Special Issue on Advanced Metal Forming Technologies in Asia
Construction of Material Constitutive Relationship Based on Surface-Layer Model at Micro/Mesoscopic Scale
2020 Volume 61 Issue 2 Pages 256-260
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2020 Volume 61 Issue 2 Pages 256-260
Micro-forming process get extensive attention in the last decades as the rapid development of micro-manufacturing technology. But the traditional metal plastic forming theory becomes no longer applicable in micro-forming process because of the so-called size effect. Many factors such as size effect, grain boundary and grain orientation should be considered to reveal the plastic deformation behaviors of metal material in micro/mesoscopic scale. In this paper, on the basis of classical Hall-Petch relationship, combining the surface layer model and composite model, a constitutive model of material in micro/mesoscopic scale was established with further consideration of grain orientation. The micro-compression experiment of pure copper specimens with different diameters and grain sizes was conducted. Based on the experiment results, the established constitutive relationship was determined and verified, and high consistence and accuracy were found. Finally, the finite element model of micro specimen was built based on the established constitutive model and Voronoi method. Then the numerical simulation of upsetting process of the pure copper was executed, which showed that the simulation results could well reflect the flow stress and deformation of the material at micro/mesoscopic scale, and further proved the availability of the proposed model.
Micro-forming process is a very promising technology of micro-manufacturing for its qualities such as high productivity, low cost, good mechanical performance and near net forming. But the traditional metal plastic forming theory becomes no longer applicable in micro-forming process because of the so-called size effect. The traditional constitutive model cannot describe the deformation behavior of material in micro/mesoscopic scale effectively, by which the application of the numerical simulation method was affected. Therefore, the construction of constitutive model of material in grain size is of great significance.
In micro/mesoscopic scale, the individual grain with different sizes, shapes and orientations has a significant impact on the overall deformation of the material, especially when the number of grains is small. In order to describe this phenomenon, many scholars have established different models to analyze and investigate the material deformation behavior in micro-scale. Geiger1) proposed the surface layer model, which divided the sample into inner region and softer surface region to explain the phenomenon that flow stress decreased with the decrease of sample size. Lai et al.2) proposed a material constitutive model by combining the surface layer model with the Hall-Petch relationship, which divided the sample into size-dependent and size-independent parts. Liu et al.3) combined the surface layer model with the composite model, and established a constitutive model considering the grain boundary hardening theory. Chan et al.4) considered the inhomogeneity of the grain deformation and studied the dispersion of materials deformation behavior caused by different grain orientations. Xu et al.5) proposed a compound material model by considering grain size, foil thickness and surface effects based on the micro tensile test of SUS304, and established a constitutive relation of SUS304 foil. Using the probability theory, Voronoi method could be used to generate polycrystalline model in the numerical simulation, which was widely used for stochastic modeling of grain aggregate. Based on this method, Zhang and Tong6) developed a 3D numerical simulation tool, which could involve the effect of grains with different orientations and grain boundaries in the finite element simulation. However, these models only took into account some factors that affect the deformation behavior of small specimens. So far, no constitutive model can fully describe the characteristics of materials deformation in micro-scale.
Therefore, based on the classical metal plastic forming theory, the surface layer model and the polycrystalline composite model were introduced, and the material constitutive model of micro-mesoscopic scale was established considering the effect of grain orientation. Then the polycrystalline aggregation was established by Voronoi method, and the proposed constitutive model was introduced into the polycrystalline aggregation to simulate the micro-compression process of pure copper. Comparing experimental results with those by the simulation, the validity and practicality of the constitutive model was verified.
Single crystal is the basic element of metal plastic deformation. The shear resolved stress on the slip plane of cylindrical single crystal can be expressed as eq. (1),7) where P is uniaxial normal pressure; θ1, θ2 are the angles between the cylinder axis and the normal direction of the slip plane, the cylinder axis and the slip plane respectively; A is the sectional area of the cylindrical single crystal.
| \begin{equation} \tau = (\mathrm{P}\cdot \cos\theta_{2})/(\mathrm{A}/{\cos\theta_{1}}) = \sigma\cdot \cos\theta_{1}\cos\theta_{2}. \end{equation} | (1) |
Therefore, the flow stress of the single crystal can be written as eq. (2), where m = (cos θ1 cos θ2)−1 is the orientation factor.
| \begin{equation} \sigma_{\text{sig}} = \mathrm{m}\tau. \end{equation} | (2) |
When θ1 = θ2 = 45°, the orientation factor m takes the minimum value of 2, which is most beneficial to the slip of single crystal. For polycrystal, grain orientation factors are different in different orientations. In the classical Taylor model, the orientation of polycrystal often takes 3.06 for the face centered cubic metal.8)
Similar to the single crystal, slipping and twinning are the main modes in the plastic deformation of polycrystal. But in the polycrystal, dislocations accumulate at grain boundaries due to the obstruction of surrounding grains, and the material exhibits obvious fine-grained strengthening phenomenon. Flow stress of the polycrystal is linear with the reciprocal of the square root of grain size, which could be described by the Hall-Petch equation.9)
| \begin{equation} \sigma_{\text{poly}}(\varepsilon) = \sigma_{0}(\varepsilon) + \mathrm{k}(\varepsilon)\mathrm{d}^{-1/2}. \end{equation} | (3) |
In eq. (3), d is the average grain diameter; k is the barrier coefficient of grain boundary; σ0(ε) is the slip resistance needs to be overcome when individual dislocation moves in the isolated grain, which is also related to the critical shear resolved stress.
2.2 Surface layer modelThe size effect refers to that the flow stress of materials decreases with the specimen size in the micro-scale. The decrease of the flow stress could be explained by the surface layer model, which assumes the grains in specimens could be divided into interior grains and surface grains with smaller constraints, as shown in Fig. 1. The volume fraction of softer surface grains becomes larger when the size of specimen gets smaller, which makes the deformation easier.
Surface layer model of micro-cylinder specimen.
If the volume fraction of surface grains is δ, the flow stress of the whole material is composed of the stresses of interior grains and surface grains in corresponding proportion, as shown in eq. (4).3)
| \begin{equation} \sigma = (1-\delta)\sigma_{\text{in}} + \delta\sigma_{\text{surf}}. \end{equation} | (4) |
For the micro-compression process which belongs to the bulk forming, it is not reasonable for the upper and lower surfaces of cylindrical specimen to be classified as surface layer, because they are difficult to deform due to the constraints of the tools. So the proportion of the surface layer in the cylindrical specimen could be calculated as eq. (5), where D is the diameter of the specimen and H is the height of the specimen.
| \begin{equation} \delta = \mathrm{V}_{\text{surf}}/\mathrm{V}_{\text{total}} = 4\mathrm{d}(\mathrm{D}-\mathrm{d})/\mathrm{D}^{2}. \end{equation} | (5) |
When the grain number is small, the individual grain plays a more significant role on the material deformation behavior. The composited model proposed by Kocks10) assumes that the deformation of polycrystal is composed of the hard grain boundary and soft grain interior, which could reflect preferably the influence of the grain size on the flow stress and deformation behavior of materials. In composited model, the flow stress of polycrystal could be written as eq. (6), where η is the volume fraction of grain boundary, σGB and σGI are the flow stresses of grain boundary and grain interior respectively.
| \begin{equation} \sigma_{\text{poly}} = \eta \sigma_{\text{GB}} + (1-\eta)\sigma_{\text{GI}}. \end{equation} | (6) |
The grain shape parameters ω1 and ω2 are introduced to describe the effect of grain shape on the grain volume Vgrain and the grain surface area Sgrain.
| \begin{equation} \left\{ \begin{array}{l} \mathrm{V}_{\text{grain}} = \omega_{1}\mathrm{d}^{3}\\ \mathrm{S}_{\text{grain}} = \omega_{2}\mathrm{d}^{2} \end{array} \right.. \end{equation} | (7) |
According to the surface layer model, the specimen in micro/mesoscopic scale could be divided into the surface layer region and interior region. The interior region of the specimen is similar to the polycrystal in macroscopic. Simplifying the grains in the polycrystal into the hexahedron model, the volume fraction of grain boundary in the interior region could be expressed as eq. (8), where t is the thickness of the grain boundary.
| \begin{equation} \eta = 1-\omega_{1}(\mathrm{d}-2\mathrm{t})^{3}/\omega_{1}\mathrm{d}^{3} = 1-(\mathrm{d}-2\mathrm{t})^{3}/\mathrm{d}^{3}. \end{equation} | (8) |
According to the research of Fu,11) the thickness of the grain boundary in the face-centered cubic copper could be formulated as eq. (9).
| \begin{equation} \mathrm{t} \approx 0.133\mathrm{d}^{0.7}. \end{equation} | (9) |
Due to the free surface in the surface layer, the volume fraction of the grain boundary is lower than the interior region. The lower part is the total free surface area, namely the lateral area of the specimen.
| \begin{equation} \mathrm{S}_{\text{surf}} = \pi \mathrm{DH}. \end{equation} | (10) |
The number of grains in the surface layer Nsurf could be calculated as:
| \begin{align} \mathrm{N}_{\text{surf}} &= \mathrm{V}_{\text{surf}}/\mathrm{V}_{\text{grain}} \\ &= \pi \mathrm{D}^{2}\cdot \mathrm{H}-\pi (\mathrm{D}-2\mathrm{d})^{2}\cdot \mathrm{H}/4\omega_{1}\mathrm{d}^{3} \end{align} | (11) |
Then the volume fraction of the grain boundary in the surface layer is:
| \begin{align} \eta' &= (\mathrm{N}_{\text{surf}}\mathrm{S}_{\text{grain}} - \mathrm{S}_{\text{surf}})\mathrm{t}/\mathrm{V}_{\text{surf}} \\ &= \mathrm{t}(\mathrm{D}-\omega \mathrm{D}-\mathrm{d})/\mathrm{d}\omega (\mathrm{D}-\mathrm{d}) \end{align} | (12) |
In eq. (12), ω = ω1/ω2 is the factor related to the grain shape. Grains with different geometries could be described using regular polyhedron models. The shape factor ω takes the maximum value of 0.167 when the geometry of the grain is close to the spherical shape.
Then the model considering the surface layer and grain boundary strengthening can be expressed as eq. (13), where σin-GI is the flow stress of the grain interior in interior of the specimen; σsurf-GI is the flow stress of the grain interior in surface of the specimen.
| \begin{align} \sigma& = (1-\delta)[(1-\eta)\sigma_{\text{in-GI}} + \eta\sigma_{\text{GB}}] \\ &\quad + \delta[(1-\eta')\sigma_{\text{surf-GI}}+\eta'\sigma_{\text{GB}}]. \end{align} | (13) |
According to the crystal plasticity theory, the grain interior in surface could be regarded as the single crystal, the orientation factor of which is the lower bound value of 2. The interior grain could be regarded as the polycrystal, the orientation factor of which is the average value of 3.06.
| \begin{equation} \sigma_{\text{surf-GI}} = \mathrm{m}_{\text{surf}}\tau_{\text{i}}(\varepsilon). \end{equation} | (14) |
| \begin{equation} \sigma_{\text{in-GI}} = \mathrm{m}_{\text{in}}\tau_{\text{i}}(\varepsilon). \end{equation} | (15) |
Combining formulas (3), (6) and (15), the strain-stress relationship of σGB could be determined.
| \begin{equation} \sigma_{\text{GB}} = \mathrm{m}_{\text{in}}\tau_{\text{i}}(\varepsilon) + \mathrm{K}(\varepsilon)\mathrm{d}^{- \frac{1}{2}}/\eta. \end{equation} | (16) |
In micro/mesoscopic scale, the orientation and shape of individual grain have a significant impact on the deformation behavior of materials. Due to the randomness of grain properties, the deformation behavior of materials shows greater uncertainty, which is the main reason for the increased scatter of flow stresses and increased surface roughness. So the model was further amended considering different orientation of every grain.
The strain-stress relationships of different grains could be determined by their corresponding orientation factors m and shear resolved stresses τi. Huang et al.12) investigated the deformation microstructure and crystallographic orientation of polycrystalline copper grains, and found that there were 3 main grain orientations in the copper: [110], [100] and [111], as shown in Fig. 2. The orientation factor, volume fraction and elastic modulus of each orientation were listed in Table 1.
Three main orientations of grains in the tensile test of pure copper.
The contribution of each grain is determined by its proportion fi in polycrystalline materials.13) The constitutive model considering grain orientation, surface layer and grain boundary strengthening can be obtained, as shown in eq. (17).
| \begin{align} \sigma & = (1-\delta)\Big[(1-\eta)\left(\sum\nolimits_{\text{n} = 1}^{\text{N}}\mathrm{f}_{\text{n}}\mathrm{m}_{\text{n}}\tau_{\text{i}}(\varepsilon) \right) \\ &\quad + \eta(\mathrm{m}_{\text{in}}\tau_{\text{i}}(\varepsilon) + \mathrm{K}(\varepsilon)\mathrm{d}^{-\frac{1}{2}}/\eta) \Big]\\ &\quad + \delta[(1-\eta')\mathrm{m}_{\text{surf}}\tau_{\text{i}}(\varepsilon)+\eta'(\mathrm{m}_{\text{in}}\tau_{\text{i}}(\varepsilon) + \mathrm{K}(\varepsilon)\mathrm{d}^{-\frac{1}{2}}/\eta)] \end{align} | (17) |
To solve and validate the proposed constitutive model, compression tests of the copper cylindrical specimens with different diameters and grain sizes were carried out. The diameters of the specimen were 1, 1.5, 2 and 3 mm respectively. The grain sizes were 22, 83 and 122 µm respectively. In the constitutive model, only τi(ε) and K(ε) are unknown variables, so they could be determined by the strain-stress relationships of the materials with 2 different grain sizes. The calculation results on the basis of relationships of the specimens with the grain sizes of 26 and 83 µm (a), 83 and 122 µm (b), 122 and 26 µm (c) were shown in eq. (18).
| \begin{align} &\text{(a)}\ \left\{ \begin{array}{l} \mathrm{K}_{\text{i}}(\varepsilon) = 237.92\varepsilon^{0.2735}\\ \tau_{\text{i}}(\varepsilon) = 141.99\varepsilon^{0.3220} \end{array} \right.\quad \text{(b)}\ \left\{ \begin{array}{l} \mathrm{K}_{\text{i}}(\varepsilon) = 233.31\varepsilon^{0.2507}\\ \tau_{\text{i}}(\varepsilon) = 136.52\varepsilon^{0.3253} \end{array} \right. \\ &\text{(c)}\ \left\{ \begin{array}{l} \mathrm{K}_{\text{i}}(\varepsilon) = 189.98\varepsilon^{0.2235}\\ \tau_{\text{i}}(\varepsilon) = 139.93\varepsilon^{0.3232} \end{array} \right. \end{align} | (18) |
The curves of τi(ε) varying with the strain were drawn in Fig. 3. It can be found that the values of τi(ε) calculated by different combinations of grain sizes were almost the same, which indicated that the values of τi(ε) was independent of the grain size and specimen diameter. Therefore, the values of τi(ε) and K(ε) can be accurately obtained by the proposed model, and then the constitutive equations of the grain boundary and different grains can be determined.
The values of τi(ε) obtained from different combinations of grain sizes.
Based on the constitutive equation of the grain and grain boundary as well as the orientation factors of different grains, the flow stress of the grain with different orientations can be determined. Using the constitutive model, the flow stresses of pure copper with different specimen sizes and grain sizes can be obtained. In this way, the flow stress of the specimen with diameter of 1.5 mm and grain size of 83 µm was calculated and compared with the experiment result, as shown in Fig. 4. It can be seen that the calculated result was in good agreement with the experiment result, especially when the strain reached 0.4, the calculated result and the experiment result were basically in coincidence.
Comparison between calculated result and experiment result.
In order to describe the actual grain shape in pure copper, based on the Voronoi method, the two-dimensional polygons were generated to represent different grains. Utilizing the developed program, the corresponding grain boundary regions were generated, and finally the polycrystalline material model including grain and grain boundary regions was obtained. Then, according to the constitutive model, the attributes of three types of grains, surface layer region and grain boundary region were defined by random traversal instructions. The FEM models of specimens with different geometric sizes were shown in Fig. 5. In order to ensure the accuracy of the simulation results, the grain boundary region was divided into at least three layers of grid.
Stochastic FEM models of samples with different geometric sizes (grain size 122 µm).
By comparing the simulation results with the experimental results, it can be seen that the simulation results under different grain sizes were in good agreement with the experimental stress-strain curves, as shown in Fig. 6, which further verified the accuracy of the constitutive model established in this paper.
Comparison of strain-stress curves between the simulation and experiment (grain size 122 µm).
In this paper, the plastic theories in micro-scale were analyzed, and the calculation method of the proportion of the surface layer was optimized. Then, on the basis of Hall-Petch relationship and the surface layer model, considering the influence of grain boundary and grain orientation, the constitutive model of materials at micro-scale was established. Compression experiments of pure copper samples with different diameters and grains were carried out to determine and verify the proposed model.
Based on the constitutive model and Voronoi method, a polycrystalline model of material in micro-scale was established. The grain boundary and surface layer regions were set up and different properties of the grain orientation were randomly assigned to the specimen. The compression process of pure copper was simulated then. Both the simulation results and experiment results showed that the proposed model was reasonable and could work with high accuracy.
The research work was supported by the National Natural Science Foundation of China (51675307, 51375269).
