**2. Influence of size effects on deformation behavior in microforming processes**

#### **2.1. Flow stress**

Structural sources:

6 Modeling and Simulation in Engineering Sciences

**•** Grain size to element thickness size effect—the grain size of metallic materials results from the material properties and determined by the casting condition, the thermal and mechan‐ ical treatments. It is impossible to obtain each material with each grain size, thus the grain size cannot be scaled down in parallel to the element dimension. **Figure 4** schematically

In conventional metal forming operations all treated materials are considered as homogene‐ ous. In microforming processes they are heterogeneous, because of relatively large grains size to the billet volume. As the grain size increases, the grain structure of the billet becomes to be more heterogeneous and the material shows anisotropic behavior in the submillimeter dimensional range. The anisotropic behavior of the workpiece material can directly change

**•** Surface structure scalability (SSS) size effect—similarly to grain size, it is often not possible to reduce the die and billet surface roughness with the element dimension, due to that the surface structure scalability is the source of size effects. In lubrication as well as in dry

shows microformability of polycrystalline and amorphous materials.

**Figure 4.** Microformability of polycrystalline and amorphous material [10].

local deformation mechanisms as shown in **Figure 5**.

conditions, SSS leads to a size-dependent friction behavior.

**Figure 5.** Backward extrusion of a billet composed of heterogeneous grain structure [3].

Flow stress decreases as workpiece size decreases [5, 11, 12]. Surface grains of the deformed material have lesser constraints compared to internal ones. Distribution of dislocations in surface grain boundaries is different from that of internal. The surface grains have a lower flow stress. When the plastically treated detail size is decreased to microscale and the grain size is relatively large, there are small number of grains constituting the workpiece and the volume fraction of surface grains increases significantly—surface layer model (**Figure 6**). Analogously, increase of grain size in small element reduces their number in its volume. This leads to decrease of grainy material flow stress in microscale (**Figure 7**).

**Figure 6.** Change of surface grains volume fraction with decreasing of specimen size—surface layer model [13, 14].

**Figure 7.** Grain and specimen flow stress size effects in: (a) tensile; (b) compression tests [13, 15].

#### **2.2. Fracture**

Similarly to flow stress, fracture strain decreases with the decrease of specimen size [16, 17] and the increase of grain size [12]. This phenomenon was found in tensile tests of wire and thin details. Fracture takes place through the localized shearing in the individual grain (**Figure 8**) [18–20]. The fracture strain increases also with the decrease of workpiece size in compression of bulk metal [21]. The tensile-tested material in microscale can be considered as a chain and each specimen's perpendicular section acts as a chain link (**Figure 9**). Material yields when all the grains in the section yield and the initial yielding occurs at the weakest section consisting of the soft grains [23]. The smaller the number of grains in the section, the higher the probability to find one with a significantly large fraction of soft grains decreasing the fracture strain.

**Figure 8.** Scanning electron micrograph of a tensile-tested aluminum wire with a diameter of 25 μm [18].

**Figure 9.** Schematic illustration of yielding at the weakest section of extended material [22].

#### **2.3. Flow behavior**

**2.2. Fracture**

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the fracture strain.

Similarly to flow stress, fracture strain decreases with the decrease of specimen size [16, 17] and the increase of grain size [12]. This phenomenon was found in tensile tests of wire and thin details. Fracture takes place through the localized shearing in the individual grain (**Figure 8**) [18–20]. The fracture strain increases also with the decrease of workpiece size in compression of bulk metal [21]. The tensile-tested material in microscale can be considered as a chain and each specimen's perpendicular section acts as a chain link (**Figure 9**). Material yields when all the grains in the section yield and the initial yielding occurs at the weakest section consisting of the soft grains [23]. The smaller the number of grains in the section, the higher the probability to find one with a significantly large fraction of soft grains decreasing

**Figure 8.** Scanning electron micrograph of a tensile-tested aluminum wire with a diameter of 25 μm [18].

**Figure 9.** Schematic illustration of yielding at the weakest section of extended material [22].

Inhomogeneous and anisotropic material behavior causes the irregular flow of formed workpiece geometry with the decrease of its size and the increase of grain size [24]. Chan et al. [12] experimentally presented decrease of flow stress, scatter of obtained data, and inho‐ mogeneous specimen deformation with decrease of the ratio of element size to its grain size. Flow stress decrease because of grain boundary surface area to billet volume ratio gets small and the grain boundary strengthening effect is reduced. In addition, when the workpiece volume contains only a few grains (**Figure 10**), their size, shape, and orientation affect the deformation process significantly, leading to the scatter of the obtained data, for example stress-strain curves (**Figure 11**), and to the inhomogeneous billet deformation.

**Figure 10.** Microstructures of compressed specimens annealed in different temperatures [12].

**Figure 11.** Flow stress curves of the billets with different grain size [12].

Flow stress size effect (**Figure 12**) and inhomogeneous deformation behavior were included in FE simulation of microcompression tests. Grain properties composite model was proposed. Grains were divided into several groups with assigned corresponding stress-strain curves. The flow stress curves of the testing samples and the scatter effect can be then estimated based on Eq. (1):

$$\sigma(\boldsymbol{\varepsilon}) = \sum\_{i=1}^{n} V\_i \cdot \sigma\_i(\boldsymbol{\varepsilon}) \tag{1}$$

where *n* is the total number of grains in the billet, *Vi* is the volume fraction (area fraction in two-dimensional case) of the *i*th grain, and *σ<sup>i</sup>* (*ε*) is the flow stress of the *i*th grain.

**Figure 12.** Illustration of flow stress size effect [12].

The volume/area fraction of each grain in the testing sample was estimated based on metal‐ lographies. The flow stress curves were then assigned to the grains randomly, as it is shown in **Figure 13**.

**Figure 13.** Spread *σ*(*ε*) of the grains flow stress-strain curves [12].

Numerical Simulation in Microforming for Very Small Metal Elements http://dx.doi.org/10.5772/64275 11

**Figure 14.** Schematic illustration of the 3D full-scale model with cubic grain.

Flow stress size effect (**Figure 12**) and inhomogeneous deformation behavior were included in FE simulation of microcompression tests. Grain properties composite model was proposed. Grains were divided into several groups with assigned corresponding stress-strain curves. The flow stress curves of the testing samples and the scatter effect can be then estimated based on

> ( ) ( ) <sup>1</sup> *n i i i*

 s e

The volume/area fraction of each grain in the testing sample was estimated based on metal‐ lographies. The flow stress curves were then assigned to the grains randomly, as it is shown

*<sup>V</sup>*<sup>=</sup> = × å (1)

(*ε*) is the flow stress of the *i*th grain.

is the volume fraction (area fraction in

se

where *n* is the total number of grains in the billet, *Vi*

two-dimensional case) of the *i*th grain, and *σ<sup>i</sup>*

10 Modeling and Simulation in Engineering Sciences

**Figure 12.** Illustration of flow stress size effect [12].

**Figure 13.** Spread *σ*(*ε*) of the grains flow stress-strain curves [12].

in **Figure 13**.

Eq. (1):

**Figure 15.** Stress distribution in the 3D full-scale models with surface-to-grain ratio of 2.8 [12].

**Figure 16.** Stress distribution in the 3D full-scale models with surface-to-grain ratio 12.5 [12].

3D full-scale simulations using a commercial CAE system, ABACUS, were conducted with eight-node linear brick element. Grain was represented as a cube for simplification

(**Figure 14**). The grain properties identified by Eq. (1) were employed into the models. The grain interior and grain boundary were considered as a single body, while the change of grain boundary strengthening effect due to the change of surface-to-grain ratio was taken into account with varying the mean flow stress of grains. FE simulation of compression of billets consisting of individual grains results showed the inhomogeneous workpiece deformation with the decreasing ratio of specimen size to grain size (**Figures 15** and **16**). In 3D full-scale model, the asymmetry of the material deformation behavior and grain properties was taken into account. The results showed that the scatter effect of the flow stress curve increases with increasing grain size and the achieved scatter range had a good agreement with the experimental result.


**Figure 17.** Microstructure of specimens and grain size, *λ* = specimen radius/grain size [25].

**Figure 18.** Aggregates of polycrystalline (a) tessellation in 140 Vornoi polyhedral; (b) freely meshed model with 140 subdomains; (c–f) Microstructure mapping on a 1313 × 60 cubic element mesh and domain with subdomain numbers 140, 26, 385, and 2520, respectively [25].

Voronoi tessellation can be adopted to describe the polycrystalline aggregate [25–27]. Lu et al. [25] proposed a mixed material model based on modified Hall-Peth relationship, surface layer model, and grained heterogeneity (e.g., grain size, shape, and deformability). Pure copper compression experiment was performed (**Figure 17**). Generated virtual microstructures which can be implemented in the commercial FE code are presented in **Figure 18**.

(**Figure 14**). The grain properties identified by Eq. (1) were employed into the models. The grain interior and grain boundary were considered as a single body, while the change of grain boundary strengthening effect due to the change of surface-to-grain ratio was taken into account with varying the mean flow stress of grains. FE simulation of compression of billets consisting of individual grains results showed the inhomogeneous workpiece deformation with the decreasing ratio of specimen size to grain size (**Figures 15** and **16**). In 3D full-scale model, the asymmetry of the material deformation behavior and grain properties was taken into account. The results showed that the scatter effect of the flow stress curve increases with increasing grain size and the achieved scatter range had a good

agreement with the experimental result.

12 Modeling and Simulation in Engineering Sciences

140, 26, 385, and 2520, respectively [25].

**Figure 17.** Microstructure of specimens and grain size, *λ* = specimen radius/grain size [25].

**Figure 18.** Aggregates of polycrystalline (a) tessellation in 140 Vornoi polyhedral; (b) freely meshed model with 140 subdomains; (c–f) Microstructure mapping on a 1313 × 60 cubic element mesh and domain with subdomain numbers

Voronoi tessellation can be adopted to describe the polycrystalline aggregate [25–27]. Lu et al. [25] proposed a mixed material model based on modified Hall-Peth relationship, surface layer The deformed part can be represented by a space tessellation into 3D Voronoi diagram which describes grain boundary [25]. Voronoi diagram is defined as *n* distinct regions, *Vi* based on an open set Ω, and *n* different seeds *zi* (*i* = 0, 1,…, *n* − 1) such as:

$$W\_i = \left\{ w \in \Omega \quad d\left(w, z\_{\mid}\right) j = 0, 1, \dots, j \neq i \right\} \tag{2}$$

where *d* is the function of distance. Voronoi polyhedra are the influence zones of these grains that are their mass center.

**Figure 19.** Profile of specimens after compression, (a) homogenous model with *λ* = 1.6, (b) *λ* = 7.5, (c) *λ* = 3, (d) *λ* = 1.6, (e) *λ* = 0.85 [25].

A 3D domain is decomposed into a certain number of subdomains. Then, the seeds are spread into these subdomains. Assuming the center of a cube is (*x*, *y*, *z*), the seed will lie *λδa*/2 away from the central point. The coordinate of the seed position should be (*x* + *λδa*/2, *y* + *λδa*/2, *z* + *λδa*/2). *δ* is the maximum distance between the seed and the cubic center, and *λ* is a random number between −1 and 1. Both *δ* and *λ* are the so-called shape factors which will determine the shapes of the Voronoi cells. A critical distance *d* between different seeds is defined to prevent each seed from being too close to another. After that, 3D Voronoi polyhedra can be generated by taking these seeds as the generating points [25].

Obtained from nanoindentation hardness of grains was used to identify the deformability of individual grain inside the billet. Applying developed material model, the microcompression test of pure copper was numerically simulated (**Figure 19**) using commercial the FEM software ANSYS/LS-DYNA. Numerical analyses results show that the scatter of deformation behavior becomes significant with decreasing factor *λ*, where λ is the specimen radius/grain size.

#### **2.4. Elastic recovery**

Springback increases with decreasing ratio of workpiece thickness to grain size [15, 28]. Liu et al. [15] used the pure copper sheet foils with the thickness from 0.1 to 0.6 mm as the testing material for tensile test and microbending (**Figure 20**). It was founded that the springback angle increases with the decreasing metal element thickness. This is consistent with the conventional bending knowledge specific for macroscale bending. Increase in the scatter of springback angle with decreasing sheet thickness was also found. It was concluded that the main reason for that is elastic anisotropy of surface grain due to grain orientation difference.

**Figure 20.** Left: Three-point bending test device and tooling; Right: Influence of grain size on springback angle [15].

**Figure 21.** Grained heterogeneity in workpiece with: (a) 13 grains; (b) 25 grains; and (c) 75 grains [29].

Fang et al. [29] conducted numerical simulation to investigate the size effect of the springback, which occurs after the micro-V-bending in terms of Voronoi tessellation. A finite element model of the micro-V-bending has been designed by using the ABAQUS/Standard commercial software. The workpiece grain sizes of 98, 152, and 201 μm have been adopted in the FE model (**Figure 21**) to recognize the relationship between the size effect and springback angle during the V-bending process. Voronoi tessellation can imitate the microstructure of materials, and represents grained heterogeneity graphically in FE models. Grains were divided into three groups with assigned corresponding stress-strain curves (**Figure 22**). Simulation results display the inhomogeneous deformation behavior during micro-V-bending process and also show that springback effect increases with grain size (**Figure 23**).

**Figure 22.** Left: Flow stress curves of the grains in workpiece with grain size 152 μm; Right: Micro-V-bending FE mod‐ el with workpiece grain size 152 μm [29].

**Figure 23.** Left: Micro-V-bending result after unloading; Right: Springback angle with different grain size [29].

#### **2.5. Frictional behavior**

**2.4. Elastic recovery**

14 Modeling and Simulation in Engineering Sciences

Springback increases with decreasing ratio of workpiece thickness to grain size [15, 28]. Liu et al. [15] used the pure copper sheet foils with the thickness from 0.1 to 0.6 mm as the testing material for tensile test and microbending (**Figure 20**). It was founded that the springback angle increases with the decreasing metal element thickness. This is consistent with the conventional bending knowledge specific for macroscale bending. Increase in the scatter of springback angle with decreasing sheet thickness was also found. It was concluded that the main reason for that

**Figure 20.** Left: Three-point bending test device and tooling; Right: Influence of grain size on springback angle [15].

**Figure 21.** Grained heterogeneity in workpiece with: (a) 13 grains; (b) 25 grains; and (c) 75 grains [29].

Fang et al. [29] conducted numerical simulation to investigate the size effect of the springback, which occurs after the micro-V-bending in terms of Voronoi tessellation. A finite element

is elastic anisotropy of surface grain due to grain orientation difference.

#### *2.5.1. Open and closed lubricant pockets (CLPs)*

Changes of interfacial friction with the decrease of element size in microforming process can be estimated based on the size-scaled ring compression test [5] and double-cup extrusion (DCE) [30]. The friction is recognized based on the comparison between the geometry of the deformed sample and the finite element simulations results. Change of friction factor with decreasing workpiece size in the DCE process in lubricated conditions is presented in **Figure 24**. It can be seen that the friction increases with the scaled down workpiece size [30].

To recognize the frictional behavior in microforming, the asperity deformation process needs to be taken into account. Surface topography evolution process is shown in **Figure 25**. Before the tool touches the workpiece surface, there is a layer of lubricant on the whole con‐ tact surface (**Figure 25a**). When the tool presses the workpiece surface, some lubricant is trapped in the roughness valleys, which results in the formation of the so-called closed lu‐ bricant pockets (**Figure 25b**). In the CLPs, a hydrostatic pressure is thus generated and part of the deformation load is shared by the lubricant. Under this condition, material slides along the tooling surface with low friction. At the workpiece edges the lubricant is squeezed out from the roughness valleys and the so-called open lubricant pockets (OLPs) are then cre‐ ated. At the OLPs, the flattened asperities support deformation load and they become to be the real contact areas (RCAs), which increases the interfacial friction (**Figure 25c**). Open and closed pockets model application in FE simulation is presented in **Figure 26** [32]. Presence of CLP and OLP results in the nonuniform deformation of material asperities surface affects the surface properties of the formed microelement.

**Figure 24.** The formed geometries in different size-scaled achieved in double-cup extrusion tests [31].

Numerical Simulation in Microforming for Very Small Metal Elements http://dx.doi.org/10.5772/64275 17

**Figure 25.** The evolution of workpiece surface asperities in deformation process [32].

**Figure 26.** 2D sketch of OLPs, RCAs, and CLPs [32].

(DCE) [30]. The friction is recognized based on the comparison between the geometry of the deformed sample and the finite element simulations results. Change of friction factor with decreasing workpiece size in the DCE process in lubricated conditions is presented in **Figure 24**. It can be seen that the friction increases with the scaled down workpiece size [30].

To recognize the frictional behavior in microforming, the asperity deformation process needs to be taken into account. Surface topography evolution process is shown in **Figure 25**. Before the tool touches the workpiece surface, there is a layer of lubricant on the whole con‐ tact surface (**Figure 25a**). When the tool presses the workpiece surface, some lubricant is trapped in the roughness valleys, which results in the formation of the so-called closed lu‐ bricant pockets (**Figure 25b**). In the CLPs, a hydrostatic pressure is thus generated and part of the deformation load is shared by the lubricant. Under this condition, material slides along the tooling surface with low friction. At the workpiece edges the lubricant is squeezed out from the roughness valleys and the so-called open lubricant pockets (OLPs) are then cre‐ ated. At the OLPs, the flattened asperities support deformation load and they become to be the real contact areas (RCAs), which increases the interfacial friction (**Figure 25c**). Open and closed pockets model application in FE simulation is presented in **Figure 26** [32]. Presence of CLP and OLP results in the nonuniform deformation of material asperities surface affects

the surface properties of the formed microelement.

16 Modeling and Simulation in Engineering Sciences

**Figure 24.** The formed geometries in different size-scaled achieved in double-cup extrusion tests [31].

**Figure 27.** The increase of OLP fraction with the decrease of workpiece size [7].

**Figure 28.** End surface topographies of the specimens (Ø1 × 1.5 mm) with the asperity size of: (a) 25 μm; and (b) 10 μm [32].

Increase of friction with miniaturization is related to the increase of OLP fraction (**Figure 27**). The influence of different sizes of asperity on the change of the compressed surface topography is shown in **Figure 28**. The thickness of OLP rim decreases with the increase of asperity size. This reveals that the efficiency of lubricant in load supporting is higher in the surface with large asperities. More lubricant can be then trapped in the roughness valleys, resulting in the increase of the CLP area fraction.

## *2.5.2. Friction in forward and backward cup extrusion*

Material tends to flow backwards (higher cup) with the decrease of the workpiece size when the fine-grained material is used [31]. Coarse grain size (**Figure 29**, case 0.5 mm) is larger than the cup wall thickness, which makes the material flow forward easily (longer shaft). Defor‐ mation behavior of fine- and coarse-grained samples in microdouble-cup extrusion is showed in **Figure 30**.

**Figure 29.** The experimental results of the combined forward rod—backward cup extrusion [31].

**Figure 30.** Microstructures of the microformed parts in microdouble-cup extrusion with different grain sizes. (a) 20 and (b) 150 μm [33].

#### *2.5.3. Surface roughening*

**Figure 27.** The increase of OLP fraction with the decrease of workpiece size [7].

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[32].

increase of the CLP area fraction.

**Figure 28.** End surface topographies of the specimens (Ø1 × 1.5 mm) with the asperity size of: (a) 25 μm; and (b) 10 μm

Increase of friction with miniaturization is related to the increase of OLP fraction (**Figure 27**). The influence of different sizes of asperity on the change of the compressed surface topography is shown in **Figure 28**. The thickness of OLP rim decreases with the increase of asperity size. This reveals that the efficiency of lubricant in load supporting is higher in the surface with large asperities. More lubricant can be then trapped in the roughness valleys, resulting in the

The ratio of tool asperity size to the formed element size increases with decrease of workpiece size and the increase of grain size [22, 34, 35]. Die workpiece interface effects become significant in microforming. Geiger et al. [31] has shown that the use of the traditional friction coefficient or friction factor can lead to erroneous results in microscale. Some of the researchers have taken into account the impact of degree of a tool roughness on the material deformation process in numerical analysis. Challen et al. [36] presented the friction model for rough contact of plastic material with a rigid tool. Model is based on a simplified geometry of the tool surface, whose actual profile reflects a triangular wave (**Figure 31**). The surface roughness used in the model is the average of the asperities parameters measured in the experiment.

**Figure 31.** Triangular wave surface roughness model [36].

In that way, using a commercial FE system DEFORM, geometry of the tool surface was modeled in experimentally verified simulations of the forward microextrusion processes [37, 38] (**Figure 32**), where a significant influence of container roughness on the material flow was shown.

**Figure 32.** Flow nets of microextruded metal rods with a tool surface characterized by: (a) constant friction factor; (b) and (c) rigid triangular waves [39].

Vidal-Sallé et al. [40, 41] modeled die roughness in the form of a rigid triangular wave and the wave of interconnected arcs—model machined by turning surfaces in FE simulations of cylinder compression (**Figure 33**).

**Figure 33.** Plastic strain distribution in the compressed billet for perfect sliping [41].

Jeon et al. [42, 43] modeled die surfaces in the form of rigid sinusoidal curve in numerical simulation of the ring compression (**Figure 34**) as opposed to the use of the traditional empirical friction coefficient or factor. This finite-element-based model has been validated experimen‐ tally in terms of loads and metal flow using the ring test and actual surface measurements. The curve was referenced to the parameter *Ra* and the friction factor was determined as *m = f(a,t).*

**Figure 34.** Tool surface geometric model with using an elliptical profile [42].

## **2.6. Repeatability**

in microforming. Geiger et al. [31] has shown that the use of the traditional friction coefficient or friction factor can lead to erroneous results in microscale. Some of the researchers have taken into account the impact of degree of a tool roughness on the material deformation process in numerical analysis. Challen et al. [36] presented the friction model for rough contact of plastic material with a rigid tool. Model is based on a simplified geometry of the tool surface, whose actual profile reflects a triangular wave (**Figure 31**). The surface roughness used in the model

In that way, using a commercial FE system DEFORM, geometry of the tool surface was modeled in experimentally verified simulations of the forward microextrusion processes [37, 38] (**Figure 32**), where a significant influence of container roughness on the material flow was

**Figure 32.** Flow nets of microextruded metal rods with a tool surface characterized by: (a) constant friction factor; (b)

is the average of the asperities parameters measured in the experiment.

**Figure 31.** Triangular wave surface roughness model [36].

20 Modeling and Simulation in Engineering Sciences

shown.

and (c) rigid triangular waves [39].

The scatter of the measured material properties (**Figure 35**) (e.g., flow stress, material defor‐ mation behavior) increases with the increase of grain size and the decrease of the workpiece size. This phenomenon is the resulted of different properties of individual grains and their less number in the workpiece volume [11, 44]. The deformation behavior of a single grain has an anisotropic nature. When the specimen size decreases and the grain size stays constant, there will be a small number of grains in the specimen volume and so the number of microstructural features decreases. The uniform distribution of different grains no longer exists. Each grain with each property plays a significant role to the overall material deformation behavior. Different crystallographic orientations, different shapes, and sizes of neighboring grains lead to inhomogeneous deformation, which result in the scatter of achieved material properties.

**Figure 35.** Schematic illustration of the modeling of the scatter effect with a normal distribution function [12].
