Deformation behavior and forming process simulation of semi-solid powder rolling based on the combined constitutive model

Semi-solid powder forming (SPF) as a novel technology has been widely used to prepare composite materials. However, most of the present models used to simulate the complicated SPF process are based on the constitutive relationship of single dense or porous material, which cannot well satisfy the actual conditions. In this study, the process of semi-solid powder rolling was simulated by the combined constitutive formula of sintered and dense Al-Cu-Mg alloy materials obtained from semi-solid compression experiments. The results show that the semi-solid compression curves of dense and sintered materials are similar, and their peak stresses increase with the decreasing porosities. The grain or particle size of sintered materials after semi-solid compression becomes finer as powders crushed, but the grain size of dense materials becomes larger due to grain coarsening and deformation, which is the key advantage of SPF with fine microstructures. The combined constitutive model of semi-solid powder materials was established and verified, and then the numerical simulation of semi-solid powder rolling based on the model was proved that can well describe the rolling process, which provides a guidance for process optimization.


Introduction
Semi-solid powder forming (SPF) is a novel near-net forming technology that has been widely used in preparing composites, due to its advantages of low forming pressure and temperature, low energy consumption, uniform and fine microstructures, and good capability of proportioning the number and type of enhanced phases at will [1][2][3][4][5][6]. During SPF, powders are deformed and broken into fragments under an external force, and then flow, rearrange and fill pores with liquid and fragments, while liquid solidifies to generate metallurgical bonding between powders and fragments, which consequently forms fine microstructures, makes the material dense, and simultaneously produces new pores [7,8]. Therefore, the process of SPF is very complicated [9,10], and necessary to be further and deeply investigated to improve the product performance.
The numerical simulation is the best method to study the forming process of SPF but the present simulated constitutive relationships were mostly obtained from dense or porous materials [7,[11][12][13]. However, the material during SPF changes from discontinuous semi-solid particles with an apparent relative-density of 30%-50% into a continuous porous solid body with a relative-density of 90%-100% [14,15]. Single constitutive model of dense or porous materials cannot well meet the actual forming situations of SPF.
Therefore, in this study, the semi-solid compressions of sintered and dense Al-Cu-Mg alloy materials were performed and their stress-strain curves were compared and analyzed, finally the constitutive model combined two materials was fitted based on the compression data. The microstructure of sintered and dense materials before and after semi-solid compression was also compared with that obtained by semi-solid powder rolling. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Based on the fitted model, the process of semi-solid powder rolling was simulated by MARC software using the elliptic yield criterion, and then the calculated results were compared with experimental results.

Experimental methods
The powder used in this experiment is the Al-Cu-Mg alloy (the chemical composition is Al-4.09Cu-1.18Mg-0.54Mn-0.21Fe wt%) produced by gas atomization. The sintered compression samples (Φ8 mm×H12 mm) with different porosities of 36%, 30%, 16% and 10% were prepared by Spark Plasma Sintering system at a vacuum environment under a low pressure (details shown in [8]). The dense compression samples as a porosity of 0% (Φ8 mm×H12 mm) were cut from commercial Al-Cu-Mg alloy extrusion rods, and then were heated to a semi-solid temperature of 873K and held for 1440 s to obtain the same microscopic size with sintered compression samples. Using the Gleeble-3500 thermal simulation tester (as shown in figure 1), the compression samples with different porosities were heated to 853, 863 and 873k (corresponding liquid fraction of 12%, 16%, and 21% [16]) and soaked for 300s, and then compressed to half of their height at rates of 0.1,1 and 5 1/s, respectively, finally cooled by water.
The powders with the same composition were put into the thermal resistance furnace, heated to a semi-solid temperature of 853k and held for 1800s in the Argon atmosphere, and then poured into the seam of two rollers which were preheated to 573k. The roller width and diameter are respectively 100 mm and 150 mm, and its rolling speed is 7.5 mm s −1 . The experiment was performed by suddenly stopping the rolling process to determine the relative-density of billets at different rolling stages, as shown in figure 13.
The relative-density of water-cooled samples after compression and rolling was measured by Archimedes' method. The cross section of samples was corroded by Keller's reagent, and the microstructure of powders, holes and grains was observed by LcicaDM1500M. The second phase morphology and distribution and the fracture surface of samples were analyzed by Quanta2000 SEM. The composition was measured by EDS and the EPMA elemental mapping analysis was determined by ZEISS Merlin.

FEM simulation models
Semi-solid powder rolling is a complex process with the temperature varying from a semi-solid state to a solid state, and with the morphology changing from non-continuous powders to a successive billet. Therefore, in order to simplify the model and save calculation time, a two-dimensional model was built by Marc software which is good at handling nonlinear problems. Considering the symmetry of rolling system, the model was made up of a billet, a roller, a push plate, and a symmetric rigid body, as shown in figure 2. The roller with 603 elements is heterogeneously divided by #39 unit, while the sheet with 160 elements is uniformly divided by #11 unit.

Results and discussions
3.1. Microstructures analysis 3.1.1. Microstructures comparison before and after semi-solid compression Figure 3 is the microstructures of samples with different porosities before semi-solid compression. As shown in figure 3(a), the powder of sintered samples basically maintains the initial spherical shape, and its average particle diameter is 53.5 μm. Seen from figure 3(b), the microstructure of dense samples consists of fine equiaxed grains, and its average grain diameter is 55 μm, which is close to the average particle diameter of powders. Figure 4 shows the microstructures after semi-solid compression at 853K with a rate of 1 1/s (equal to 6 mm s −1 ) that is close to the actual rolling speed. Compared figures 4(a)-(k) with figure 3(a), the particles become finer after semi-solid compression, and a few spherical powder boundaries are in the center but more   boundaries at the edge. As the porosity decreases, the particle diameter becomes smaller and its distribution becomes denser and more uniform.
When the porosities are 36% and 30% (figures 4(a)-(d)), the powder is basically spherical and randomly stacks. A small amount of powders are crushed, and the black areas are holes. When the porosity is 17% (figures 4(e) and (f)), the number of broken powders increases both in the center and at the edge, while the particle diameter decreases. When the porosity is 10% (figures 4(h) and (i)), few spherical powder boundaries are observed and fragments distribute more tightly. Thus, it is concluded that more powders crush with the decreasing porosity. Because the sintered sample with a porosity of 10% has less holes and thus its powders directly bear the load during compression, so it is easier to be crushed. In addition, a small amount of powders may be broken during Spark Plasma Sintering, due to the higher sintered temperature.
Compared figures 4(j)-(k) with figure 3(b), the grain of dense samples grows significantly after semi-solid compression, which has a polygonous or irregular shape and is obviously deformed (perpendicular to the compression direction). While little grain or particle deformation is observed after semi-solid compression of sintered samples. The central microstructure of dense samples is denser than the edge microstructure, which is the same with the sintered samples. This is because the particle breakup and flowing occur during semi-solid compression of both dense samples and sintered samples [17], but it is the grain breakup in dense samples, and the powder crushing in sintered samples.
Compared between figure 4(a)-(k), the grain size of dense materials is greater than the particle size of all sintered materials, which are basically equal before compression. The reason for this interesting phenomenon is that the grain will grow larger and become globular at a semi-solid state, whether the sintered material or dense material it belongs to. But as the powder of sintered materials bonds less tightly, its coarsening rate is lower than that of dense materials [18], resulting in the finer particle of sintered samples. In addition, the powder breakup of sintered materials makes the particles further finer after semi-solid compression. Therefore, the product prepared by SPF has obvious finer microstructures, which is a major advantage of SPF and verified by [1, 3, 4 and 6]. Figure 5 shows the fracture surfaces of samples after semi-solid compression. Seen from figure 5(a), the fracture surface of sintered sample with a porosity of 16% is mostly made up of spherical particles, which is consistent with its spherical microstructure at the edge (shown in figure 4(f)). It can also be obtained at a low temperature, or a low strain, or a high strain rate [19]. As shown in figure 5(b), the fracture surface of the sintered sample with a porosity of 10% has an irregular-shaped and glassy surface, which can also be obtained at a high temperature, or a large strain, or a low strain rate. This glassy fracture has been reported in [20 and 21], which is different from a ductile fracture nor a brittle fracture. As described in [21], the surface of glassy fracture is a layer of solidified liquid film. The irregular-shaped surface is attributed to fragments broken along the grain boundaries occupied by liquid in powders, resulting in finer particles in figure 3(i). As shown in figure 5(c), the fracture surface of dense materials also has a smooth surface like figure 5(b), but their particles are significantly larger. Karagadde and Lee considered it as a transgranular fracture [17]. Therefore, it can be deduced that the dense material fractures from the grain boundary, while the sintered material fractures from the sintered neck between powders or the grain boundary inside powders. Figure 6 is the metallurgical and SEM images of Al-Cu-Mg alloy powders rolled at a temperature of 853K. Compared figure 6 with figure 4, it was discovered that the microstructure of semi-solid powder rolled billets is similar to that of semi-solid powder compression. There are also some spherical powder boundaries in figure 6, but most of powders are crushed into small particles. Although holes also exist, the microstructure is significantly denser than that of sintered materials in figure 4(a)-(i). A small amount of deformation grains are observed, but which are far less than those of dense materials in figure 4(j)-(k). Moreover, the grains in figure 6 are finer than those of dense materials. Figure 7 shows the SEM images of samples after semi-solid compression with porosities of 36% and 10%. It is discovered that there are obvious grain boundaries in powders, and the white phases are distributed in grains or between grains in both figures 7 and 6(b). The element content of white phase and gray phase of sintered materials after semi-solid compression was analyzed by EDS (shown in table 1), while the elemental distribution of samples after semi-solid powder rolling was analyzed by EPMA elemental mapping (shown in figure 8). It is discovered that the copper content of the white phase is much greater than that of the gray phase. According to the references and the XRD results (shown in figure 9), the white phase in SEM images is Al 2 Cu, and the gray phase is the aluminum matrix. Therefore, it can be concluded that the microstructure of semi-solid powder rolling is similar to that of semi-solid powder compression, and the generated phase is the same. In other words, the material characteristics during semi-solid powder rolling can be represented by the deformation data of semi-solid powder compression.

Stress strain curves of semi-solid compression
As the deformation ratio of semi-solid powder rolling is generally 40%-70%, the strain of semi-solid compression was chosen as 50% (true strain 0.69). Namely, the sample was compressed to half of its original height. Besides, as there are holes and liquid in the semi-solid material, its stress strain curve fluctuates.  As shown in figure 10, the stress of sintered materials increases rapidly with the increasing strain, and then decreases greatly, afterwards basically keeps constant when the strain is large, which is similar to that of dense materials. Especially at a higher temperature or a larger strain, the curves of materials basically merge together. The stress of materials increases with the decreasing porosity, while it decreases with the increasing temperature. The differences of their peak stresses between different porosities are larger at lower temperature. This is because that the value of peak stresses depends strongly on the strength of solid-phase skeletons, which is closely related to both liquid fraction and the porosity [22]. The curve of sintered samples with a large porosity changes slowly and fluctuates greatly, as its sintered neck area is small and thus the strength of solid-phase skeletons is low, while the curve with a small porosity is close to that of dense materials.

Constitutive formula for semi-solid compression
It is considered that the stress of dense materials is the power function with the strain rate and depends exponentially on the temperature [17]. As discussed above, the microstructure, fracture surface and stress strain curve of sintered materials after semi-solid compression are similar to those of dense materials. Therefore, the following formula was attempted to characterize the deformation behavior of sintered and dense materials at a semi-solid state:  Where Z is Zener Hollomon parameter; σ is the stress, MPa; e  is strain rate, 1/s; Q is semi-solid deformation activation energy, J mol −1 ; R 1 is gas parameter, generally 8.31 J mol −1 K −1 ; T −1 is absolute temperature, K; H and m are material constants. Formula (2) was obtained by calculating the natural logarithm of Formula (1): was the simplification of Formula (2): The parameters a 1 , a 2 , and a 3 were used to represent the material constant and relevant parameters lne  and 1/T, respectively, as shown in Formula (4), (5) and (6). Then Formula (7) was deduced, and lnσ and lne¢ are linear with 1/T. In order to simplify the calculation process and maximize the variation characteristics of rheological stress, the peak stress σ p was taken as the stress σ to be substituted into formulas. The values of lnσ p , ln e¢ and 1/T were calculated from the experiment data of σ p , e  and T. Data analysis and regression calculation were performed by using SPSS software through Formula (7), and the fitting results are shown in table 2. ( ) The parameter a 2 characterizes the sensitive degree of the strain rate on peak stresses. As holes disrupt the continuity of samples, the value of a 2 basically increases with the decreasing porosity and reaches the maximum of 0.139 at a porosity of 0%, but it is still less than 0.15 in [23]. The activation energy Q can be calculated from a 3 in table 2 as 817, 1941, 1065, 1414 and 1684 kJ mol −1 with porosities of 36%, 30%, 16%, 10% and 0%, respectively, which are the same magnitude order with those of semi-solid aluminum alloy in [24] and extremely larger than those of hot deformed aluminum alloy in [25 and 26]. This is because there are porosities and liquid during semi-solid compression of sintered samples, making the value of m larger than 10 (while it is between 2 and 5 during hot deformation). The fitted and experimental peak stresses of samples with difference porosities were plotted in figure 11. It was discovered that with the decrease of porosity, the experimental values agree better with the fitted values, and the experimental points of samples with a low porosity are basically around the fitting line. As shown in table 2, the goodness-of-fit increases with the decrease of porosity and the highest value is close to 1, indicating that the denser the material is, the more accurately Formula (1) can describe this relationship. Based on above discussions, the liquid fraction and porosity as parameters were introduced into the following constitutive Formula.
Where Φ is the porosity; f L is liquid mass fraction; b 2 is the strain rate sensitivity parameter; b 1 , b 3 , and b 4 are material constants. The material constants of Formula (8) were obtained by the same method, as shown in table 3. The value of b 2 is equal to 0.081, slightly larger than that of a 2 at a porosity of 10%. As f L is influenced by T, the value of b 3 in Formula (8) decreases and it no longer represents the semi-solid compression activation energy. The goodnessof-fit of all experimental data is 0.95, which means that the stress of semi-solid Al-Cu-Mg alloy powders is still exponentially related to the deformation temperature considering the influence of liquid fraction and the porosity. It provides a theoretical foundation for studying the process of SPF.

Finite element method simulation and experimental validation
The established constitutive formula was input to the rolling model, and the updated Lagrangian algorithm thermodynamic coupling analysis was used to simulate the semi-solid rolling process of Al-Cu-Mg alloy powders based on the elliptic yield criterion. Afterward, the simulated results were compared with the experimental results. The simulation parameters were shown in table 4.   Figure 12 is the distribution contours of displacement, equivalent strain, relative-density, temperature, and contact normal force at the front of the simulated billet. As shown in figure 12, the X-direction displacement, equivalent strain and relative-density gradually increase while the billet temperature decreases as the billet gradually contacts with rollers. And the latter two basically remain constant and uniformly distribute at the area of rolling out. The contact normal force of the roller as a rigid body is considered as the rolling force because that of the billet as a flexible body is not accurate. The rolling force is very small at the area where the billet and rollers just contact, and it increases as the contact area increases and reaches the maximum at the middle line of the roller. Figure 13 shows the curves that the simulated and measured rolling force and simulated billet temperature change with time. As shown in figure 13, the process of semi-solid powder rolling was divided into three stages. When the billet has not yet contacted with the roller, the rolling force is zero at the beginning stage, which is the feeding stage I. With the time prolonging, when the billet is gradually bitten into the roller seam after 0.43s, it enters into the stage II. The rolling force firstly increases slowly at the biting area and then rapidly increases to the maximum at the compaction area. When the billet is rolled out and separated from the roller at 2.1s (entering into the stage III), the rolling force is gradually stable. At the stage I, the measured rolling force is not equal to zero but slightly higher than the simulated value, due to the clamp-holding force and so on. At the stage II., the simulated rolling force basically coincides with the measured value. The former smoothly decreases and then slightly increases, while the latter fluctuates significantly after reaching the maximum value. This is because the rolling force versus time curve covers multiple rolling processes, and both the measured and simulated values are affected by many factors, such as the instable powder supply [27]. As mentioned in [28], the 1°-2º angle before the billet being rolled out is the compaction area, where the rolling force reaches the maximum in a very short time. Thus, the maximum rolling force at 2.1s was selected as the rolling force of semi-solid powder rolling. The simulated maximum rolling force is 13,288N, while the measured value is 11,390 N, with an error of 16.66%. Therefore, it can be considered that the simulated rolling force is basically consistent with the measured value.
The changing trend of billet temperature is opposite to that of the rolling force. First, The billet that is placed in the air at the stage I has heat exchanges with the surrounding environment, resulting in a slight decrease of the temperature. At the stage II, the billet temperature greatly reduces and reaches 760.2 K after running out of the roller seam, which is lower than the Al-Cu-Mg alloy solid phase temperature of 782 K. When entering the stage III, the billet temperature continues to decrease but its reduction rate becomes slow. Figure 14 shows the relative-density versus time curves determined by the simulation and experiments. As shown in figure 14, there are few differences of the billet relative densities between the center and upper surface, indicating that the relative-density distribution is basically uniform during the rolling process. The relativedensity at the stage I remains unchanged, with the initial relative-density of 40% in the model and the powder apparent relative-density of 41% measured in the experiment. At the stage II, the relative-density gradually increases to the maximum. The relative densities of the stuck wedge billet obtained by stopping the roller rotation were measured as 50.1% and 89.6%, respectively, which is close to the simulated value of 88.3%. At the stage III , the relative-density keeps basically constant and is 88.6% at 3.8 s, which is almost equal to the measured value of 88.5%. Therefore, it can be considered that the simulated relative-density agrees well with the measured value.
The above analysis shows that the model based on the combined constitutive formula of semi-solid compression can better simulate the process of Al-Cu-Mg alloy powder rolling at a semi-solid temperature of 853K. For three important variables of relative-density, billet temperature and rolling force, the simulation results are acceptable, which can be used to comprehensively analyze and investigate the influence of various rolling parameters on the semi-solid powder rolling process in order to improve its product performance. 4. Conclusions 1. The particle of sintered materials after semi-solid compression becomes finer and its size decreases with the reduction of porosity, but the grain of dense materials becomes larger and changes from a circular shape to an irregular shape. The sintered material fractures from the liquid-rich grain boundary or powders, while the dense material fractures from the partially melted grain boundary. This is because grain coarsening, powder or grain crushing and flowing occur simultaneously during the semi-solid compression process, but powder crushing dominates during the deformation of sintered material, making its microstructure much smaller than that of dense material, which is the major advantage of SPF. While the grain coarsening and deformation dominate during the deformation of dense material. 2. The microstructure of billets during semi-solid powder rolling is basically similar to that of sintered and dense samples during semi-solid compression, which is composed of many fine particles, some powders and holes. But the former is significantly smaller than the latter, because the rolling force is much greater than the compressing force. The phase and its distribution in three microstructures are basically the same, the gray phase is the aluminum matrix and the white phase is Al 2 Cu.
3. The changing tendency of strain stress curve of dense materials during semi-solid compression is similar to that of sintered materials, which are almost the same especially at a higher temperature. Their peak stresses decrease with the increasing temperature, but the gap is larger at a lower temperature. In addition, the peak stress of sintered samples increases with the decrease of porosity.
4. Based on the experimental data of sintered and dense Al-Cu-Mg alloy samples, considering effects of the porosity and liquid fraction, the combined constitutive model of semi-solid powder materials was established and verified to be accurate, with a goodness-of-fit of 0.95.
5. The numerical simulation of semi-solid powder rolling based on the combined constitutive model can well describe the rolling process at 853K. The simulated results of relative-density, billet temperature and rolling force are acceptable, which can provide a guidance for process control and optimization.