K*<sup>2</sup> for forward slip zone and backward slip zone, respectively; *ki* is a positive constant with *k*<sup>1</sup> and *k*<sup>2</sup> for forward slip zone and backward slip zone, respectively; *Vg* is relative slip velocity between the strip and the roll and can be obtained by:

$$\text{relative slip velocity} \overset{\circ}{\text{between the strip and the roll and can be obtained by:}}$$

$$V\_g = \sqrt{\upsilon\_\text{v} \sec \theta - V\_\text{rl})^2 + \upsilon\_y^2} \tag{2}$$

where *vx* and *vy* are the velocity components in the *x* and *y* directions, respectively, *β* is the angular position of the node, *VR* is the tangential velocity of the roll, and the distribution of these frictional shear stress models is shown in **Figure 1**.

As shown in **Figure 2**, a quarter of the strip was studied. Isoparametric hexahedral elements were applied with eight Gauss points throughout the deformed workpiece. The element number in *x*, *y*, and *z* directions are 10, 8, and 5, respectively, and totally there are 594 nodes and 400 elements.

From the simulation with low carbon steel, **Figures 3** and **4** show the effect of reduction on rolling pressure and spread of strip for different *k*<sup>2</sup> and constant *k*<sup>1</sup> = 0.1. *k*<sup>2</sup> influences the simulation results significantly where the rolling pressure increases with decreased *k*<sup>2</sup> . When *k*2 value is below 0.1, the rolling pressure calculation value is in agreement with the measured one. The spread calculation value for *k*<sup>2</sup> = 0.1 is also close to the measured one when the reduction is less than 43%. For *k*<sup>2</sup> = 0.1, the change of *k*<sup>1</sup> also has an effect on the simulation results, as shown in **Figures 5** and **6**. It can be seen that the calculated results are in good agreement with the measured values for *k*<sup>1</sup> = 0.1. Therefore, the simulation results are close to measured values when *k*<sup>1</sup> and *k*<sup>2</sup> are less than 0.1.

The rolling of copper strip is simulated with work roll diameter 158.76 mm, width of strip 76.2 mm, rolling speed 0.16 m/s, and friction factor *m*<sup>1</sup> = 0.4. For case 1, *K*<sup>1</sup> = *K*<sup>2</sup> = 1.0 and *k*<sup>1</sup> = *k*<sup>2</sup> = 0.1

**Figure 1.** Frictional shear stress models.

results where the friction is changeable in the roll bite [1], and the rolling pressure and model control accuracy will be influenced significantly. The deformation mechanics of thin foil [2] and the foil rolling with constant friction during cold rolling [3] have been investigated. The finite element method has been proposed in special-shaped strip rolling [4–6], particularly with variable friction models [5, 6]. Considering modeling accuracy, a friction variation model

The application of crystal plasticity finite element method (CPFEM) has been introduced in the simulation of surface asperity flattening in cold quasistatic uniaxial planar compression process. Rate-dependent crystal plasticity constitutive models have been established on the basis of experimental conditions [7], and the influences of the reduction and strain rate on the surface roughness are investigated using the 3D crystal plasticity finite element method [8]. The experimental results are also employed in the 3D CPFEM model and compared with the

Microforming differs from the conventional forming technology in terms of materials, processes, tools, and machines and equipment due to the miniaturization nature of the whole microforming system [9]. It is impossible to scale down all parameters in the microforming process according to the theory of similarity due to the existence of size effects in microforming processes. A number of unexpected problems in key aspects of mechanical behavior, tribology, and scatter of material behavior are encountered [10, 11]. Challenges remain in the high efficiency manufacturing of high-quality microproducts due to the common problem of microscale size effects [9, 11], complexity of processes for making microproducts, and the ever

In Section 4, novel material model with grained heterogeneity in 3D Voronoi tessellation has been developed in the simulation of micro cross wedge rolling, springback analysis in micro flexible rolling and the micro V-bending processes considering grain boundary and generation process of grains in the workpiece [12–16]. The modified FE model in microforming has been applied with the consideration of size effects including material characterization, friction/contact characterization, and other size-related factors presented in Section 5. Open and closed lubricate pocket (OCLP) theory and size-dependent friction coefficient are proposed in micro deep drawing (MDD) and micro hydromechanical deep drawing (MHDD) [17–19]. Real microstructures and Voronoi structures are applied in microstructural models through

The three-dimensional (3D) finite element method (FEM) has been used in the analysis of strip rolling, shape rolling, and slab rolling, and Jiang et al. [4–6, 22] used this finite element method to solve special-shaped strip rolling. This is a major drawback to producing accurate and reliable models for the cold rolling of thin strip due to the lack of well-defined friction boundary conditions. The 3D rigid plastic FEM has been proposed to solve the thin strip rolling considering friction variation in the deformation zone, and the comparison between the

should be introduced in the cold rolling simulation of thin strip.

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

increasing requirement to improve product quality and performance.

the image-based modeling method [20, 21].

**2. FEM analysis applied to thin strip rolling**

computed results and measured values has also been made.

simulation results.

**Figure 4.** Effect of *K*<sup>2</sup>

**Figure 5.** Effect of *K*<sup>1</sup>

**Figure 6.** Effect of *K*<sup>1</sup>

on spread.

on spread.

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on rolling pressure.

**Figure 2.** One-quarter of the deforming workpiece.

**Figure 3.** Effect of *K*<sup>2</sup> on rolling pressure.

and case 2, *K*<sup>1</sup> = 0.7, *K*<sup>2</sup> = 1.4, and *k*<sup>1</sup> = *k*<sup>2</sup> = 0.1. The friction variation in the roll bite has a significant effect on the spread as shown in **Figure 7**, where the spread calculated through the constant friction model is greater than the result obtained from the friction variation one, and the spread increases with an increase of reduction. The spread decreases when *K*<sup>1</sup> increases and *K*<sup>2</sup> decreases (in case 2) due to the increased forward slip as shown in **Figure 8**, more metal flows along the rolling direction, resulting in a decrease of the transverse flow of metal. It is found in **Figure 7** that the effect of friction variation on spread is not significant for reduction < 25%.

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**Figure 4.** Effect of *K*<sup>2</sup> on spread.

**Figure 5.** Effect of *K*<sup>1</sup> on rolling pressure.

**Figure 6.** Effect of *K*<sup>1</sup> on spread.

and case 2, *K*<sup>1</sup>

**Figure 3.** Effect of *K*<sup>2</sup>

= 0.7, *K*<sup>2</sup>

**Figure 2.** One-quarter of the deforming workpiece.

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

= 1.4, and *k*<sup>1</sup>

on rolling pressure.

= *k*<sup>2</sup>

spread increases with an increase of reduction. The spread decreases when *K*<sup>1</sup>

cant effect on the spread as shown in **Figure 7**, where the spread calculated through the constant friction model is greater than the result obtained from the friction variation one, and the

decreases (in case 2) due to the increased forward slip as shown in **Figure 8**, more metal flows along the rolling direction, resulting in a decrease of the transverse flow of metal. It is found in **Figure 7** that the effect of friction variation on spread is not significant for reduction < 25%.

= 0.1. The friction variation in the roll bite has a signifi-

increases and *K*<sup>2</sup>

**Figure 7.** Effect of reduction on spread.

experimental results. Furthermore, the relationship between the surface asperity flattening process (surface roughness) and the above-mentioned parameters will be investigated. The

can be expressed [7]:

⨂*m<sup>α</sup>*

̇ and the resolving shear stress *τ*(*α*)

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1/*m*

̇ (3)

*<sup>h</sup>αβ*|*γ*(*β*) ´ <sup>|</sup> (5)

is formulated

(4)

mechanism of surface asperity flattening will be analyzed.

**Figure 9.** Methodology of crystal plasticity finite element modeling.

̇ is the plastic shear rate of the *α*th slip system.

*<sup>p</sup> Fp*<sup>−</sup><sup>1</sup>

̇ = *γ*<sup>0</sup>

For cubic metal, the hardening equation of the slip system can be simplified as [7]:

where *h* is the hardening matric of the slip system *α* led by the slip system *β*.

´ =∑ *β*=1 *n*

̇ *sgn* (*τ*(*α*))


= ∑ *α*=1 *n*

*γ*(*α*) *S*(*α*)

Flow rule of plastic deformation gradient *F<sup>p</sup>*

The relationship between the shear rate *γ*(*α*)

*γ*(*α*)

*S*(*α*)

*F*̇

where *γ*(*α*)

below [7]:

**Figure 8.** Effect of reduction on forward slip.

### **3. Application of crystal plasticity finite element method (CPFEM)**

Little research has been done on the surface development of constraint surface (surface asperity flattening process) with CPFEM. Most current CPFEM research focus on the development of free surface (surface roughening) by uniaxial and biaxial tensile deformation. In particular, there are almost no reports that mention the relationship between the orientation of surface grains and surface roughness. The texture development of the constraint surface is also a very interesting topic. In metal forming, the strain rate contributes significantly to the workpiece work hardening, but there is little research on how the strain influences the surface roughness. A physical simulation has been conducted on an INSTRON servo-hydraulic testing machine by using a channel die. The relationship between the surface roughness and related parameters such as gauged reduction, friction, texture (grain orientation), and grain size and strain rate has been identified.

The methodology of crystal plasticity finite element modeling (**Figure 9**) follows the rules as: rate-dependent crystal plasticity constitutive models will be written into the UMAT and then used in the ABAQUS main program (geometric model). The geometric model is established based on experimental conditions (reduction, strain rate, friction, original surface roughness, and original texture information). The modeling results will be compared with the Application of Finite Element Analysis in Multiscale Metal Forming Process http://dx.doi.org/10.5772/intechopen.71880 103

**Figure 9.** Methodology of crystal plasticity finite element modeling.

**3. Application of crystal plasticity finite element method (CPFEM)**

**Figure 8.** Effect of reduction on forward slip.

**Figure 7.** Effect of reduction on spread.

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

Little research has been done on the surface development of constraint surface (surface asperity flattening process) with CPFEM. Most current CPFEM research focus on the development of free surface (surface roughening) by uniaxial and biaxial tensile deformation. In particular, there are almost no reports that mention the relationship between the orientation of surface grains and surface roughness. The texture development of the constraint surface is also a very interesting topic. In metal forming, the strain rate contributes significantly to the workpiece work hardening, but there is little research on how the strain influences the surface roughness. A physical simulation has been conducted on an INSTRON servo-hydraulic testing machine by using a channel die. The relationship between the surface roughness and related parameters such as gauged reduction, friction, texture (grain orientation), and grain size and strain rate has been identified. The methodology of crystal plasticity finite element modeling (**Figure 9**) follows the rules as: rate-dependent crystal plasticity constitutive models will be written into the UMAT and then used in the ABAQUS main program (geometric model). The geometric model is established based on experimental conditions (reduction, strain rate, friction, original surface roughness, and original texture information). The modeling results will be compared with the experimental results. Furthermore, the relationship between the surface asperity flattening process (surface roughness) and the above-mentioned parameters will be investigated. The mechanism of surface asperity flattening will be analyzed.

Flow rule of plastic deformation gradient *F<sup>p</sup>* can be expressed [7]:

$$\dot{F}^{p}F^{p^{e}} = \sum\_{a=1}^{n} \mathcal{V}\_{(a)} \mathcal{S}\_{(a)} \dot{\bigotimes} m\_a \tag{3}$$

where *γ*(*α*) ̇ is the plastic shear rate of the *α*th slip system.

The relationship between the shear rate *γ*(*α*) ̇ and the resolving shear stress *τ*(*α*) is formulated below [7]:

$$\dot{\boldsymbol{\gamma}}\_{(a)}^{\cdot} = \dot{\boldsymbol{\gamma}}\_{0} \text{sgn}\left(\boldsymbol{\tau}\_{(a)}\right) \left| \frac{\boldsymbol{\tau}\_{(a)}}{S\_{(a)}} \right|^{1/n} \tag{4}$$

For cubic metal, the hardening equation of the slip system can be simplified as [7]:

$$\mathcal{S}'\_{\alpha\beta} = \sum\_{\beta=1}^{\mu} \hbar\_{\alpha\beta} \left| \gamma'\_{\{\beta\}} \right| \tag{5}$$

where *h* is the hardening matric of the slip system *α* led by the slip system *β*.

### **3.1. Three-dimensional (3D) model**

A three-dimensional model based on crystal plasticity finite element (CPFE) is proposed according to the atomic force microscopy (AFM) experimental values where the results are sorted and applied in MATLAB for modeling the surface morphology. Every four neighboring elements at the top surface have one orientation for keeping the weight function of orientation in the model. Some elements on the top surface are refined. There are 840 C3D8R integration elements; among them 280 elements are with 70 Euler angle triplets and the others are featured by one element with one orientation. Both the tool and mold have 460 discrete rigid elements. A spatial orientation distribution has been assigned for the workpiece based on the electron backscatter diffraction (EBSD) experimental results.

The relationship between the AFM measured results, the MATLAB calculated results, and 3D CPFE model is shown in **Figure 10**. Direction 1 corresponds to the rolling direction, direction 2 to the normal, and direction 3 to the transverse direction. The three-dimensional model is 100 μm × 100 μm × 100 μm in size. Due to the small size of the sample, only a quarter of practical samples were chosen for simulation. It is considered that during the modeling, the combined slip system includes 12 {110} <111> slip systems (slip planes and slip directions). A total of 630 Euler angle triplets from the experimental results were input into ABAQUS as the initial crystallographic condition of the 3D model [7, 8]. All the parameters of simulation are taken from **Table 1** as a reference.

**3.2. Results and discussion**

*γ*̇

*3.2.1. Influence on surface roughness*

**Table 1.** Material parameter of aluminum.

*<sup>C</sup>*<sup>11</sup> 106,750 MPa *<sup>s</sup>*

*<sup>C</sup>*<sup>22</sup> 60,410 MPa *<sup>h</sup>*

*<sup>C</sup>*<sup>44</sup> 28,340 MPa *<sup>s</sup>*

*m* 0.02 *q*

*3.2.2. Influence of the strain rate on hardness*

**Figure 11.** Influence of strain rate on surface roughness.

**Figure 11** shows that the surface asperity of the samples tends to be flattened with an increase of reduction. With an increase in reduction, the sample with a higher strain rate has a higher flattened rate of surface asperity than the sample with a lower strain rate. Increasing the applied macroscopic strain rate will increase the shear rate of lip systems in the surface area. Then under the same reduction, the sample deformed at a higher strain rate will activate more slip

<sup>0</sup> 12.5 MPa

<sup>0</sup> <sup>60</sup> MPa

*<sup>s</sup>* <sup>75</sup> MPa

<sup>1</sup> 1.0 (coplanar)

1.4 (no coplanar)

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with a higher strain rate is 0.16 μm, while the sample with a lower strain rate is only 0.09 μm.

**Figure 12** shows the influence of the strain rate on the hardness of the sample, and the influence is nonlinear. There are different stages in the evolution of hardness because when the

of the sample

systems in the surface area. When the reduction is 40%, the surface roughness *Ra*

**Parameter Value Parameter Value**

<sup>0</sup> 0.001 *<sup>a</sup>* 2.25

**Figure 10.** Relationship between AFM, MATLAB, and the 3D CPFE model.


**Table 1.** Material parameter of aluminum.

#### **3.2. Results and discussion**

**3.1. Three-dimensional (3D) model**

taken from **Table 1** as a reference.

A three-dimensional model based on crystal plasticity finite element (CPFE) is proposed according to the atomic force microscopy (AFM) experimental values where the results are sorted and applied in MATLAB for modeling the surface morphology. Every four neighboring elements at the top surface have one orientation for keeping the weight function of orientation in the model. Some elements on the top surface are refined. There are 840 C3D8R integration elements; among them 280 elements are with 70 Euler angle triplets and the others are featured by one element with one orientation. Both the tool and mold have 460 discrete rigid elements. A spatial orientation distribution has been assigned for the workpiece based

The relationship between the AFM measured results, the MATLAB calculated results, and 3D CPFE model is shown in **Figure 10**. Direction 1 corresponds to the rolling direction, direction 2 to the normal, and direction 3 to the transverse direction. The three-dimensional model is 100 μm × 100 μm × 100 μm in size. Due to the small size of the sample, only a quarter of practical samples were chosen for simulation. It is considered that during the modeling, the combined slip system includes 12 {110} <111> slip systems (slip planes and slip directions). A total of 630 Euler angle triplets from the experimental results were input into ABAQUS as the initial crystallographic condition of the 3D model [7, 8]. All the parameters of simulation are

on the electron backscatter diffraction (EBSD) experimental results.

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

**Figure 10.** Relationship between AFM, MATLAB, and the 3D CPFE model.

#### *3.2.1. Influence on surface roughness*

**Figure 11** shows that the surface asperity of the samples tends to be flattened with an increase of reduction. With an increase in reduction, the sample with a higher strain rate has a higher flattened rate of surface asperity than the sample with a lower strain rate. Increasing the applied macroscopic strain rate will increase the shear rate of lip systems in the surface area. Then under the same reduction, the sample deformed at a higher strain rate will activate more slip systems in the surface area. When the reduction is 40%, the surface roughness *Ra* of the sample with a higher strain rate is 0.16 μm, while the sample with a lower strain rate is only 0.09 μm.

#### *3.2.2. Influence of the strain rate on hardness*

**Figure 12** shows the influence of the strain rate on the hardness of the sample, and the influence is nonlinear. There are different stages in the evolution of hardness because when the

**Figure 11.** Influence of strain rate on surface roughness.

**Figure 12.** Influence of the strain rate on hardness (a) valley and (b) ridge.

reduction is lower (less than 60%), increasing the strain rate generally increases the hardness. At a larger reduction, increasing the strain rate will decrease the hardness under the same reduction [23]. When the reduction is lower, increasing the strain rate can increase the shearing rate of slip systems and also increase the density of dislocation. However, when reduction exceeds a certain value, the dislocation motion will overcome the barrier of grain boundary. In some areas, the density of dislocation decreases.

**Figure 13.** Effect of the strain rate on surface roughness *Ra*

**Figure 14.** Effect of the strain rate on texture (reduction 60% without lubrication).

: (a) experimental and (b) simulation.

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#### *3.2.3. Effect of strain on surface roughness (R<sup>a</sup> )*

In **Figure 13**, both the experimental and simulation results show the same tendency that increasing the strain rate can lead to a decrease in surface roughness under the same reduction. When reduction is less than 10%, the effect of the strain rate on surface roughness is insignificant, where mostly elastic deformation influences the flattening behavior of surface asperity. In this case, the increase of strain rate affects insignificantly the elastic deformation surface roughness. Plastic deformation plays an important role on surface area when the reduction exceeds 10%. When slip is the only deformation mode, the increased strain rate can result in more slip through the increased slip shear rate. Therefore, the surface roughness will decrease greatly with an increase in the strain rate.

#### *3.2.4. Effect of the strain rate on texture*

**Figure 14** shows that the influence of strain rate on the pole figures with at strain rate of 0.001 s−1 and 0.01 s−1 is not significant. In this case, every experiment has been carried out at room temperature, and the two applied strain rates are quite small. Deformation under the two strain rates belongs to the quasistatic deformation, and the difference between the two applied strain rates is small compared to the other dynamic deformation.

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**Figure 13.** Effect of the strain rate on surface roughness *Ra* : (a) experimental and (b) simulation.

reduction is lower (less than 60%), increasing the strain rate generally increases the hardness. At a larger reduction, increasing the strain rate will decrease the hardness under the same reduction [23]. When the reduction is lower, increasing the strain rate can increase the shearing rate of slip systems and also increase the density of dislocation. However, when reduction exceeds a certain value, the dislocation motion will overcome the barrier of grain boundary.

*)*

In **Figure 13**, both the experimental and simulation results show the same tendency that increasing the strain rate can lead to a decrease in surface roughness under the same reduction. When reduction is less than 10%, the effect of the strain rate on surface roughness is insignificant, where mostly elastic deformation influences the flattening behavior of surface asperity. In this case, the increase of strain rate affects insignificantly the elastic deformation surface roughness. Plastic deformation plays an important role on surface area when the reduction exceeds 10%. When slip is the only deformation mode, the increased strain rate can result in more slip through the increased slip shear rate. Therefore, the surface roughness will

**Figure 14** shows that the influence of strain rate on the pole figures with at strain rate of 0.001 s−1 and 0.01 s−1 is not significant. In this case, every experiment has been carried out at room temperature, and the two applied strain rates are quite small. Deformation under the two strain rates belongs to the quasistatic deformation, and the difference between the two

applied strain rates is small compared to the other dynamic deformation.

In some areas, the density of dislocation decreases.

**Figure 12.** Influence of the strain rate on hardness (a) valley and (b) ridge.

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

decrease greatly with an increase in the strain rate.

*3.2.4. Effect of the strain rate on texture*

*3.2.3. Effect of strain on surface roughness (R<sup>a</sup>*

**Figure 14.** Effect of the strain rate on texture (reduction 60% without lubrication).

*3.2.5. Analysis of pole figure*

**Figure 16.** FE model in grained heterogeneities of workpiece in MCWR.

**microforming**

Normally, the close-packed plane in FCC metal is {111}. In this case, the pole figure {111} is used for the analysis. Before compression, the sample has a cubic texture {111}<001> as shown in **Figure 16**. The predicted result has been compared to the experimental result; both of them show the same texture development. In the pole figure {111}, with an increased reduction, the brass orientation {110}<112> of silk texture becomes obvious while the cubic texture {001}<100> gets weaker. When the reduction reaches 60%, the brass orientation {110}<112> of silk texture shows extreme strong around a and d areas shown in **Figure 15**. Additionally, some S orientations {123}<634> can be seen in b and c areas. These results are basically in the agreement with the Sarma and Dawson's

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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

results [7, 8], which show a consistent development in hardness and grain size.

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

**Figure 15.** Comparison of the experimental pole figures with the simulation results.

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**Figure 16.** FE model in grained heterogeneities of workpiece in MCWR.

### *3.2.5. Analysis of pole figure*

**Figure 15.** Comparison of the experimental pole figures with the simulation results.

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Normally, the close-packed plane in FCC metal is {111}. In this case, the pole figure {111} is used for the analysis. Before compression, the sample has a cubic texture {111}<001> as shown in **Figure 16**. The predicted result has been compared to the experimental result; both of them show the same texture development. In the pole figure {111}, with an increased reduction, the brass orientation {110}<112> of silk texture becomes obvious while the cubic texture {001}<100> gets weaker. When the reduction reaches 60%, the brass orientation {110}<112> of silk texture shows extreme strong around a and d areas shown in **Figure 15**. Additionally, some S orientations {123}<634> can be seen in b and c areas. These results are basically in the agreement with the Sarma and Dawson's results [7, 8], which show a consistent development in hardness and grain size.
